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ECMOR XVI - 16th European Conference on the Mathematics of Oil Recovery
- Conference date: September 3-6, 2018
- Location: Barcelona, Spain
- Published: 03 September 2018
1 - 50 of 172 results
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Reduced Variables Method for Four-Phase Equilibrium Calculations of Hydrocarbon-H2O-CO2 Mixtures at a Low Temperature
Authors M. Imai, H. Pan, M. Connolly, H. Tchelepi and M. KuriharaSummaryCarbon dioxide (CO2) flooding has been widely applied to enhance oil recovery. In low temperature CO2 injection cases, three hydrocarbon phases may be formed at equilibrium. Given the fact that connate water always exists in formations, and in many cases water injection precedes CO2 injection, four-phase equilibrium may arise where one aqueous phase plus three hydrocarbon phases coexist. In cases where CO2 dissolution into water cannot be ignored, a robust and efficient four-phase equilibrium calculation framework is necessary for a compositional reservoir simulator. This is challenging not only because the number of variables increases but also because stability analysis becomes much complicated as the number of phases increases.
In this research, a novel four-phase equilibrium calculation framework is proposed for a compositional reservoir simulator. A reduced variables method is adopted to solve four-phase flash problems efficiently and robustly. Multiphase flash calculations using reduced variables (RV) can converge to the equilibrium solution faster than formulations using conventional variables ( Petitfrere and Nichita, 2015 ). Also RV solves numerical problem in Newton iterations with trace components in aqueous phase. In addition to the implementation of the RV formulation, a systematic procedure consisting of stability analysis and flash calculations is proposed without any prior knowledge of initial K-values. Sets of different initial K-values are appropriately tested in each stability analysis.
We perform comprehensive testing using characterized fluids found in publications, in order to validate robustness of the proposed procedure. The four-phase regions in pressure-composition (PX) space can be accurately identified using our procedure. On the other hand, some points are mistakenly evaluated as the three-phase state if the existing approaches such as Li and Firoozabadi (2012) are used, as in the case of previously published articles on four-phase equilibrium calculations. Our framework proposed in this paper is a promising four-phase equilibrium calculation framework for a compositional reservoir simulator. The procedure achieves excellent robustness and efficiency with minimal modification to the conventional two or three-phase equilibrium calculation framework.
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Production Optimization Of Thermodynamically Rigorous Isothermal And Compositional Models
Authors T.K.S. Ritschel and J.B. JørgensenSummaryIn this work, we consider algorithms for solving production optimization problems that involve isothermal (constant temperature) and compositional oil production processes. The purpose of production optimization is to compute a long-term production strategy that is economically optimal. We present a thermodynamically rigorous model of isothermal oil production processes. We derive the model from first principles by applying a number of assumptions including the assumption of constant temperature. The model is based on two key principles, namely phase equilibrium and conservation of mass and energy. The conservation equations are expressed as partial differential equations, and we model the phase equilibrium as a VT flash process. It is common to formulate the phase equilibrium conditions in oil reservoir flow models as the fugacities being equal. We describe how to derive that condition from the phase equilibrium conditions from the VT flash problem. The VT flash is an adaption of the second law of thermodynamics, i.e. the entropy of a closed system in equilibrium is maximal, to isothermal systems. The VT flash can therefore be formulated as an inner optimization problem that needs to be solved for each grid cell in the discretized reservoir in the forward simulation of the oil production process. We demonstrate that it is natural to model such isothermal production processes with differential-algebraic equations in a semi-explicit index-1 form. We describe a single-shooting algorithm for solving the production optimization problem efficiently. It is key to the efficiency of such algorithms to compute gradients. For that purpose, we use an adjoint algorithm. We implement the singleshooting algorithm in C/C++ using the open-source software DUNE, the open-source thermodynamic software ThermoLib, and the numerical optimization software KNITRO. Finally, we present a numerical example that involves optimal waterflooding.
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Parameterization Of Element Balance Formulation In Reactive Compositional Flow And Transport
More LessSummaryParameterization of element balance formulation in reactive compositional flow and transport
K. Kala1, D. Voskov1,2
1 Department of Geoscience and Engineering, TU Delft
2 Department of Energy Resources Engineering, Stanford University
We present a novel nonlinear formulation for modeling reactive-compositional transport in the presence of complex phase behavior related to dissolution and precipitation in multi-phase systems. This formulation is based on the consistent element balance reduction of the molar (overall composition) formulation. To predict a complex phase behavior in such systems, we include the chemical equilibrium constraints to the multiphase multicomponent negative flash calculations and solve the thermodynamic and chemical phase equilibrium simultaneously. In this solution, the phase equilibrium is represented by the partition coefficients whereas the chemical equilibrium reaction is represented by the activity coefficients model. This provides a generic treatment of chemical and thermodynamic equilibrium within an EOS SSI loop by modification of the multiphase flash to accommodate chemical equilibrium. Using the Equilibrium Rate Annihilation matrix allows us to reduce the governing unknowns to the primary set only while the coupling between chemical and thermodynamic equilibrium is captured by a simultaneous solution of modified multiphase flash equations. An input in this thermodynamic computation is an element composition of the mixture when an output contains fractions of components in each phase, including solids. This element balance molar formulation along with the modified formulation for multiphase flash has been tested in a simple transport model with dissolution and precipitation reactions. The same approach will be later used to model a system involving kinetic reactions. The simulation of more general practical models is performed using the recently developed Operator-Based Linearization (OBL) technique. In the modified version of the OBL, the nonlinear element based governing equations are formulated in terms of space and state-dependent parameters constrained by the solution of the extended multiphase flash based on molar element compositions. This approach helps us to add equilibrium reaction capabilities to the computationally efficient OBL technique.
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Compositional Reservoir Simulation Using (T,V) Variables-Based Flash Calculation
Authors D. Paterson, M.L. Michelsen, E.H. Stenby and W. YanSummaryPhase equilibrium calculation is at the heart of compositional reservoir simulation. The conventional example is the isothermal isobaric flash (T,P) which must be solved in every grid-block at each iteration during simulation. This is conventionally solved through the repeated use of stability analysis and phase-split calculation. To accurately represent the fluid properties it is useful to use an equation of state.
Most compositional reservoir simulations use the cubic equations of state (SRK or PR). More advanced equations of state which, for example, take account of association are attractive alternatives (e.g. CPA or PC-SAFT) for some simulations. Each of these models are functions for the Helmholtz energy with the natural variables (T,V,n), using the conventional flash framework it is necessary to solve the equation for volume (at a given pressure) at each iteration. Though simple for the cubic equations of state this is a more significant issue when using advanced equations of state where the association equations must also be solved iteratively at each volume iteration. This increase in computational cost is one reason that the more advanced equations of state are not yet in common use.
An alternative framework to solve the isothermal isobaric flash problem is possible. Instead of solving the equation of state for volume at each iteration the pressure of the phases is matched only at the final equilibrium point. This allows for the natural variables, (T,V,n), of the equation of state to be used to co-solve the equation of state with the equilibrium equations. Using this framework means that the equation of state does not need to be solved for volume at each iteration. This means that the more complex equations of state are only marginally more computationally expensive than the simple cubics.
In this work we will present a method to solve the (T,P) flash problem using the natural variables of the equation of state, (T,V,n). The resulting framework will be used in a multiphase, compositional, 3D reservoir simulator and demonstrated using a number of examples. The computational cost of the proposed method will be compared with the conventional method for solving the (T,P) flash problem when solving the same simulation problem.
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Theoretical Investigation Of Two-Ends-Open Free Spontaneous Imbibition
More LessSummaryTwo-ends-open free spontaneous imbibition refers to a laboratory core experiment, with one end face exposed to the wetting phase and the other end exposed to the non-wetting phase. Spontaneous imbibition leads to the production of non-wetting phase both co-currently and counter-currently. This paper extends previous work on systems of infinite length and presents the exact one-dimensional semi-analytic solution for such a system, and validates the solution with numerical simulation.
The methodology solves the partial differential equation of unsteady state immiscible, incompressible flow with arbitrary relative permeability and capillary pressure functions using a fractional flow concept as a function of saturation and time. The solution strategy uses backward finite differences on both the temporal variable and water saturation to solve for the instantaneous and average normalized water fluxes. The approach avoids the evaluation of implicit integral solutions and applies iterations on the flux ratio to satisfy both the flow and pressure boundary conditions. The solutions are obtained through two shooting processes for both normalized water flux and the pressure profile at all temporal steps.
The wetting phase is continuously imbibed into the core with initial flux being close to infinity. As the imbibition front propagates, the ratio of the co-current non-wetting phase flux to the inlet water flux increases from zero to a finite value below one which is dependent on the intrinsic properties of the system. This indicates that the production of non-wetting phase at the inlet will not cease before the front reaches the outlet, irrespective of the length of the system. The time for the front to reach the outlet, the ending ratio of co-current oil flux to inlet water flux and the variation of total produced volume from both ends, along with their sensitivities with respect to the absolute permeability, system length and wettability are analyzed in this study as well. The results also indicate the solution is independent of system length and permeability in its dimensionless form.
Unlike previous literature, we have not assumed self-similar solutions or treated the flow as purely co-current or counter-current. The boundary conditions for the system analyzed here are easily achievable in the lab and have been discussed in the literature. The results from this study could be used to serve as a benchmark for numerical simulations, in future applications such as relative permeability and capillary pressure estimation, or improved interpretation of lab to field relationships through scaling group analysis.
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Field Scale Modeling Of Bio-Reactions During Underground Hydrogen Storage
Authors B. Hagemann, L. Ganzer and M. PanfilovSummaryThe energy transition from fossil and nuclear energy towards an energy supply system from renewable sources will require an enormous extension of the existing energy storage capacity. For this intention depleted oil and gas reservoirs could play a key role when they are used as storage reservoirs for hydrogen or other energy carriers in a seasonal or more frequent cycle.
In previous studies it was shown that chemical reactions catalyzed by anaerobic microorganisms and mixing phenomena between gases with different composition have important influences in underground storage of hydrogen. In particular hydrogenotrophic microorganisms could produce methane by metabolizing hydrogen and carbon dioxide. To describe these effects a model was developed which couples the compositional two-phase transport of gas and water to microbial population dynamics and bio-chemical reactions.
In this work the numerical model was applied to a field scale storage scenario using a real geological model and several storage operation wells. The complex multi-physical model applied on around 200.000 grid cells results in approximately 2 million degrees of freedom. In addition the strong coupling between the microbial population dynamics and transport of chemical components is numerically difficult to handle and consequently small times steps not larger than one or two days have to be calculated. To overcome the computational effort the simulation study was executed on a high performance computing cluster.
The interpretation of the simulation results shows that a significant amount of the stored hydrogen was transformed into methane due to the bio-chemical reactions. In addition it was demonstrated that the produced gas contains H2S in the range of some parts per thousand when sulfate reducing bacteria were present.
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Well Modelling By Means Of Coupled 1D-3D Flow Models
Authors I.G. Gjerde, K. Kumar and J.M. NordbottenSummaryIn this work we present a new numerical method for solving coupled 1D-3D flow models, which can be used, for example, to model the flow in a well and the surrounding reservoir. Assuming 1D Poiseuille flow in the well, we use Stirling’s law of filtration to couple it to the 3D flow model used for the reservoir. The presence of a line source in the governing equations of the reservoir is known to cause a logarithmic type singularity in the solution around the well. For this reason, the solution is difficult to approximate numerically. We therefore introduce a decomposition technique where the solution is split into an explicitly known logarithmic term capturing the singularity and a well-behaved correction term. After a decoupling, the coupled 1D-3D model can then be reformulated as a fixed point iteration scheme iterating over the 1D pressure in the well and the 3D correction term for the reservoir. The iteration scheme can be implemented using both Galerkin and mixed finite element methods, the former of which leads to mass conservative solutions. The advantage of the decoupling and reformulation is twofold: Firstly, it recasts the model into a system for which the discretization schemes and solution methods are readily available. Secondly, it recovers optimal convergence rates without needing to perform a mesh-refinement around the well.
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Vanishing Artificial Capillary Pressure For Improved Real-Size Reservoir Simulations
Authors P. Salinas-Cortes, C.C. Pain, H. Osman, C. Heaney, D. Pavlidis and M.D. JacksonSummaryA common approach to stabilise the system arising from the discretisation of the advection equation is to introduce artificial diffusion. However, introducing artificial diffusion affects the result, therefore a balance has to be found so that the introduced artificial diffusion does not affects the final result.
Recently, a vanishing artificial diffusion was presented. In that method, the diffusion was controlled by the convergence of the non-linear solver by multiplying the artificial diffusion term by the difference between the most recent saturation estimation and the one obtained in the previous non-linear iteration. This approach showed that it is capable to help to reduce the computational effort required by the non-linear solver, as classical artificial diffusions do. However, this approach could lead to an introduction of an artificial source/sink in the system, therefore not conserving mass.
A conservative vanishing artificial diffusion is presented here. It improves the convergence and convergence rate of the non-linear solver by reducing the non-linearity of the equations. Moreover, it is tailored to specially help to deal with the capillary pressure. The vanishing artificial diffusion is introduced using the same model employed to introduce the capillary pressure, obtaining a vanishing artificial capillary pressure diffusion term. By solving this term implicitly in the saturation equation, a very efficient method to model multiphase porous media flow with physical capillary pressure is obtained. This is tested in real-size reservoir simulations with realistically high capillary numbers to prove its efficiency.
The presented method provides accurate results and significantly reduces the effort required by the non-linear solver to achieve convergence. It enables to carry out very demanding numerical simulations, e.g. when the physical capillary pressure effects are dominant, with Courant numbers that are at least two orders of magnitude bigger than without it.
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Application of a quasi-implicit scheme to the simulation of non-Newtonian flows through porous media
Authors R. de Loubens, L. Léon and L. PatacchiniSummaryWhen modeling non-Newtonian flows through porous media, numerical difficulties arise due to the velocity-dependence of phase mobilities. While a fully implicit treatment is unconditionally stable in the von Neumann sense, it leads to at least a 19-point stencil on 3D hexahedral grids; its implementation is therefore complex and its computational cost is high. Simpler semi-implicit schemes are often encountered, whereby the pressure gradient driving the flow is treated implicitly while the velocity-argument of mobilities is evaluated explicitly. These are however only conditionally stable, and in some practical situations of interest, clearly non-monotone. In this context, a quasi-implicit discretization has recently been proposed, where the velocity-argument of the mobilities is evaluated at cell edges with an implicit normal component, and an explicit transverse component. It has better stability and monotonicity properties than conventional semi-implicit schemes, while requiring only a 7-point stencil on 3D hexahedral grids.
This quasi-implicit scheme was implemented in our parallel in-house research reservoir simulator to model non-Newtonian polymer flows. A detailed description of its implementation is provided, accommodating different rheological models. Extensive numerical tests in various geometries are then performed to validate the implementation and illustrate the advantages of the new scheme.
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Robust And Accurate Formulation For Modeling Of Acid Stimulation
More LessSummaryAccurate representation of processes associated with energy extraction from subsurface formations often requires models which account for chemical interactions between different species in the presence of multiphase flow. In this study, we focus on modeling of acid stimulation in the near-well region. For the chemical processes which include a dissolution of rock material, an issue arises with the predictive representation of flow. Taking into account the spatial scale of discretization, some of simulation control volumes can have values of porosity close to 1, which makes an application of Darcy’s law inconsistent and requires employing a true momentum equation such as the Darcy-Brinkman-Stokes (DBS) equation. The DBS equation automatically switches the description between Darcy equation in control volumes with low porosity and Stokes equation in grid blocks with high porosity. For chemical reactions, we propose a local nonlinear solution technique that allows solving the balance of solid species separately yet retaining the full coupling with rest of the equations. Finally, we study the impact of multiphase flow. The DBS approach is not well established for multiphase flow description. Therefore we employ a hybrid approach, where we assume that the single-phase DBS flow and the multiphase Darcy flow occur in separate regions. We test the accuracy and performance of both approaches on realistic models of practical interest.
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A Novel AMG Approach Based On Adaptive Smoothing And Prolongation For Reservoir Simulations
Authors V.A. Paludetto Magri, M. Ferronato, A. Franceschini and C. JannaSummaryReservoir models can easily incorporate millions or even billions of unknowns. Algebraic multigrid (AMG) methods are often the standard choice as iterative solvers or preconditioners for the solution of the resulting linear systems. These comprise a family of techniques built on a hierarchy of levels associated with decreasingsize problems. In this way, optimality and efficiency are achieved by combining two complementary processes, i.e. relaxation and coarse-grid correction. One of the key factors defining a fast AMG method consists of capturing accurately the near-null space of the system matrix for the construction of suitable prolongation operators.
In this work, we propose a novel AMG package, aSP-AMG, where aSP means “adaptive Smoothing and Prolongation” and the “adaptive” attribute implies that we follow the perspective of adaptive and bootstrap AMG. We construct a space of smooth vectors of limited size (test space) using the Lanczos method and introduce the factorized sparse approximate inverse (FSAI) as a smoother. This improves the smoothing capabilities of the aSP-AMG as FSAI is more effective than Jacobi and much sparser than Gauss-Seidel. Moreover, FSAI has been shown to be strongly scalable. The coarsening phase is carried out as in classical AMG, but the strength of connection is computed by means of the affinity based on the test space. Finally, three new techniques are developed for building the prolongation operator: i) ABF, running few iterations of the aFSAI algorithm; ii) LS-ABF, updating the ABF coefficients with a least squares minimization; iii) DPLS, considering a least-squares process only.
The aSP-AMG performance is assessed through the solution of reservoir engineering problems including both fluid flow and geomechanical test cases. Comparisons are made with the FSAI and BoomerAMG preconditioners, showing that the new method is generally superior to both approaches.
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On The Acceleration Of Ill-Conditioned Linear Systems: A Pod-Based Deflation Method For The Simulation Of Two-Phase Flow
Authors G.B. Diaz Cortes, J.D. Jansen and C. VuikSummaryWe explore and develop POD-based deflation methods to accelerate the solution of large-scale linear systems resulting from two-phase flow simulation.
The techniques here presented collect information from the system in a POD basis, which is later used in a deflation scheme.
The snapshots required to obtain the POD basis are captured in two ways: a moving window approach, where the most recently computed solutions are used, and a training phase approach, where a full pre-simulation is run. We test this methodology in highly heterogeneous porous media: a full SPE 10 model containing O(10^6) cells, and in an academic layered problem presenting a contrast in permeability layers up to 10^6. Among the experiments, we study cases including gravity and capillary pressure terms.
With the POD-based deflated procedure, we accelerate the convergence of a Preconditioned Conjugate Gradient (PCG) method, reducing the required number of iterations to around 10–30 %, i.e., we achieve speed-ups of factors three to ten.
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Localized Linear Systems For Fully-Implicit Simulation Of Multiphase Multicomponent Flow In Porous Media
Authors S. Sheth, A. Moncorgé and R.M. YounisSummaryDuring the solution of fully-implicit reservoir simulation time-steps, it is often observed that the computed Newton updates may be very sparse, considering computer precision. This sparsity can be as high as 95% and can vary largely from one iteration to the next.
In recent work, a mathematically sound framework was developed to predict the sparsity pattern before the full linear system is solved. The theory is restricted to general, scalar nonlinear advection-diffusion-reaction problems in multidimensional and heterogeneous settings. This theory had been applied to reduce the size of the linear systems that were computed during sequential implicit timesteps for two-phase flow. The results confirmed that the linearization computations and the linear solution processes may be localized by as much as 95% while retaining the exact Newton convergence behavior and final solution. Inspired by the great success of that methodology, this work develops algorithmic extensions in order to devise localization algorithms for fully-implicit coupled multicomponent problems.
We propose, apply, and test a novel algorithm to resolve a system of hyperbolic equations obtained from an Equation of State (EOS) based compositional simulator. When applied to various fully-implicit flow and multicomponent transport simulations, involving six thermodynamic species, on the full SPE 10 geological model, the observed reduction in computational effort ranges between six to forty-nine fold depending on the level of locality present in the system. We apply this algorithm to several injection and depletion scenarios with and without gravity and capillarity in order to investigate the adaptivity and robustness of the proposed method to the underlying heterogeneity and complexity. We demonstrate that the algorithm enables efficient and robust full-resolution fully implicit simulation without the errors introduced by adaptive discretization methods or the stability concerns of semi-implicit approaches.
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Transition Of Algebraic Multiscale To Algebraic Multigrid
Authors S. Ehrmann, S. Gries and M.A. SchweitzerSummaryAlgebraic Multiscale (AMS) is a recent development for the construction of efficient linear solvers in certain reservoir simulations. It employs analytical upscaling ideas to coarsen the respective linear system and provides a high amount of inherent parallelism.
However, it has the drawback that it can currently only be applied to problems for a single scalar physical unknown, e.g., a pressure sub-problem. Moreover, typical AMS exploits a structured grid and results in a two-level scheme only. Generalizing the AMS approach to overcome these limitations requires substantial efforts and is not straightforward.
To exploit the benefits of AMS, however, we integrate its core ideas in an Algebraic Multigrid (AMG) method. Thus, all results and techniques from the well-established AMG are directly available in conjunction with (core ingredients of) AMS. This holds regarding multilevel usage and applicability for unstructured problems. But it also holds for the System-AMG approach that allows to consider additional thermal and mechanical unknowns. In this paper, we identify the algorithmic similarities between both approaches, AMS and AMG. In fact, the basic coarsening idea of AMS corresponds to the so-called aggregative AMG approach. However, plain aggregative AMG suffers from the drawback of simplified transfer operation, or interpolation, within the hierarchy. By the integration of the AMS transfer we overcome this limitation and significantly improve the robustness of the aggregative AMG; especially in cases with inhomogeneous material coefficients. Yet, the method is not as robust as classical AMG, though. However, the setup phase is significantly simplified.
Moreover, we adjust the AMS-like interpolation to work purely algebraically, independent of any grid structure. This involves certain compromises to the original AMS idea. However, it can now be used as an interpolation in any aggregative AMG approach.
An additional advantage of our approach is the improved controllability of the hierarchy’s operator complexity, i.e., its memory consumption. This is especially important with increasing density of the matrix stencils, e.g., in (geo)mechanics or data science.
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On the Development of a Relative Permeability Equation of State
Authors P. Purswani, M.S. Tawfik, Z.T. Karpyn and R.T. JohnsSummaryStandard compositional simulators use composition-dependent cubic equations-of-state (EoS), but saturationdependent relative permeability (kr). This discrepancy causes discontinuities, increasing computational time and reduced accuracy. To rectify this problem, kr has been recently defined as a state function, so that it becomes compositional dependent. Such a form of the kr EoS can significantly improve the convergence in compositional simulation, in that time step sizes are near the IMPEC stability limit and flash calculation convergence is improved.
This paper revisits the development of kr EoS by defining relevant state variables and deriving functional forms of the state function via a methodical approach. The state variables include phase saturation, phase connectivity, wettability, capillary number, and pore topology. The developed EoS is constrained to physical boundary conditions. The model coefficients are estimated through linear regression on data collected from a pore-scale simulation study that estimates kr based on micro-CT image analysis. The results show that a simple quadratic expression gives an excellent match with simulation measurements from the literature. The goodness of fit (R2) value is 0.97 for kr at variable phase saturation and phase connectivity, and constant wettability, pore structure, and capillary number (∼10-4). The quadratic response for kr also shows excellent predictive capabilities.
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Efficient Modeling Of Near Wellbore Phenomena For Large Scale Gas-Condensate Systems In Massively Parallel Reservoir Sim
Authors S. Manzoor, U. Middya, T.J. Byer and P.I. CrumptonSummaryGas condensate reservoirs exhibit complex behaviour when they are produced below dew point pressure under isothermal conditions; this is due to the appearance of a two-phase gas-condensate in the near wellbore region. In addition, at high flow rates in the near wellbore region, inertial forces counteract with velocity dependence of relative permeability. This behaviour can be resolved using local grid refinements; however, the computation burden becomes excessive, especially in a full field simulation. Alternatively pseudo-pressure approach can be used which iteratively solves a non-linear equation at each integration point, and is also computationally expensive. Furthermore, the conventional pseudo-pressure method lacks efficient coupling of the complex interaction of fluid composition, liquid dropout rate, gas-oil relative permeability, gas-oil interfacial tensions, and non-Darcy flow effects. Development of a computationally efficient and accurate approach to model near wellbore phenomena without increasing grid resolution is presented. An adaptive piecewise representation of pseudo-pressure is used, replacing non-linear equation solved at each integration point, thus drastically reducing computational cost without undue loss of accuracy. Non-Darcy flow effects and interaction of rock and fluid properties are captured in the pseudo-pressure integrand in a unified manner. The results are validated against a commercial simulator, and fine grid results, which demonstrate the accuracy and consistency of the approach. Finally the efficiency of the approach is demonstrated by simulating a giant gas-condensate model with thousands of wells and millions of cells, all solved on a massively parallel computer.
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Phase behavior computations using Gibbs free energy minimization on GPUs for speeding up compositional simulations
Authors S. Shiozawa, A. Venkatraman and B. DindorukSummaryCO2 flooding in a preferred method of enhancing oil recovery as it has the dual benefit of sequestration and increasing oil recovery. In order to evaluate and design these processes, compositional simulations are used to track component changes during the course of flow. However, the phase behavior, as well as the stability computations associated with compositional simulations are time-consuming. In a compositional reservoir simulator, EOS, typically a cubic EOS (i.e., Peng-Robinson) for hydrocarbons, must be solved in all the grid blocks at every time step in order to accurately predict the phase behavior. In typical field-scale simulations, millions of grid blocks are used, and hence solving EOS for multicomponent hydrocarbon systems really slows down the simulation. There is a need to increase the speed and improve the efficiency of phase equilibrium computation for field-scale simulations. In this research, we develop a Gibbs free energy framework optimized for graphic processing unit (GPU) implementation. The Gibbs free energy minimization is preferred as it is a unifying function to combine components described using different thermodynamic models - Equation of State (EOS) as well as activity coefficient models. We use a combination of CPUs and GPUs to solve a constrained minimization problem and the solution is the equilibrium composition at a fixed temperature and pressure (PT flash). The NVIDIA Tesla GPUs help parallelize multiple functions evaluations simultaneously, which results in a significant speedup in computation times. A comparison of computation times of our approach with other approaches to compute and also algorithms to obtain equilibrium compositions is presented -1. CPU versus GPU. 2. K-value versus Gibbs free energy approach. The proposed model can easily be incorporated into existing reservoir simulators to decrease computational times.
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Nonlinear Domain Decomposition Scheme For Sequential Fully Implicit Formulation Of Compositional Multiphase Flow
Authors O. Møyner and A. MoncorgéSummaryNew Sequential Fully Implicit (SFI) methods for compositional flow simulation have been recently investigated. These SFI schemes decouple the fully coupled problem into separate pressure and transport problems and have convergence properties comparable to the Fully Implicit (FI) method. The pressure system is a parabolic problem with fixed overall-compositions and the transport system is a hyperbolic problem with fixed pressure and total-velocity. We discuss some aspects of how to design optimal SFI schemes for compositional flow with general equation-of-states by localizing the computations. The different systems are solved sequentially and the Fully Implicit solution is recovered by controlling the a-posteriori splitting errors due to the choice of decoupling.
When the parabolic and the hyperbolic operators are separated, it is possible to design nonlinear domain decomposition schemes taking the advantage of the specific properties of each operator. Usually, for reservoir simulation models, most of the reservoir is converged with SFI methods in one outer-iteration. However, in some localized regions with strong coupling between the pressure and the compositions, the SFI algorithms may need several outer-iterations. Here we propose a domain decomposition method based on a predictor-corrector strategy. As a first step, the nonlinear parabolic pressure equation is solved on the whole domain with the Multiscale Restriction-Smooth Basis (MsRSB) method used as a linear domain decomposition solver. In a second step the compositions system is solved. At the end of this first outer-iteration, most of the reservoir is converged. Based on a-posteriori splitting-errors of the SFI scheme in volume and velocity, we define local regions where additional global outer-iterations would be required in the conventional SFI scheme. We then fix Dirichlet boundary conditions for the pressure and the compositions and solve local problems in these non-converged regions. After convergence of these smaller nonlinear problems, if the boundary conditions are changed by the updated regions, the global pressure problem is revisited. An additional post-processing of local transport iterations makes sure mass is conserved everywhere. The resulting algorithm converges to the same solution as the FI solver, with all simultaneous updates to composition and pressure in localized regions.
We demonstrate the robustness of this nonlinear domain decomposition algorithm across a wide parameter range. Realistic compositional models with gas and water injection are presented and discussed.
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Consistent Upwinding for Sequential Fully Implicit Compositional Simulation
Authors A. Moncorge, H.A. Tchelepi and P. JennySummaryThere is strong interest to design Sequential Fully Implicit (SFI) methods for compositional flow simulations with convergence properties that are comparable to Fully Implicit (FI) methods. SFI methods decompose the fully coupled system into a pressure equation and a transport system of the components. During the pressure update, the compositions are frozen, and during the transport calculations, both the pressure and total-velocity are kept constant. The two systems are solved sequentially, and the solution, which is a fully implicit one, is obtained by controlling the splitting errors due to the decoupling. Having an SFI scheme that enjoys a convergence rate similar to FI makes it possible to design specialized numerical methods optimized for the different parabolic and the hyperbolic operators, as well as the use of high-order spatial and temporal discretization schemes. Here, we show that phase-potential upwinding is incompatible with the total-velocity formulation of the fluxes, which is common in SFI schemes. We observe that in cases with strong gravity or capillary pressure, it is possible to have flow reversals. These reversals can strongly affect the convergence rate of SFI methods. In this work, we employ implicit hybrid upwinding (IHU) with a SFI method. IHU determines the upwinding direction differently for the viscous, buoyancy, and capillary pressure terms in the phase velocity expressions. The use of IHU leads to a consistent SFI scheme in terms of both pressure and compositions, and it improves the SFI convergence significantly in settings with strong buoyancy or capillarity. We demonstrate the robustness of the IHU-based SFI algorithm across a wide parameter range. Realistic compositional models with gas and water injection are presented and discussed.
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Nonlinear Gauss-Seidel Solvers With Higher Order For Black-Oil Models
Authors Ø.S. Klemetsdal, A.F. Rasmussen, O. Møyner and K.-A LieSummaryThe fully implicit method is the most commonly used approach to solve black-oil problems in reservoir simulation. The method requires repeated linearization of large nonlinear systems and produces ill-conditioned linear systems. We present a strategy to reduce computational time that relies on two key ideas: i) a sequential formulation that decouples flow and transport into separate subproblems, and ii) a highly efficient Gauss-Seidel solver for the transport problems. This solver uses intercell fluxes to find all cells that only depend on their upstream neighbors and groups all remaining cells into local clusters of cells that are mutually dependent because of counter-current flow. The single cells and local clusters can then be sorted and solved in sequence, starting from the inflow and moving gradually downstream, since each new cell or local cluster will only depend on upstream neighbors that have already been computed. Altogether, this gives optimal localization and control of the nonlinear solution process.
This method has been successfully applied to real-field problems using the standard first-order finite volume discretization. Here, we extend the idea to first-order dG methods on fully unstructured grids. We also demonstrate proof of concept for the reordering idea by applying it to the full simulation model of the Norne oil field, using a prototype variant of open source OPM Flow simulator.
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Numerical Modelling Of Electromagnetic-Based Hybrid Eor Techniques For Bitumen Recovery
Authors A. Sadeghi, H. Hassanzadeh, T. Harding, B. MacFarlane and P. HaghighatSummaryIn recent years, electromagnetic heating (EMH) has been the focus of ever-increasing theoretical and experimental studies both in the laboratory and field scale to examine if it can be used to heat up the geomaterials in field scale. EM-solvent based bitumen recovery methods, such as ESEIEH pilot in Athabasca oil field, mainly use radiofrequency waves to generate heat in reservoir, and thereby reduce the viscosity of the bitumen to mobilize it.
EM wave propagation in a reservoir poses a coupled multi-physical process that involves not only the heat transfer and fluid flow, but also EM field distribution, which currently, a non-coupled approach is followed by industry using a conventional thermal simulator and an external electromagnetic wave solver where both are linked through an interface. To address the mentioned issues, the present study presents a coupled compositional numerical modeling approach to explore the EM heating phenomena pertinent to fluid flow in oil sand reservoirs. Generic field equations governing the coupling between energy equation and EM wave propagation are derived using the Maxwell’s equations.
The developed in-house numerical simulator is used to study the importance of EM-induced volumetric heat generation in a multiphasic heterogeneous oil sand reservoir. Results reveals that electromagnetic heating can be a promising method for the development of low quality oil sands. EMH moderates the amount of needed energy and also cuts the emitted CO2 compare to SAGD process. Furthermore, the operating temperature of vapor chamber is less than 160 °C for the optimized EM-solvent scenario, while it is more than 220 °C for the SAGD method.
We presented a coupled approach for modeling of electromagnetic heating of oil sands. The developed model can be used as a toolbox to perform sensitivity analysis, design of experimental setups and pilot scale implementation of electromagnetic heating of oil sands.
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Fully-Implicit Solvers For Coupled Poromechanics Of Fractured Reservoirs
Authors N. Castelletto, M. Ferronato, A. Franceschini, R.R. Settgast and J.A. WhiteSummaryIn this work we present a family of preconditioners for accelerating the fully-implicit solution of linear systems encountered in two practical applications: (i) Lagrange multiplier-based fault mechanics simulations using a mixed finite element approach, and (ii) multiphase poromechanics based on a mixed finite element-finite volume formulation. We consider block preconditioning strategies and focus on various Schur complement approximations that are based on a combination of physical and algebraic arguments. The performance of the proposed framework is illustrated using two challenging numerical examples---a synthetic fault mechanics test with manufactured solution and a large-scale water flooding problem.
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Conditioning Reservoir Models To Pressure Transients By Iterative Ensemble Smoother
Authors A. Khrulenko, A. Shchipanov and R. BerenblyumSummaryRecent developments in permanent reservoir monitoring and well surveillance enable accurate, real-time downhole pressure measurements, providing enormous amount of data.
A number of previous studies had shown that integrating well test data into reservoir models improves significantly their predicting capability.
In our study, the iterative Ensemble Smoother was applied to condition permeability field to well test data. 1D and 2D synthetic reservoir models were used to investigate the method performance with respect to measurement error and localization which we consider important from practical point of view.
At first, we analyzed the influence of measurement error. Despite of high accuracy of the modern pressure gauges, the pressure data are often quite noisy. In practice, various filtering, denoising and smoothing techniques are employed in order to clarify the reservoir response and reduce data uncertainty. We evaluated several cases with different variance of measurement error. The comparison revealed that the pressure data noise has strong impact on the parameter estimation and the method convergence. In many cases, the noise caused the ensemble drifting away from the true solution.
Another important practical aspect is localization of model updates. During well test, pressure transients reflect pressure propagation away from the well. The propagation dynamic is governed by the formation properties within the disturbed reservoir domain. Therefore, the pressure measurements at a given time may be used for updating formation properties in the model only within the disturbed domain around the well. This would lead to the conclusion that a localization technique may be employed to relate model updates to relevant observations representing response from different reservoir areas. A time/distance dependent localization technique was tested to address this problem. The testing results showed that the proposed localization technique allowed for better estimation of permeability distribution (in terms of discrepancy with the true case). Validation by a blind test showed a better uncertainty propagation as the final ensemble retained significant diversity in areas remoted from test wells, in contrast to the non-localized case.
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Simulation Of Gaussian Random Fields Using The Fast Fourier Transform (Fft)
Authors P. Abrahamsen, V. Kvernelv and D. BarkerSummaryWe generate independent Gaussian random variables on a regular grid and use a spatial filter to smooth the independent random variables to obtain a spatially correlated Gaussian random field. The FFT is used to speed up the smoothing since convolution is a simple cell by-cell multiplication in the Fourier domain. A representation of the spatial convolution filter in the Fourier domain is efficiently obtained from the FFT of any stationary correlation function. Since FFT is cyclic, the grid must be padded to ensure that opposite sides are uncorrelated. The size of the padding is discussed in detail. Most standard covariance functions fail to be positive definite on finite cyclic domains. This causes striping artifacts in the final simulated realizations and failure to meet statistical properties such as variogram reproduction in the simulated realizations. These problems are addressed and solutions are provided to ensure near perfect statistical properties of the generated realizations. The method is fast and can generate a hundred million grid cell realization in approximately 1.5 minutes on a standard laptop PC. The method scales approximately linearly in the number of grid cells.
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Determining Nonsynchrony Between PDG Transient Temperature, Pressure, And Flow-Rate Data With Wavelet Transform
By F. WangSummaryTransient temperature, another source to provide reservoir information besides transient pressure, has been received much attention in recent years. Different from the immediate response of transient pressure due to flow rate change, there are obvious time lags between the transient temperature change and flow rate change. In this paper, the nonsynchrony between PDG transient temperature, pressure and flow-rate data was quantitatively investigated with wavelet transform (WT). Field PDG data analysis show that transient pressure changes nearly at the same time of flow rate change. However, due to the adiabatic expansion/compression effect, the time lags between transient temperature and flow rate are significant and cannot be neglected. Field PDG data demonstrates that averagely transient temperature changes about 0.225 hours later than the flow rate. Accurately identifying the time of flow events from PDG temperature data is difficult. The identified flow event time from PDG temperature is the time when Joule-Thomson effect dominates the reservoir temperature change. The flow events with time periods less than the time lags cannot be identified from PDG temperature data due to the adiabatic expansion/compression effect. Compared with the Haar wavelet, the the second derivative of the Gaussian wavelet can more accurately identify flow events from PDG temperature data. This study will be useful for improving the accuracy of transient identification from PDG data, and can benefit transient temperature analysis by clarifying the time lags due to adiabatic expansion/compression effect, and synchronize PDG transient temperature, pressure and flow-rate data for better data processing and analysis.
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Geomodeling Concept For The Reservoir With High Heterogeneity
Authors M. Nikolovski, M. Pesalj and M. TcukanovaSummaryThe “STC” oilfield was discovered in 1959., it is in production since 1960. and it is one of the most important oil field in Serbia. Up to 2017. on the field were drilled 87 wells.
Geological structure is represented by Sarmatian, Badenian and Jurassic formation. From the aspect of the current field development and quantity of the remaining balance reserves, the most important stratigraphic unit is Badenian which is made of terrigenous reservoirs of different composition and characteristics.
The main goal of this study is to identify the causes and control factors of the existence of different well production and water-cut reduction from Badenian. Badenian reservoir is characterized by high lithological heterogeneity and the rocks are composed from several lithofacies, such as limestone, sandstone and their variation. Each lithofacie has own petrophysical properties. Due to the frequent vertical and horizontal lithological heterogeneity, it was necessary to create a new geological model, including a detailed lithological characterization of badenian’s reservoirs and localize it.
In order to achieve this, it was necessary to include a detailed sedimentological and petrophysical interpretation of the badenia reservoirs, where all cores and well logs data were analyzed. The result of the interpretation is the separation of dominant lithofacies in the reservoir of Badenian. The next step was to define the diffusion of interesting litofacies, which required detailed analysis of seismic data and surface attributes analysis.
The result of all the above-mentioned modeling phases is a geological model that can help us in further development of the field, define the most prosperous zones for effective production wells allocation and necessary operations on existing wells. Also, oil reserves have been estimated for each of the facies, what will enable better field development and planning further production. The first positive results are seen in the overlapping of zones with a lower degree of water-cut in boreholes with increased content of sandstones, next to limestone, which leads to the conclusion that these zones have less water-cut of during the production. The research helped us to create search criteria for geological modeling of similar objects.
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The Ability of Multiple-Point Geostatistics for Modelling Complex Fracture Networks in Tight and Shale Reservoirs
Authors Hojjat Khani, Hamidreza Hamdi, Long Nghiem, Zhangxing Chen and Mario Costa SousaSummaryFracturing horizontal wells is an important technology that can make production from tight and shale formations economical. The fractured tight and shale formations are recognized by complex fracture networks around the primary hydraulic fractures. Microseismic mapping is a technique which can shed light on the activities happen around the main fractures which can direct us towards the extent of the fracture half-length and the secondary fracture networks in the stimulated reservoir volume (SRV). However, microseismic mapping does not necessarily indicate if the observed events can be directly related to the increased conductivities around the wellbore. There is rather a large uncertainty about the interpretation of the extent of effective (reopened) fracture network which can have a large impact on the performance of the flow simulations.
In this paper, a quantitative workflow is attempted to model the discrete fracture networks using multiple-point geostatistical algorithms to account for the uncertainty in the interpretation of the microseismic events. Uncertainty in microseismic data interpretation is also included in the algorithm (in terms of secondary probability maps) to account for the variability in the extent of the discrete fracture network within the stimulated reservoir volume (SRV). A sensitivity study is performed to understand the effect of different parameters on the well flow performance given different fracture network models. The results show that the connectivity of the fracture networks generated by the MPS method in this study is rather poor. Consequently, the permeability of the natural fractures has a dominant effect on the flow performance. In fact, the poor connectivity of fracture network does not allow to observe the effect of porosity of natural fracture and the permeability of hydraulic fracture on the flow performance. This research restresses that the MPS algorithm is not a push-a-button method to always generate reliable realizations. This work provides a guideline to better screen the generated geostatistical realization.
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Geologically Plausible History Matching With Structural Uncertainty
Authors G. De Paola, P. Koryuzlov, R. Rodriguez Torrado, A. Fernandez, E. Reding, M. Bartnik and M. SeignoleSummaryA geologically plausible history matching workflow has been applied to a complex reservoir to improve reservoir characterization. Different structural interpretations have also been included in the formulation which allow with a single workflow to match petrophysical properties, structural interpretations, fluid properties and fault transmissibilities avoiding any regional multipliers or inconsistent discontinuities. A multiobjective optimization was formulated to assimilate production and pressure timeseries as well as well test data. An inhouse implementation of a Particle Swarm Optimization allows to efficiently solve the optimization problem and provide multiple matching solutions for an improved uncertainty quantification. The multiobjective formulation allows the decision maker to screen the matching realizations based on the degree of confidence on the difference data type as well have better control selected the most representative realizations. The workflow proposed shows good match with the observed quantities and allows a review of the initial model of the field based on the improved understanding of the dynamic response of the reservoir. It is the first step before a field development plan optimization with structural uncertainty.
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Accelerating CMA-ES In History Matching Problems Using An Ensemble Of Surrogates With Generation-Based Management
Authors M. Sayyafzadeh, R. Koochak and M. BarleySummaryBecause of the quasi-gradient update embedded in CMA-ES algorithm, it can outperform most of the population-based algorithms, from a convergence speed standpoint. However, due to the computationally expensive fitness function associated with history matching, the reduction of function (simulation) calls can be favourable.
In this study, an ensemble of surrogates (proxies) with generation-based model-management is proposed to reduce the number of simulation calls efficaciously. Since the fitness function is highly nonlinear, an ensemble of surrogates (Gaussian process) is utilised. The likelihood term is divided into multiple functions, and each is represented via a separate surrogate. This improved the response surface fitting.
In generation-based management, a stochastically selected measure (surrogate or reservoir-simulation) should be used to evaluate all the individuals of each generation. CMA-ES requires ranking of the individuals to select the parents. Therefore, the generation-based model-management fits well in CMA-ES, as surrogates are normally better in ranking the individuals than approximating the fitness.
History matching for a real problem with 59 variables and PUNQ-S3 with eight variables was conducted via a standard CMA-ES and the proposed surrogate-assisted CMA-ES. The results showed that up to 65% and 50% less simulation calls for case#1 and case#2 were required.
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A Two-Level MCMC Based On The Distributed Gauss-Newton Method For Uncertainty Quantification
Authors J. Rafiee and A.C. ReynoldsSummaryWe present a methodology to obtain a correct sampling of the posterior probability density function (pdf) conditional to observations where this posterior pdf can be formally expressed using Bayes’ theorem. Generating a correct sampling of a multimodal posterior pdf is a challenging task which can only be achieved with Markov chain Monte Carlo (MCMC) methods. In standard MCMC such as random-walk MCMC, evaluation of acceptance probability for a proposed state requires a forward model run (a reservoir simulation run). When the forward model run is computationally expensive, we cannot afford to generate a long Markov chains with tens of thousands or more states. Therefore, it is critically important to design the MCMC such that it converges to the posterior pdf after generating a few thousand or less states.
Here, a two-level MCMC procedure which can sample multimodal posteriors relatively efficiently is developed and applied. In the first step, we use the distributed Gauss-Newton (DGN) method to generate many modes of the posterior pdf in parallel; this procedure estimates sensitivity matrices without the need of an adjoint solution. A Gaussian mixture model (GMM) is then constructed based on the distinct modes that we find in the first step. In the second step, the constructed GMM is used as the proposal distribution for our MCMC algorithm. Because the proposal distribution is constructed as a direct approximation of the target pdf (without the normalizing constant), the Markov chain(s) constructed should converge relatively quickly to the posterior distribution and applications of the two-level MCMC algorithm to test problems show that our proposed two-level MCMC is far more efficient than the random-walk MCMC.
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Revealing Hidden Reservoir Features During History Matching Using An Adjoint Method
Authors D.D. Awofodu, L. Ganzer and H. AlmuallimSummaryOur work focuses on developing a novel approach to improving reservoir characterization using the Adjoint method applied to history matching and optimization workflows. The developed approach, termed the “Adjointbased model screening method”, can be used to reveal hidden reservoir features not captured in reservoir models. The need for the development of our model screening method is necessitated by reservoir simulation models that miss important reservoir behaviours occurring beneath the surface. The impact of such modelling practice on history matching is the extreme tweaking of reservoir parameters to fit such models to available measured data. This paper demonstrates the strength of our approach in revealing the location of hidden faults and channels using synthetic homogeneous models.
Over the course of our research, an efficient model screening approach capable of revealing hidden reservoir behaviour has been developed and subjected to synthetic homogeneous blind tests. Our model screening approach utilizes reservoir permeabilities as input to screen our synthetic homogeneous models for hidden reservoir features like faults and channels. Observed data are generated from cases containing faults and/or channels and cases without these faults/channels are defined as the starting case. The Adjoint method is then used to reveal the location of these hidden reservoir features on a grid block basis.
In order to ascertain the superiority of our model screening approach, we compared the performance of our approach to other approaches reported in literature [Capacitance Resistance Model (CRM) and Interwell Numerical Simulation Model with Front Tracking (INSIM-FT)]. Results obtained demonstrate that our Adjoint-based model screening method is capable of handling varying and constant water injection rates as opposed to other approaches mentioned that can handle only varying injection rates. In addition, besides revealing the location of channels and faults, our approach infers the degree of transmissivity of faults. The developed approach was tested with 2-D and 3-D homogeneous models and results obtained proved that regardless of model dimensionality, hidden reservoir features can be revealed.
The most significant finding is that the accuracy of the adjoint-based model screening method in revealing the location of hidden reservoir features depends on the number of wells existing in the reservoir and their arrangement.
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Geostatistically-Consistent History Matching Of Lithofacies And Reservoir Properties Applied To Synthetic and Real-Field Cases
Authors E.S. Zakirov, I.M. Shiryaev, I.M. Indrupskiy, O.V. Lyubimova, E.Yu Arkhipova and D.P. AnikeevSummaryIn previous studies of the authors presented at the ECMOR-XIV and ECMOR-XV, an adjoint-based geostatistically-consistent approach was proposed for automated history matching of reservoir models. In the approach, reservoir property and facies distributions in a 3D models are consistently modified in the efficient iterative history matching procedure, with control parameters being anisotropic variogram parameters for the facies distribution and for reservoir properties distributions within each facie, as well as parameters of poroperm petrophysical correlations and values at pilot points. Derivatives of the objective function are obtained with adjoint method taking into account geostatistical relations between reservoir properties and variogram parameters. A concept of continuous "facies" and a weighting method for reservoir property calculations were also developed.
In this study we present the results of validation and further development of the geostatistically-consistent procedures for history matching.
In the first part of the study, we show and analyze the results of multivariant synthetic validation of the approach with simultaneous identification of anisotropic variogram parameters and reservoir properties at pilot points.
Then we describe the application of the continuous-facies approach to modeling of lithofacies and reservoir properties distributions for a real massive terrigenous gas reservoir in Western Siberia. We show that it proved successful in reflecting highly-heterogeneous distribution of reservoir properties with lens-like inclusions while preserving the overall geological consistency and continuous nature of sedimentation in the 3D model.
It is interesting to note that previous 3D dynamic flow model for field development planning was manually history matched through simulation of reservoir bodies' discontinuity. In other words, variogram ranges were artificially lowered for achieving desired level of reservoir disconnectivity. From production data analysis it is clear that gas-water contact advanced locally in vicinities of producing wells instead of an overall global contact elevation in accordance with global pressure distribution. Almost 80% of initial gas in place has been already produced, and pressure declined to almost 20% of its initial value. At this stage of development one could expect global displacement of gas by water, but reservoir heterogeneity played an important role forming hard-to-recover gas reserves at distant reservoir zones. All these peculiarities were successfully taken into account within the new 3D model built for a geostatistically-consistent history matching to production data. The results of this study would be presented in a conference paper and presentation.
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Petrophysical Parameters Inversion From Seismic Data Using An Ensemble-Based Method - A Case Study From A Compacting Reservoir
Authors T. Bhakta, E. Tolstukhin, C. Pacheco, X. Luo and G. NævdalSummaryWe implement an ensemble based petrophysical parameter inversion framework to estimate static as well as dynamic reservoir/ petrophysical parameters such as saturations, pressure and / or porosity fields using seismic data. Here, we consider acoustic impedance (Ip) data as the seismic data. The suggested approach is solved as a Bayesian inversion problem where the prior is provided as an ensemble of pressure-saturation and porosity fields. Here, the realizations of porosity and permeability fields of the prior model are generated using geostatistical methods and are further used in a reservoir simulator to obtain the realizations of pressures-saturations fields at the time of the seismic acquisition. The pressure-saturations and porosities are then changed to account for the information available from acoustic impedances using an iterative ensemble smoother. The outcome is a new ensemble of pressure-saturation and porosity fields that honor the seismic data.
The new approach differs from conventional deterministic petrophysical parameter inversion algorithms using seismic data by being stochastic, and more importantly, it pays more attention to the uncertainty quantification. Our results show that the suggested ensemble-based method is suitable to handle the nonlinear inverse problem and has the capacity of providing quantification of the uncertainty of the result.
We apply the proposed framework to a field-like 3D synthetic reservoir model, based on a compacting field scenario. The reservoir model consists of three fluid phases (water, oil and gas), and exhibits production related compaction. The numerical results from study indicates that the proposed framework can integrate the reservoir-engineering data as prior knowledge with the seismic data, achieving reasonable estimates of both the static and dynamic reservoir parameters.
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A Novel Multilevel Method For Assimilating Spatially Dense Data
Authors K. Fossum and T. MannsethSummaryInverted time-lapse seismic data are rich with respect to information about reservoir fluid flows. Ensemblebased data assimilation (EDA) of such spatially dense data into reservoir models requires a sufficiently large number of degrees of freedom (DOF). The DOF in straightforward EDA equals the ensemble size, E. Only a moderately sized E is, however, computationally feasible for large reservoir models. To increase the DOF, localization is routinely applied, but successful localization requires preconceived knowledge of the specific case and substantial manual effort. Alternative methods for increasing the DOF are therefore desirable. The large imbalance between data-space size (DSS) and DOF for problems with spatially dense data emphasizes this further.
We have considered generic methods for better balancing DSS and DOF. To decrease the DSS we used coarse data representation (CDR) of spatially dense data, that is, we map the data onto a regularly coarsened grid using averaging. To increase E (and, hence, the DOF) without increasing the computational cost of an ensemble forward run, we used simulations on a regularly coarsened grid with simple upscaling of reservoir properties (CGU). Results obtained with a combination of CDR and CGU, where the data and simulation grids were coarsened to the same level, were very good, but the optimal level varied from one case to another.
To avoid manual selection of an optimal level, we consider multilevel EDA using the novel Multilevel Hybrid EnKF (MlHEnKF) in combination with multilevel data representation (MDR) on a sequence of regularly coarsened grids. The resulting EDA method - MlHEnKF with MDR - can be applied in conjunction with localization, if desired. Assimilating inverted time-lapse seismic data in a reservoir-history-matching example, we assess the performance of the MlHEnKF with MDR by comparing the results to those obtained with a standard EDA approach.
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Evaluation of Regions of Influence for Dimensionality Reduction in Emulation of Production Data
Authors C.J. Ferrerra, G. Avansi, I. Vernon, D.J. Schiozer and M. GoldsteinSummaryOil and gas companies use reservoir simulation models for production forecasting and for business and technical decisions at the various stages of field management. The size and complexity of the reservoirs often requires reservoir models with a high resolution (number of grid blocks) to improve the reservoir behaviour prediction. As a consequence, simulation time becomes a limiting factor for routine workflows such as probabilistic history-matching, production optimization or uncertainty quantification, which requires a higher number of reservoir simulations. One possible solution to this problem is to use Bayesian statistic techniques known as emulation to substitute the simulator in parts of the workflow. An emulator is an approximate representation of a complex physical model; it is usually several orders of magnitude faster to evaluate than simulation, hence facilitating previously intractable calculations because of its speed. However, the challenge to incorporate spatial attributes, such as geostatistical realizations, as inputs remains. It is unfeasible to consider the reservoir spatial property value from each grid cell as a single input, so it is necessary to perform a dimensionality reduction to handle spatial properties as inputs in the emulation process. The use of region of influence is a way to deal with a high-dimensional model in an emulation setting and reduce the spatial properties space. Therefore, we evaluate different types of region of influence during the dimensionality reduction process to emulate production data of a complex numerical model. The regions of influence evaluated were defined using: streamlines, producer-injector pairs, Voronoi based on injection wells and Voronoi based on production wells. The dimensionality method considered were Principal Variables and Stepwise AIC. Our goal is to present and discuss alternatives to treat the high-dimensional input space, i.e., spatial reservoir properties instead of multipliers, to build effective emulators for production history data to use in oil industry workflows, which typically are time-consuming.
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Efficient Multi-Objective History-Matching Using Gaussian Processes
Authors H. Hamdi, I. Couckuyt, T. Dhaene and M. Costa SousaSummaryIn a multiobjective optimization approach, a trade-off is sought to balance between the optimality of different objectives. In this paper, we introduce a new efficient multiobjective optimization approach using sequential Gaussian Process (GP) modeling that can quickly find the Pareto solutions in a minimal number of model evaluations. This is the first time that we present this approach for history-matching. The difference with other optimization algorithms is elucidated for the cases where we can only afford to run a limited number of simulations. Unlike other surrogate-based methods, we do not aim for a greedy approach by minimizing the surface itself as there can be a large uncertainty in the surrogate approximations. Instead, statistical criteria are introduced to account for both proxy model uncertainty as well as its extrema.
This multiobjective optimization approach has been successfully applied for the first time to history match the production data (i.e. pressure, water and hydrocarbon rates) from a multi-fractured horizontal well in a tight formation. A GP surface is constructed for each misfit, to provide the predictions and the associated uncertainty for any unknown location. Multiobjective criteria, i.e., the hypervolume-based Probability of Improvement (PoI) and Expected Improvement (EI), are developed to account for the uncertainty of the misfit surfaces. The maximization of these statistical criteria ensures to balance between exploration and exploitation, even in higher dimensions. As such, a new point is selected whose values in different objectives are predicted to hopefully extend or dominate the solutions in the current Pareto set.
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Use Of Subsea Technologies For Produced Water Management In Mature Offshore Fields Using Integrated Asset Modeling
Authors O.J. Peña Piraneque, J.C. Hohendorff Filho and D.J. SchiozerSummaryThe results of the created methodology for evaluating the economic viability of installing subsea technologies for Oil-Water (O-W) Separation and Produced Water Re-Injection (PWRI) in mature offshore fields as a solution for water production management by using Integrated Asset Modeling (IAM) are presented. The methodology was tested in the benchmark case UNISIM-I-D showing excellent results, nevertheless, its application can be extensible to any other field where the installation of this kind of subsea systems is being analyzed.
Through the explicit coupling of specialized simulators of reservoir, multiphase flow in the tubing, production network, and economic modeling is possible both forecasting the production behavior of the field and generating the economic scenarios in a more realistic manner when the complex subsea systems are included to the production network. The equipment modeled consists of a subsea O-W separator located at the producer wellhead and a subsea pump that directly re-injects the separated water to the injector wellhead. Although the model has some simplifications, it permitted evaluating the implementation from a reservoir engineering perspective and knowing the production response without losing the representativeness of phenomena occurring in the field.
Besides being an economically attractive solution, it is also environmentally friendly because of the water used for injection is the produced water from the wells. Separating the water from the hydrocarbon stream has other additional benefits that favor the oil production from the reservoir and hence, positively influence the Oil Recovery Factor (ORF). For instance, the relief of water-and-liquid capacity of the platform, oil production anticipation associated with high amounts of produced water and decreasing the pressure drop along from the flowline to the platform. In fact, the results obtained from the economic model shows that this solution might be viable due to the revenues anticipation that from another way would not be possible to be earned without considering the implementation of these technologies.
This work can be considered as an interdisciplinary approach where including this kind of subsea technologies in the production network and its influence in the production of the reservoir have to be analyzed from a holistic point of view. Several disciplines as reservoir engineering, production engineering, and economic calculations are involved in building a coupled model that permits analyzing multiple production scenarios and network configurations, determining the best arrangement of the components, evaluating the economic viability of the project and supporting the making-decision process during field management.
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A Triangular Element-Based Finite Volume Formulation For Solving Problems Of Heat Transfer In Oil Reservoirs
Authors J.V. Contreras Sandia and C. AraujoSummaryCurrently, most of reservoir simulators have been developed using a finite volume method (FVM) as the numerical scheme to discretize the domain. However, FVM faces some issues to handle appropriately complex domains and boundaries. An element-based finite volume method (EbFVM) numerical scheme combines the FVM advantages and the ability of finite element methods to tackle complex reservoir domains.
The purpose of this work is to obtain a numerical formulation where EbFVM is applied to discretize the differential equation that describe diffusive situation for incompressible flow in a two-dimensional domain with problems of nonlinear characteristics in a transient regime.
The spatial discretization was performed by using a structured grid with triangular elements, which are very convenient to represent any two-dimensional complex domain with good accuracy. The conservation laws are locally applied in a secondary control volume grid, which was built around a node by connecting the centroid of each triangle with the midpoints of the triangle’s sides ( Minkowycs, 2006 ). The equation of the element was obtained from interpolation function depending on element coordinates and nodal values, as proposed in the work of Baliga and Patankar. Fluid and rock properties remain constant inside each element, but these properties may vary from element to element, and can be calculated according to the pressure and temperature values prevailing in each element. In the case of single-phase flow, the equation of state used was the fluid compressibility definition. The discretization of the time was developed with the implicit scheme, which is more numerically stable when solving problems with larger time steps, resulting in less computer time. The algorithm employed to discretize the conservation equation, was used to handle all conserved properties, in a sequential manner.
In this work, two examples were compared with solutions obtained from commercial simulation programs that employ the traditional FVM. One example involves a single phase flow and the other consists of heat injection by using a bottom hole heater. Numerical performance were studied with good accuracy in results.
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Symmetric Positive Definite Control-volume Distributed Multi-point Flux Approximation On Tetrahedral Grids
Authors R. Ahmed and M.G. EdwardsSummaryA three-dimensional symmetric positive definite (SPD) cell-centred control-volume distributed multi-point flux approximation (CVD-MPFA) is presented for porous media flow simulation on unstructured tetrahedral grids. The scheme depends on a single degree of freedom per control-volume and is derived in physical space, where the continuous fluxes are resolved directly along the face normals of the tetrahedra, maintaining exact grid geometry. Analysis and properties of the method will be presented.
Comparisons with the standard MPFA scheme shows that the new CVD-MPFA scheme yields well resolved pressure fields and improved convergence for homogeneous and heterogeneous as well as both isotropic and anisotropic full-tensor permeability fields.
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Unstructured CVD-MPFA Reduced-Dimensional Discrete Fracture Models For Two-Phase Flow
Authors Y. Xie and M.G. EdwardsSummaryControl-volume distributed multi-point flux approximation (CVD-MPFA) coupled with single-phase reduceddimensional discrete fracture models [1], are extended to two-phase flow, including gravity and capillary pressure.
Both continuous and discontinuous fracture models are considered coupled with higher resolution methods, leading to novel finite-volume schemes for flow in subsurface fractured porous media on unstructured grids. Performance comparisons are presented for tracer and two-phase flow problems on a number of 2D fractured media test cases including hybrid gravity and capillary pressure effects on unstructured meshes.
[1] R. Ahmed, M.G. Edwards, S. Lamine, B.A.H. Huisman and M. Pal
“Control Volume Distributed Multi-Point Flux Approximation coupled with a lower-dimensional fracture model” J. Comput. Phys vol 284 pp 462–489 March 2015
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Higher Resolution Hybrid And Vt Finite-Volume Formulations For 3-Component 2-Phase Flow On Unstructured Grids
Authors Y. Xie and M.G. EdwardsSummaryNovel fractional-step higher resolution hybrid cell-centred finite-volume formulations are presented for twophase and three component two-phase flow with gravity on structured and unstructured grids. We note that previous hybrid methods [1] are first order and presented for structured 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 approximations involving pressure and gravity flux operators, leading to a novel formulation for two-phase and three-component two-phase flow on unstructured grids.
Comparisons with both higher resolution and standard first order characteristic based upwind methods and classical phase upwinding is presented.
Results demonstrate the benefits of the new methods for a range of problems including channel flow and shale-barrier problems.
[1] S. Lee, Y. Efendiev, H. Tchelepi, Hybrid upwind discretization of nonlinear two phase flow with gravity, Advances in Water Resources 82 (2015) 27–38.
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Complex-Structured Reservoir Modeling On Dynamically Adaptive PEBI-Grids
Authors D.D. Filippov, I.Yu. Kudryashov, D.Yu. Maksimov, D.A. Mitrushkin and A.P. RosSummaryThe paper considers generation and application of dynamically adaptive unstructured PEBI-grids for adequate multiphase flow modeling of complex-structured reservoirs exploited by horizontal and deviated wells, including wells with single or multistage hydraulic fractures.
We developed methods for constructing PEBI-grids which are more detailed near the reservoir structural features (wells, natural faults, hydraulic fractures and reservoir boundaries) and sparse away from them. These grids make it possible to increase the accuracy of flow treatment in the vicinity of operational objects, without significant slowing down of the entire calculation. A number of algorithms for constructing the PEBI-grid are developed to account for non-vertical geological faults with a complex structure (normal fault, reverse fault), the real geometry of hydraulic fracturing and hydraulically connected natural fractures obtained from a geomechanical simulator.
On grids under consideration, three-phase flow problem, accounting for gravitational, viscous and capillary forces and phase transition of hydrocarbon components is solved numerically. Within the framework of the paper, the approach of the direct calculation of the inflow to and the flow inside hydraulic fractures is developed and implemented. Two-point flux approximation and mimetic finite difference are used for solving three-phase flow problem.
We developed the algorithm of local grid rearrangement due to newly opened wells and hydraulic fractures growth (including waterflood-induced fracturing) that reduces simulation time.
Results of calculations showing the speed, accuracy and physical adequacy of the proposed approach to the reservoir modeling of complex-structured reservoirs are presented.
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Local Forward-Mode Automatic Differentiation For High Performance Parallel Pilot-Level Reservoir Simulation
Authors A. Lauser, A.F. Rasmussen, T.H. Sandve and H.M. NilsenSummaryLocal forward-mode automatic differentiation for high performance parallel pilot-level reservoir simulation. Robust reservoir simulation requires accurate linearization and involve complex property evaluations and dynamics. Handcoded Jacobian derivative calculations require significant resources to maintain and change, when taking into account all needed features for industrially relevant simulations. Automatic differentiation (AD) is a technique which gives machine precision accuracy of derivatives while requiring minimal extra effort, essentially only requiring the implementation of the residual equations. This makes extending the model simpler and less error-prone.
The optimal use of AD techniques depend on the particular grid structures and discretizations used. Here we present how local forward-mode AD can be used with a discretization based on the Distributed Uniform Numerics Enviroment (DUNE) grid interface to achieve a high performance reservoir simulator.
This paper discusses how one can exploit the structure of the full reservoir equations to obtain a dense data representation with only local evaluations in the AD framework, thereby avoiding excessive treatment of sparse sets or matrices. We highlight aspects of the C++ implementation which contribute to giving clean code, parallel performance and efficient use of modern microprocessors. Finally, the OPM Flow simulator is used to demonstrate the approach on field case examples.
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Design Unstructured Grids For Modeling Complex Fields Of Oil Deposits
Authors O.N. Turar, D.Zh. Akhmed-Zaki and D.V. LebedevSummaryMost of the existing algorithms of unstructured grid construction are designed based on the simple geometry and graph paradigms. Such approach may lead to close placement of different size cells sometimes. But considering the fact that it is continuous physical values to be discretized on constructed grids scientists mostly need deliberate changing of cell size.
The paper offers paradigm of using of differential equations to construct unstructured grids to keep valuable characteristics of cells as suitable for physical value discretization as possible. Such methods are being widely used for adaptive structured grid construction in many branches of computational physics industry. One of the main reasons of such prevalence is its physicality together with its intuitiveness and convenience for finite differential computations. In case of finite volume and finite element methods the role of physicality is also very important. It can be provided by using of differential equations to determine the positions of the cells and nodes of the grid. The smoothness of grid cell size changes can be granted by tension of diffusion to bring values to average. Consequently, using of elliptic and parabolic differential equations will lead to unstructured grids smooth in terms of cell size.
In this paper we used the method of construction based on solving Beltrami equation. This equation represents a diffuse spread of coordinate values on some specific metric. The nodes and cell centers in this situation would evenly spread over some abstract surface with this metric. On simple cartesian space it leads to curvilinearly adapted grid. The smoothness of the metric further guarantees smoothness of the grid.
At the same time cells keep basic criteria of the unstructured computational grids such as Delaunay criterion. It’s due to the fact that solving the equation only defines spread of points’ set and grid itself constructed using standard methods. This approach allows to construct physically adequate computational grids in case of geo-modelling with complicated structure and complex form of the field.
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Improving Peaceman Well Flow Modelling With 1D Radial Simulation
By H. CuiSummaryIn reservoir simulation, Peaceman’s method is the most popular solution for simulating the flow from the reservoir to the well by knowing the gridblock and wellbore pressures, and the fluid phase properties in the well block. The Peaceman method is easy to implement and works well if there are no phase changes in the wellblock. A gas-condensate field is an example of a phase-change situation. When the pressure falls below the dewpoint, condensate drops out in the reservoir and accumulates near the wellbore. The gas relative permeability falls and the well productivity is affected. Evaluating the reservoir fluid phase behaviour using the gridblock pressure will give an incorrect value for well flow productivity. Rigorous well flow simulation requires a very fine grid level that demands a large computational effort. There are various methods for obtaining a detailed pressure profile around wells, for example, the pseudo-pressure method, local grid refinement and hybrid gridding. However, these methods either require many assumptions to estimate the phase properties (as in the pseudo pressure method) or a dedicated gridding technique (as in the local grid refinement method and hybrid gridding), or perform poorly.
In the present work, the Peaceman well flow modelling method has been improved with 1D radial simulation to mimics the local physics around a well accurately with little computational effort. For that goal, a local 1D cylindrical radial flow from the equivalent radius to the well bore was considered. Within the wellblock, cylindrical co-ordinates are used, with the well’s axis being the z-axis. Each radial node represents an annulus around the z-axis extending from the entry point to the exit point of the well in the wellblock in 3D space. Because the grid and internode properties of the radial nodes can be calculated analytically, there is no need to generate these nodes in the real grid space. In our present work, we have mapped the blocks generated by the local grid refinement to the 1D radial nodes. As these radial nodes are real model blocks, relative permeability and pressure–volume–temperature modelling are simulated in these blocks without extra effort. The new method is easily adapted by most reservoir simulators that implement the Peaceman method. Numerical experiments show that, with a few extra 1D radial nodes, the method can accurately and efficiently simulate the rapid pressure profile and phase changing within a wellblock.
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Finite Element With Embedded Discontinuities Analysis Of Well Production Decline Due To Fracture Closure In Naturally Fr
Authors L. Beserra, L. Guimarães, O.L. Manzoli and L. BerrioSummaryIn naturally fractured reservoirs, the fractures may represent the main pathway for fluid flow. Therefore, the magnitude of fracture permeability plays a fundamental role in the productivity of such type of reservoir. In reservoirs sensitive to the stress state, the depletion due to production can lead to the closure of the fractures, as a function of the increase of effective confining stress, promoting a significant decrease in the overall permeability of the reservoir. Thus, understanding the hydraulic characteristics of the fracture network as a function of the effective confining stress is fundamental for the design of reservoir development, besides the predictability of its behavior.
In this paper, a strong discontinuity approach to embed discontinuities into finite elements was adopted to represent the behavior of fractures in rock formations. To properly derive embedded discontinuity finite element formulations, fundamental aspects regarding to the kinematics and statics of the discontinuity must be considered. The kinematic enrichment must correctly reflect the position of the interface in the element as well as the relative displacement (opening and sliding) between the two opposite faces of the interface. Furthermore, the traction continuity condition must be properly imposed to ensure a correct relationship between the tractions in the internal interface and the stresses in the surrounding continuum portion.
The simulation of the problem of closing natural fractures by reservoir depletion was carried out. A numerical tool was used to embed the natural fracture network into the finite element mesh, with respect to the geological mapping of these fractures. To model the mechanical behavior of material, it was adopted a hyperbolic model of fracture closure proposed by Barton & Bandis. It was possible to observe a decrease in the rate of production due to the collapse of the existing fractures that decreased the permeability around the production wells.
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Modeling Of Stimulated Reservoir Volume By Multistage Hydraulic Fracturing In Formation With Pre-Existing Natural Fractu
Authors A. Erofeev, V. Vostrikova, R. Sitdikov, R. Nikitin and D. MitrushkinSummaryThis article describes a software tool developed by authors for modeling the process of SRV forming. The calculation of the hydraulic fractures growth and the flow of a mixture of liquid and proppant in a network of fractures is carried out within the cell-based Pseudo-3D model. Developed model also takes into account the interaction of hydraulic fractures with natural fractures and «stress shadow» effect. In addition, the implemented tool allows to simulate the deposition of the proppant and the process of fracture closing after stopping the injection. The influence of the main input parameters on SRV formation is investigated, and the simulation results using real data are presented in the article.
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Proppant Transport Modeling With Effects Of Suspension Yield Stress, Jamming, And Filtration Through The Proppant Pack
Authors A.A. Osiptsov, S.A Boronin and E.V DontsovSummaryWe present further development of the 2D two-continua model of proppant transport derived from first principles using the lubrication approximation [1]. The model includes effects associated with presence of the slip velocity between particles and fluid, as well the yield stress of carrying fluid. These features are typically missing in standard, effective-medium models of proppant transport, though power-law rheology is often included. Predictions of model [1] have gone through a thorough validation against a set of carefully selected lab data [2].
It is important to stress the presence of a fundamental issue with standard semi-empirical relationships for suspension rheology, which predict singular behavior near the particle packing limit. One possibility to resolve the issue is to introduce an ad-hoc regularization by stepping out from the singularity at a small epsilon to mimic the transition from flowing suspension to Darcy filtration through the packed bed. Clearly, such a regularization is unable to accurately describe the physics for all possible scenarios since the flux has a different dependence on the channel width for Poiseuille and Darcy flows. Alternatively, one may utilize a recently developed suspension flow model [3], in which the issue of singular behavior is resolved by developing the model from first principles. The latter approach is physics-based, self-consistent, and covers the entire range of variation of the particle volume fraction, from dilute through dense to granular pack, and in particular predicts Darcy filtration at the packing limit.
Here, we will present a proppant transport model that accounts for the combined effects of particle jamming due to bridging, dehydration, and transition to close packing, combined with Bingham rheology of the suspension (induced by cross-linking of the polymer-based fracturing fluid, presence of fibers, and suspension rheology itself near the packing limit). We use a unified closure relation for the suspension rheology proposed recently in [3] to cover the whole range of proppant concentration, from dilute suspension, to dense and close packing. Numerical results are given to illustrate the newly introduced effects.
References:
- Osiptsov, A.A., 2017. Fluid Mechanics of Hydraulic Fracturing: a Review. J. Petrol. Sci. Eng. V. 156, July 2017, pp. 513–535.
- Boronin, S.A., Osiptsov, A.A. and Desroches, J., 2015. Displacement of yield-stress fluids in a fracture. International Journal of Multiphase Flow, 76, pp.47–63.
- Dontsov, E.V., and Peirce, A.P., 2014. Slurry flow, gravitational settling, and a proppant transport model for hydraulic fractures. Journal of Fluid Mechanics , 760, 567–590.
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Assimilation Of Microseismic Data Into Coupled Flow And Geomechanics Models
Authors S Hakim-Elahi and B. JafarpourSummaryGeologic CO2 storage in deep saline aquifers requires reliable risk assessment to evaluate and minimize unintended consequences such as potential CO2 leakage and induced seismicity. To mitigate such risks continuous monitoring and model updating is needed to improve future predictions and risk assessment. Injection-induced microseismicity has been proposed as a monitoring technique that can be used to constrain rock flow and mechanical properties. We present our ongoing work to develop a framework for assimilation of microseismic monitoring data for estimation of rock mechanical properties using coupled flow and geomechanics simulation as a forward model. Coupled flow and geomechanics simulation is combined with Mohr-Coulomb failure criterion and a stochastic measurement model, to provide a rigorous approach for prediction and interpretation of spatiotemporal distribution of discrete microseismic events in the formation. The focus of the paper is on building a geomechanics-based stochastic framework that can be used to establish physical correlation among rock mechanical properties and microseismic response data. The resulting correlations can then be used to estimate rock properties from observed microseismic clouds. In this paper, we present the developed framework and preliminary results to evaluate its performance for integration of microseismic data.
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Characterization Of Liquid Bridge Formed During Gas-Oil Gravity Drainage In Fractured Porous Media
Authors B. Harimi, M. Masihi, M.H. Ghazanfari and A. ShoushtariSummaryGas–oil gravity drainage that takes place in the gas-invaded zone of fractured reservoirs is the main production mechanism of gas-cap drive fractured reservoirs as well as fractured reservoirs subjected to gas injection. Interaction of neighboring matrix blocks through reinfiltration and capillary continuity effects controls the efficiency of gravity drainage. Existence of capillary continuity between adjacent matrix block is likely to increase the ultimate recovery significantly. Liquid bridge formed in fractures has a significant role in maintaining the capillary continuity between two neighboring matrix blocks. The degree of capillary continuity is proportional to capillary pressure in the fracture due to the presence of formed liquid bridge. Only a handful of studies have focused on the subject of liquid bridge in fractures and related capillary pressure. The main contribution is to develop a numerical procedure to predict liquid bridge characteristics (e.g. its shape, its stability and its capillary pressure). Accurate determination of gas-liquid interface profile of liquid bridge is crucial to predict fracture capillary pressure precisely. To this end, numerical solution of Young-Laplace equation in the absence and in the presence of gravitational effects is found and the obtained results are verified by the experimental data. Computation of fracture capillary pressure as a function of liquid bridge volume for different contact angles revealed that the fracture capillary pressure-liquid saturation curve has a shape similar to that of a matrix. Therefore, the capillary pressure of porous media can be applied directly for fractures considering proper modifications. Furthermore, the stability of liquid bridge has been investigated using the concept of critical fracture aperture. Critical fracture aperture is defined as the maximum fracture aperture that a liquid bridge with specific volume can exist. Finally, an empirical relation has been developed that correlates the critical fracture aperture to both the liquid bridge volume and the contact angle. Results of this study emphasize the importance of capillary continuity created by liquid bridges and therefore, incorporation of liquid bridges in the study of gas-oil gravity drainage will lead to more realistic performance prediction of fractured porous media.
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