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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 - 20 of 163 results
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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