A sequential discontinuous Galerkin method for two-phase flow in deformable porous media
Introduction
The field of poromechanics pertains to the study of coupled fluid flows and mechanical deformations in porous media. Applications include the prediction of land subsidence due to extraction of water and/or hydrocarbons from subsurface [1]. Mathematical models of the poroelastic two-phase flow problem can be found in [2] and were derived by Biot [3], [4] using a phenomenological approach. In the case of single phase flow, the poroelasticity equations have been extensively studied by applied mathematicians and engineers in the scientific literature [5], [6], [7], [8], [9], [10]. In contrast, there are very few works on the design of efficient numerical methods for multiphase flows in deformable porous media. The main contribution of this work is the formulation of a numerical method that employs discontinuous piecewise polynomial approximations for the wetting and non-wetting phase pressures and the displacement of the medium. Discontinuous Galerkin methods (DG) have been successfully applied to multiphase flows in rigid porous media because of their flexibility resulting from the lack of continuity constraint between mesh elements. DG methods are locally mass conservative, they easily handle local mesh refinement and local high order of approximation; and they are well suited for the solution of convection-dominated problems because they exhibit little numerical diffusion. At each time step, we propose to solve the mass balance equations and the momentum equation sequentially by a DG method. Because of the decoupling of the equations, a first order term is added to the discrete momentum equation to stabilize the method. The idea of decoupling flow and displacement equation and using a stabilization term was introduced in [11] for single phase flow in deformable porous media. The term is needed to prove convergence of the method for single phase Biot problem, in particular the term helps control the time derivative of the divergence of the displacement in the flow equations.
In this work, we focus on isothermal flows where inertial forces are neglected. The resulting coupled partial differential equations can be solved fully implicit, iteratively or sequentially [12]. Fully implicit finite element methods are the most stable ones but also the most computationally expensive. In [13], finite element methods in space are combined with the theta method in time and the resulting system is solved by Newton–Raphson’s method at each time step. The method is applied to one-dimensional and two-dimensional problems. In [14], fully implicit mixed finite element methods combined with standard finite element methods are applied to solve for pressure, saturation, displacement and their gradients in two-dimensional problems. The iterative approach (fixed-stress split) is combined with finite volume methods in [15] for different choices of primary unknowns and for one-dimensional problems. Our approach for solving the two-phase Biot problem is novel in the sense that no iterations are needed for stability. At each time step, each equation is solved separately and the computational cost is smaller than the one for fully implicit methods. We apply the proposed method to three-dimensional problems and we study the impact of heterogeneities (regions with different capillary pressures) and loading on the propagation of the fluid phases in the medium. Finally, we point out that the fully implicit finite element method has been applied to more complex dynamic and non-isothermal flows in [16], [17], [18], [19].
An outline of the paper follows. Section 2 introduces the mathematical model and the assumptions on the input data. The numerical algorithm is described and analyzed in Section 3. Numerical results, including convergence rates and validation of the method by benchmark problems, can be found in Section 4. Conclusions follow.
Section snippets
Model problem
Mathematical models for compressible two-phase flow poroelasticity are described by two mass conservation equations coupled by a momentum conservation equation [2]. Let (resp. ) denote the wetting (resp. non-wetting) phase pressure and saturation respectively and let denote the displacement of the porous medium . By definition, , and we use this relation to eliminate the non-wetting phase saturation from the system of equations. The difference between phase pressures is
Discontinuous Galerkin scheme
The equations are discretized by the interior penalty discontinuous Galerkin method. Let be a partition of the domain made of tetrahedral elements of maximum diameter . Let denote the set of interior faces. For any interior face , we fix a unit normal vector and we denote by and the two tetrahedra that share the face such that the vector points from into . The jump and average of a function across an interior face are denoted by and respectively:
Numerical results
We first verify the optimal rate of convergence of our proposed numerical method for smooth solutions and then we apply our scheme to various porous media problems: the McWorther problem, a non-homogeneous medium with different capillary pressures, a medium subjected to load, and a medium with highly varying permeability and porosity. Unless explicitly stated in the text, all examples use the following physical parameters.
Conclusions
We have presented an accurate and robust numerical method for solving the coupled two-phase flow and geomechanics equations in porous media. The method is sequentially implicit, therefore computationally less expensive than a fully implicit scheme. The sequential scheme is stable due to stabilization terms added to the displacement equation. The method is validated on three-dimensional benchmark problems and the numerical results confirm the stability, robustness and accuracy of the proposed
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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