Fully implicit mixed-hybrid finite-element discretization for general purpose subsurface reservoir simulation
Introduction
Management of subsurface resources, including water aquifers and oil/gas reservoirs relies on numerical reservoir simulation. Flow simulation entails discretization of the equations and constitutive relations that describe multi-component, multiphase flow in subsurface porous formations. The reservoir characterization model (RCM) that serves as input to the flow simulator can be quite complex, both in terms of the grid geometry and the formation properties (e.g., permeability). Each gridblock is assigned material properties and, in most cases, the properties are different in each block. For highly-heterogeneous fields, comprised of many stratigraphic compartments, the number of gridblocks can be extremely large, and the computational domain can be highly unstructured. Therefore, accurate modeling of the local flow dynamics, especially the fluxes between the gridblocks is of great importance. Moreover, the highly nonlinear nature of the coupled conservation equations and constitutive relations make honoring local conservation a fundamental requirement.
Traditionally, a two-point flux approximation (TPFA) technique is employed to approximate the phase flux in the Darcy equation [1]. As the name implies, two points are used to approximate the flux at the interface between the control-volumes (gridblocks). The TPFA scheme requires the mesh of the domain to be structured, such that the interfaces are orthogonal to the interface between control-volumes, and the grid is aligned with the principal directions of any anisotropic material property, e.g., the permeability. There is strong interest in the community to use high-resolution, geometrically complex RCMs that represent the large-scale subsurface formations as accurately as possible. Accommodating three-dimensional unstructured grids with full-tensor properties is a difficult – practically impossible – task using TPFA. Multi-point flux-approximation (MPFA) discretization schemes have been developed to address issues related to nonorthogonal grids and full-tensor heterogeneous permeability [2], [3]. Similar to TPFA schemes, divergence-free conditions are imposed on the control-volume to approximate the flux. MPFA schemes – subject to limitations – allow for using unstructured grids and anisotropic material properties [4]. Several variations of the method have been developed [5], [6] with corresponding convergence studies [7], [8], [9]. In practice, the ‘MPFA-O’ method is the most widely used. More recently, nonlinear flux approximation schemes have been developed [10], [11], [12].
Another flux approximation scheme for unstructured meshes is the mixed-finite-element method (MFEM), where the mass and momentum equations are coupled and solved simultaneously [13]. This method is locally conservative, and it can accommodate high-order approximations and anisotropic material properties. However, in its original form, MFEM leads to an algebraic system of saddle-point type, whereby the linearized matrix is indefinite. An efficient variant for the method was developed to overcome the challenges associated with saddle-point problems, in which Lagrange multipliers for the primary variables are placed on the elements interfaces [14]. This approach is called the mixed-hybrid finite-element (MHFE) method. Here, we present a new fully-implicit method for the MHFE formulation of general-purpose reservoir simulation. Another approach to overcome the saddle-point problem is to eliminate the momentum balance equation from the linear system and replace it with a particular MPFA scheme in the mass balance equation using special quadrature rules [15]; this elimination procedure can be considered a special case for the MPFA-O method [16], [17]. Previous MHFE-based efforts include single-phase flow [18]; [19] used the MHFE method for modeling two-phase incompressible fluids; [20] extended the method for compositional equation-of-state models, and gravity was included by [21]. However, all these schemes were designed for specific problems and have not been applied for general-purpose reservoir simulation where all the main physics are modeled and simulated in-conjunction. The existing literature on MHFE has employed explicit, or partially explicit, temporal discretization. In these schemes, the governing equations for the system are decoupled, and the explicit treatment greatly restricts the time steps that can be used. More recently, sequential temporal discretization has been applied for MHFE without capillary pressure [22].
In this work, we employ the MHFE method for general-purpose compositional reservoir simulation using a fully-implicit method. The velocities are approximated with Raviart–Thomas vector basis functions [23], which ensure local conservation of momentum. The scalar variables: pressure, phase saturations, and component concentrations, are discretized using the locally conservative finite volume method. We employ a fully implicit scheme whereby the conservation equations are coupled, and the system is solved using the Newton–Raphson method. The nonlinear variables are spatially discretized on the elements and on element interfaces by the MHFE process. We also introduce a new continuous flux approach that guarantees that the momentum is conservative for the mass balance equations. This is the first fully-implicit temporal discretization and implementation for the MHFE for isothermal general-purpose reservoir simulation.
The MHFE framework is integrated into the Automatic Differentiation General Purpose Research Simulator (AD-GPRS) [24], [25]. This framework includes EOS-based multiphase multi-component formulation [26], [27], thermal-compositional capabilities [28], [29], fully coupled geomechanics [30], chemical reactions [31] and adjoint gradients, [32] which makes it suitable for solution of a wide range of subsurface problems. To validate the proposed discretization approach, we perform extensive numerical tests highlighting the convergence and accuracy for the different physical phenomena. We also compare the discretization to other advanced schemes. Moreover, we apply the method for problems with the emphasis on unstructured grids and tensor permeability.
The paper is organized as follows. In Section 2, the governing equations are discussed. In Section 3, the MHFE formulation is described, and in Section 4, the nonlinear solution is detailed. We present the numerical tests and examples in Sections 5 and 6. Finally, in Section 7, we present our conclusions.
Section snippets
Governing equations
In this section, we present the equations that govern the fluid flow and the thermodynamic phase equilibrium for isothermal compositional reservoirs.
Mixed hybrid finite element method
In this work, we use the MHFE method to discretize the conservation equations. In this method, we use vector basis functions of having a flux of one at interface and zero elsewhere (see Fig. 1). where is the vector basis function of interface i, is the normal outward oriented vector on interface j and is the area of interface j of element E. Another property of the mixed formulation is that is constant over an element, where is volume of
Nonlinear solver
In this section, we discuss the procedure for the nonlinear solution for the unknowns described in the previous sections. We use the backward Euler method for the temporal discretization and the Newton–Raphson method to linearize the system, in which a Jacobian system is assembled and solved for each nonlinear iteration for our fully-implicit system:
Here, is residual for the thermodynamic equilibrium constraint and the mass balance
Numerical convergence
We present numerical tests which demonstrate the accuracy and convergence of the fully-implicit MHFE method for several problems. First, we test the convergence of the method to an existing analytical solution for a single-phase flow in heterogeneous domain in the presence of a barrier and a full permeability tensor. Then, we validate the applicability of the method for flow in the presence of strong capillarity and gravity forces, and we compare the results with analytical solution. In
Applied computational cases
In this section, we present applied computational cases to illustrate the capability and the robustness of our method for general purpose reservoir simulation. First, we test the capability of the method for modeling multiphase flow in discontinuous anisotropic domains. Then, we study the effect of geological faults for the migration of hydrocarbon components between reservoirs. Finally, we present a challenging heterogeneous case which is a modification of the SPE10 benchmark test [56] with
Concluding remarks
In this paper, we presented a fully implicit discretization scheme for the mixed hybrid finite element (MHFE) method for general-purpose compositional reservoir simulation. The scheme couples the momentum and mass balance equations and solves them simultaneously, and the mass balance is discretized using the finite volume method. We introduced a new flux approach for the mass balance equations that is conservative. The system was linearized using the Newton–Raphson method, and the non linear
Acknowledgements
We thank the Qatar Foundation – Research & Development for funding this research. We also thank the SUPRI-B Reservoir Simulation consortium at Stanford University.
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