- Home
- Conferences
- Conference Proceedings
- Conferences
ECMOR XV - 15th European Conference on the Mathematics of Oil Recovery
- Conference date: 29 Aug 2016 - 01 Sep 2016
- Location: Amsterdam, Netherlands
- ISBN: 978-94-6282-193-4
- Published: 29 August 2016
1 - 100 of 163 results
-
-
Pore to Pore Validation of Pore Network Modelling against Micromodel Experiment Results
Authors J. Yang, I. Bondino, M. Regaieg and A. MoncorgéPore network modeling (PNM) has been widely used to study the multiphase flow and transport in porous media. Although a number of recent papers discussed the PNM validation on core scale parameters such as permeability, relative permeability, capillary pressure etc; quantitative predictive potential of PNM on pore by pore basis is rarely been studied. In this article, A PNM validation workflow against micro model experiment on pore scale is firstly discussed. A glass etched micro model is used to quantify the accuracy of a dynamic PNM solver on pore and core level. Two phase drainage micro fluidic experiments at different flow conditions are performed on micro models. PNM simulations are performed on the same pattern and flow conditions as used in micro model experiments. The two phase distribution extracted from experiment images are registered onto results of PNM simulations for direct pore to pore comparison. An image processing tool is developed to extract pore to pore oil/water distribution from micro model images for further repeatability check and pore by pore comparisons to PNM simulations. Pore scale matching level is found around 75% for all three test cases, which indicates the oil/water displacement in 75% pores can be predicted by PNM. Compared to the matching level of repeated experiments around 84%, the agreement between PNM and experiments on pore by pore is considered as reasonable. The matching level of core scale parameters such as Swc and oil phase permeability varies from case to case; the relative error to micro model experiment measurements varies from 15% to 60%. Possible reasons leading to discrepancies on core scale parameters are discussed. Imperfection of micro model fabrication and lack of consideration of experimental uncertainty in validation can be two principal factors. In general PNM simulations produced positive results against micro model experiments and PNM is a promising tool for complex flow study in porous media.
-
-
-
Model Comparison for High-pressure Adsorption in Shale and its Influence on Phase Equilibria
Authors D. Sandoval, W. Yan, M. Michelsen and E. StenbyShale formations are of great importance in the last decades since they are large potential sources of oil and natural gas. The shale reservoirs are characterized for its low permeable and low porosity nature, due to these characteristics, adsorption plays a major role in the storage and phase equilibria of the hydrocarbons within the rock. The aim of this work is to provide a comparison and analysis of different models for pure and multicomponent adsorption at high pressure in shale found in recent literature. Additionally, an insight of the phase equilibria calculations under a capillary pressure difference coupled with the adsorption film thickness is given. The models used for pure component are: Langmuir, the modified Toth-Langmuir, and the Potential Theory of Multicomponent Adsorption using Dubinin-Radushkevich potential (MPTA-DRA). The three models show similar deviations close to 10%. For the multicomponent adsorption comparison, Multicomponent Langmuir (ML), Ideal Adsorbed Solution Theory (IAST) and MPTA were evaluated. MPTA shows the lowest deviation with 17.9%. In connection to the phase equilibria, the influence of the adsorption film on the phase envelope was studied. The adsorption film thickness modifies the effective capillary radius enhancing the capillary pressure of the system. These combined effects modify the saturation pressure in the whole temperature range except in the critical point. Having its biggest influence on the bubble point branch, away from the critical point.
-
-
-
On Modification of Relative Permeability in Compositional Simulation of Near-miscible Processes
Authors A. Alzayer, D. Voskov and H. TchelepiMiscible gas injection is one of the most effective enhanced oil recovery (EOR) techniques. There are several challenges in accurately modeling this process that mostly occur in the near-miscible region. The adjustment of relative permeability for near-miscible processes is the main focus of this work. The dependence of relative permeability on phase identification can lead to significant complications while simulating near-miscible displacements. We present an analysis of how existing methods incorporate compositional dependence in relative permeability functions. The sensitivity of the different methods to the choice of reference points is presented with possible guidelines to limit the modification of the relative permeabilities to physically reasonable values. We distinguish between the objectives of reflecting near miscible behavior and ensuring smooth transitions across phase changes in the existing methods. We highlight an important link that combines the two objectives in a more general framework. We make use of Gibbs free energy as a compositional indicator to honor the generalized framework. The new approach was implemented in the Automatic Differentiation General Purpose Research Simulator (ADGPRS) and tested on a set of near-miscible gas injection problems. We show that including compositional dependencies in the relative permeability near the critical point impacts the simulation results with significant improvements in nonlinear convergence.
-
-
-
Modelling Relative Permeability Hysteresis Based on Subphase Evolution
Authors K. Khayrat and P. JennyA recently introduced subphase framework for modeling immiscible two phase flow in porous media has been extended. In this framework the nonwetting and wetting phases are divided into subphases distinguished by their connectivity. The nonwetting phase is divided into three subphases: backbone, dendritic, and trapped subphases. Similarly, the wetting phase is divided into four subphases; backbone, dendritic, film, and isolated subphases. The subphase saturations evolve according to volume transfer terms, which require modeling. Within this framework, relative permeability models can be developed, which take into account the contributions of the different subphases appropriately. For example, only the nonwetting backbone subphase contributes to nonwetting relative permeability. Quasi-static flow network simulations of several drainage-imbibition cycles are conducted to analyze the evolution of the subphases in three different pore-networks. Furthermore, a relative permeability model for the wetting phase as a function of the subphase saturations is proposed. The resulting model can capture complex hysteretic behavior present in relative permeability-saturation curves.
-
-
-
Tangent Plane Criteria for Phase Stability Computation for System with Hydrocarbon and Aqueous Phase Components
Authors A. Venkatraman, G. Singh and M. WheelerCO2 injection in depleting oil reservoirs provides the dual benefit of increasing oil recovery as well as sequestration. Phase changes occur during the injection and the injected CO2 can react with aqueous phase components, especially high ionic concentration brines typically found in the Middle East hydrocarbon reservoirs. Compositional simulations using equation of state (EOS) models are used to represent changes in phase behavior of hydrocarbon components accompanying CO2 injection. However, most of these models account for CO2 dissolution in the aqueous phase using a Henry’s law approximation valid only at low pressures. These models do provide the ability to track the changes in CO2 as well as other ions in the aqueous phase following the injection of CO2. A new tangent plane criterion to evaluate stability of each phase at equilibrium conditions has been developed using the unified Gibbs free energy function. This criteria accounts for both hydrocarbon and aqueous components described using EOS and activity coefficient models, respectively. A Gibbs free energy module for compositional simulations has been developed that includes this stability criterion along with a free energy minimization algorithm. This can be integrated in any reservoir simulator for varying reservoir fluid compositions.
-
-
-
Three-phase EoS-based Reservoir Simulation with Salinity Dependent Phase-equilibrium Calculations
Authors M. Petitfrere, L. Patacchini and R. de LoubensLight gases (CO2, H2S, …) are soluble in water at typical reservoir conditions, particularly when salinity is low; besides, at high enough temperatures water can significantly vaporize into the gas phase. Such gas-water mass exchanges can play an important role in the recovery mechanisms and need to be appropriately accounted for in reservoir simulation. In this context, the Søreide and Whitson equation of state (SW-EoS) is an attractive option to model those phase equilibria since it is reasonably predictive and incorporates salinity effects, while being simpler than more advanced EoS such as CPA or SAFT. Most publicly available reservoir simulators treating gas-water mass exchanges either use K-values or two-phase EoS models combined with Henry’s law, or the SW-EoS but limited to a two-phase (gas/water) environment. In this work, the SW-EoS has been implemented in our fully compositional research reservoir simulator. The model is first tested with constant salinity. Two-phase simulations of an Asian gas field with high CO2 content are presented, and results benchmarked against a commercial simulator. Three-phase simulations of a French CO2 sequestration pilot with significant gas-water exchanges are then discussed, matching the experimental production data. The model is further extended to account for salt transport and precipitation. To the best of our knowledge, this is the first time in the literature that the SW-EoS is coupled to dynamic salinity. The phase equilibrium algorithm is modified to reach quadratic convergence even when the attractive term of the cubic EoS depends on dynamic salinity. The algorithm is tested on 3D simulations of a tertiary gas injection process with dry-out and salt precipitation effects in the near-well region, and comparisons with simulations based on the Peng-Robinson EoS or considering static salinity with the SW-EoS are presented, showing significant impact on oil recovery.
-
-
-
Non-equilibrium Phase Behavior of Hydrocarbons in Compositional Simulations and Upscaling
Authors I.M. Indrupskiy, O.A. Lobanova and V.R. ZubovNumerical models widely used for multiphase flow simulations are based on the assumption of equilibrium phase behavior of hydrocarbons. However, it is not uncommon for oil and gas-condensate reservoirs to exhibit essentially non-equilibrium phase behavior, e.g., in the processes of secondary recovery after primary pressure depletion below saturation pressure. Also, the ability to match field data with an equilibrium model depends on simulation scale. The only method to account for non-equilibrium phase behavior adopted by the majority of flow simulators is the option of limited gas dissolution (condensate evaporation) rate in black oil models. For compositional simulations, no practical method has been presented so far, except for some upscaling techniques in problems of oil recovery by gas injection. Rarely presented previous models for non-equilibrium compositional simulations have a common problem of doubling the number of flow equations and unknowns at each simulation step compared to equilibrium formulation. This significant drawback obstructs their incorporation to existing simulators and widely adopted simulation algorithms. We suggest a physically-consistent formulation for the problem of compositional flow simulation with non-equilibrium phase behavior which has yet shown the following benefits. 1. The new formulation is based on the same form of flow equations in as in the equilibrium case, with modified flash equations. This makes it possible to incorporate the non-equilibrium model in existing simulators without significant extra computational costs and principal modifications to flow simulation modules. 2. A consistent technique has been developed for upscaling an equilibrium or non-equilibrium model into a coarse-scale non-equilibrium model with account for influence of scale effect on non-equilibrium phase behavior. 3. A model for component interphase mass transfer rate in the non-equilibrium flash equations has been presented which considers phase composition relaxation dynamics under changing pressure and overall composition and provides an efficient and robust tool for history matching in non-equilibrium phase behavior conditions. A number of simulation cases for real oil and gas-condensate mixtures are to be presented. Main contributions: 1. A new formulation has been presented for multiphase compositional flow model with account for non-equilibrium phase behavior suitable for practical reservoir simulations and implementation in existing compositional simulators. 2. Relation between problem scale and relevance of non-equilibrium effects has been physically justified. An upscaling technique has been developed for equilibrium and non-equilibrium compositional models based on the proposed non-equilibrium model formulation. 3. A model has been derived for component interphase mass transfer rates in non-equilibrium flash equations to provide robust, efficient and physically-consistent history matching of compositional models.
-
-
-
High-resolution Numerical Modelling to Resolve the Dynamics of Pipe Structures in Porous Media
Authors L. Räss, V. Yarushina, T. Duretz and Y. PodladchikovCommon features visible on a large majority of the seismic cross sections are chimneys or pipe structures. They represent regions of focused fluid flow in porous media. As seismic surveys are widely performed in many regions where the subsurface is of economic interest, a better understanding of the formation and evolution process of these chimneys is vital. They should be considered when performing risk assessment linked to leakage within subsurface waste storage projects. They might also lead to a better understanding of fluid migration pathways and subsurface fluid localization. In that context, we propose a new physical model that predicts the formation and the evolution in space and time of these chimneys. We use a two-dimensional (2D) implicit solver and the three-dimensional (3D) high-resolution iterative parallel GPU code to solve a thermodynamically consistent system of nonlinear equations for two-phase flow in deforming porous media. We will show that the different 2D implicit and 3D iterative methods used to solve the fully coupled system of nonlinear equations are in good agreement. They predict the formation and the propagation of a nonlinear solitary wave, resulting in a chimney formation. These chimneys, with order of magnitude permeability increase, are a natural outcome of an interplay between buoyancy forces driving upward fluid propagation and a resistance of a deformable rock to locally increasing fluid pressure. We will also discuss and highlight the importance of a proper coupling between the geomechanics (Stokes solver) and the reservoir fluid flow (nonlinear Darcy solver).
-
-
-
GPU Acceleration of Equation of State Calculations in Compositional Reservoir Simulation
Authors R. Gandham, K. Esler, K. Mukundakrishnan, Y.P. Zhang, C. Fang and V. NatoliEquation-of-state (EOS) based compositional simulations accurately capture the dynamics of reservoirs with strong compositional effects. One of the major computational bottlenecks in such simulations is the need to enforce the phase equilibrium constraint for the hydrocarbon system for every grid block in the model. These constraints must be enforced at every time step and possibly, at every nonlinear iteration level within each time step for implicit methods. Hence, detailed simulations of models with many millions of cells and a large number of hydrocarbon components are prohibitively time-consuming. However, the high computational intensity and parallelism exhibited by these calculations make them ideal for significant acceleration using high throughput devices such as Graphics Processing Units (GPUs). In this study, we propose new techniques for accelerating the EOS-based phase equilibrium calculations on the GPUs. First, we make full use of the large number of fast registers and floating point units available on GPUs for the double-precision arithmetic , thereby significantly accelerating the equilibrium calculations. Second, we exploit the fast hardware intrinsics available for single precision to further increase the performance. By iteratively combining the single and double-precision calculations, we not only achieve the full accuracy of double-precision but also gain an order-of-magnitude speedup over using double-precision arithmetic alone. Accuracy and performance results from several benchmark problems available in the literature will be provided to demonstrate the speedup achieved using our proposed techniques. The performance results will then be compared with the recently published timings generated using highly optimized code on the CPUs. We will discuss the implications of such performance gains on the selection of implicit algorithms for the full compositional flow simulation.
-
-
-
Parallel Fully Implicit Smoothed Particle Hydrodynamics Based Multiscale Method
Authors A. Lukyanov and C. VuikPreconditioning can be used to damp slowly varying error modes in the linear solver residuals, corresponding to extreme eigenvalues. Existing multiscale solvers use a sequence of aggressive restriction, coarse-grid correction and prolongation operators to handle low-frequency modes on the coarse grid. High-frequency errors are then resolved by employing a smoother on fine grid. In reservoir simulations, the Jacobian system is usually solved by FGMRES method with two-level Constrained Pressure Residual (CPR) preconditioner. In this paper, a parallel fully implicit smoothed particle hydrodynamics (SPH) based multiscale method for solving pressure system is presented. The prolongation and restriction operators in this method are based on a SPH gradient approximation (instead of solving localized flow problems) commonly used in the meshless community for thermal, viscous, and pressure projection problems. This method has been prototyped in a commercially available simulator. This method does not require a coarse partition and can be applied to general unstructured topology of the fine scale. The SPH based multiscale method provides a reasonably good approximation to the pressure system and speeds up the convergence when used as a preconditioner for an iterative fine-scale solver. In addition, it exhibits expected good scalability during parallel simulations. Numerical results are presented and discussed.
-
-
-
A Parallel Framework for a Multipoint Flux Mixed Finite Element Equation of State Compositional Flow Simulator
Authors G. Singh, B. Ganis and M. WheelerMathematical models of physical problems are becoming increasingly complex and computationally intensive. At the same time, computing hardware is becoming more parallelized with an increasing number of cores promoting simultaneous tasks. In this work we present a parallel, equation of state (EOS), compositional flow simulator for evaluating CO$_2$ sequestration, enhanced oil recovery techniques such as gas flooding, and other subsurface porous media applications. Using the multipoint flux mixed finite element (MFMFE) method for spatial discretization, it can handle complex reservoir geometries using general distorted hexahedral grid elements, as well as satisfy local mass conservation and compute accurate phase fluxes. A parallel framework for the MFMFE is presented that has been extended to the highly non-linear, EOS, compositional flow model. Much of the non-linearity is due to the local flash and stability calculations associated with interphase mass transfer and phase behavior. Parallel multigrid linear solver libraries such as HYPRE are utilized to solve the algebraic problems on each Newton step. We perform a variety of strong and weak parallel scaling studies up to 10 million elements and 1024 processors, and discuss possible load balancing issues.
-
-
-
Interpretation of Well-cell Pressures on Hexagonal K-orthogonal Grids in Numerical Reservoir Simulation
More LessPeaceman's equivalent well-cell radius for 2D Cartesian grids has been generalized to 2D uniform hexagonal K-orthogonal grids in an anisotropic medium. An analytic expression for the equivalent well-cell radius for infinitely fine grids is derived. The derivation is performed by comparison of analytical and numerical solution for boundary value problems with one or two wells. The derivation for the anisotropic case is based on a transformation to an isotropic image space and follows Peaceman's derivation closely. Since the well-cell radius varies slowly with the grid fineness, the found formula can be considered representative for all grid sizes. Since 2D seven-point stencils are more rotationally invariant than five-point stencils, they are often preferred to reduce grid-orientation problems. The formula can be applied to calculate the correct difference between the bottomhole pressure and the numerical well-cell pressure for 2D hexagonal grids. Such a formula is necessary in case of pressure-controlled wells. It is also useful for rate-controlled injection wells with an upper pressure bound. The formula is easy to implement in a reservoir simulator.
-
-
-
Dynamic Unstructured Mesh Adaptivity for Improved Simulation of Bear Wellbore Flow in Reservoir Scale Models
Authors P.S. Salinas, D. Pavlidis, A. Adam, Z. Xie, C.C. Pain and M.D. JacksonIt is well known that the pressure gradient into a production well increases with decreasing distance to the well and may cause downwards coning of the gaswater interface, or upwards coning of wateroil interface, into oil production wells; it can also cause downwards coning of the water table, or upwards coning of a saline interface, into water abstraction wells. To properly capture the local pressure drawdown into the well, and its effect on coning, requires high grid or mesh resolution in numerical models; moreover, the location of the well must be captured accurately. In conventional simulation models, the user must interact with the model to modify grid resolution around wells of interest, and the well location is approximated on a grid defined early in the modelling process. We report a new approach for improved simulation of nearwellbore flow in reservoirscale models through the use of dynamic unstructured adaptive meshing. The method is novel for two reasons. First, a fully unstructured tetrahedral mesh is used to discretize space, and the spatial location of the well is specified via a line vector. Mesh nodes are placed along the line vector, so the geometry of the mesh conforms to the well trajectory. The well location is therefore accurately captured, and the approach allows complex well trajectories and wells with many laterals to be modelled. Second, the mesh automatically adapts during a simulation to key solution fields of interest such as pressure and/or saturation, placing higher resolution where required to reduce an error metric based on the Hessian of the field. This allows the local pressure drawdown and associated coning to be captured without userdriven modification of the mesh. We demonstrate that the method has wide application in reservoirscale models of oil and gas fields, and regional models of groundwater resources.
-
-
-
Hybrid Dimensional Modelling and Discretization of Two Phase Darcy Flow through DFN in Porous Media
Authors K. Brenner, J. Hennicker, R. Masson and P. SamierWe provide a model for two phase Darcy flow through discrete fracture networks (DFN) in porous media, in which the d−1 dimensional flow in the fractures is coupled with the d dimensional flow in the matrix, leading to the so called hybrid dimensional Darcy flow model. It accounts for fractures acting either as drains or as barriers, since it allows pressure jumps at the matrix-fracture interfaces. The model also permits to treat discontinuous capillary pressure at the material interfaces as well as gravity dominated flow. In particular, it incorporates upwind normal fluxes that are needed to reproduce gravitational segregation inside the DFN. We adapt the Vertex Approximate Gradient (VAG) scheme to this problem, in order to account for anisotropy and heterogeneity aspects as well as for applicability on general meshes. For diphasic flow, we present several test cases, and use VAG to compare our hybrid dimensional model to a hybrid dimensional model that assumes continuous pressure at the matrix fracture interfaces and to the generic equidimensional model, in which fractures have the same dimension as the matrix. This does not only provide quantitative evidence about computational gain, but also leads to deep insight about the quality of the reduced models.
-
-
-
An Efficient Fully-implicit High-resolution MFD-MUSCL Method for Two-phase Flow with Gravity in Discrete Fractured Media
Authors J.M. Jiang and R.M. YounisStandard reservoir simulation schemes employ single-point upstream weighting for approximation of the convective fluxes when multiple phases or components are present. These schemes are only first-order and give a poor approximation and induce high viscosity effect. A second-order scheme provides a better approximation and manages to reduce the viscous smoothing effect in the vicinity of the shocks. In reservoir simulation practice, implicit discretisations capable of taking large time steps are preferred in practical computations. However, assembling and solving a large nonlinear system is often very expensive, even for a simple first-order method, and using a higher-order spatial discretisation introduces extra couplings and increases the nonlinearity of the discretised equations. It has been shown that the strong nonlinearity as well as the lack of continuous differentiability in numerical flux function and flux limiter can cause serious nonlinear convergence problem. Cell-centered finite-volume (CCFV) discretizations may offer several attractive features, especially for the fluid flow in discrete fractured media. The objectives of this work are to develop a novel cell-centered multislope MUSCL method and an adaptive limiting strategy that have improved computational efficiency, smoothness properties, and accuracy. The reconstruction scheme interpolates the required values at the edge centroids in a more straightforward and effective way by making better use of the triangular mesh geometry. Because optimal second-order accuracy can be reached at the edge centroids, the numerical diffusion caused by mesh skewness is also significantly reduced. An improved gradually-switching piecewise-linear flux-limiter is introduced according to mesh non-uniformity in order to prevent spurious oscillations. The developed smooth flux-limiter can achieve high accuracy without degrading nonlinear convergence behavior. For the discretization of pressure and Darcy velocities, a mimetic finite difference method that provides flux-continuity and an accurate total velocity field is used. The developed fully-coupled MFD-MUSCL CCFV framework is adapted to accommodate a lower-dimensional discrete fracture-matrix model. Several numerical tests with discrete fractured system are carried out to demonstrate the efficiency and robustness of the numerical model. The results show that the high-order MUSCL method effectively reduces numerical diffusion, leading to improved resolution of saturation fronts compared with the first-order method. In addition, it is shown that the developed multislope scheme and adaptive flux limiter exhibit superior nonlinear convergence compared with other alternatives.
-
-
-
Discrete Fracture Model based on Multiple Interacting Continua Proximity Function for Unconventional Reservoirs
More LessShale formations presents multi-scale heterogeneities, including stimulated and non-stimulated natural fractures. Besides, a hydraulic stimulation is required in order to increase production from unconventional low permeability reservoirs. However, this kind of operation increases the complexity of the fracture network and modeling a complex discrete fracture network (DFN) become crucial for simulating production from unconventional reservoirs. This paper propose a methodology, taking into account various size of fractures with different locations and orientations in a low permeability reservoir, in order to suggest a unique model as simple as possible. A typical discrete Fracture Model (DFM) rely on unstructured grids to conform the fracture geometry and location. This kind of model discretizes all types of fractures leading to a complicated and often non tractable numerical system to solve. To overcome these limitations, hierarchical methods such as Embedded Discrete Fracture Models (EDFM) are usually used to deal with this multi-scales problem. However, the matrix-fracture interaction is not properly handled with the EDFM due to the very low matrix permeability and the large matrix grid cells. In this paper, we will present a DFM based on Multiple INteracting Continua (MINC) approach to improve the EDFM. This approach rely on a triple-porosity model: matrix media, large hydraulic/propped fractures, and unpropped stimulated/non-stimulated natural fractures. Large propped fractures are explicitly discretized, natural fractures are homogenized, and their connections are based on a proximity function obtained with an integral representation. The connections between the matrix and fractures are computed with the MINC method based on a proximity function using a stochastic process. The implementation of the MINC method improves the flow exchange between the matrix and fracture media. Thus, the matrix grid cell is subdivided according to a MINC proximity function based on the distance from all sort of fractures, by using randomly sampled points. The proposed approach is particularly useful for multi-phase flow simulations in a low permeability unconventional reservoir such as a tight-oil reservoir. Several numerical examples will be presented to illustrate the accuracy of this improved DFM for a single-phase flow case and a multi-phase flow case with gas liberation from a tight-oil formation.
-
-
-
Modelling of Flow Induced Shear Failure in Poro-elastic Fractured Media
More LessA finite volume based numerical modeling framework using a hierarchical fracture representation has been developed to compute flow induced shear failure. To accurately capture the mechanics near fracture manifolds, discontinuous basis functions are employed which ensure continuity of the displacement gradient across fractures. With these special basis functions, traction and compressive forces on the fracture segment can be calculated without any additional constraints, which is extremely useful for estimating the irreversible slip based on a constitutive friction law. Unlike other models, here asymptotic dilation of fracture aperture due to shear failure is considered. To solve the resulting linear system, a sequential approach is used, that is, first the flow- and then the mechanics problems are solved. The new modeling framework is very useful to predict seismicity, permeability- and flow evolution in geological reservoirs. This is demonstrated with numerical simulations of enhancing a geothermal system. Novelties of this approach are (i) that due to the special basis functions only one additional degree of freedom per fracture segment is introduced (opposed to four in the XFEM method) and (ii) achieving consistent coupling with the flow solver using asymptotic aperture relaxation.
-
-
-
Methodology to Compute Mathieu Functions for Arbitrary Large Parameter q and Its Application to Pressure Transient Analysis
More LessMathieu functions are widely used for the solution of boundary value problems in elliptical systems. In spite of their common use, they are notorious for their inherent instability at high (absolute) values of the parameter q. In this paper we present robust solution methodologies enabling the computation of Mathieu functions for values of q that can go to infinity. First, we present a methodology dealing with inherent instability in the recurrence relationships for Fourier coefficients, thus enabling their accurate computation for arbitrarily high q. Secondly, we overcame the ‘subtraction error’ in the computation of Mathieu functions as infinite sums of Bessel function products by defining asymptotic approximations for ratios between Mathieu function values for different value of the elliptical coordinate. The accuracy of these asymptotic approximations was extensively tested over large ranges of the relevant parameters, and excellent agreement was found. Thirdly, we computed accurate early-time transient pressure profiles for particular sets of boundary conditions by expressing these as linear combinations of Mathieu function ratios (instead of Mathieu functions directly). We illustrate our methodology by applying it to two well-known problems in the area of Pressure Transient Analysis, the limitations of some concepts that are well-accepted in literature are demonstrated.
-
-
-
Combining the Modified Discrete Element Method with the Virtual Element Method for Fracturing of Porous Media
Authors H.M. Nilsen, X. Raynaud and I. LarsenSimulation of fracturing processes in porous rocks can be divided in two main branches: (i) modeling the rock as a continuum enhanced with special features to account for fractures, or (ii) modeling the rock by a discrete (or discontinuous) modeling technique that describes the material directly as an assembly of separate blocks or particles, e.g., as in the discrete element method (DEM). In the modified discrete element (MDEM) method, the effective forces between virtual particles are modified in all regions without failing elements so that they reproduce the discretization of linear FEM for linear elasticity. This provides an expression of the virtual forces in terms of general Hook's macro-parameters. Previously, MDEM has been formulated through an analogy with linear elements for FEM. We show the connection between MDEM and the virtual element method (VEM), which is a generalization of traditional FEM to polyhedral grids. Unlike standard FEM, which computes strain-states in reference space, MDEM and VEM compute stress-states directly in real space. This connection leads us to a new derivation of the MDEM method. Moreover, it gives the basis for coupling (M)DEM to domains with linear elasticity described by polyhedral grids, which makes it easier to apply realistic boundary conditions in hydraulic-fracturing simulations. This approach also makes it possible to combine fine-scale (M)DEM behavior near the fracturing region with linear elasticity on complex reservoir grids in the far-field region without regridding. To demonstrate simulation of hydraulic fracturing, the coupled (M)DEM-VEM method is implemented in the Matlab Reservoir Simulation Toolbox (MRST) and linked to an industry-standard reservoir simulator. Similar approaches have been presented previously using standard FEM, but due to the similarities in the approaches of VEM and MDEM, our work is a more uniform approach and extends previous work to general polyhedral grids for the non-fracturing domain.
-
-
-
Simulation of a High-velocity Jet to Predict Its Influence on Crack Initiation and Propagation in Jet-assisted Drilling
By P. EisnerAn effective method to significantly increase the rate of penetration (ROP) in hard formations is the application of ultra-high pressure jet-assisted drilling. To obtain a better understanding of the impact of the high-velocity jet on the hole bottom, the velocity, pressure and shear stress distributions are evaluated both analytically and numerically. These distributions are of great interest, since they particularly influence the fracture initiation and propagation of the rock to be drilled. The analytical approach is based on the mechanics of a turbulent impinging jet. This is mainly done to better comprehend the interaction of the governing parameters downhole and to find an optimum configuration for such a jet-assisted drilling system. The computational fluid dynamics (CFD) simulation is conducted by an open source software for turbulent, steady-state, incompressible and isothermal flow. Nozzle diameter, impingement height and exit velocity are varied in several simulation runs. Moreover, the shear stress distribution at the bottom is determined in the CFD simulation, which will contribute to a better understanding of the crack initiation downhole. The contribution of this work is the evaluation of an optimum configuration to obtain the maximum impact of the high-velocity jet.
-
-
-
XFEM-EDFM-MINC for Coupled Geomechanics and Flow in Fractured Reservoirs
Authors G.T. Ren, J.M. Jiang and R.M. YounisUnconventional reservoirs are often naturally and hydraulically fractured with characteristically small pores and low permeability within the matrix. The underlying fracture networks can have a wide range of length scales and complex geometries. While hydraulic fractures may be propped, natural fractures are predominantly supported by pore pressure. A timely topic in the simulation of unconventional petroleum resources is in devising models that can accurately capture the coupling between the geomechanics of the fractured media and the multiphase fluid flow and transport. We develop a mixed discretization approach to adequately resolve the fracture system while accurately and efficiently modeling both flow and geomechanics. An extended finite element method (XFEM) is applied to approximate the geomechanics, and an embedded-discrete-fracture model (EDFM) is used for the multiphase flow equations. The two schemes are fully coupled, and the time discretization for flow is fully-implicit. Moreover, a hybrid fracture representation concept is employed where the multiple interacting continua (MINC) approach is used in conjunction with the embedded discrete representation in order to capture small-scale fracture networks efficiently. Several validation and computational results are presented. We also apply the proposed method to production scenarios with horizontal wells and hydraulic fractures in reservoirs with secondary fractures.
-
-
-
Modelling Geomechanical Impact of CO2 Injection and Migration Using Precomputed Response Functions
Authors O.A. Andersen, H.M. Nilsen and S.E. GasdaWhen injecting CO2 or other fluids into a geological formation, pressure plays an important role both as a driver of flow and as a risk factor for mechanical integrity. The full effect of geomechanics on aquifer flow can only be captured using a coupled flow-geomechanics model. In order to solve this computationally expensive system, various strategies have been put forward over the years, with some of the best current methods based on sequential splitting. In this present work, we seek to approximate the full geomechanics effect on flow without the need of coupling with a geomechanics solver during simulation. We do this by means of precomputed pressure response functions. At grid model generation time, a geomechanics solver is used to compute the mechanical response of the aquifer for a set of pressure fields. The relevant information from these responses is then stored in a compact form and embedded with the grid model. We test the accuracy and computational performance of our approach on a simple 2D model and a more complex 3D model, and compare the results with those produced by a fully coupled approach as well as from as simple decoupled method based on Geertsma's uniaxial expansion coefficient.
-
-
-
Micro-continuum Formulation for Modelling Dissolution in Natural Porous Media
Authors CS Soulaine and H.A. TchelepiAdvances in imaging technologies and high-performance computing are making it possible to perform Direct Numerical Simulation (DNS) of flow processes at the pore scale; nevertheless, the restrictions on the physical size of the sample (porous rock) that can be fully resolved using Navier-Stokes-based DNS are quite severe. Even for samples on the order of a cm3, the complexity of the spatial heterogeneity of the pore space precludes Navier-Stokes-based DNS. To deal with this challenge of having a wide range of length scales – even for `small’ systems, we describe a micro-continuum formalism, whereby locally averaged equations and associated coefficients can be used to model the effects of scales that are below instrument resolution and/or DNS capability. A hybrid modeling framework based on the Darcy-Brinkman-Stokes (DBS) equation is employed. In this approach, a single equation is applicable for flow in `channels’ (so-called `free flow') and in porous media (solid-fluid aggregates). A unified simulation framework for multi-physics problems of mass and heat transport in natural porous media based on a hybrid Darcy-Brinkman-Stokes approach is described. We discuss two specific applications: minerals dissolution in CO2-brine systems, and dissolution instabilities (worm-holing phenomena) associated with the acidizing treatment of carbonate formations.
-
-
-
Estimation of In-situ Compositions in Lean Gas Condensate Reservoirs
Authors H.R. Nasriani, E. Asadi, C. Johnson, M. Nasriani and A. ChamchineThere is a high degree of complexity in both fluid flow and phase-behavior of gas condensate reservoirs during the depletion period due to retrograde condensation. Natural depletion in gas condensate reservoirs results in low condensate recoveries at surface due to in-situ condensation and accumulation of condensate in the reservoir especially in vicinity of wellbore. During reservoir depletion, the overall composition of reservoir fluid varies and becomes different from initial reservoir composition as pressure decline to values less than dew-point pressure. In-situ gas and condensate composition in the reservoir are changing accordingly. This paper develops a novel approach to obtain initial gas condensate reservoir composition from gas and liquid compositions taken from separator tests during several depletion stages. Based on the composition of mixed sample and initial reservoir composition, a set of novel correlations is developed for estimating initial gas condensate reservoir composition. The generalized reduced gradient (GRG) algorithm of iteration was used to tune the constant and exponents of the correlation based on available field data. The convergence criteria were to minimize the value of the squared sum, SS, of the difference between the real data and the estimated one.
-
-
-
The Role of the Y-function for Checking the Reliability of PVT-data
Authors K. Potsch, P. Toplack and T. GumpenbergerPVT experiments are not free of systematic and random errors, therefore checking the validity of the results is a must. Textbooks use the Y-function as a consistency checks for constant compositions experiment (CCE) of black oil (BO). This paper extends the application of the Y-function. As it turns out, the Y-function can not only be applied for the CCE of BO, but also for the same experiment with a gas-condensate (GC). Furthermore, it will be demonstrated that the Y-function is a useful tool for evaluation of the differential liberation experiment (DLE, black oil) and the constant volume depletion (CVD, gas-condensate) as well. So far this aspect has not been used in the consistency checks of laboratory measurements. Textbooks claim that the Y-function is a straight line. Modeling the fluid system (BO or GC) with an equation of state (EOS) shows that the shape of this function is only close to a straight line. Having set up the Y-function properly, it is possible to estimate the quality of the CCE and the DLE for the black oil case and CCE and CVD for the gas-condensate by comparing the results of the experiments with the EOS calculations.
-
-
-
Compositional Numerical Simulation of CO2-enhanced Gas Recovery in very Low-pressure Partially-depleted Reservoirs
Authors S. Goudarzi, S.A. Mathias and J.G. GluyasInjection of CO2 into depleted gas reservoirs offers the potential to securely store carbon dioxide while enhancing hydrocarbon recovery. Mathias et al. (2014) developed a two-layer vertical-equilibrium model for the injection of carbon dioxide into a low-pressure (<1 MPa) reservoir. In contrast to previous vertical-equilibrium models, the compressibility of all components was fully accounted for and non-Darcy effects were also considered. However, they ignored the effects of compositional changes and mixing between phases. This study seeks to extend the study of Mathias et al. (2014) by incorporating compositional effects, using a Method of Lines (MOL) three-component two-phase numerical simulator. MOL requires formulation of derivatives of the Primary Variables (PV) with respect to time. This often gives rise to the need for evaluating partial derivatives of some of the flow variables (in particular the bulk fluid density per unit volume of rock) with respect to the PVs. In this work, it will be shown how it is possible to evaluate these partial derivatives analytically (as opposed to e.g. conventional finite differencing). Moreover, following an assumption of constant equilibrium ratios for CO2 and CH4, a ternary flash calculator is developed providing closed-form relationships for exact interpolation between equations-of-state for CO2+H2O and CH4+H2O binary mixtures. Overall, the numerical results confirm the finding of Mathias et al. (2014), that pressure build-up, during CO2 injection into low pressure depleted gas reservoirs, can be expected to be largely unaffected by heat transport and other associated thermodynamic effects. Reference: Mathias, S. A., McElwaine, J. N., & Gluyas, J. G. (2014). Heat transport and pressure buildup during carbon dioxide injection into depleted gas reservoirs. Journal of Fluid Mechanics, 756, 89-109.
-
-
-
Computer Modelling of Non-isothermal, Multiphase and Multicomponent Flow by Using Combined EOR Technologies
Authors T.S. Imankulov and D. Akhmed-ZakiDevelopment of new highly efficient oil production technologies require a deep analysis of complex mechanisms in real reservoir processes. To improve existing methods of enhanced oil recovery (EOR), including chemical, microbial (MEOR) and thermal methods, it is necessary to study the processes that are accompanied by phase state change of reservoir fluids. In the development of oil and gas fields using various EOR methods occurs complex three-phase flow of multicomponent mixtures with intensive phase transformations. Using known simplified models, which are not takes into account the phase transitions, to describe such processes can lead to significant quantitative errors and qualitative distortion of real process nature in reservoir. Similar changes in reservoir flow can occur when using thermal methods. This paper presents a numerical study of EOR methods by chemical (polymer or surfactant, polymer and surfactant) injection, MEOR and thermal methods, which considers, that flow is multiphase and fluids are multicomponent. Developed numerical algorithm to solve the problem using a fully implicit method. Considered different flooding system, different sequence of injection agents in combined flooding case and different temperature of injected water. Distributions of main technological parameters are obtained and efficiency of methods mentioned before is shown. The main results of numerical experiments are compared with calculations of reservoir simulator Eclipse 300 (PVTi).
-
-
-
Shale Gas Lithofacies Classification and Gas Content Prediction Using Artificial Neural Network
Authors S. Ouadfeul and L. AliouaneHere, we show the contribution of the artificial intelligence such as neural network to predict the lithofacies in the lower Barnett shale gas reservoir. The Multilayer Perceptron (MLP) neural network with Hidden Weight Optimization Algorithm is used. The input is raw well-logs data recorded in a horizontal well drilled in the Lower Barnett shale formation, however the output is the concentration of the Clay and the Quartz calculated using the ELAN model and confirmed with the core rock measurement. After training of the MLP machine weights of connection are calculated, the raw well-logs data of two other horizontal wells drilled in the same reservoir are propagated though the neural machine and an output is calculated. Comparison between the predicted and measured clay and Quartz concentrations in these two horizontal wells shows the ability of neural network to improve shale gas reservoirs characterization. The present paper is limited only to the lithofacies prediction however the MLP neural network will be used also for the prediction of the gas content form well-logs data in the Barnett shale gas reservoirs which is an extended work of the present research.
-
-
-
Analytical Modelling of Low-salinity Waterflooding with Fines Migration
Authors S. Borazjani, A. Behr, L. Genolet, A. Van der net, Z. You and P. BedrikovetskyLow-salinity waterflood is presently one of the most cost-effective EOR methods. The governing system of equations for oil and water flow with changing water salinity is derived. So-called mechanistic model is developed. This model accounts for two EOR factors: a) fines migration, which is induced by salinity variation and reduces the relative permeability of water, and b) wettability alteration, which affects the relative permeability and capillary pressure. The “lump-salt” description is used, i.e. the ionic water composition is represented by the total ionic concentration. The basic equations are simplified for three basic asymptotic cases: 1) large length scales, 2) low-velocities, 3) high-velocities. An exact analytical solution is derived for 1d displacement large-scale-approximation problem of oil using high-salinity water followed by the injection of low-salinity-slug and high-salinity water chase drive. The solution is obtained by so-called splitting method, where the Lagrangian coordinate is used as a free variable instead of time. The effects of wettability alteration and fines migration on oil recovery as two distinct physical mechanisms are analyses using the analytical model. The significant EOR-effects of both mechanisms are observed for the typical oil-reservoir conditions.
-
-
-
Modelling Bio-reactive Transport in Underground Hydrogen Storages - Spatial Separation of Gaseous Components
Authors B. Hagemann, M. Panfilov and L. GanzerIn the context of energy revolution large quantities of storage capacity are required for the integration of strongly fluctuating energy production from wind and solar power plants. The conversion of electrical energy into chemical energy in the form of hydrogen is one of the technical possibilities. This technology, where hydrogen is stored in subsurface formations similar to the storage of natural gas, is currently in the exploratory focus of several European countries. Despite the deviating hydrodynamic behavior of hydrogen compared to natural gas, bio-chemical reactions can have an imported role in underground hydrogen storage. The fact that hydrogen is a favored substrate for several anaerobic microorganisms induces their growth and results in a degradation of hydrogen. In particular the activity methanogenic archaea can lead to drastic variations in the gas composition which were observed in some former town gas storages. To describe this behavior a mathematical and numerical model was developed in preliminary work which couples compositional two-phase flow with bio-chemical reactions and the dynamics of microbial growth and decay. In the present paper the stability of the coupled dynamic system was analyzed. Dependent on the parameter space it was shown that the system can undergo a limit cycle behavior or diffusion-driven (Turing) instability. The numerical solutions within these parameter regions show different oscillatory regimes. The instability leads to the formation of alternating spots with either a high concentration of H2 or a high concentration of CH4. The injection rate is a decisive factor which controls the behavior of the dynamic system.
-
-
-
A Network Model for the Kinetics of Bioclogged Flow Diversion for Enhanced Oil Recovery
Authors L.A. López Peña, B. Meulenbroek and F.J. VermolenAfter the primary extraction in oil reservoirs up to 60 % of the oil remains trapped in the reservoir (Sen, 2008). Therefore, different mechanisms have been developed to get the oil out to the reservoir. One of these techniques is Microbial Enhanced Oil Recovery (MEOR) which is a technique used to produce more oil in a secondary extraction by using microbes in the reservoir. The main effects caused by microbes in oil recovery is the reduction of the interfacial tension between oil and water, wettability change of the rock and bioclogging caused by the growth and development of biofilm. Among these mechanisms, interfacial tension reduction and biclogging is thought to have the greatest impact on recovery (Sen, 2008). In this work, we describe the growth of biofilm, the growth of the microbial population and the transport of nutrients using a pore network model. We follow the previous models of Thullner et al. (Thullner, 2008) and Ezeuko et al. (Ezeuko, 2011) in which the biofilm is considered as a permeable layer. We consider the biofilm and the bacteria separately. Additionally, we assume that once a tube is full with biofilm, this biofilm can spread to the neighboring tubes. Finally, we study the changes in the hydrodynamic properties of the medium caused by the plugging of the pores and we study the flow diversion of water caused by plugging of the high permeability zones.
-
-
-
The HLD Model for Optimum Phase Behavior Formulation of Ternary Surfactant Mixtures
Authors M. Mavaddat, S. Riahi and A. BahramianBecause many surfactant flooding formulations involve surfactant mixtures, mixtures are generally used to obtain the best formulation to increase the oil recovery. This study is focused on an experimental study to investigate the phase behavior of the mixture of two and three anionic surfactants in the Winsor type III condition. The HLD model has been used as an equation of state to predict the properties of the water-oil-surfactant systems. The model was applied to conduct single and double salinity scans to find the optimal salinity for an unknown surfactant and for the mixture of two and three surfactants at the same total surfactant concentration. Results show that in a binary anionic surfactant mixture, the concentration of the second anionic surfactant (in molar fraction) has a linear relation to the logarithm of the optimal salinity at the same system, which follows the ideal mixing rule in the system. Ternary anionic surfactant mixtures are found to exhibit a slight deviation from ideal at some surfactant compositions. The deviations were fitted using a three-parameter Margules equation. The results suggest that the deviations are due to the change in the surfactant molar fraction on the surface, which is caused by presence of highly branched surfactants.
-
-
-
A Model for Non-Newtonian Flow in Porous Media at Different Flow Regimes
Authors O.M. Nødland, A. Lohne, A. Stavland and A. HiorthThe EOR potential of polymer flooding is well documented in the scientific literature. However, it has remained a challenge to create good simulation tools that can be used for predictive purposes. A main limitation with the current models is the insufficient description of the transition between the different flow regimes that characterize the polymer rheology. Typically, Newtonian behaviour is observed at low shear rates, followed by shear-thinning, shear-thickening and shear-degradation regimes at increasing shear rates. Furthermore this is complicated by the fact that the apparent viscosity of the polymer is influenced by a combination of factors, such as adsorption, brine salinity, polymer concentration and molecular weight. In this work we present a core scale simulation model that is capable of describing all the aforementioned flow regimes. The novel feature of the proposed model is the inclusion of an equation to describe polymer (mechanical) degradation. The polymer degradation rate is linked to the effective pore radius (via permeability through a Kozeny-Carman type equation), wall shear stress, and polymer molecular weight, Mw. The degradation results in a lower Mw, while the polymer volumetric concentration is unaffected. The change in Mw over a time step is found using an implicit chord method at the end of each transport time step, and the solution is then used to update the effective polymer properties. The main flow field is computed using a standard sequential algorithm, where a linear pressure equation is solved first, followed by an implicit saturation equation formulated in a fractional flow approach. The model is applied to a series of laboratory experiments. Our model explains the core data very well, taking into account that several experimental factors have been varied such as synthetic polymer types, core length and permeability.
-
-
-
Wettability Alteration with Smart Water on Carbonate Reservoirs - Effect of Phosphate and Sulfate Multivalent Anions
Authors H. Moradi, A.H. Saeedi Dehaghani, A. Bahramian and M.N. KardaniIncreasing evidence has shown that the adjusting of ionic composition of water injection in the reservoir can improve oil recovery. The carbonate reservoirs comprise about the 50% of oil reservoirs and most of them are neutral to oil wet. More than 65% of carbonate reservoirs are oil wet and 12% are intermediate wet. Wettability is a very important factor in the oil recovery mechanisms. Because it has drastic effect on the distribution and flowing of the oil and water in the reservoirs. Ionic composition adjusting of water which is referred as smart water is known as one of wettability alteration method in the carbonate reservoirs. Although the mechanisms behind of this technic don’t have recognized as well but wettability alteration is a result of these mechanisms. The effect of individual ions must be investigate for better recognition of mechanisms causing improving oil recovery. For complete investigation of mechanisms of smart water technic it must be studied about the crude oil/formation water/injection water/rock system but this system is very complex and for better understanding it must become simpler. In this study, the aim is the wettability alteration deal with smart water using sessile drop method for contact angle measurement. The effect of phosphate and sulfate multivalent anions in different temperature, pH and concentration of them was studied. Modified brines containing phosphate salt were more effective than with sulfate salt. In acidic pH both anions have not good effect but in upper pH they can alter wettability to more water wetting. The phosphate was more soluble in upper temperatures and both anions have better wettability alteration properties to water wetting.
-
-
-
Numerical Modelling of Non-newtonian Fluid Flow in Fractures and Porous Media
Authors K. Bao, A. Lavrov and H.M. NilsenNon-Newtonian fluids having Bingham or power-law rheology are common in many applications within drilling and reservoir engineering. Examples of such fluids are drilling muds, foams, heavy oil, hydraulic-fracturing and other stimulation fluids, and cement slurries. Despite the importance of non-Newtonian rheology, it is rarely used in reservoir simulators and fracture flow codes. We study two types of non-Newtonian rheology: the truncated power-law (Ostwald-de Waele) fluid and the Bingham fluid. For either of the two types of non-Newtonian rheology, we construct relationships between the superficial fluid velocity and the pressure gradient in fractures and porous media. The Bingham fluid is regularized by means of Papanastasiou-type regularization for porous media and by means of a simple hyperbolic function for fracture flow. Approximation by Taylor expansion is used to evaluate the fluid velocity for small pressure gradients to reduce rounding errors. We report simulations of flow in rough-walled fractures for different rheologies and study the effect of fluid parameters on the flow channelization in rough-walled fractures. This effect is known from experiments and from previous numerical studies. We demonstrate how different rheologies on different domains can be included in a fully-unstructured reservoir simulation that incorporates discrete fracture modeling (DFM). The above formulation was implemented in the open-source Matlab Reservoir Simulation Toolbox (MRST), which uses fully implicit discretization on general polyhedral grids, including industry-standard grids with DFM. This robust implementation is an important step towards hydro-mechanically coupled simulation on hydraulic fracturing on reservoirs with realistic non-Newtonian fluid rheology.
-
-
-
On the Coupling of Two-phase Free Flow and Porous Flow
Authors Z.Q. Huang, B. Gao, X.Y. Zhang and J. YaoThe coupling of free flow with porous flow is of special interest in a wide range of environmental phenomena and industrial applications. In this work, we extend the classical single-phase two-domain model to a laminar two-phase coupling flow system. The free fluid region can be considered as separated two-phase flow for simplicity, which is modeled by using the Navier-Stokes and Cahn-Hilliard equations. And the mathematical model of two-phase flow in porous media is based on Darcy’s theory. The main challenge is how to introduce specific interface conditions to couple these two models. To this end, the normal continuity conditions of flux and forces are developed, and an extended Beavers-Joseph-Saffman condition for two-phase flow system is also proposed as a Cauchy boundary condition based on consistent phenomenological explanations. These lead to a simple and solvable coupling model, and an efficient finite element numerical scheme is developed. The numerical results show that the developed model is capable to capture the macroscopic flow characteristics of laminar two-phase coupling flow system by comparing to the experimental results. Our model can be used to model the related two-phase flow process in karstic aquifers and fractured reservoirs, and wind-driven evaporation from soil.
-
-
-
Pseudo-3D Hydraulic Fracture Model with Complex Mechanism of Proppant Transport and Tip Screen Out
Authors A. Bochkarev, S. Budennyy, R. Nikitin and D. Mitrushkintip screen-out (TSO), packed (final) fracture geometry, fluid rheology changing caused by proppant presence. Proppant transport governs fracture shape and effective fracture area. The key objective of present work is to develop a joint solution of fracture growth within the multi-layered lithology and multiphase flow inside to present a complete fracture model, full enough for design-execution-evaluation cycle. The project is carried-out with financial support of the Ministry of Education and the Ministry of Education and Science of the Russian Federation in the framework of the project №14.581.21.0008 from 03.10.2014 г (ID RFMEFI58114X0008)
-
-
-
Accelerating Coupled Dynamic Flow and Geomechanical Systems for Complex Reservoir Simulations
Authors A. Pearce, H. Mustapha, S. Kisra, P. Welsh and K.H. LeeCoupled reservoir and geomechanical simulations are required to account accurately for pressure variations and induced stress changes during the lifecycle of oil and gas fields. This understanding is critical for mitigating drilling and completion hazards and for predicting compaction or subsidence in reservoirs containing weak and fractured rocks. 3D full-field models with accurate rock and fluid properties have become increasingly possible due to advances in the hardware and software involved in determining property distribution. This has subsequently led to a rise in the size and complexity of the simulations and computational efforts. Typically, geomechanics simulations are computationally more expensive than reservoir simulations due to the grid size required to model a geomechanical problem. This paper introduces a new workflow based on the latest multiparallel high-resolution unstructured reservoir simulator coupled to a geomechanics simulator using enhanced linear solvers with advanced deflation algorithms. A performance analysis reveals that the proposed coupled system is significantly faster than the existing workflow. The accelerated coupled system brings the solution of full 3D geomechanics problems within the reach of a wider community of engineers using laptop or desktop computers, enabling them to deliver answers to field problems faster and without the need to access high-end supercomputers.
-
-
-
The Impact of Layering and Heterogeneity on Stresses around Boreholes
Authors H. Agheshlui and S.K. MatthaiAn accurate understanding of orientation and magnitude of the stresses surrounding a borehole is decisive for the identification of a stable well path and the successful design of completions and stimulation measures including fracking stages. However, measurement of the in situ stress is challenging: Current engineering practice favours two approaches: 1) borehole break-out – drilling-induced fracture interpretation complemented by extended leak-off tests; 2) stress estimation from shear-wave slowness measured with advanced wireline tools. Both methods rely on the applicability of Kirsch’s (1898) stress perturbation equations and laboratory measurements of elastic moduli that are correlated with rock properties that can be logged. Method (2) can be applied where breakouts are absent. Yet little is known how lithologic layering and spatial variations in the mechanical properties of the rock affect its results and the detection limit of stress anisotropy. Here we present finite-element simulations of borehole-related stress perturbations in multi-layer composites with realistic scattering and spatial variations in elastic moduli. Using stress magnitude-shear wave correlations from the literature, travel times are calculated for borehole-parallel trajectories. These results are interpreted in terms of the minimum differential stress that needs to exist in order to be able to detect stress directions. The uncertainty of measurements of stress magnitudes obtained with this method is analysed as well. Our results show that the stress field around boreholes is strongly affected by lithological variations. “Ideal” Kirsch-compatible conditions where the well is aligned with one of the eigenvectors of the stress field, layers are perpendicular to the well, and far-field stresses are Andersonian is rare. More common scenarios are going to be illustrated with a series of simulations including deviated wells. These will also elucidate how rock stress responds to fluid pressure changes in the well.
-
-
-
Virtual Element Method for Geomechanical Simulations of Reservoir Models
Authors X. Raynaud, H. Nilsen and O. AndersenIn this paper we study the use of Virtual Element method for geomechanics. Our emphasis is on applications to reservoir simulations. The physical processes that form the reservoirs, such as sedimentation, erosion and faulting, lead to complex geometrical structures. A minimal representation, with respect to the physical parameters of the system, then naturally leads to general polyhedral grids. Numerical methods which can directly handle this representation will be highly favorable, in particular in the setting of advanced work-flows. The Virtual Element method is a promising candidate to solve the linear elasticity equations on such models. In this paper, we investigate some of the limits of the VEM method when used on reservoir models. First, we demonstrate that care must be taken to make the method robust for highly elongated cells, which is common in these applications, and show the importance of calculating forces in terms of traction on the boundary of the elements for elongated distorted cells. Second, we study the effect of triangulations on the surfaces of curved faces, which also naturally occur in subsurface models. We also demonstrate how a more stable regularization term for reservoir application can be derived.
-
-
-
The Interplay of Capillary and Viscous Forces Driving Flow through Layered Porous Media
Authors Y. Debbabi, M.D. Jackson, G.J. Hampson, P.J.R. Fitch and P. SalinasWe examine the impact of viscous and capillary forces on immiscible, two-phase flow parallel and perpendicular to continuous layers of contrasting material properties. We consider layers of contrasting porosity and relative permeability, in addition to the contrasts in absolute permeability investigated previously. We define a set of dimensionless numbers which characterize flow. Some of these are common to flow both parallel and perpendicular to layering, such as the longitudinal permeability ratio σ and the ratio Rs of the moveable pore volumes (MPV) in each layer. Others are specific to a given flow direction, such as the dimensionless capillary to viscous ratio Ncv, and the effective aspect ratio RL that quantifies crossflow for layer-parallel flow. We examine how variations in the dimensionless numbers affect the trapping/recovery efficiency, defined as the fraction of the model MPV occupied by the injected phase after 1 MPV injected, and which is numerically equivalent to the fraction of the displaced phase recovered from the model after 1 MPV injected. The results are directly applicable to geological carbon storage and hydrocarbon production. We find that the trapping efficiency is clearly controlled by the dimensionless numbers. When flow is perpendicular to layering, heterogeneity only influences flow when capillary forces are significant (Ncv>0). As Ncv is increased, a larger fraction of the non-wetting phase is trapped if the layers have contrasting capillary pressure curves. When flow is parallel to layering, both viscous and capillary forces are important. In the viscous limit (Ncv=0), heterogeneity reduces trapping efficiency if σ≠Rs. As capillary forces become more significant (Ncv increases) and if crossflow between layers can occur (RL>0), the trapping efficiency also increases in response to capillary crossflow and reaches a maximum at a given Ncv. At higher Ncv, the benefit of crossflow is outweighed by along layer diffusion of the injected phase.
-
-
-
Low Salinity Carbonated Waterflooding
Authors T. Blom, A.C. Alvarez, W.J. Lambert, D. Marchesin and J. BruiningIt has been shown in the literature that a secondary low salinity waterflood can improve the oil recovery by 5-20%. A possible mechanism is that the low salinity causes desorption of organic material, which may increase water-wetness and lead to more favorable relative permeability behavior. A less well-known mechanism is enhanced solvent (e.g., carbonated water) recovery as low salinity enhances the aqueous solubility of neutral components, which after injection will be transferred from the aqueous phase to oleic phase thus decreasing the oil concentration in the oleic phase and diluting the residual oil. By way of example we consider a low salinity carbonated waterflood into a reservoir containing oil equilibrated with high salinity carbonated water. For a given pH, the CO2 equilibrium concentration in low salinity injection water is higher than in the high salinity initial water. PHREEQC, a geochemical aqueous equilibrium programme, can be extended to obtain the accurate partition coefficient of neutral species that are soluble both in the oleic and the aqueous phase. For this we use the Krichevsky-Ilinskaya extension of Henry’s law for solubility of gases in liquids. Gibbs phase rule shows that the phase behavior only depends on the pH and the chloride concentration. In PHREEQC, we use Pitzer’s activity coefficients to extend the validity up to 6M. The output of PHREEQC can only be successfully incorporated in multiphase flow simulation programmes, e.g. COMSOL(TM), after applying a smoothing procedure for which we choose symbolic regression (EUREQA(TM)). An optimal formulation avoids spurious broadening of the concentration profiles in contact “discontinuities”. We obtain the saturation, composition and the total Darcy velocity profiles. The significant new insight is that by changing the salinity at constant pH the oil recovery by carbonated water flooding can be enhanced. This insight can be applied to optimize enhanced oil recovery with a low salinity waterflood.
-
-
-
Analyses of Geo-statistical Modelling, in the Wavenumber Domain, for Multi-dimensional Models
By J. LeguijtThe probabilistic seismic inversion program, Promise, has been equipped with a module that is able to account for lateral continuity. In this module, the prior probability density function (pdf) is generated using two point statistics. Bayes rule is used to account for the observations; the result is a posterior pdf. The observations consist of seismic traces and arrival times of interpreted horizons. An iterative algorithm is deployed to sample the posterior pdf. This visits all the locations in its model, numerous times in succession and effectively creates a Metropolis algorithm. Unexpected results are sometimes generated by this. Consequently, a closer examination of the algorithm and its theoretical background has been partaken. The models that are used in Promise are multi-dimensional; a one to one mapping between the model parameters and the observations does not always exist. This makes the analyses more complicated. The uncertainty in the observations is represented by the seismic noise. For convenience, it is often assumed that the seismic noise at different locations is statistically independent. This is often an erroneous assertion and causes the modeling results to be adversely affected. To make the analyses comprehensible, they are carried out in the wavenumber domain, for models with a linear relation between the model parameters and the observations. Some of the learnings will be discussed for models that are based on a 1D grid.
-
-
-
Oil Flow Rate History Reconstruction using Downhole Transient Temperature Data with Wavelet Transform
More LessFlow rate history is important for pressure-transient analysis and history matching, but due to high cost of metering equipment, the accurate real-time flow rate history for individual well is not always available. This paper presents a method of reconstructing unknown flow rate history from downhole transient temperature data with wavelet transform (WT). Firstly, the time of flow rate change was identified from transient temperature data with the Haar wavelet. As reservoir temperature changes dramatically at the time of flow rate change due to the Joule-Thomson effect, and the time of flow rate change can be identified in the WT detailed signal of the transient temperature data. Then, the proportional relationship between the amplitude of WT coefficients and flow rate change was discovered and theoretically proved. Based on this relationship, the flow rate for each flow event was allocated back from the cumulative production. Finally, with the identified time and calculated flow rate for each flow event, the unknown flow rate history was reconstructed from downhole transient temperature data. This method works well in the oil reservoirs with constant reservoir and fluid properties, and the reliability of this method was demonstrated with a field case study.
-
-
-
Discrete Fracture Method for Petroleum Reservoir Simulation Using an Element-based Finite Volume Method
Authors C.R. Maliska, B.T. do Vale and F. MarcondesAny porous media flow is inherently complex to model due to the impossibility of giving the real flow geometry to the simulation model. If heterogeneous media is involved, as is the case of naturally fractured petroleum reservoirs, the difficulty increases even more, since large fractures can be seen as discontinuities, having as background the porous matrix, in which many smaller sized fractures are present. The porous matrix can be treated by a stochastic procedure and are, normally, large deposits of oil, while the large fractures are better solved through a deterministic treatment. A network of connected large fractures linked to the porous matrix may be the most important flow path for oil production. In the other hand, depending on the physical properties, capillarity and permeability, the fractures and porous matrix combination may lead to a undesirable secondary oil recovery, leaving considerable amount of oil in the porous matrix. Therefore, the prediction of this combined flow (fractures +porous matrix) is of utmost importance for the oil industry. There are several approaches to solve this combined flow, all of them based, of course, on a idealized fracture configuration, which gets more and more realistic as the characterization methods evolves, due to the specialization of well-logging, 4D seismic and other methods. The final goal would be to solve the local flow for any single fracture, irrespective its size, nowadays an impossible task due to the lack of characterization methods and computer capacity. However, as computational power and characterization techniques evolve, methods able to solve the details of the flow should be devised. This paper follows this route and presents a DFM (Discrete Fracture Method) in the framework of an Element-based Finite Volume Method (EbFVM) using unstructured grids. The EbFVM is per se a multi-point flux approximation, avoiding the usual two-point approach, which is conceptually wrong, since the errors do not vanish as the grid is refined. The EbFVM also avoids the need of more complex MPFA algorithms for having correct flux evaluation. Additionally, the EbFVM framework allows the use of truly directional upwind and higher order schemes with no extra efforts. The 2D oil-water flow using the DFM method with superposition for connecting the fractures and the porous matrix is solved. The fractures are assumed to be 1D considering its real thickness. Several aspects of the model are investigated, as the capillarity effects, especially in the situation in which imbibition of the porous matrix occurs and the anisotropy of the coefficients of the linear system resulting from the superposition of the equations. Since IMPES method is used, comments on the time step adaption is also given.
-
-
-
Simulation of Densely Fractured Tight Oil Reservoirs Using a New Simulator with Automatic Differentiation
Authors W.C. Fang, H.Q. Jiang, J. Killough, J.J. Li, W.C. Teng, L.K. Li and L. ZhaoDevelopment of tight oil reservoirs mainly relies on massive hydraulic fracturing, which can generate complex fracture networks in the reservoirs. These highly discrete fracture networks bring great challenges for reservoir simulation. An efficient model applying an unstructured discretization method and automatic differentiation is proposed. The flexibility in unstructured control volume shapes enables the gridding of complex fracture systems. By introducing the concept of half-transmissibility for each grid, transmissibility list including connections of matrix-matrix, matrix-fracture, fracture-fracture is established. Nonlinear flow and transport equation system is solved by a modified Newton’s method, in which the Jacobian matrix is computed by automatic differentiation (AD). Accuracy of the model was validated by performing simulations using a commercial simulator. We implemented our model in several cases with a uniform physical domain (radial model with diameter=1 km) but different fracture properties. Results show that fracture configuration and property have significant impacts on the production. Moreover, our model shows a high efficiency in the densely fractured system with fracture density up to 1018/km2. The novelty of the model is in the ability to represent complex fracture systems individually and explicitly, and in the application of automatic differentiation, which greatly facilitates the model establishment and improves computational efficiency.
-
-
-
Mixed Hybrid Finite-element Formulation for General Purpose Reservoir Simulation
Authors A.S. Abushaikha, D.V. Voskov and H.A. TchelepiWe present a mixed hybrid finite-element (FE) formulation for modeling subsurface flow and transport for general-purpose compositional reservoir simulation. The formulation is fully implicit in time and employs a hybrid FE method for the spatial discretization of the conservation equations. The hybrid FE formulation is implemented in the Automatic Differentiation General Purpose Research Simulator (ADGPRS); consequently, the new FE-based methodology inherits all the `physics’ capabilities of ADGPRS, including compositional EOR models. The high-order mixed hybrid FE discretization scheme works for many types of finite elements and can handle highly anisotropic material properties. The formulation is locally conservative. The momentum and mass balance equations are solved simultaneously, including Lagrange multipliers on element interfaces. The fully implicit scheme uses the automatic differentiation capability to construct the Jacobian matrix. The hybrid FE approach accommodates unstructured grids, which are needed for honouring the complex geometry of the subsurface, in a straightforward manner. We present compositional test cases with full permeability tensors, and we discuss the accuracy and computational efficiency of the formulation. We also compare the performance of the hybrid FE-based scheme with finite-volume based Multi-Point Flux Approximation (MPFA) methods.
-
-
-
3D Geological Feature Honored Cell-centered and Vertex-centered Unstructured Grid Generation, and CVD-MPFA Performance
Authors S. Manzoor, M.G. Edwards, A.H. Dogru and T.M. Al-ShaalanGrid generation for reservoir simulation, must honor classical key geological features and multilateral wells. For the purpose of grid generation, the geological features are classified into two groups; 1) involving layers, faults, pinchouts and fractures, and 2) involving well distributions. In the former, control-volume boundary aligned grids(BAGs) are required, while in the latter, control-point well aligned grids(WAGs) are required. In reservoir simulation a choice of grid type and consequent control-volume type is made, i.e. either primal or dual-cells are selected as control-volumes. Regardless of control-volume type, the control-point is defined as the centroid of the control-volume. Three-dimensional unstructured grid generation methods are proposed that automate control-volume boundary alignment to geological features and control point alignment to wells, yielding essentially PEBI-meshes either with respect to primal or dual-cells depending on grid type. In the grid generation methods presented, for both primal and dual-cell feature based meshes, both frameworks use primal-cells (tetrahedra, pyramids, prisms and hexahedra) as grid elements. Dual-cell feature honored grids are derived from underlying primal-meshes such that features are recovered in the dual-setting. Geological features are honored by using the idea of protection spheres, and protection halos around key geological features. Halo construction requires the use of prisms and/or hexahedra. Pyramids are used as transition elements providing interfaces between quad faces of the halo elements and triangular faces of the main tetra-mesh. A novel method for constructing pyramids as transition elements in an unstructured mesh together with a novel technique for ensuring fully constrained recovery of geological features is proposed. The grids generated are employed to study comparative performance of cell-vertex versus cell-centred CVD-MPFA finite-volume formulations using equivalent degrees of freedom. The benefits of both types of approximation are presented in terms of flow resolution relative to the respective degrees of freedom employed. The cell vertex method proves to be the most beneficial with respect to accuracy and efficiency.
-
-
-
Immiscible Two-phase Darcy Flow in Fractured Porous Media - New Robust Formulation and Application to the Tight Gas Recovery
Authors K. Brenner, M. Groza, L. Jeannin, R. Masson and J. PellerinNumerical simulations of two-phase Darcy flows in heterogeneous porous media requires choosing an appropriate set of primary unknowns, which may be challenging, especially when dealing with very flat capillary pressure curves, dry regions or saturation jumps at the rock type interfaces. The classical approaches fail to cope with all of those difficulties. In particular the two-pressure formulation allows handling saturation jumps, but beaks down if the capillary pressure doesn’t depend on saturation. It also lacks robustness when dealing with nearly residual water saturations. On the other hand, for homogeneous medium, the pressure – saturation formulation is known to be robust when dealing with dry media and can handle vanishing or constant capillary pressure curves. Unfortunately it is not always possible to extend it to the case of discontinuous capillary pressure curves. In this paper, a new formulation based on parametrization techniques for the capillary pressure monotone graph extension is proposed which handles all the above mentioned difficulties while still using only two unknowns by degree of freedom. We illustrate the efficiency of our approach by numerous numerical experiments dealing with water gas flow in fractured tight gas reservoirs using the data set presented in [2]. Following [1], the fractures are modelled as interfaces of codimension one with continuous pressure at the matrix fracture interfaces. During the injection phase of the simulation, water penetrates only a few tens of centimetres deep in the matrix rock. Therefore, in order to obtain an accurate numerical approximation, an anisotropic refinement of the mesh is used in the neighbourhood of the fractures using prismatic elements. The connection with the surrounding tetrahedral mesh in the matrix domain is achieved using pyramids. Following [1], the model is discretized using the Vertex Approximate Gradient scheme which allows for general polyhedral cells. The numerical performance of the new approach is evaluated for various choices of capillary pressures curves. The comparison with classical formulations shows that the new approach is more efficient both in terms of Newton iterations and CPU time. [1] K.Brenner, M.Groza, C.Guichard, R.Masson: Vertex Approximate Gradient Scheme for Hybrid Dimensional Two-Phase Darcy Flows in Fractured Porous Media, M2AN, pp.49 2, 303-330, 2015. [2] D.Y.Ding, H.Langouet, L.Jeannin: Simulation of Fracturing Induced Formation Damage and Gas from Fractured Wells in Tight Gas Reservoirs, SPE 153255, 2012.
-
-
-
Enhanced Nonlinear Finite Volume Scheme for Multiphase Flows
Authors K. Nikitin, V. Kramarenko and Y. VassilevskiWe present the latest enhancement of the nonlinear monotone finite volume method for the near-well regions. The original nonlinear method is applicable for diffusion, advection-diffusion and multiphase flow model equations with full anisotropic discontinuous permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which reduces to the conventional two-point flux approximation (TPFA) on cubic meshes but has much better accuracy for the general case of non-orthogonal grids and anisotropic media. The latest enhancement of the nonlinear method takes into account the nonlinear (e.g. logarithmic) singularity of the pressure in the near-well region and introduces the nonlinear correction to improve accuracy of the pressure and the flux calculation. The new method is generalized for anisotropic media, polyhedral grids and nontrivial wells cases. Numerical experiments show the noticeable reduction of the numerical errors compared to the original monotone nonlinear FV scheme with the conventional Peaceman well model and even with the given analytical well rate.
-
-
-
A Topological Approach for Automated Unstructured Meshing of Complex Reservoir
Authors V. Gauthier, A. Arnould, H. Belhaouari, S. Horna, M. Perrin, M. Poudret and J.F. RainaudEstimations of petroleum reserves rest on finite volumes computational simulations of the reservoir fluid dynamics. These simulations are operated on 3D meshed reservoir models produced by a complex and poorly automated chain of operations. This paper proposes a mesh building methodology, which uses geological rules for building reservoir meshes in a more automated way. We start from a surface structural model and from a description of the stratigraphy both packed thanks to the industry standard RESQML. We construct a volume structural framework based on generalized map topological structures. These structures include topological boundary relations between the represented geological objects (horizons, faults, units) and some dedicated data attached to the topological cells (vertices, faces, volumes, etc.), such as geometry or geological labels (e.g. names, relative ages, deposit methods). In particular, on a single topological representation, we can attach two different geometric representations respectively describing the present day layer geometry (“folded model”) and the original positions of the various layers in their “deposition space” ("unfolded model"). Thanks to a dedicated rule-based language, we deduce from the geological interpretation, a set of topological and geometric operations that allow an automated building of the structural framework, on which the reservoir meshes will be implemented. This language allows a fast prototyping of complex operations (boolean operations for instance) and it guarantees the geological and topological consistency of the model. Using this consistent and fully informed structural framework, we can create in an automated way various conformal unstructured 3D meshes organized in layers. These meshes agree both with the topology induced by the succession of deposition, erosion and tectonic events that constitute the local geological history and with the peculiarities of the used fluid flow simulators. A use-case is presented to demonstrate the feasibility of our method.
-
-
-
Numerical Simulation of Three-dimensional Complex Fracture Geometries Using an Unstructured Voronoi Mesh
Authors Y. Wang and S.A. AryanaTo date, most numerical simulators developed to simulate production from unconventional gas reservoirs focus on two-dimensional (2d) representations of geological properties. Fracture geometry is, however, highly uncertain and three-dimensional models would enable more robust representations while considering gravity effects. In this paper, a Discrete Fracture Model (DFM) of 3d complex fracture geometries is developed where unstructured PEBI grids are used to discretize flow equations. The mesh generation algorithm, implemented in 3d space, is based on the equilibrium state of forces in a truss system. Each fracture is modeled as a 2d plane with an arbitrary orientation and shape. Nodes along the fracture plane are populated uniformly while the proposed dynamic force-based model is applied in the matrix region. A distance function is developed to assist node population in the matrix region. Fractures are regarded as bounded plane-constraints and the distance function is expressed in terms of geometrical information from nodes in the matrix and fracture planes in 3d space. Assuming isotropic media, the generated mesh guarantees local orthogonality, and two-point flux approximations (TPFA) are used in the numerical discretization scheme. Hydraulic and induced fractures and their intersections are accounted for explicitly without invoking transformations. A sensitivity study is performed to investigate the effect of grid size in regards to embedded fractures and optimal node density to balance accuracy and efficiency. The fully 3d model is validated against results from a commercial simulator using a synthetic case. Inclusion of gravity is shown to have a significant impact on gas and water production rates. In summary, the proposed approach enables modeling complex fractures in fully 3d space using an unstructured Voronoi mesh with gravity effects and provides a better understanding of two-phase systems representing recovery from fractured unconventional gas reservoirs.
-
-
-
C1-PPU Schemes for Efficient Simulation of Coupled Flow and Transport with Gravity
Authors J.M. Jiang and R.M. YounisIt is revealed by recent studies that in the presence of counter-current flow due to buoyancy, nonlinear convergence problems may be pronounced when the popular PPU scheme is used to approximate the numerical flux. The PPU numerical flux is non-differentiable across the co-current/counter-current flow regimes and thus may lead to oscillations or even divergence in the Newton iterations. Recently proposed methods address improved smoothness of the numerical flux. In this paper we devise and analyze an alternative numerical flux scheme called C1-PPU that allows a smooth variation between the co-current/counter-current flow regimes as well as an optimal balance between the scalar nonlinearity and accuracy of the flux function. The C1-PPU scheme involves a novel use of the flux limiter concept from the context of high-resolution methods. Numerical examples including 1D scalar transport problem and 2D heterogeneous problem with fully-coupled flow and transport are presented. The results indicate that in addition to smoothness, nonlinearity may also be critical for convergence behavior and thus needs to be considered in the design of an efficient numerical flux scheme. Moreover, the results show that our C1-PPU scheme exhibits superior convergence properties for large time steps compared to the other alternatives.
-
-
-
Simulation of Immiscible Viscous Fingering Using Adaptive Unstructured Meshes and Control-volume Galerkin Interpolation
Authors A.G. Adam, D. Pavlidis, J.R. Percival, P. Salinas, Z. Xie, C.C. Pain, A.H. Muggeridge and M.D. JacksonDisplacement of one fluid by another in porous media occurs in various settings including hydrocarbon recovery, CO2 storage and water purification. When the invading fluid is of lower viscosity than the resident fluid, the displacement front is subject to a Saffman-Taylor instability and is unstable to transverse perturbations. These instabilities can grow, leading to fingering of the invading fluid. Numerical simulation of viscous fingering is challenging. The physics is controlled by a complex interplay of viscous and diffusive forces and it is necessary to ensure physical diffusion dominates numerical diffusion to obtain converged solutions. This typically requires the use of high mesh resolution and high order numerical methods. This is computationally expensive. We demonstrate here the use of a novel control volume - finite element (CVFE) method along with dynamic unstructured mesh adaptivity to simulate viscous fingering with higher accuracy and lower computational cost than conventional methods. Our CVFE method employs a discontinuous representation for both pressure and velocity, allowing the use of smaller control volumes (CVs). This yields higher resolution of the saturation field which is represented CV-wise. Moreover, dynamic mesh adaptivity allows high mesh resolution to be employed where it is required to resolve the fingers and lower resolution elsewhere. We use our results to re-examine the existing criteria that have been proposed to govern the onset of instability. Mesh adaptivity requires the mapping of data from one mesh to another. Conventional methods such as collocation interpolation do not readily generalise to discontinuous fields and are non-conservative. We further contribute a general framework for interpolation of CV fields by Galerkin projection. The method is conservative, higher order and yields improved results, particularly with higher order or discontinuous elements where existing approaches are often excessively diffusive.
-
-
-
Cell-centred Higher Resolution Finite-volume Total Velocity Vt and Va Formulations on Structured and Unstructured grids
Authors Y. Xie and M.G. EdwardsY.Xie [email protected], [email protected] Novel cell-centred finite-volume formulations are presented for two-phase flow with gravity and capillary pressure on structured and unstructured grids. The Darcy-flux is approximated by a control-volume distributed multipoint flux approximation (CVD-MPFA) coupled with a higher resolution approximation for convective transport. The CVD-MPFA method is used for Darcy-flux approximation involving pressure, gravity and capillary pressure flux operators. Two formulations for coupling the pressure equation with fluid transport are presented. The first is based on the classical total velocity Vt fractional flow (Buckley Leverett) formulation, and the second is based on a more recent Va formulation. The CVD-MPFA method is employed for both the Vt and the Va formulations. The advantages of both coupled formulations are contrasted. The methods are tested on a range of structured and unstructured quadrilateral and triangular grids. The tests show that the resulting methods are found to be comparable for a number of classical cases, including channel flow problems. However when gravity is present, flow regimes are identified where the Va method becomes locally unstable, in contrast to the total velocity formulation. The test cases also show the advantages of the higher resolution method compared to standard first order single point upstream weighting.
-
-
-
Compositional Dual Mesh Method for Single Phase Flow in Heterogeneous Porous Media - Application to CO2 Storage
Authors D. Guerillot and J. BruyelleThe geological static models of realistic contexts are described with high resolution meshes (HRM) and cannot be directly used as input for fluid flow reservoir simulators due to memory and/or running time constraints. The pragmatic approach consists in averaging the high resolution petrophysical values to assign to a low resolution mesh (LRM) used to perform reservoir simulations. Hence, predictions made with these coarser meshes are inevitably less accurate than those that would have been obtained on HRM. For compositional modelling, the loss of accuracy due to upscaling processes will come not only for the component displacements but also from the solution of the thermodynamic and/or geochemical equilibrium equations. For example, a chemical reaction of an acid on carbonated rock may highly depends on its concentration. Therefore, our main motivation here is to keep an HRM for calculating those chemical equilibriums. We propose to name this innovative approach “Compositional Dual Mesh Method” (CDMM). The CDMM is a formulation with two different meshes: The pressure equation is solved on a LRM using upscaled properties and the transport equation and chemical equilibrium are solved on a HRM.
-
-
-
Modelling Near-well Flow Performance for Horizontal Wells in Anisotropic Media
Authors J. Cao, L.A. James and T.E. JohansenThis paper presents a novel methodology to model flow performance in an anisotropic reservoir in the near-well region with an arbitrary well trajectory. It is based on an analytical productivity model describing coupled axial reservoir flow and radial well inflow. To apply this model in an anisotropic reservoir, the permeability field relative to the radial direction perpendicular to the well trajectory and the axial direction along the well trajectory must first be determined. A classical transformation is used to obtain a virtual isotropic model. The transformation preserves the volumes and average pressures. It is applied in the near-well region without modifying the outer boundary conditions. The use of this virtual isotropic model requires the Dietz shape factor for an ellipse, which is determined numerically. For example, in a circular-cylindrical near-well region, this transformation method maps the anisotropic reservoir onto an equivalent virtual isotropic media which is an elliptical cylinder. The coupled axial and radial productivity model is implemented in a numerical simulator incorporating formation anisotropy and wellbore hydraulics. The specific productivity index along the well trajectory is generated using the virtual configuration. Numerical results for different anisotropy ratios and also incorporating frictional losses in the well are presented.
-
-
-
FV-MHMM Methods for Reservoir Modelling
Authors J. Franc, L. Jeannin, R. Masson, P. Egermann and G. DebenestThe present paper proposes a new family of multiscale finite volume methods. These methods usually deal with a dual mesh resolution, where the pressure field is solved on a coarse mesh, while the saturation fields, which may have discontinuities, are solved on a finer reservoir grid, on which petrophysical heterogeneities are defined. Unfortunately, the efficiency of dual mesh methods is strongly related to the definition of up-gridding and down-gridding steps, allowing to define accurately pressure and saturation fields on both fine and coarse meshes and the ability of the approach to be parallelized. In the new dual mesh formulation we developed, the pressure is solved on a coarse grid using a new hybrid formulation of the parabolic problem. This type of multiscale method for pressure equation called Multiscale Hybrid-Mixed method (MHMM) has been recently proposed for finite elements and mixed-finite element approach [1]. We extend here the MH-Mixed Method to a Finite Volume discretization, in order to deal with large multiphase reservoir models. The pressure solution is obtained by solving a hybrid form of the pressure problem on the coarse mesh, for which unknowns are fluxes defined on the coarse mesh faces. Basis flux functions are defined through the resolution of a local finite volume problem, which accounts for local heterogeneity, whereas pressure continuity between cells is weakly imposed through flux basis functions, regarded as Lagrange multipliers. Such an approach is conservative both on the coarse and local scales and can be easily parallelized, which is an advantage compared to other existing finite volume multiscale approaches. It has also a high flexibility to refine the coarse discretization just by refinement of the Lagrange multiplier space defined on the coarse faces without changing nor the coarse nor the fine meshes. This refinement can also be done adaptively with respect to a posteriori error estimators. The method is illustrated by the application of single phase (well-testing) and multiphase flow in heterogeneous porous media at the field scale. [1] R. Araya, C. harder, D. Parades, F. Valentin, Multiscale Hybrid-Mixed Method, SIAM J. Numer. Anal. 51(6), 3505-3531, 2013.
-
-
-
Modelling of the Waterflooding Process in the Presence of Discontinuities in the Reservoir
Authors E.V. Andriyanova, V.I. Astafev and A.E. KasatkinThe knowledge of the nature of the fluid motion in the reservoir allows us to optimize the system of oilfield development. Thus, the study of the filtration process in reservoirs with discontinuities, such as fractures, has a great importance for the oilfield development. For instance, the hydraulic fracturing is one of the most common recovery methods for the unconventional reserves. But the modern level of geophysics can show that mostly reservoirs have the tectonic faults with various permeability, and that has a great impact on well productivity. This article will show the impact of inclusions of different permeability in the reservoir on the waterflooding process. The steady-state flow process of incompressible fluid to the production well in a reservoir of constant height and permeability is considered. There is a thin area in the reservoir with constant permeability, which might be a highly permeable crack or low permeable barrier. The production and injection wells are placed inside the reservoir’s external boundary. The characteristics of waterflooding process are studied for various permeability values and different locations of the fracture and a pair of wells. Finally, flow lines of the fluid flow will be analyzed for every considered case.
-
-
-
Analysis of Sparse Matrix-vector Multiply for Large Sparse Linear Systems
Authors M. Grossman, M. Araya-Polo, F.O. Alpak, F. Frank, J. Limbeck and V. SarkarDiscretization of the partial differential equations that govern the physics of multi-phase multi-component fluid flow and transport gives rise to large sparse linear systems for practical pore-scale simulation. In this work, we focus on a linear system arising from the discretization of the Cahn-Hilliard equation that governs the separation of a two-component mixture in the pore space. The discretization is performed using the discontinuous Galerkin method. The resulting nonlinear system is solved by use of the Newton's method, which entails multiple large sparse linear systems over Newton iterations course. The sparse linear systems are solved by use of an iterative linear solver. Iterative linear solvers approach the solution process by the computation of sparse matrix-vector (SpMV) products. SpMV products are computational bottlenecks for the simulation of large problems, since they are extremely memory bound. In this work, we contribute with a quantitative and qualitative evaluation of techniques for performing SpMV on large matrices. We evaluate different SpMV software implementations (frameworks and kernels) across a range of state-of-the-art hardware platforms. For example, we find that for a 5GB matrix wrt to a naive multi-threaded x86 baseline the highest performing GPU kernel runs 1.75x faster, and the highest performing x86 kernel runs 1.29x faster.
-
-
-
Reservoir Simulation with High Volume of Tracers whilst Retaining Performance
Authors H. Mustapha and T. JonsthovelTracking fluid movement in a reservoir using single-well tracer and inter-well tracer tests is an important mechanism for optimizing recovery. In full-field simulations with a high number of tracers, the solution to tracer systems can be computationally expensive because each tracer requires the solution of a linear system with dimension equal to the grid size. Hence, for an increasing number of hydrocarbon components, grid size, and number of tracers, the performance of the simulator degrades. Relying on the nature of the material balance equations governing the tracer flow, we propose to utilize a convenient feature of the underlying partial differential equations (PDEs) in that the matrix-form equations of all tracers carried by a given fluid phase or component are the same, and only the right-hand sides of the corresponding linear systems change. We provide an overview of the optimal linear solver for solving tracer linear systems in different scenarios (e.g. high volume of tracers). Moreover, we show that by overlapping computation on the central processing unit (CPU) and the graphic processing unit (GPU), we can significantly reduce the impact of solving the tracer equations on the overall simulator performance, which enables running a simulation with a high volume of tracers.
-
-
-
Operator-based Linearization for Non-isothermal Multiphase Compositional Flow in Porous Media
More LessNon-isothermal multiphase compositional simulation is based on the solution of governing equations describing mass and energy transfer in the subsurface. The solution strategy requires a linearization of strongly nonlinear governing equations describing the process. Usually, a Newton-based method is used for the linearization that demands an assembly of a Jacobian matrix and residuals for a fully coupled system of equations. Recently, a new linearization approach was proposed for compositional problems and tested for simulation of binary compositional and low-enthalpy geothermal flow. The key idea of the approach is the transformation of discretised mass conservation equations to an operator form with separate space-dependent and state-dependent components. This transformation provides an opportunity for an approximate representation of exact physics (physical properties) of the problem. Specifically, each term of conservation equations is represented as a product of two different operators. The first operator depends on a current physical state of a system and contains different properties such as density, viscosity, relative permeability, etc. The second operator captures both spatially altered properties such as permeability and the rest of state variables such as pressure in the discrete approximation of the gradient. At the pre-processing stage, all state-dependent operators are uniformly parametrized within the physical space of the problem (pressure-composition intervals). During the simulation process, a multi-linear interpolation is applied to approximate the first type of operators, while the second type of operators is processed based on the conventional approach. In this work, we have extended this approach to general purpose simulation. We introduced the operator-based parametrization of mass and energy conservations equation based on the pressure, composition, temperature, and porosity. In addition, the approach has been extended and tested on truly multi-component systems of practical interest. The accuracy and robustness of the new method have been tested against the results of simulations based on the conventional approach.
-
-
-
Physics-based Pre-conditioners for Large-scale Subsurface Flow Simulation
Authors G.B. Diaz Cortes, C. Vuik and J.D. JansenWe consider deflation-based pre-conditioning of the pressure equation for large-scale reservoir models with strong spatial variations in the permeabilities. The use of deflation techniques involves the search for good deflation vectors, which usually are problem-dependent. We propose the use of proper orthogonal decomposition (POD) to generate physics-based problem-specific deflation vectors. The use of POD to construct pre-conditioners has been attempted before but in those applications, a snap-shot-based reduced-order basis was used as pre-conditioner directly whereas we propose the use of basis vectors as deflation vectors. We investigate the effectiveness of the method with numerical experiments using the conjugate gradient iterative method in combination with Incomplete Cholesky preconditioning (ICCG) and POD-based deflation (DICCG). We consider incompressible and compressible single-phase flow in a layered model with large variations in the permeability coefficients, and the SPE10 benchmark model. We obtain an important reduction for the number of iterations with our proposed DICCG method in comparison with the ICCG method. In some test problems, we achieve convergence within one DICCG iteration. However, our method requires a number of preparatory reservoir simulations proportional to the number of wells and the solution of an eigenvalue problem to compute the deflation vectors. This overhead will be justified in case of a large number of subsequent simulations with different control settings as typically required in numerical optimization or sensitivity studies.
-
-
-
Comparison of Linear Reconstructions for Second Order Finite Volume Schemes on Polyhedral Grids
Authors R. Klöfkorn, A. Kvashchuk and M. NolteImproved and enhanced oil recovery methods require sophisticated simulation tools to predict the injected flow pass together with the chemical reactions inside it. One approach is the application of higher order numerical schemes to avoid excessive numerical diffusion that is very typical for transport processes. In this work we provide a first step towards higher order schemes applicable on general polyhedral and corner-point grids typically used in reservoir simulation. We compare two possible approaches of linear reconstruction and slope limiting techniques on a variety of different meshes in two and three space dimensions and discuss advantages and disadvantages.
-
-
-
Improving the Computational Efficiency of a Dynamic Pore Network Model - A Hybrid Approach for a Better Performance
Authors M. Regaieg and A. MoncorgePore scale simulation is more and more used to study various pore scale phenomena that cannot be reproduced by conventional Darcy-based simulators. Dynamic pore network models are a method to study the flow at the pore scale without having to use the very precise and time consuming direct numerical simulators. However, these models are still very slow when applied to 3D core scale simulations. In fact, to reproduce the competition between viscous and capillary forces governing the immiscible flow in porous media, these models require computing many expensive pressure gradients. However, at low rates the displacement tends to become dominated by capillary forces and this means that, during drainage, the pores having the lowest capillary entry pressure are filled first. In this case, simple flow rules can be defined thus avoiding the pressure calculations. These simplified models are named quasi-static and can be only used when viscous forces do not influence the flow. In the literature, most researchers have used either a dynamic pore network or a quasi-static model. Since quasi-static algorithms are faster and are able to reproduce similar results to dynamic models at low rates, we propose to combine these two approaches in a hybrid algorithm taking advantage of the speed of quasi-static algorithms when the flow is governed by the capillary forces and that can simulate the viscous effects when they are important. We propose a criterion to localize the pressure solution to the important areas to enhance the computational efficiency of the algorithm even in viscous dominated regimes. In this paper, we first show that using the classical definition of the capillary number as a switching criterion is not good enough to characterize the domain where the flow is controlled by capillary forces. Therefore, we use the macroscopic capillary number as a criterion to switch between the dynamic and quasi-static flow regimes. Finally, we present several test cases where we show that the hybrid algorithm can considerably improve the computational performance of the pore network simulator without losing the accuracy of the solution. For capillary dominated regimes, the observed speed-up on 3D networks can reach 500 and 16000 for our industrial networks of 43000 and 1 million nodes, respectively. For viscous dominated regimes the speed-up on 3D networks can reach 5 and 30 for 43000 and 1 million nodes, respectively. This approach is compatible with a multiscale method for the pressure computations and will provide an additional speed-up.
-
-
-
Damping of Newton Iterations Using Automatic Error-control Step-length Selection
Authors G. Lutidze and R.M. YounisConsiderable recent interest in the improvement of the robustness of Newton-like methods for implicit simulation has led to the development of a number of successful safeguarding strategies and alternate discretization methods with improved differentiability. To date however, the focus of these reported efforts has been on specific sets of physics and canonical multiphase flow problems. Motivated by the success of these problem-specific efforts, this work proposes a general safeguarding strategy for Newton-like methods that are applied to implicit time-dependent simulation. The proposed method combines ideas from ad hoc and classical safeguarding strategies. The proposed method is applicable to any level of complexity of physical problem. The safeguarding approach proposed in this work is based on the idea of treating Newton’s method as an explicit Euler integration process of the continuous Newton flow Ordinary Differential Equations. For well-posed simulation problems, the Newton flow prescribes a continuous and locally differentiable path from the initial guess to the solution. We propose to tailor a local error control ODE step-length selection algorithm in order to determine a diagonal damping matrix. The result is that the Newton iteration traces a single Newton flow path more accurately. Two novel aspects are developed in order to achieve the objective. First, we develop two sets of local discretization error estimates as a function of damping-factor size. The first set is an a posteriori estimate, and is independent of the form of the particular simulation problem. The second estimate is derived specifically for transport equations by analyzing the backward Euler discretization error of Newton’s method. The second development is a proposed ad hoc strategy to loosen the error tolerance that is enforced as the iteration progresses in order to improve performance. Computational results are presented for a series of simulation problems with increasing complexity. The results for two phase flow simulations demonstrate that the proposed method is competitive with, but not superior to recently proposed strategies. These comparisons show that absent of tailored ad hoc strategies such as those recently proposed for black oil simulation, the proposed strategy is more robust and efficient than the current state-of-the art.
-
-
-
Phase Switching Algorithm for Slug Flows in Wellbores
Authors A.B. Starostin, B.I. Krasnopolsky and A.A. LukyanovWell test data might update reservoir model to precise reserves evaluation and field development plan. We consider the cases of gas-liquid flows in a deviated borehole. In order to evaluate the properties of reservoir, one has to be able to filter out the non-linear input of trajectory. Particularly the wellbore flow may have pulses and phase change even under constant reservoir inflow. The evaluation of well tests demands the development of fast and robust numerical techniques for modelling capabilities. The study presents a multi-fluid model and its implementation using the Jacobian-Free Newton-Krylov method. The fully implicit formulation framework described in this work enables to efficiently solve governing fluid flow equations. A reduction of a multi-fluid model in zones of phase disappearance is based on the phase state distribution over the cells. The numerical method implement a novel phase switching algorithm when the single phase cells and multiphase cells are distinguished by solving reduced set of governing equations. A transient two-fluid model is used to verify and validate the phase switching algorithm for conditions of terrain-induced slug flow regime. The algorithm results are in a good quantitative agreement with other multi-fluid simulator, experimental data and well tests.
-
-
-
Percolation-based Effective Permeability Estimation in Real Heterogeneous Porous Media
More LessIt has long been understood that flow behavior in heterogeneous porous media is largely controlled by the continuity of permeability contrasts. With this in mind, we are looking in new methods for a fast estimation of the effective permeability which concentrates on the properties of the percolating cluster. From percolation concepts we use a threshold permeability value (K_th) by which the gridblocks with the highest permeability values connect two opposite side of the system in the direction of the flow. Those methods can be applied to heterogeneous media of a range of permeabilities distribution and various underlying structures. We use power law relations and weighted power averages that can be inferred either from the statistics and the properties of percolation sub-networks at the threshold point. This approach does not need fitting to the experimental data of conductivity measurements to estimate the model parameter as is done in empirical methods. We examine the order of accuracy of these methods on some layers of 10th SPE model and found very good agreements with the values determined from the commercial flow simulators. The results of this work open insights on new methods in estimating the effective permeability using percolation concepts.
-
-
-
How Fracture Capillary Pressure Affects Ensemble Relative Permeability of Naturally Fractured Reservoirs
Authors M. Sedaghat, S. Azizmohammadi and S.K. MatthaiThis work presents a significant advance over earlier methods because it employs a surface-roughness based fracture dilation model to compute aperture distributions. From these, fracture capillary pressure is computed before saturation functions are extracted. This upscaling is performed using an unsteady state approach to evaluate the impact of fracture capillary pressure on ensemble relative permeability and ultimate recovery. The simulation approach is applied to outcrop-based meter- and kilometre-scale DFM models. For these fracture geometries, aperture attributes are computed for plausible regimes of in situ stress. Corresponding capillary pressure values are assigned to individual fractures. The capillary pressure of the rock matrix is parameterized with representative data for siliciclastics and carbonates. The two-phase flow simulations are performed with the Finite Element-Centered Finite Volume Method (FECFVM). Flow-based upscaling establishes ensemble relative permeability between capillary and viscous limits. Based on results, for a water-wet rock matrix, there is more fracture-matrix transfer and oil recovery is higher. Counter-current-imbibition flux is diminished gradually since the small fractures that dominate the fracture-matrix interface area have drastically smaller fracture-matrix pressure differentials. These differences become more pronounced near the capillary limit. As the wettability tends to the oil, two phase flow occurs within a narrower range of saturation.
-
-
-
An Efficient Multiscale Mixed Finite Element Method for Modelling Flow in Discrete Fractured Reservoirs
More LessFractures can significantly impact the flow patterns of carbonate reservoirs and should be accurately accounted for in a geological model. Accurate modeling of flow in fractured media is usually done by discrete fracture model (DFM), as it provides a detailed representation of flow characteristic. However, DFM poses a particular challenge to traditional numerical method with regard to computational efficiency and accuracy. In this study, a multiscale mixed finite element method (MsMFEM) has been proposed for detailed modeling of two-phase oil-water flow in fractured reservoirs. The MsMFEM uses a standard Darcy model to approximate pressure and fluxes on a coarse grid. Fine-scale effects of fractured media are captured through basis functions constructed numerically by solving local DFM on the fine-scale grid. In our approach, we consider arbitrary fracture orientations and use triangular fine grid. Through multiscale basis functions, we can maintain the efficiency of an upscaling technology, while at the same time generate a more accurate and conservative velocity field on the full fine-scale grid. Comparisons of the multiscale solutions to the fine-scale discrete fracture model solutions indicate that the fine-scale flow in fracture networks can be represented within a coarse-scale Darcy flow model. The results demonstrate that the MsMFE technology is a promising method toward fine flow simulation of high-resolution geological models of fractured reservoirs.
-
-
-
Bayesian Experimental Design for the Influence Identification of Uncertain Geological Parameters on the CO2-GAGD Process
Authors W.J.M. Al-Mudhafar, D.N. Rao and J. TangDetermining the most influential reservoir parameters on the GAGD process is an essential step to understanding the EOR process efficiency. In this paper, we introduce Bayesian Model Averaging (BMA) as a stochastic linear modelling approach to select the most influential parameters affecting the Gas Assisted Gravity Drainage (GAGD) Process performance in a multilayer heterogeneous sandstone oil reservoir. Lithofacies and petrophysical property model was reconstructed considering multiple-point geostatistics for 3D property distribution. CO2 is injected through vertical injectors at the top two layers. The 2nd three layers were left as a transition to allow a vertical depth interval for gas gravity drainage. Horizontal producers were set up through the sixth, seventh, and eighth layers where the oil saturation has the highest levels. The last four layers were left with no injection/production activity. The studies reservoir factors are horizontal permeability, anisotropy ratio (Kv/Kh), and porosity. Latin Hypercube Design created many simulation jobs and the elimination was conducted by the BMA stochastic approach, which adopts posterior probability to choose the best model among a set of candidate models. Moreover, the accurate determining of influential factors through BMA has led to better understanding of the effect of heterogeneity and anisotropy on the GAGD process.
-
-
-
Study the Effect of Connectivity between Two Wells on Secondary Recovery Efficiency Using Percolation Approach
Authors S. Sadeghnejad, M. Masihi, P.R. King and P.A. GagoEstimating available hydrocarbon to be produced during secondary oil recovery is an ongoing activity in field development. The primary plan is normally scheduled during early stage of field’s life through master development plan studies. During this period, due to the lake of certain data, estimation of the field efficiency is usually based on rules of thumb and not detailed field characterization. Hence, there is a great motivation to produce simpler physically-based methodologies. The minimum necessity inputs of percolation approach make it a useful tool for foration performance prediction. This approach enables us to attain a better assessment of the efficiency of secondary recovery methods at early production time. The main contribution of this study is to establish a continuum percolation model based on Monte Carlo simulation that can estimate the connectivity of good sands between two wells. In the classical percolation, the connectivity is considered between two lines and two faces of the system in 2- and 3-D; whereas, hydrocarbon production is achieved through wells with the shape of lines (e.g., vertical, horizontal, or deviated wells). In addition, the results showed that not implementation of the correct geometry of wells can alter the estimated results from the percolation approach.
-
-
-
Hydrocarbon Formation Evaluation Using an Efficient Genetic Algorithm-based Factor Analysis Method
By N.P. SzaboA global optimization approach for the factor analysis of wireline logging data sets is presented. Oilfield well logs are processed together to give an estimate to factor logs by using an adaptive genetic algorithm. Nonlinear relations between the first factor and essential petrophysical parameters of shaly-sand reservoirs are revealed, which are used to predict the values of shale volume and permeability directly from the factor scores. Independent values of the relevant petrophysical properties are given by inverse modeling and well-known deterministic methods. Case studies including the evaluation of hydrocarbon formations demonstrate the feasibility of the improved algorithm of factor analysis. Comparative numerical analysis made between the genetic algorithm-based factor analysis procedure and the independent well log analsis methods shows consistent results. By factor analysis, an independent in-situ estimate to shale content and permeability is given, which may improve the reservoir model and refine the results of the reserve calculation.
-
-
-
Advanced Geologically-consistent History Matching and Uncertainty Evaluation
Authors E.S. Zakirov, I.M. Indrupskiy, I.M. Shiryaev, O.V. Lyubimova and D.P. AnikeevAt the ECMOR-14 conference we proposed a method for automated geologically-consistent history matching of a 3D reservoir model and presented the results of its implementation as a numerical algorithm. In the approach considered, control parameters to be evaluated through dynamic data assimilation were geostatistical parameters of a 3D reservoir model. It was assumed that porosity within inter-well space was calculated by the kriging procedure based on measured values on wells. Permeability distribution was supposed to be calculated through its correlation dependence on porosity. Through the inverse problem solution, the most uncertain parameters of a geostatistical model were estimated, namely, parameters of an anisotropic semivariogram and the porosity-to-permeability dependence for each facies. The inverse problem solution algorithm was based on efficient methods of the optimal control theory (the adjoint methods). In the present study, the approach is further advanced by the transition from the deterministic kriging procedure to stochastic geostatistical methods such as sequential Gaussian simulation. The control parameters evaluated through the inverse problem solution – parameters of the anisotropic semivariogram and porosity-to-permeability dependence - are supplemented by the values in pilot points chosen with a special algorithm. For some common depositional environments, original algorithm has been also developed to adjust facies distribution within the inverse problem. Thus, firstly, it becomes possible to effectively identify heterogeneities of parameter distributions in the inter-well space. Secondly, efficient assessment of uncertainty in the parameters obtained by the inverse problem solution can be carried out. As an alternative implementation to the computationally-intensive approach based on the group analysis of an ensemble of history matched models, a simplified procedure has been implemented based on the linearization of the objective function at the optimal point. To obtain the covariance matrix, the sensitivity matrix is calculated by means of a special computationally-efficient procedure using the state variables variation problem - one of the subproblems of the adjoint-based algorithm for inverse problem solution. The paper provides a number of examples demonstrating performance of the proposed approaches and algorithms. Main contributions: - Geologically-consistent history matching based on adjoint methods has been extended to stochastic geostatistical formulations for evaluation of anisotropic semivariogram and porosity-to-permeability relation parameters for each facies and values in pilot points chosen with a special algorithm. - Original algorithm has been developed to constrain facies distribution to dynamic data. - Uncertainty assessment procedure for inverse problem solution has been implemented based on a subproblem of the adjoint algorithm for sensitivity and covariance matrix computation.
-
-
-
An Ensemble 4D Seismic History Matching Framework with Wavelet Multiresolution Analysis - A 3D Benchmark Case Study
Authors X. Luo, T. Bhakta, M. Jakobsen and G. NævdalIn a previous work (Luo et al., 2016), we proposed an ensemble 4D seismic history matching framework, which has some relatively new ingredients, in terms of the type of seismic data in choice, the way to handle big seismic data and related data noise estimation, and the use of a recently developed iterative ensemble history matching algorithm. In seismic history matching, it is customary to use inverted seismic parameters as the observations. In doing so, extra uncertainties may arise during the inversion processes. We avoid such intermediate inversion processes by adopting amplitude versus angle (AVA) data. To handle the big-data problem in seismic history matching, we adopt a wavelet-base sparse representation procedure. Concretely, we apply a discrete wavelet transform to seismic data, and estimate noise in resulting wavelet coefficients. We then use an iterative ensemble smoother to history-match leading wavelet coefficients above a certain threshold value. In the previous work (Luo et al., 2016), we applied the proposed framework to a 2D synthetic case. In the current study, we extend our investigation to the 3D Brugge benchmark case. Numerical results indicate that, the proposed framework is very efficient in handling big seismic data, while achieving reasonably good history matching performance.
-
-
-
Assisted History Matching for Multi-facies Channelized Reservoir Using ES-MDA with Common Basis DCT
Authors Y. Zhao, F. Forouzanfar and A.C. ReynoldsHistory matching a reservoir with multiple facies has always posed a great challenge to researchers. Most traditional history-matching techniques were designed to work with Gaussian distributed continuous variables instead of non-Gaussian distributed categorical variables like facies. Inspired by the previous researchers, we develop a workflow which combines Ensemble Smoother with Multiple Data Assimilation (ES-MDA) algorithm with common basis Discrete Cosine Transform (DCT) to conduct assisted history matching for multi-facies channelized reservoir, especially the 3D problems which has rarely been investigated before. In this work, an ensemble of geological realizations is first generated by using multi-point statistics (for 2D case) or object-based modeling (for 3D case). Then the DCT is implemented for each realization (facies field) to get the particular basis functions and their corresponding coefficients. For the purpose of extracting the general geological features among different realizations, we retain a series of common basis functions which are identical and fixed for all realizations. The corresponding coefficients of each realization are recomputed with respect to the common basis set in order to reconstruct the original facies field by minimizing the least square residual. Through history matching the observed data using ES-MDA, the DCT coefficients are updated and the facies field is renewed with the updated coefficients and the common basis set. The discrete facies field is obtained by applying an optimization algorithm to truncate the continuous values at the end of each ES-MDA iteration. We apply this procedure to both 2D and 3D synthetic problems considering complex three facies (shale, levee and sand) channelized reservoir. The results show that the proposed algorithm can provide good data matches and reduce the uncertainty in the prior ensemble significantly. Moreover, the posterior estimation of model parameters properly reflects the main geological features of the true model. Compared to previous studies, this work not only applies the ensemble-based method with common basis DCT for history matching and uncertainty quantification for 2D and 3D multi-facies reservoirs, but provides a robust and relatively easy approach to handle 3D cases which has very limited report in the literature.
-
-
-
Ensemble-based Seismic History Matching with Distance Parameterization for Complex Grids
Authors Y. Zhang and O. LeeuwenburghRecently, a distance parameterization of flood fronts derived from seismic anomalies was proposed as a solution in combination with the ensemble Kalman filter (EnKF), which is known as an efficient method for conditioning of multiple reservoir models to observed data. Even though the distance parameterization in terms of front positions is efficient and effective in reducing both nonlinearity and the effective number of seismic data, which improves the performance of the EnKF, the method adopted for distance computation therein is only applicable for reservoir models with regular Cartesian grids because large errors will be introduced otherwise. In this paper, we improve the applicability of the distance parameterization in terms of front positions by extending the fast marching method for solution of the Eikonal equation to complex simulation grids. This is realized by taking advantage of a diagonal stencil in the fast-marching implementation which allows more accurate calculations of distances between observed and simulated fronts, and by an isoparametric mapping which provides a transformation from the Cartesian to curvilinear coordinates. The improvements of the proposed methods are demonstrated through a number of numerical experiments on corner-point grid including a 3D synthetic case of Norne full-field model.
-
-
-
Application of Ensemble Smoother and Multiple-data Assimilation for Estimating Relative Permeability from Coreflood Experiments
Authors A. Jahanbakhsh, A. ElSheikh and M. SohrabiRelative permeability curves (kr) are flow functions governing multiphase flow in porous media. These functions are an essential component of any large-scale simulator of porous media flow of different phases (oil, water, and gas) with several applications in environmental and petroleum engineering. Unsteady state methods are commonly performed on core samples taken from subsurface reservoirs to obtain the relative permeability curves experimentally. The obtained measurements are then used to calibrate analytical functions (to be embedded in the flow simulator) through automatic history matching. In this study, we evaluate iterative ensemble-based history matching techniques based on the Ensemble-Smoother (ES) formulation. Mainly, the Ensemble Smoother with Multiple-Data Assimilation (ES-MDA) is used for calibrating the parametric relative permeability models using data from unsteady-state core flood experiments. An Ensemble-Smoother updates the model parameters globally by assimilating all the time depended data at once, from the start to the end of the experiment. This is to be contrasted with online updating scheme adopted in Ensemble Kalman Filtering methods. Recently, ES-MDA was developed to improve on ES and to provide reliable uncertainty quantification of the unknown parameters with low computational cost. In the current work, ES-MDA is compared to global optimization methods for calibrating the relative permeability curves. The results of estimating two and three-phase relative permeability curves from three-phase coreflood experiments are presented. The experiments were performed on 65 mD mixed-wet Clashach sandstone core and cumulative productions and pressure drop across the core were measured during the course of experiments. ES-MDA was able to find the global optimum parameters at much faster convergence rates in comparison to genetic algorithm (GA), a widely used global search method. This was evident for the history matching of three-phase unsteady state experiments where optimal solutions were obtained efficiently while preserving uncertainties in the estimated parameters.
-
-
-
Different Parameterizations of the Initial Ensemble for a Channelized Reservoir in an Assisted History Matching Context
Authors B. Sebacher, A.S. Stordal and R.G. HaneaIn this paper we present a comparison of three parameterizations of channelized reservoirs generated using multipoint geostatistics (MPS) in combination with a training image. In a previous study, we suggested estimating the facies probability fields from an ensemble generated with MPS and linked, marginally, the facies probability fields with the standard Gaussian variables by means of the normal score transform. We have parameterized the facies fields with random fields, marginally Gaussian, using the conditional mean of the Gaussian variables. This parameterization keeps a possible dependence structure inherited from the training image, but marginally the sampling from the Gaussian distribution is discrete and bi-modal. Here, we extend this parameterization in two directions. First, we do not take into account the dependence structure and parameterize by random sampling from the conditional distribution. The second idea is to draw samples from the conditional distribution, but using the same random seed for each grid cell within each ensemble member, but different random seeds across the ensemble members. This would preserve the dependence structure within each ensemble member while increasing the variability between the ensemble members. Both parameterizations have the property that, marginally, samples correctly from the standard Gaussian distribution. We compare the behavior of the parameterizations within a history matching process assimilating the production data. The comparison has two main directions: to prove the impact of the stochastic forcing on the history matching of geological properties and to prove the stochastic forcing on the predictive power of the models. We have used the iterative adaptive Gaussian mixture filter (IAGM) for history matching because the IAGM is suited for highly nonlinear problems and has a re-sampling step that allow us to use the already existing technique of re-sampling from the training image using updated probability fields. The re-sampling step is necessary to re-position the facies geometry, lost after a cycle of data assimilation.
-
-
-
Large-ensemble Data Assimilation Using an Upscaled Model
Authors K. Fossum and T. MannsethWhen performing ensemble based data assimilation (DA) one can, due to the high inherent computational cost of running a complex reservoir simulator, only afford to apply a moderate number of ensemble members. Without modification it is well known that the DA procedure will fail when assimilating data in high-dimensional geophysical systems. Distance-based localization mitigate the effects of few ensemble members, but, for many cases, it is difficult to define localization in a suitable manner, this is especially the case for problems with a nonlocal relationship between data and parameters. An alternative to localization is to increase the ensemble size. With fixed computational resources, an increase in the ensemble size must be compensated for by a decrease in the cost of each reservoir simulation. This can be achieved by replacing the reservoir simulator with a proxy model. In this work, we investigate the use of a proxy model that is constructed by discretizing the reservoir model equations on a coarser grid than the original reservoir simulation model. A modest reduction in the number of grid cells should be sufficient to compensate for the increase in computation from using a large ensemble size. This reduction is achieved by a flexible and adaptive upscaling procedure, capable of handling all grids, which is constructed based on a second generation wavelet transform. Since the update step of the DA algorithm is much less computationally demanding than the forecast step, we consider a DA method where the forecast is performed on the coarse model while the update is performed on the the fine grid. This is formulated without the need for error structure correction. The large ensemble proxy DA method is compared with a localization methods on several numerical examples where localization is traditionally required.
-
-
-
Distinguishing Signal from Noise in History Matching - Analysis of Ensemble Collapse on a Synthetic Data Set
Authors P. Roe, A. Almendral Vazquez and R. HaneaUnderestimation of posterior parameter uncertainty is one of the main problems encountered when doing history matching using ensemble based methods. In history matching results with the partial or full ensembles collapse, it is very hard to distinguish updates due to spurious correlation with noise in the data from the actual updates attributed to information in the data. History matching of porosity and permeability based on well production data using the ensemble smoother with multiple data assimilation has been performed on a synthetic data set. The presence of ensemble collapse has been evaluated by different means: by looking at the stability of the update based on the starting ensemble, by adding dummy parameters to the update which do not affect the forward model, and by examining how well the data set used to generate the production data matches the posterior distributions of the parameters. Ensemble collapse can be avoided by increasing the number of ensembles. This is however a prohibitively expensive strategy for cases with a large number of history data. Localization methods have been proposed in the literature as a way to increase the ensemble spread and hence avoid collapse, by for example limiting the analysis update to regions of influence of the data, while at the same time keeping the number of ensembles low. A local analysis was performed to reduce the problems related to ensemble collapse. The results from the localized history matching produce a posterior distribution that better matches the original data set. Since our test data set is synthetic, we may perform measures of posterior uncertainty estimation by comparing with the true solution, with and without localization.
-
-
-
Estimating Observation Error Covariance Matrix of Seismic Data from a Perspective of Image Processing
More LessEstimating observation error covariance matrix properly is a key towards successful seismic history matching. Observation errors of seismic data are usually correlated, therefore the observation error covariance matrix is non-diagonal. Estimating such a non-diagonal covariance matrix is the focus of the current study. We decompose the estimation into two steps: (1) estimate observation errors; and (2) construct covariance matrix based on the estimated observation errors. Our focus is on step (1), whereas at step (2) we use a procedure similar to that in Aanonsen et al., 2003. In Aanonsen et al., 2003, step (1) is carried out using a local moving average algorithm. By treating seismic data as an image, this algorithm can be interpreted as a discrete convolution between an image and a rectangular window function. Following the perspective of image processing, we consider three types of image denoising methods, namely, local moving average with different window functions (as an extension of the method in Aanonsen et al., 2003), non-local means denoising and wavelet denoising. The performance of these three algorithms is compared using both synthetic and field seismic data, and it is found that the wavelet denoising method leads to the best performance in our investigated cases.
-
-
-
Pilot Design Analysis Using Proxies and Markov Chain Monte Carlo Method
Authors B. Chen, J. He, X. Wen, W. Chen and A. ReynoldsA pilot project is a crucial step of reservoir management that enables the minimization of subsurface risks and improves the quality of decisions on full-field development. Selecting a pilot project involves evaluating the expected uncertainty reduction and the value of information (VOI) attainable from a set of plausible pilot projects. Proxy-based pilot analysis (PBPA) represents a promising approach for characterizing the uncertainty reduction and VOI from each of a set of feasible pilot projects. In the PBPA method, multiple plausible realizations of observed data from a pilot are generated and probabilistic history matching (based on filtering) is performed for each realization of the vector of observed data in order to obtain the corresponding posterior distribution. The multiple history-matching runs are accomplished with a manageable number of simulations with the help of proxies. Previously, PBPA was successfully applied to quantify the expected value of uncertainty reduction in cases where the history-matching tolerance is high, but as shown here, the filtering-based history-matching procedure can fail when the tolerance is low. Moreover, it has not been demonstrated previously that the PBPA method can quantify VOI. In this paper, enhancements to PBPA that eliminate these two PBPA shortcomings are introduced. First, a Markov chain Monte Carlo (MCMC) method is used in place of the filtering procedure to calculate the posterior distribution. The combined MCMC-PBPA procedure is shown to outperform the filtering-based PBPA when the history-matching tolerance is low. Secondly, we define a framework that combines the MCMC-PBPA method with decision tree analysis in order to calculate the VOI. The proposed framework is demonstrated for a synthetic waterflooding pilot in the Brugge reservoir where it successfully quantifies the VOI for different pilot designs.
-
-
-
Uncertainty quantification using a self-supervised surrogate-assisted parallel Metropolis-Hastings algorithm
More LessParallel tempered algorithm is a Markov-chain Monte Carlo technique, applied to propagate uncertainty in the parameters of interest, using multiple Metropolis-Hastings algorithms. Despite the effectiveness, it is a computationally intensive method, since, at every step, a numerical reservoir simulation should be executed. To reduce the CPU-time of such a process, an online-learning surrogate-assisted algorithm is proposed in which two surrogates (one for high-temperature chains and one for low-temperature chains) are utilised together with the exact function (numerical simulation). After each swap step, both surrogates are re-trained, and their fidelity is estimated. According to the estimated fidelities, the frequency of use, for each surrogate, is defined with a heuristic fuzzy rule. The algorithm stochastically decides between the simulation and surrogates, based on the frequencies. This creates a self-supervised strategy, which can optimise the use of the numerical simulation, through the sampling process. The robustness of the proposed algorithm is analysed using IC-fault model. The outcomes are compared with the results achieved by a typical (unassisted) parallel tempered Metropolis-Hastings algorithm, over a range of chain lengths. The comparison indicates that the proposed algorithm can deliver a significantly better approximation of the probability density function, with the same amount of computation.
-
-
-
On Obtaining Optimal Well Rates and Placement for CO2 Storage
Authors R.D. Allen, H.M. Nilsen, O. Andersen and K.A. LieLarge-scale storage of CO2 in saline aquifers is considered an essential technology to mitigate CO2 emissions. Storage potential has mainly been estimated based on volumetrics or detailed simulations for specific injection scenarios. In practice, achievable storage capacity will depend on engineering, economical, and political restrictions and be limited by the length of the injection period, costs associated with potential CO2 leakage, pressure management, etc. We show how achievable storage volumes can be estimated and maximized using adjoint-based optimization and a hierarchy of simulation methods. In particular, vertical equilibrium models provide the simplest possible description of the flow dynamics during the injection and early post-injection period, while percolation type methods provide effective means for forecasting the long-term fate of CO2 during the later migration stages. We investigate the storage volumes that can be achieved for several formations found along the Norwegian Continental Shelf by optimizing well placement and injection rates, using production wells for pressure management when necessary. Optimal strategies are obtained under various objectives and simple but realistic constraints, namely: penalization of CO2 leakage, minimization of well cost, and restriction of pressure buildup.
-
-
-
Gradient-based Production Optimization with Economic Constraints
Authors O. Volkov and M.C. BelloutThis work develops an analytical framework to study the effects of enforcing simulator-based constraints while performing gradient-based production optimization. In particular, this work studies how the enforcement of this type of constraints affects the consistency of the adjoint-based gradient and the performance of a gradient-based algorithm. In reservoir management, production optimization is commonly performed using gradient-based algorithms that rely on the efficient computation of control gradients through an adjoint formulation. Often, production optimization is implicitly coupled with economic constraints, which typically are implemented through well performance limits enforced within the reservoir simulator. We show that enforcing simulator-based economic constraints triggers non-differentiable unscheduled changes in both the well model equations and in the functions defining the economic criteria. These discontinuities lead to inconsistencies within the adjoint gradient formulation that eventually translate into decreased algorithmic performance. The analytical framework developed in this study allows us to devise an efficient implementation of the simulator-based constraints that provides gradients that are consistent with the adjoint formulation. These analytical results are described using the theoretical framework developed in this paper, and implemented for a production optimization test case where they are shown to outperform common modes of economic constraint enforcement.
-
-
-
Well Placement Plan Optimization by Dynamic Pattern Adjustment with Existing Wells and Streamlines
More LessWe present an original method that automatically generates an optimal well placement plan (WPP) by dynamic pattern aberration with existing wells and streamlines. The new method is a pattern-based optimization of a field development plan, where WPP is optimized using a constrained downhill-simplex approach. In existing methods, a secondary production pattern, e.g. 5-spot waterflood, is generated from five control variables. We extend this approach, while still honoring the topology of a pattern. We distort the pattern geometry to consider existing wells and to honor the underlying fluid and reservoir heterogeneities, which is essential for mature brownfields and geologically complex reservoirs. A fast streamline simulator is used to identify allocation factors for each producer-injector pair, which is then used to guide the optimizer to discover the optimal irregular geometry of the pattern. This is analogous to fast cloth simulation algorithms heavily employed in various industries. Existing wells are dealt with by introducing a radius constraint. This constraint defines maximum distance within which existing well, when found, could be re-used either as producer or injector, with producer/injector conversion allowed. Modeling production with streamline simulations and combining many individual drilling decisions into a consolidated WPP does not significantly affect convergence time and optimization problem complexity. At the same time, the algorithm is capable to produce realistic plans with much higher NPV and sweep efficiency, when compared to a basic repatterning approach. Results are illustrated on the Brugge synthetic field, widely known for history match and prediction benchmarking.
-
-
-
Manifold-mapping Optimization Applied to Oil Field Operations
Authors D. Echeverria Ciaurri and C. Wagenaarhas been reported as an efficient surrogate-based optimization approach in several engineering applications (e.g., design of electromagnetic devices, microwave structures and antennas, friction-stir welding and analysis of fluid-structure interaction). Manifold Mapping relies on the iterative solving of a “coarse” (approximate, fast-to-evaluate) formulation of the optimization problem of interest. This formulation is progressively corrected using scarce evaluations of a satisfactorily accurate (and often computationally more expensive) “fine” formulation of the same problem. In order to obtain optimized solutions of acceptable quality by means of Manifold Mapping we typically need to perform as many evaluations of the fine formulation as the number of degrees of freedom that the optimization problem really has (as can be expected, this behavior depends strongly on the particular coarse formulation considered). Manifold Mapping can be combined with gradient information but it is normally used in a derivative-free manner. We will first present results of Manifold Mapping for a relatively simple production optimization case based on a small reservoir model with eight wells. The fine formulation is constructed by means of reservoir flow simulation and the coarse formulation through approximation/interpolation using simulation output. Manifold Mapping yields speed-up factors of 2-3 with respect to direct optimization of the fine formulation. We also introduce geological uncertainty in that example by means of a fine formulation with 20 realizations. A coarse formulation built on approximation using simulation output from only one realization accelerates again 2-3 times the optimization of the fine formulation. We will conclude the paper with additional experiments that include different number of control variables and nonlinear optimization constraints.
-
-
-
Inter-well Connectivity in Waterfloods - Modelling, Uncertainty Quantification, and Production Optimization
Authors T. Wen, X. Zhai and S.F. MatringeTo improve performance of waterfloods with minimal capital investment is important as the crude price is low. Adjusting the well controls to achieve a more efficiency sweep pattern is more economic than side tracking or infill drilling. This paper presents a methodology designed to guide well controls and maximize the recovery of remained oil in large and mature waterfloods by modeling and optimizing the inter-well connectivities. The workflow includes three steps: modeling, uncertainty quantification (UQ), and production optimization. Firstly, the reservoir is modeled as a connected network characterized by the strength and efficiency of each injector-producer connection. The concept is similar to the flux pattern derived by streamlines (Thiele and Batycky, 2006). But the presented approach does not use streamlines, and instead simulates tracer concentration between each well pair to quantify the strength of energy support from injectors to producers. The technique is a generalized form of the work by Shahvali, et al. (2012). The efficiency of connections measures the oil contribution of each connection, which identifies the water cycling. It is history matched by a data-driven technique. In the UQ step, the method estimates the possible range of efficiency due to the non-uniqueness of the history matching solution. The efficiency of the connection carrying less flux in the entire history tends to be more uncertain. We quantify the uncertainty by evaluate the upper and lower bound of the efficiency subject to similarly good history matching. The formulation of the maximization/minimization was inspired by the work of Van Essen, et al. (2010), but the optimization algorithm differs and is a non-linear constrained pattern search method. For production optimization, a nonlinear optimization problem is formulated based on the connectivity model to find well controls strengthening efficient connections and weakening inefficient connections. The optimization algorithm takes advantage of the linearity of the network model to achieve faster performance than pattern search. Here UQ regulates the risk of the recommended well control strategy. The methodology was tested based on a full simulation model of a real field with 200+ wells, which was regarded as the true reservoir in this study. We trained our network model for 3 years then started to optimize the waterflooding strategy for six months. The results demonstrated that the optimized strategy maintained oil production and reduced water production by 50% without adding new well, while the historical operation satisfied the oil target by drilling tens of new wells and scarifying water-cut.
-
-
-
On Solving Large-scale Well Positioning Problems
Authors A.M. Kuvichko, N.Y. Andrianov and A.I. ErmolaevWe consider several mathematical aspects of technological objects positioning applied to oil and gas fields. Best results of these algorithms obtained while dealing with non-great fields of complex structure; these algorithms are suitable for non-regular well location problems, when one cannot define a strong pattern for well positions. Proposed problems are formulated in terms of Boolean programming. We present a new iterative method of solving described Boolean problems finding optimum within a reasonable time. This method is a clustering algorithm designed to deal with Boolean parameters. We introduce a Monte-Carlo-like approach forecasting the optimal value for a certain problem. Several applications of formulated algorithms applied to test field models with properties similar to real field models. We present both Boolean problems solutions. Comparing different scales, we show complexity of proposed algorithm. We also present comparisons between computational time and number of iterations for different initialization schemes. Optimal solutions of formulated problems are good as initial points for other iterative (gradient-based) algorithms of well location problems. Proposed algorithms are practical for field development cases applied to complex oil and gas fields.
-
-
-
Optimization of Cyclic CO2 Flooding through the Gas Assisted Gravity Drainage Process under Geological Uncertainties
Authors W.J. Al-Mudhafar, D.N. Rao and S.M. Hosseini NasabThe purpose of this research is to determine an actual optimal solution through cyclic optimization of CO2-Gas Assisted Gravity Drainage (GAGD) process in a heterogeneous sandstone reservoir under geological uncertainties. We propose an integrated approach to optimize durations of gas injection, soaking, and oil production under geological uncertainties. Therefore, 100 stochastic reservoir realizations of the 3D permeability and porosity distributions were created honouring geological constraints. Ranking was applied through quantifying of reservoir oil response to select P10, P50, and, P90 that represent the overall reservoir uncertainty. More than 400 training simulation runs were created including the durations and geological uncertainty parameters through Latin Hypercube Design to build the second-order proxy model along with approximately 200 extra verification runs. The verification runs led to keep the solutions in global optima and obtain satisfactory proxy model through an iterative validation procedure. The cyclic optimization has shown its feasibility to increase oil recovery through the GAGD process from 71.5% to 75.5% with incremental cumulative oil production of 225 million barrels. The presented robust optimization workflow under geological uncertainties led to higher recovery factor than nominal realization optimization with providing degrees of freedom for the decision-maker to significantly reduce the project risk.
-
-
-
Multi-fidelity Proxy Models for Reservoir Engineering
Authors A. Thenon, V. Gervais and M. Le RavalecProxy models are built to approximate outputs that depend on many uncertain parameters using few evaluations. In reservoir engineering, they can strongly reduce the number of flow simulations required for sensitivity analysis, history matching or production optimization. However, a large number of simulations can still be necessary to compute predictive proxy models. We propose to build multi-fidelity proxy models based on co-kriging to speed up the process. This approach introduces coarser resolution levels for the reservoir model that are less informative, but faster to estimate. These coarser levels can be obtained using a fluid flow simulator with simplified physics or an upscaled reservoir model. Then, the fine and coarse level evaluations are combined to build a proxy model of the reference – or fine – level. The objective is to retrieve as much information as possible from the faster levels in order to limit the calls to the fine, but most expensive ones. Sequential design strategies can also help reduce the number of simulations required to get predictive proxy models by iteratively defining the appropriate location of the added point or the appropriate fidelity level to consider when in multi-fidelity context. Sequential design strategies that take advantage of some kriging/co-kriging features (kriging variance and cross-validation predictions) were thus introduced to fully exploit the potential of the proposed approach. Comparisons of time saving between the simple and multi-fidelity proxy modeling methodologies were then performed through a sensitivity analysis for the Brugge field.
-
-
-
Organization of High-performance Computing on the Mobile Platforms to Calculate the Oil Recovery of Reservoir
Authors T.S. Imankulov, D. Akhmed-Zaki, B.S. Daribayev and O.N. TurarRecently variety of novel parallelization technologies and approaches is rapidly developing. Distribution of high-powered graphical processors and parallelization tools on mobile platforms required a detailed comparison of computational capabilities on practical problems. This paper considers numerical investigation of oil displacement process by polymer/surfactant flooding taking into account water salinity and temperature effects, study parallel algorithms for it on several platforms and describes the organization of high-performance computing by using GPU of mobile devices. Calculation times, efficiency and speed up of these algorithms on various platforms were compared in order to identify optimal technology or combination of technologies for effective and well-optimized developments of industrial scale simulator. Basic parallelization technologies being used: MPI and CUDA. The calculations are performed on a mobile devices Xiaomi MiPad with NVIDIA Tegra K1 and on a personal computers with NVIDIA GeForce and Tesla K20. Besides the comparison of different version’s working time profound analysis of different platforms’ using expedience presented considering of the received working time benefits. Standard and proven parallelization tools are being used by way of samples for justification competitiveness of new types of parallelization. The research output has led the authors to come to the main conclusion: mobile devices can be used as computers to solve problems in oil industry.
-
-
-
A Hybrid DFN with Elastic Properties to Construct a Seismic Forward Model
Fractured reservoirs present significant modeling uncertainties. This paper describes a methodology for generating a hybrid Discrete Fracture Network (DFN) model to be employed in a dynamic fluid flow simulation, and then introduces a mathematical construct to create elastic properties from the simulation results. The hybrid DFN model is constructed by extracting in-situ, meso to macro-scale DFN population of a fracture system from 3D seismic data, and stochastically modeling micro to meso-scale DFN population of the same fracture system using population properties of the extracted 3D DFN from seismic data and 1D fracture data from multiple boreholes. This hybrid model enables to better constrain 3D network connectivity with a multi-scale fracture system and thereby to carry out more realistic flow simulation. The hybrid model also provides the basis for the computation of anisotropic elastic properties which are then used for rock physics/fluid substitution modeling and computation of representative seismic data. The seismic-simulation loop is then closed by comparing modeled seismic data to an acquired seismic survey, and using the observed discrepancies between these data to achieve improvements in the match to observed production data through guided adjustments to the computer modeling approach. The calibrated model is used to optimize a surface network and allow for the introduction of other reservoirs into the gathering facility.
-
-
-
Asphaltene Flow Assurance Risk Mitigation through Emerging Approach of Inhibitor Numerical Modelling
More LessThis work was motivated for establishing a comprehensive evaluation method of asphaltene mitigation using inhibitor in an oil field that has a high risk of asphaltene precipitation in tubing. Application of asphaltene inhibitor is a typical counter measure and widely applied in many fields; however, most of the applications are temporary relief to mitigate problems. During an entire field life, the production operating condition has varied such as pressure decline, water cut and GOR increase, and so on. According to these variation, the inhibitor formulation that was once selected as the best effective one, its efficiency fades away. Then, another screening process is required to select alternative one and/or to modify the original formulation to adapt effectiveness to the new operating condition. This paper demonstrates a comprehensive estimation of its inhibiting efficiency during a whole field life by generating a numerical model based on results of asphaltene dispersant test (ADT) that was performed to experimentally select inhibitor for our one of oil field asset. The best asphaltene inhibitor IB-23 was selected through the two staged ADT from total nineteen samples because the IB-23 revealed highest inhibiting efficiency more than 80 % at 200 ppm concentration and maintained its efficiency more than 90 % even at 12.5 ppm. Based on this testing result, an emerging technique was applied to generate a numerical model to reproduce inhibiting efficiency. This special technique treated asphaltene inhibitor as pseudo-component defined using physical data that was available in public accessible material safety data sheet (MSDS). To date, any commercial software is not available for modelling of asphaltene inhibitor due to confidentiality for inhibitor’s physical data; however, our approach achieves to express the inhibiting efficiency as size-reduction of asphaltene precipitation envelope (APE) on thermodynamic plot. The model was generated using cubic-plus-association (CPA) EoS for fluid characterization, and the model validation was confirmed by comparing with the ADT data. Assuming natural depletion, the APEs were compared with variation of vertical lifting curves (VLC) in tubing. Two VLCs were assumed to represent early and late field conditions (i.e. high wellhead/reservoir pressures and depleted ones). The no-inhibitor case revealed precipitating risk existed over most of the tubing section. In contrast, the inhibitor dosed case could significantly reduce the risks in the early stage in particular. Even in the late stage, the risks could be minimized as the interception of VLC on the APE became shorter than the no-inhibitor case.
-
-
-
Investigation of Water Diversion by a Novel Polymer Gel System for Enhancing Oil Recovery
Authors A. Jahanbani Ghahfarokhi, J. Kleppe and O. TorsaeterNumerical study of water diversion by gel treatment is presented in this paper, particularly investigating layered reservoirs where crossflow may be an important recovery mechanism. The transport of the new gel system which is, unlike many other systems, environmentally acceptable through porous medium and the mechanism of permeability reduction are evaluated. Permeability reduction causing the water diversion is mathematically modelled by interaction of the gelants or gel with the rock matrix in terms of equilibrium adsorption both reversibly and irreversibly. Long half-life of the gelants is applied to the gelation kinetics model to simulate the controlled release of the crosslinkers. Modelling the blockage for both aqueous and oil phases and including the inaccessible pore volume results in a more realistic situation. The location of the high permeability streak is analysed to closely investigate gravity and crossflow. Permeability reduction and crossflow are the main mechanisms involved. Results indicate that high permeability reduction in the thief zone should exist to improve the recovery. This is justified by observing the spread of residual resistance factor. Gel treatment is generally more efficient than polymer flooding in terms of increased oil recovery and reduced water cut, however, gel is not completely formed in the case of high crossflow between layers since some reactants are lost to the low permeability zone and cause damage and additional water crossflow. Investigation of individual layers shows a peak in oil saturation and production rate of the high permeability layer which is due to resaturation of this depleted layer by crossflow from the low permeability layer. The oil production rate of the low permeability layer adjacent to the thief zone increases after gel treatment reflecting the effectiveness of water diversion treatment. This effect is observed to be more significant in the case of low crossflow between layers, as discussed before. Study of the injection strategy in terms of alternating water, polymer and gel injected shows that a small slug of gel injected early after water breakthrough is more effective than injection following a polymer flood. In real cases, crossflow and permeability contrast between layers are beyond control and injection conditions should be optimized when designing a treatment. Therefore, effects of blocking properties, injection time and concentration and reaction rate are studied in details in this work.
-
-
-
An Application of Green's Function Technique for Computing Well Inflow without Radial Flow Assumption
Authors A. Korneev, A.V. Novikov, D.V. Posvyanskii and V.S. PosvyanskiiWell modeling plays an important role in numerical reservoir simulation. The main difficulty in well modeling is the difference in scale between the wellbore radius and the well block grid dimension used in the simulation. Peaceman’s formula is widely used in reservoir simulation in order to match the cell pressure to the local solution of the diffusivity equation describing the flow near the well. However, it was developed under the assumption of radial flow. The objective of this study is to calculate a semi-analytical expression for the well productivity index without making any assumption about radial flow, and to subsequently use it in numerical reservoir simulation. Radial flow may not occur due to boundary conditions at the top (bottom) of reservoir or well trajectory. The well inflow equation can be solved through the Green’s function method (GFM), which may take into account various boundary conditions and different well trajectories. The GFM solution of the diffusivity equation is presented as a series over the eigenvalues of the Laplace differential operator, but the series converges conditionally and its direct summation is time-consuming. This makes the GFM solution impractical for well modeling. Reference [1] presented the method of fast summation of such a series, which was successfully applied for analyzing pressure build up curves. In this paper we apply the same techniques for calculating the well productivity index for horizontal, deviated and partially penetrating wells . It is shown that for reservoirs with a gas cap (or underline water), using Peaceman’s model for the well productivity index leads to a significant discrepancy between the numerical and presented semi-analytical solution. Although local grid refinement around the well leads to a reduction in the discrepancy, it introduces its own set of numerical problems. It is demonstrated that using new expression for the well index models the well inflow with high accuracy even on a coarse grid. [1] E.S. Makarova, D.V.Posvyanskii, V.S.Posvyanskii, A.B. Starostin ECMOR XI P26 2008
-
-
-
Evaluation of Ultra Low Concentration Surfactant System for Chemical Flooding
Authors I. Sagbana, P. Diaz and M. CentenoIn order to select a surfactant formulation for chemical flooding, the surfactant has to be evaluated at reservoir conditions to determine its compatibility with the reservoir to be injected in. This is to avoid formation of gels and precipitation in the reservoir which can make surfactant enhanced oil recovery unsuccessful. In several studies, surfactants have been tested in the laboratory at room temperature using only sodium chloride salt in the brine. While in oilfield scenario, the temperature is higher and the reservoir brine contains divalent ions. In this study, very low concentration alcohol alkoxy sulfate with and without a co-surfactant in hard brine and medium crude oil has been evaluated. The results from the salinity scan, phase behaviour and core flooding experiments at 60°C shows that alcohol alkoxy sulfate is tolerant to divalent ions and its stability can be improved with the addition of methyl ester sulfonate and internal olefin sulfonate as co-surfactants. These co-surfactants were able to reduce the viscosity of microemulsion phase, create a lower interfacial tension by increasing solubilisation ratio and also increase oil recovery by at least 20%.
-
-
-
Satistical Analysis and Mapping of Oil and Gas Development Cost Based on Field Development Plan in Indonesia
Authors A. Azizurrofi, A. Asnidar, J. Simanjuntak and R. FirdausThe oil price has already sunk to its lowest level at 37.13 US$/bbl (per December 2015) over the last 11 years. In this economic downturn, the oil and gas company must take extra measures to cope with this issue. In a country that adopts a Production Sharing Contract fiscal regime such as Indonesia, there are several changes which can be proposed such as accelerating depreciation, providing incentives (Investment Credit and Interest Cost Recovery), DMO Holiday or readjusting the First Tranche Petroleum or Split Ratio. Another solution is to discover a new oil and gas area that has low cost development which also has large commercial reserves. Here, this paper will provide an insight about the development of oil and gas industry in Indonesia during the downfall of oil price and produces the bubble map of development cost and commercial reserves based on geographic area of Indonesia to show the attractiveness of investing in oil and gas industry based on statistical data analysis. There were 387 Plan of Development (POD) approved during 2003 - 2015. For the purpose of this paper, the geographic area of Indonesia shall be simplified into 6 different area (Sumatera, Natuna Sea, Java, Kalimantan, Sulawesi and Papua). From here, the total development cost and commercial reserves of each respective contract area in the POD shall be calculated and redistributed into the aforementioned clusters. Based on data analysis, the oil and gas industry in Indonesia is still considered attractive for the investors despite suffering from low oil price, this is because the maximum development cost needed in Indonesia is around 15.48 US$/bbl or smaller than the current oil price. Futhermore, there are several areas which have untapped wealth of commercial reserves such as Papua and Natuna Sea. Under the current adverse condition, this paper eventually provides a good insights for the investors and help them creating and revisiting their strategy and portfolio to invest in Indonesia's oil and gas industry.
-
-
-
An Efficient Hybrid Model for Fractured-vuggy Reservoir Based on Discrete Fracture-vug Network Model
More LessThe numerical simulation of fractured-vuggy reservoirs has received much attention in the past years, because of the significant contribution of fractured-vuggy reservoirs to the oil and gas reserves and productions. In this paper, an efficient hybrid method is proposed to simulate two phase flow in the fractured-vuggy reservoirs with multiple-length scaled fractures and vugs, which cannot be easily modeled by only the continuum models or discrete models. In this hybrid model, small fractures and vugs are modeled by a continuum model, and long fractures are modeled by the embedded discrete fracture model. Firstly, the coarse grid system is made according to the structure characteristics of fractures and vugs. Then, the discrete fracture-vug model (DFVN) is implemented in each coarse grid to calculate equivalent permeability tensor based on homogenization theory, and an analytical procedure is implemented to obtain a pseudo relative permeability curves for each grid containing fractures and cavities. The long fractures are embedded in homogenized media system. After that, an efficient numerical simulator is devised to solve the coupled system of long fracture and homogenized media based on mimetic finite difference method. At last, several numerical examples have been shown to verify the validity and accuracy of the hybrid model.
-