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Thiele, Christopher (Rice University) | Araya-Polo, Mauricio (Shell International Exploration & Production, Inc.) | Alpak, Faruk Omer (Shell International Exploration & Production, Inc.) | Riviere, Beatrice (Rice University)
Direct numerical simulation of multiphase pore-scale flow is a computationally demanding task with strong requirements on time-to-solution for the prediction of relative permeabilities. In this paper, we describe the hybrid-parallel implementation of a two-phase two-component incompressible flow simulator using MPI, OpenMP, and general-purpose graphics processing units (GPUs), and we analyze its computational performance. In particular, we evaluate the parallel performance of GPU-based iterative linear solvers for this application, and we compare them to CPUbased implementations of the same solver algorithms. Simulations on real-life Berea sandstone micro-CT images are used to assess the strong scalability and computational performance of the different solver implementations and their effect on time-to-solution. Additionally, we use a Poisson problem to further characterize achievable strong and weak scalability of the GPU-based solvers in reproducible experiments. Our experiments show that GPU-based iterative solvers can greatly reduce time-to-solution in complex pore-scale simulations. On the other hand, strong scalability is currently limited by the unbalanced computing capacities of the host and the GPUs. The experiments with the Poisson problem indicate that GPU-based iterative solvers are efficient when weak scalability is desired. Our findings show that proper utilization of GPUs can help to make our two-phase pore-scale flow simulation computationally feasible in existing workflows.
Agent-based models (ABMs) provide a fast alternative to traditional partial differential equation (PDE)- based oil reservoir models by applying localized inexpensive simulations, rather than solving a partial differential equation at every time-step. However, while there have been theoretical and numerical results obtained with ABMs in social science applications, the accuracy of ABMs has not been analyzed in the context of oil reservoir modeling.
A discontinuous Galerkin method of first order is proposed to solve the three-phase flow problem in threedimensional heterogeneous reservoirs. The formulation is based on the compositional model and the primary unknowns are the total mass fraction of gas, the aqueous phase saturation and the liquid phase pressure. The algorithm is sequential and controls the nonlinearity with a subiteration scheme. Robustness of the method is shown on reservoirs with different heterogeneities: random permeability field, reservoir with barriers and layered reservoir. The algorithm easily handles phase appearance and disappearance, as well as mass transfer between the vapor and liquid phase.
Frank, Florian (Rice University) | Liu, Chen (Rice University) | Alpak, Faruk O. (Shell International Exploration and Production) | Berg, Steffen (Rice University) | Riviere, Beatrice (Shell Global Solutions International)
Advances in pore-scale imaging, increasing availability of computational resources, and developments in numerical algorithms have started rendering direct pore-scale numerical simulations of multiphase flow on pore structures feasible. In this paper, we describe a two-phase-flow simulator that solves mass- and momentum-balance equations valid at the pore scale (i.e., at scales where the Darcy velocity homogenization starts to break down). The simulator is one of the key components of a molecule-to-reservoir truly multiscale modeling work flow.
A Helmholtz free-energy-driven, thermodynamically based diffuse-interface/phase-field method is used for the effective simulation of numerous advecting interfaces, while honoring the interfacial tension (IFT). The advective Cahn-Hilliard (CH) (mass-balance, energy dissipation) and Navier-Stokes (NS) (momentum-balance, incompressibility) equations are coupled to each other within the phase-field framework. Wettability on rock/fluid interfaces is accounted for by means of an energy-penalty-based wetting (contact-angle) boundary condition. Individual balance equations are discretized by use of a flexible discontinuous Galerkin (DG) method. The discretization of the mass-balance equation is semi-implicit in time using a convex/concave splitting of the energy term. The momentum-balance equation is split from the incompressibility constraint by a projection method and linearized with a Picard splitting. Mass- and momentum-balance equations are coupled to each other by means of operator splitting, and are solved sequentially.
We discuss the mathematical model and its DG discretization, and briefly introduce nonlinear and linear solution strategies. Numerical-validation tests show optimal convergence rates for the DG discretization, indicating the correctness of the numerical scheme and its implementation. Physical-validation tests demonstrate the consistency of the phase distribution and velocity fields simulated within our framework. Finally, two-phase-flow simulations on two real pore-scale images demonstrate the usefulness of the pore-scale simulator. The direct pore-scale numerical-simulation methodology rigorously considers the flow physics by directly acting on pore-scale images of rocks without remeshing. The proposed method is accurate, numerically robust, and exhibits the potential for tackling realistic problems.
Thiele, Christopher (Shell International E&P Inc.) | Araya-Polo, Mauricio (Shell International E&P Inc.) | Alpak, Faruk O. (Shell International E&P Inc.) | Riviere, Beatrice (Rice University) | Frank, Florian (Rice University)
Hierarchical scale separation (HSS) is a new approach to solve large sparse systems of linear equations arising from discontinuous Galerkin (DG) discretizations. We investigate its applicability to systems stemming from the nonsymmetric interior penalty DG discretization of the Cahn-Hilliard equation, discuss its hybrid parallel implementation for large-scale simulations, and compare its performance to a widely used iterative solver. The solution of the linear systems, in particular in massively parallel applications, is a known performance bottleneck in direct numerical approaches. HSS splits the linear system into a coarse-scale system of reduced size corresponding to the local mean values of the DG solution, and a set of decoupled local fine-scale systems corresponding to the higher order components of the DG solution. The scheme then alternates between coarse-scale and fine-scale system solves until both components converge, employing a standard iterative solver for the coarse-scale system and direct solves for the set of small fine-scale systems, which allow for an optimal parallelization. The motivation of HSS is to increase parallelism by decoupling the linear systems, therefore reducing the communication overhead within sparse matrix-vector multiplications of classical iterative solvers. Providing some mild assumptions on the underlying DG basis functions, the above-mentioned splitting can be done on the resulting linear systems (i.
Frank, Florian (Rice University) | Liu, Chen (Rice University) | Alpak, Faruk O. (Shell International Exploration and Production Inc.) | Araya-Polo, Mauricio (Shell International Exploration and Production Inc.) | Riviere, Beatrice (Rice University)
Advances in pore-scale imaging, increasing availability of computational resources, and developments in numerical algorithms have started rendering direct pore-scale numerical simulations of multiphase flow on pore structures feasible. In this paper, we describe a two-phase flow simulator that solves mass and momentum balance equations valid at the pore scale, i.e. at scales where the Darcy velocity homogenization starts to break down. The simulator is one of the key components of a molecule-to-reservoir truly multiscale modeling workflow.
A Helmholtz free-energy driven, thermodynamically based diffuse-interface method is used for the effective simulation of a large number of advecting interfaces, while honoring the interfacial tension. The advective Cahn–Hilliard (mass balance) and Navier–Stokes (momentum balance) equations are coupled to each other within the phase-field framework. Wettability on rock-fluid interfaces is accounted for via an energy-penalty based wetting (contact-angle) boundary condition. Individual balance equations are discretized by use of a flexible discontinuous Galerkin (DG) method. The discretization of the mass balance equation is semi-implicit in time; momentum balance equation is discretized with a fully-implicit scheme, while both equations are coupled via an iterative operator splitting approach.
We discuss the mathematical model, DG discretization, and briefly introduce nonlinear and linear solution strategies. Numerical validation tests show optimal convergence rates for the DG discretization indicating the correctness of the numerical scheme. Physical validation tests demonstrate the consistency of the mass distribution and velocity fields simulated within our framework. Finally, two-phase flow simulations on two real pore-scale images demonstrate the utility of the pore-scale simulator. The direct pore-scale numerical simulation method overcomes the limitations of pore network models by rigorously taking into account the flow physics and by directly acting on pore-scale images of rocks without requiring a network abstraction step or remeshing. The proposed method is accurate, numerically robust, and exhibits the potential for tackling realistic problems.
Finite volume (FV) methods are still the most popular methods in practical reservoir simulation. They have good accuracy when used properly and are computationally very cheap. FV can be used on unstructured grids, but grid generation becomes a very difficult task. More advanced methods, discontinuous Galerkin (DG) for example, work on more general meshes, but are computationally more expensive and are still not accepted by the practitioners. We propose coupling of FV and DG methods that can improve the accuracy of the FV with reasonable increase of the computational cost.
We developed the algorithms for coupling DG and FV for model diffusion and convection diffusion problems on Voronoi /PEBI grids. We demonstrate with examples how DG can be used to alleviate the problems with grids in 2-D, gridding around pinch-outs and patch local refinement for areas with tensor coefficients.