Multipoint Flux Approximation Method Research Articles

Simulation of two-phase flow in 3D fractured reservoirs using a projection-based Embedded Discrete Fracture Model on Unstructured tetrahedral grids (pEDFM-U)

The modeling of fluid flow in heterogeneous, anisotropic and fractured porous media is relevant in many applications, including hydrocarbon and groundwater extraction, dispersion of contaminants, hydrogen or carbon dioxide (CO2) storage. Thus, accurate and scalable simulation of fluid flow through these formations continues to be a great challenge. The presence of fractures that are explicitly modeled, ranging from flow barriers to highly conductive channels, significantly increases the complexity of the numerical simulation. The Embedded Discrete Fracture Model (EDFM) produces results that can be as accurate as those obtained using equidimensional Discrete Fracture Models (DFM) at a much lower computational cost for highly conductive fractures, but it is not adequate for impermeable barriers. In contrast, the pEDFM (projection-based Embedded Discrete Fracture Model) produces accurate results for both, channels and barriers by using additional matrix-fracture connectivities. In its current versions, it is restricted to cartesian and corner-point grid geometries and to the classical Two Point Flux Approximation (TPFA) scheme. In this work, for the first time, the pEDFM is extended to handle unstructured tetrahedral meshes (pEDFM-U). In addition, interface fluxes are approximated by using the Multipoint Flux Approximation method with a Diamond stencil (MPFA-D). This allows for simulation of highly heterogeneous and anisotropic geo-models. The developed method is robust and can deal with full permeability tensors on arbitrary tetrahedral meshes. The advective terms are discretized considering an implicit First Order Upwind (FOU) method. Our method is implemented within the DARSim (Delft Advanced Reservoir Simulation) open-source simulator framework. Through several test cases, the proposed method showed to be robust and capable to accurately capture the effects of both high and low permeability fractures for general tetrahedral meshes, under arbitrary heterogeneous and anisotropic permeability tensors for the rock matrix.

A very high-order flux reconstruction approach coupled to the MPFA-QL finite volume method for the numerical simulation of oil-water flows in 2D petroleum reservoirs

In recent years, high-order methods for the discretization of the advection dominated saturation equation, and that results from the modelling of the 2D Oil-Water displacement through porous formations, have received renewed attention from the CFD community, as a suitable alternative to the traditional first-order upwind (FOU) finite volume (FV) scheme, which is the standard in industry. A key aspect of high-order methods applied to numerical simulation of petroleum reservoirs is the lower computational cost for a comparable degree of accuracy against their lower-order counterparts. In this context, in the present paper, for the first time in literature, we propose the use of the very high order Flux Reconstruction (FR) formulation for the discretization of the advective terms in the saturation equation using unstructured quadrilateral meshes. To solve the elliptic pressure equation, we use the recently developed Multipoint Flux Approximation method with a Quasi-Local Stencil (MPFA-QL). This method is very robust and able to handle highly heterogeneous and anisotropic permeability tensors. The system of equations is solved in a segregated manner by using the classical Implicit Pressure Explicit Saturation (IMPES) procedure. To properly couple the cell-centered MPFA-QL method with the FR formulation, it is necessary to obtain an adequate velocity reconstruction throughout the control volumes of the mesh. As the cell-centered finite volume method naturally delivers fluxes across control surfaces of the primal grid, then, we use a reconstruction operator based on the first-order Brezzi–Douglas–Marini (BDM) space, and the Piola transformation, to get the complete knowledge of the velocity field throughout the domain. To eliminate spurious oscillations around the discontinuities, which are typical in high-order methods, we have used a hierarchical multidimensional limiting process (hMLP). To compute fluxes over the control surfaces we have used the Roe-E approximate Riemann solver. Finally, through some representative examples, we show the accuracy, efficiency and robustness of our new high order numerical methodology to model oil-water displacements in highly heterogeneous and anisotropic petroleum reservoirs.

An improved numerical model for groundwater flow simulation with MPFA method on arbitrary polygon grids

Groundwater models are critical for simulating subsurface hydrological processes and guiding informed policy-making for groundwater management. However, the widely applied groundwater models typically use regular-shaped grids to discretize aquifer systems and require that the directions of the grid edges are aligned with the hydraulic conductivity tensor. Such rigorous requirements for spatial discretization have constrained the models’ application in aquifer systems with anisotropic hydrogeological characteristics. To address such limitations, we develop an improved groundwater flow model based on the multipoint flux approximation (MPFA) method in this study. The new model allows us to use arbitrary-shaped polygon grids to discretize aquifer systems and relaxes the rigorous requirement of the alliance between polygon edges and hydraulic conductivity tensor. The functionality and performance of the new model are demonstrated by comparing the output between our model, MODFLOW, and analytical solution in four case studies with various hydrogeological conditions. In a real-world watershed with complex-shaped boundaries, our model outperforms the conventional groundwater model in boundaries. The modeling results show that our model can yield accurate simulation of subsurface hydrological processes in aquifer systems with complex-shaped boundaries. Furthermore, our model can provide a more flexible discretization solution to couple surface water and groundwater model in integrated hydrological model development.

Modeling variable-density flow in saturated-unsaturated porous media: An advanced numerical model

Modeling variable-density flow in unconfined aquifers is a challenging task because of the nonlinear coupling between variably saturated flow and contaminant transport. This results in a highly nonlinear system since the strongly nonlinear Richards flow equation is, in addition, coupled to the advection-dispersion transport equation by viscosity and density variation. The solution of such a nonlinear system is often subject to convergence issues and can be very expansive in terms of computational time, especially for large-scale problems. Conventional numerical algorithms based on the sequential approach and the classical finite difference or finite element methods with the first-order backward Euler time integration scheme are generally inefficient and/or do not provide satisfactory results. In this work, we develop a new efficient and accurate 2D numerical model for the transport of dense contaminants in unsaturated porous media that allows for the simulation of large-scale problems. This research describes a new model that combines advanced spatial discretization methods (mixed hybrid finite element method, discontinuous Galerkin finite element method, and multipoint flux approximation method) with higher-order time integration techniques via the method of lines (MOL). The latter allows one to adapt the time step's size and the order of the time integration to improve the computational efficiency while maintaining accuracy. The robustness and accuracy of the new model are shown by comparison against a widely used commercial code based on the standard finite element method. The applicability of the developed model to a large-scale problem is then investigated by simulating saltwater intrusion under a climate change projection and long-term pumping regimes for the Akkar coastal aquifer in Lebanon using a simplified 2D conceptual model.

A New Nonlinear Two-Point Flux Approximation Method for Solving the Anisotropic Diffusion Equation with Reduced Violations of the Discrete Maximum/Minimum Principle

Summary A class of monotone cell-centered nonlinear finite-volume methods has been proposed in the past decade to solve the anisotropic diffusion equation. The nonlinear two-point flux approximation (TPFA) (NTPFA) method preserves the nonnegativity of the solution values but can violate the discrete maximum/minimum principle (DMP). To enforce DMP, the nonlinear multipoint flux approximation (NMPFA) method ought to be used. In this work, we propose a novel NTPFA method that can reduce the severity of DMP violations significantly compared with the standard NTPFA method. The new formulation uses conormal decomposition for the construction of the one-sided fluxes. To define the unique flux through a connection between two cells, we choose a convex combination of the two one-sided fluxes such that the absolute differences of the magnitudes of the two transmissibility terms associated with the two neighboring cells are minimized, thus bringing the discrete coefficient matrix closer to having the zero row-sum property. Numerical experiments are conducted to test the performance of the new NTPFA method. The results demonstrate that the new scheme has comparable convergence order for both the solution and the flux compared with the standard NTPFA method or the classical multi-point flux approximation (MPFA-O) method. Moreover, the new NTPFA formulation shows marked improvements over the standard NTPFA in terms of reducing DMP violations. However, depending on the specific problem configuration, our new NTPFA formulation can lead to a system of nonlinear equations that is more difficult to solve.

A Fitted L-Multi-Point Flux Approximation Method for Pricing Options

In this paper, we introduce a special kind of finite volume method called Multi-Point Flux Approximation method (MPFA) to price European and American options in two dimensional domain. We focus on the L-MPFA method for space discretization of the diffusion term of Black–Scholes operator. The degeneracy of the Black-Scholes operator is tackled using the fitted finite volume method. This combination of fitted finite volume method and L-MPFA method coupled to upwind methods gives us a novel scheme, called the fitted L-MPFA method. Numerical experiments show the accuracy of the novel fitted L-MPFA method comparing to well known schemes for pricing options.

Efficient simulation of groundwater solute transport using the multipoint flux approximation method with arbitrary polygon grids

Groundwater solute transport models are critical for simulating subsurface contaminant processes, guiding policy making for groundwater management, and pollution remediation. In this study, we develop an improved groundwater solute transport model which uses arbitrary-shaped polygons for spatial discretization. This allows for the efficient simulation of solutes in complex boundary aquifer systems. The governing equation is discretized in space using the finite volume method to ensure flux conservation both locally and globally. The dispersion term is discretized by the multipoint flux approximation (MPFA) method which is a finite volume method that is suitable on arbitrary polygon grids. A total variation diminishing (TVD) scheme is used to control numerical oscillation in the advection term. The functionality and performance of the new model are demonstrated by comparing the simulated results between our model and MT3DMS model in five case studies. The comparisons show that the new model can well represent the solute transport processes. In a real-world watershed with complex-shaped boundaries, the new model outperforms the conventional groundwater solute transport model with smaller computed root-mean-square errors. These modeling results suggest that our model can provide robust modeling solutions for simulating groundwater solute transport processes, especially in aquifer systems with complex shaped boundaries. Furthermore, our model can provide a more flexible discretization solution for coupling surface water models and groundwater solute transport models.

WITHDRAWN: Efficient simulation of groundwater solute transport using the multipoint flux approximation method with arbitrary polygon grids

WITHDRAWN: Efficient simulation of groundwater solute transport using the multipoint flux approximation method with arbitrary polygon grids

Dynamic effects of fault modeling on stair-step and corner-point grids

Fault modeling has become an integral element of reservoir simulation for structurally complex reservoirs. Modeling of faults in general has major implications for the simulation grid. In turn, the grid quality control is very important in order to attain accurate simulation results. We investigate the dynamic effects of using stair-step grid (SSG) and corner-point grid (CPG) approaches for fault modeling from the perspective of dynamic reservoir performance forecasting. We have performed a number of grid convergence and grid-type sensitivity studies for a variety of simple, yet intuitive faulted flow simulation problems with gradually increasing complexity. We have also explored the added value of the multipoint flux approximation (MPFA) method over the conventional two-point flux approximation (TPFA) to increase the accuracy of reservoir simulation results obtained on CPGs. Effects of fault seal modeling on grid-resolution convergence and grid-type sensitivity have also been briefly examined. For simple geometries, both SSG and CPG can be used for fault modeling with similar accuracy in conjunction with the pillar-grid approach. This is evidenced by the fact that simulation results from SSG and CPG converge to identical solutions. SSG and CPG yield different results for more complex geometries. Simulation results approach to a converged solution for relatively fine SSGs. However, a SSG only provides an approximation to the fault geometry and reservoir volumes when the grid is coarse. On the other hand, non-orthogonality errors are increasingly evident in relatively more complex faulted models on CPGs and such errors cannot be addressed by grid refinement. It has been observed that MPFA partially addresses the discretization errors on non-orthogonal grids but only from the total flux accuracy perspective. However, transport related errors are still evident. Grid convergence behaviors and grid effects are quite similar with or without fault seal modeling (i.e., dedicated fault-zone modeling by use of scaled-up seal factors) for simple geometries. However, in more complex test cases, we have observed that it is more difficult to achieve converged results in conjunction with fault seal modeling due to increased heterogeneity of the underlying problem.

Hybrid finite volume-finite element methods for hydro-mechanical analysis in highly heterogeneous porous media

In this study, two classes of advanced finite volume schemes, including Multi-Point Flux Approximation (MPFA) and Dual Discrete Finite Volume (DDFV) method, have been employed in conjunction with the finite element (FE) geomechanics simulator to model the coupled fluid-solid system. Fully saturated porous media with poroelastic behavior, random field permeability and elastic modulus are considered as parameters. The performance of the proposed hydro-mechanical models, including MPFA O-FEM and DDFV-FEM, is examined through different test cases. First, the models are validated and compared with the closed-form solutions in the homogeneous domain. Second, the methods' stability and convergence properties are analyzed in highly heterogeneous porous media with different permeability and elastic modulus realizations. Finally, implementing the models in a large-size realistic problem of a synthetic reservoir accomplishes this comprehensive study. Although both methods agree well with the analytical solutions in homogeneous porous media, the DDFV-FEM yields higher accuracy. In the cases where the random field of permeability varies over several orders of magnitude (4th and more), the MPFA O-FEM suffers from significant oscillations and even instability. The analysis illustrates that MPFA O-FEM exhibits oscillation in the cases with high anisotropy ratio, while the DDFV-FEM demonstrates superior performance for handling anisotropy and heterogeneity.

Streamline-based simulation in highly heterogeneous and anisotropic petroleum reservoirs using a non-orthodox MPFA method and an adaptive timestep strategy with unstructured meshes

In this paper, an adaptive timestep size selection algorithm applied in streamline-based simulation for highly heterogeneous and anisotropic reservoirs is presented. To handle an arbitrarily, anisotropic, and discontinuous permeability tensor, we use a non-orthodox Multipoint Flux Approximation Method with a Diamond stencil (MPFA-D) discretize the pressure equation. For the solution of the transport equation, a classical time of flight discretization approach is used. To improve computational efficiency we use an adaption of the Modified IMPES (MIMPES), initially devised by Hurtado et al. (2006) and da Silva et al. (2016). To demonstrate and validate the efficiency and usefulness of our approach, we present various numerical simulations, including two-phase flows of oil and water in highly heterogeneous and anisotropic petroleum reservoirs and flows with high mobility ratios that present a challenge to streamline simulation. Our results are encouraging in comparison with other results found in the literature, achieving a better balance between accuracy and performance with minor grid orientation effects.

Anisotropic analysis of fibrous and woven materials part 2: Computation of effective conductivity

Micro-scale modeling enables the study of composite material properties at a fundamental level. These properties must be computed at the scale of a representative elementary volume in order to accurately inform macro-scale response simulations. The main focus of this work is to predict the effective thermal conductivity of anisotropic materials based on the three-dimensional reconstruction of their fibrous structure, obtained from X-ray micro-tomography. A computational method that accounts for multiple scales of anisotropy of woven composites is presented, including that of the constituting fibers and tows, and that of woven architectures. The finite volume formulation and implementation based on the Multi-Point Flux Approximation method is detailed. The algorithm was verified against analytical solutions and validated against experiments on both random and woven fibrous samples. In order to apply the method to real materials, the techniques presented in the first part of this study [1] are used to compute the local material orientation, necessary to align the conductivity tensor along the fibers. The method is ultimately used to reverse engineer the orthotropic conductivity of a single carbon fiber by performing parametric studies and comparing against experiments.

A Fitted Multi-point Flux Approximation Method for Pricing Two Options

In this paper, we develop novel numerical methods based on the multi-point flux approximation (MPFA) method to solve the degenerated partial differential equation (PDE) arising from pricing two-assets options. The standard MPFA is used as our first method and is coupled with a fitted finite volume in our second method to handle the degeneracy of the PDE and the corresponding scheme is called fitted MPFA method. The convection part is discretized using the upwinding methods (first and second order) that we have derived on non uniform grids. The time discretization is performed with $$\theta $$-Euler methods. Numerical simulations show that our new schemes can be more accurate than the current fitted finite volume method proposed in the literature.

A new multipoint flux approximation method with a quasi-local stencil (MPFA-QL) for the simulation of diffusion problems in anisotropic and heterogeneous media

In this paper, we present a new linear cell-centered finite volume multipoint flux approximation (MPFA-QL) scheme for discretizing diffusion problems on general polygonal meshes. This scheme uses a quasi-local stencil, based upon the conormal decomposition, to approximate the control face flux when solving the steady state diffusion problem, being able to reproduce piecewise linear solutions exactly and it is very robust when dealing with heterogeneous and highly anisotropic media and severely distorted meshes. In our linear scheme, we first construct the one-sided fluxes on each control surface independently and then a unique flux expression is obtained by a convex combination of the one-sided fluxes. The unknown values at the vertices that define a control surface are interpolated by means of a linearity-preserving interpolation procedure, considering control volumes surrounding these vertices. To show the potential of the MPFA-QL scheme, we solve some benchmark using triangular and quadrilateral meshes and we compare our scheme with other numerical formulations found in literature.

Impact of spatially varying hydraulic conductivities on tumor interstitial fluid pressure distribution

Abstract This work deals with numerical computation of the interstitial fluid pressure distribution for various synthetic tumor configurations in two spatial dimensions. In order to attend a reasonable degree of realism, we also consider asymmetric configurations and use unstructured triangular grids to accurately discretize the computational domains. A multi point flux approximation method, previously used for porous medium flow simulations, is utilized for solving the corresponding Poisson equation for the pressure. We are particularly interested in the effects of spatial variation of the hydraulic conductivity as well as vessel hydraulic conductivities on the interstitial fluid pressure. Spatially varying hydraulic conductivity was first studied by Liu and Schlesinger [17] in some special situations using analytical methods. Here we complement their work by performing a set of relatively comprehensive numerical investigations. A considerable difference in the computed pressure profiles is found when comparing the results with the more classical approach using piecewise constant hydraulic conductivities. We also explore the response of the numerical model when varying i.e. size of necrotic core and vessel hydraulic conductivities. The pressure profiles display a striking dependence on the arterial hydraulic conductivity in the peripheral region. A strong dependence on the arterial hydraulic conductivity is also observed when introducing so called “vascular hot-spots” in the model as well as looking at non-symmetric geometries. Finally, we discuss our numerical results in light of other numerical studies of TIFP distribution and experimental studies of tumor biology. It is concluded that the computed results are in accordance with the general understanding of TIFP. Although we have worked on a simplified system, our results highlight the importance of describing hydraulic conductivity as a spatially varying parameter.

A two-step finite volume method to discretize heterogeneous and anisotropic pressure equation on general grids

A novel Two-Step cell-centered Finite Volume Method (TSFVM) is developed in this work to discretize the heterogeneous and anisotropic pressure equation on triangular and quadrilateral grids in 2D and hexahedral and tetrahedral grids in 3D. Physical properties such as permeability and porosity are piece-wise constant on each grid cell. In the first step, the Galerkin Finite Element Method (FEM) is utilized to compute pressure solutions at all cell vertices. In the second step, pressure values at cell vertices are used to derive continuous two-point flux stencils for cell faces. Mass conservation equations are then written for each cell to obtain a system of linear equations that can be solved for pressure at cell centers. Extensive numerical experiments are carried out to test the performance of our TSFVM. In particular, we compare TSFVM with the classical Multipoint Flux Approximation (MPFA-O) method as well as a more recently developed MPFA method with full pressure support called enhanced MPFA (eMPFA). The results show that the TSFVM compares well with eMPFA for challenging test cases for which MPFA-O breaks down. Specifically, and as a significant step forward, our TSFVM is quite robust for challenging problems involving heterogeneous and highly anisotropic permeability tensors when both MPFA-O and eMPFA suffer from unphysical oscillations. Finally, the numerical convergence study demonstrates that TSFVM has comparable convergence behavior to MPFA-O method for both homogeneous and discontinuous permeability fields.

A multipoint flux mixed finite element method for the compressible Darcy–Forchheimer models

We develop a multipoint flux mixed finite element method for the compressible Darcy–Forchheimer models. This method is motivated by the multipoint flux approximation method. Based on the lowest order Brezzi–Douglas–Marini mixed finite element method and combined with a special quadrature rule that allows for local velocity elimination and leads to a cell-centered system for the pressures. We develop two approximation methods for the problems, the semi-discrete method and fully discrete method, where the fully discrete method includes the backward Euler scheme and the Crank–Nicolson scheme. Theoretical results show the convergences for both pressure and velocity. The numerical experiments show that the convergence rates of the method are in agreement with the theoretical analysis.

A two-step finite volume method for the simulation of multiphase fluid flow in heterogeneous and anisotropic reservoirs

Efficient and accurate simulation of multiphase flow and transport in porous media plays a key role in optimizing recovery of hydrocarbon resources. Two-Point Flux Approximation (TPFA) is widely used in the industry for discretizing the pressure equation because of its robustness and efficiency despite the fact that it introduces O(1) flux errors on non-K-orthogonal grids. Multi-Point Flux Approximation (MPFA) provides a consistent discretization of the flow equations at the cost of larger flux stencils which leads to a denser, asymmetric system of linear equations requiring more computer memory and work for linear solvers. This drawback of MPFA becomes more pronounced for generally unstructured triangular grids and is even worse in 3D applications. In this work, we develop a two-step finite volume method (TSFVM) that is consistent for the anisotropic pressure equation on triangular and tetrahedral grids with improved efficiency as compared to MPFA methods. During each time step, the pressure equation is solved implicitly and saturation is then updated explicitly (IMPES). To discretize the pressure equation with full permeability tensors, Galerkin finite element method (FEM) is employed in the first step to compute pressure at grid vertices. Pressures at grid vertices are then utilized in the second step to derive a two-point flux stencil with a constant for each grid face. Pressure at cell centers and flux across grid faces can then be solved using a TPFA approach. Finally, the flux field at the new time level is used to update saturations explicitly. Although during each time step our TSFVM needs to solve two systems of linear equations, results of the numerical experiments clearly demonstrate that it is still less expensive than the classical MPFA-O method while providing high-quality numerical solutions matching that of MPFA-O. The gained efficiency is especially significant for 3D tetrahedral grid. As the scale of the problem increases, MPFA becomes less appealing because of its large flux stencils that result in much denser matrices while our TSFVM still remains computationally tractable thanks to the flexibility of FEM in the first step and the two-point flux stencil in the second step. On the other hand, the first step of the TSFVM is reminiscent of the Control Volume Finite Element Method (CVFEM) but with important differences. Unlike CVFEM, our TSFVM does not need to construct a dual control volume mesh around grid vertices on the basis of the primary mesh and the saturation unknowns are not associated with node-centered control volumes but are piece-wise constant on grid cells in which reservoir properties are distributed. This ensures that a saturation value does not straddle across material discontinuities which CVFEM suffer from.

A MODIFIED FLOW ORIENTATION SCHEME COUPLED WITH A ROBUST MPFA-DIAMOND FOR THE SOLUTION OF TWO-PHASE FLOW IN HIGHLY ANISOTROPIC PETROLEUM RESERVOIRS

In this paperwe simulate two-phase flow in anisotropic petroleum reservoirs. The IMPES procedure is used to solve the coupling between pressure and saturation equations. The pressure equation is discretized by a robust Multipoint Flux Approximation Method with a Diamond-type support. This formulation is capable of reproducing piecewise linear solutions exactly and deals with anisotropic media. To solve the saturation equation a Modified Flow Oriented Scheme (M-FOS) is proposed. This alternative computes the multidimensional numerical fluxes using higher order accuracy in space. This formulation explicitly takes into account the angular distortion of the computational mesh by means of an adaptive weight that tunes the multidimensional character of the formulation according to the grid distortion. A recently devised Multidimensional Limiting Process is adopted in this paper to control the spurious oscillations in higher order approximation. This strategy guarantees monotone solutions and can be used with any polygonal mesh. Finally, an efficient entropy fix strategy, originally proposed in magneto-dynamics context, is also employed in order to produce convergent solutions. The performance of this set of numerical schemes is verified by solving some relevant benchmark problems, where we observe that the Grid Orientation Effects are clearly diminished by using this M-FOS framework. Keywords: Oil and Water displacements, Anisotropic Porous Media, MPFA-D, MLP, Modified Flow Oriented Scheme.

Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem

We present two new cell-centered nonlinear finite-volume methods for the heterogeneous, anisotropic diffusion problem. The schemes split the interfacial flux into harmonic and transversal components. Specifically, linear combinations of the transversal vector and the co-normal are used that lead to significant improvements in terms of the mesh-locking effects. The harmonic component of the flux is represented using a conventional monotone two-point flux approximation; the component along the parameterized direction is treated nonlinearly to satisfy either positivity of the solution as in [29], or the discrete maximum principle as in [9]. In order to make the method purely cell-centered, we derive a homogenization function that allows for seamless interpolation in the presence of heterogeneity following a strategy similar to [46]. The performance of the new schemes is compared with existing multi-point flux approximation methods [3,5]. The robustness of the scheme with respect to the mesh-locking problem is demonstrated using several challenging test cases.