An Introduction to Multi-point Flux (MPFA) and Stress (MPSA) Finite Volume Methods for Thermo-poroelasticity

Abstract

We give a unified introduction to the MPFA-and MPSA-type finite volume methods for Darcy flow and poro-elasticity, applicable to general polyhedral grids. This leads to a more systematic perspective of these methods than has been exposed in previous texts, and we therefore refer to this discretization family as the MPxA methods. We apply this MPxA framework to also define a consistent finite-volume discretization of thermo-poro-elasticity. In order to make the exposition accessible to a wide audience, we avoid much of the technical notation which is used in the research literature, and compensate for this by an expanded summary and literature review of the main properties of the MPxA methods. We close the chapter by a section containing applications to problems with complex geometries and nonlinear physics.

Nonlinear finite volume discretization of geomechanical problem

“…Below we use 𝑞 𝛿1 , 𝑞 𝛿2 to denote semi-fluxes, but the same statements remain in power for the transversal semi-fluxes 𝑞 𝜏1 , 𝑞 𝜏2 used in Equation (B13). Once the approximation of pressure gradients substituted to either Equation (B4) or to Equation (B1) we can represent the approximation for two semi-fluxes as follows 24 𝑞 𝛿1 = 𝑐 11 (𝑝 1 − 𝑝 2 ) + 𝑐 12 (𝑝 1 − 𝑝 3 ) + 𝑐 13 24 and the positiveness of coefficients 𝑐 𝑖𝑗 > 0 is guaranteed by the proper choice of the triplet of equations in Equations (B5), (B7), (B10), (B11) for the reconstruction of gradients. Substitution of Equations (B14), (B15) to Equation (B12) (or the same expressions for 𝑞 𝜏1 , 𝑞 𝜏2 to Equation (B13)) gives the approximation of fluid flux 𝑞 12 .…”

“…Another advantage is that the use of collocated grids simplifies the formulation, solution and implementation of coupled fluid mass, energy and momentum balances within a unified FVM framework. [13][14][15] While MPFA gives a representative solution to the flow problem, it is known to be conditionally monotone 7,16 and may violate the discrete maximum principle. This non-monotone behavior often takes the form of spurious oscillations in numerical solution across the grid.…”

“…Although the research in the field of simulation of momentum equations by Finite Elements (FE) is more mature, the advantage of using an FVM for mechanics is the support of a wide range of polyhedral meshes including Corner Point Geometry 12 that is ubiquitously used in reservoir modelling. Another advantage is that the use of collocated grids simplifies the formulation, solution and implementation of coupled fluid mass, energy and momentum balances within a unified FVM framework 13–15 …”

Parallel efficiency of monolithic and fixed-strain solution strategies for poroelasticity problems

“…Criterion (a) makes use of traditional finite element method (FEM) problematic, since meshes for subsurface domains can contain cells which are general polyhedra. While FVM for geomechanics exists [7][8] and is applied in poroelastic case [10] and more complex ones [9], it is still somewhat new and not so widely used option. Recent developments include FVM scheme achieving improved robustness by avoiding decoupling into subproblems and introducing stable approximation of vector fluxes [11].…”

Two-level simulation of injection-induced fracture slip and wing-crack propagation in poroelastic media

“…19,33,59 The conservation equations for flow in Ω M and Ω F i as well as momentum in Ω M are discretized by a family of cell-centered multipoint finite volume methods developed for poroelasticity. 60,61 The methods are based on constructing discrete representations of stresses (respective fluxes) over cell faces regarding displacements (respectively pressures) in nearby cell centers. The balance of momentum and mass is enforced on the cells.…”