Coupled Problem

Mathematical Model

Hooke's law, mass conservation law and Darcy's law are used to descrip the solid deformation and fluid flow, where the Biot theory is introduced to capture the coupling process between them. The real-stress (effective stress) of solid is strongly effected by the pressure of fluid. Note that the primary assumptions of small deformation and single-phase are valid.

To complete the proposal of well-posed problem of coupled problem, inital and boundary condition should be imposed. In flow problem, pressure or flux are used. In deformation problem, displacement or force should be specified.

The initial displacements and strains are defined to zeros. The pressure and stress equal to specified values.

Space Discretization for Governing Equations

We devised a method that coupling the finite volume method (FVM) and finite element method (FEM), so-called FVM-FEM scheme.

In FVM for flow problem, the pressure is located at cell center. In FEM for deformation problem, the displacement is located at node of each cell. This discretized scheme has local mass conservation at the cell level, and has the capacity of continous displacement field that allows the tracking of the deformation, and good convergence property at a low order discretization (Jha B, Juanes, 2007).

Sequential Method

Fully Coupled and Sequential Coupled

There are two basic strategies to solve coupled problem, fully method (FM) and sequential method (SM).

SM can be classified into two different types regarding either flow or mechanics solved first, and the other subpropblem can be handled through the intermediate solution information. Four different operator split strategies are considered: (1) the mechanical subproblem is sovle d first (drained split and undrained split), and (2) the flow subproblem is solved first (fixed-strain split and fixed-stress split).

Two methods are reduced from SM, explicitly coupled and loosely coupled.

Sequential Methods (mechanics then flow)

In the SM-based schemes that the mechanics subproblem is solved first, where the quasi-static deformation assumption is applied and the inertia can be neglected so the governing equation is a elliptic PDE. Boit theory is used to obtain the classical parabolic-type flow problem.

There are two classical SM in mechanics first and then flow, drained split and undrained split. A converged sollution are calculated by iteration for instance the Newton-Raphson method. The corresponding schemes in theromelasticity are isothermal split and adiabatic split.

The prons of SM is that it can make use of the existing solvers of flow and mechaincs, then only the interface between these two solvers be implemented. Not only the desirable features of SM, it maybe limited by the bad stability and convergence of the over operator split.

Drained split (fixed-pressure). The drained method freezes the pressure during the mechanical process. It is conditionally stable in which even then stability but non-convergence. It has bad properties when the coupling between flow-stress is strong.
Undrained split (fixed-fluid mass). The undrained split freezes the fluid mass during content when solving mechanical process. This method follows the dissipative effects and can be applied to linear and non-linear coupled problems. It is unconditionally stable.

Sequential Methods (flow then mechanics)

In contrast to solving mechanics first, we have another option that flow solved first and then mechanics. In this case, the SM can be considered as two different schemes again, fixed-strain and fixed-stress.

Fixed-strain split. To fix the rate of total strain during solving the flow sub-problem. It is conditionally stable in which even then stability but non-convergence. It has bad properties when the coupling between flow-stress is strong.
Fixed-stress split. To fix the rate of total stress during solving the flow sub-problem.It is unconditionally stable.

Some Remarks

The drained split and fixed-strain split are commomly used and only conditionally stable. Their stability limits are independent of time step size and depend only on the coupling of flow-stress weak connected. Oscillations would also be occured when the stability is good.
The undrained split and fixed-stress split are unconditionally stable regardless of the strong coupling of flow-stress. Fixed-stress strategy is more accury than undrained split.
The fixed-stress metho is recommended.

Appendices

References

Kim J, Tchelepi H A, Juanes R. Stability and convergence of sequential methods for coupled flow and geomechanics: Fixed-stress and fixed-strain splits[J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(13-16): 1591-1606.
Kim J, Tchelepi H A, Juanes R. Stability and convergence of sequential methods for coupled flow and geomechanics: Drained and undrained splits[J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(23-24): 2094-2116.
Kim J, Tchelepi H A, Juanes R. Stability, accuracy and efficiency of sequential methods for coupled flow and geomechanics[C]//SPE reservoir simulation symposium. Society of Petroleum Engineers, 2009.
Jha B, Juanes R. Coupled multiphase flow and poromechanics: A computational model of pore pressure effects on fault slip and earthquake triggering[J]. Water Resources Research, 2014, 50(5): 3776-3808.
Asadollahi M. Finite Volume Method for Poroelasticity[J]. 2017. Master Thesis of TU Delft.
Garipov T T, Karimi-Fard M, Tchelepi H A. Discrete fracture model for coupled flow and geomechanics[J]. Computational Geosciences, 2016, 20(1): 149-160.
Jha B, Juanes R. A locally conservative finite element framework for the simulation of coupled flow and reservoir geomechanics[J]. Acta Geotechnica, 2007, 2(3): 139-153.