One-Way versus Two-Way Coupling in Reservoir-Geomechanical Models

Abstract

Procedures to couple reservoir and geomechanical models are reviewed. The focus is on immiscible compressible non-compositional reservoir -geomechanical models. Such models require the solution to: coupled stress, pressure, saturation and temperature equations. Although the couplings between saturation and temperature with stress and fluid pressure are "weak" and can be adequately captured thru staggered (fixed point) iterations, the coupling between stress and pressure equations are "strong" and require special procedures for accurate integration. Issues related to both simultaneous integration and sequential integration of pressure and stress equations are investigated. It is shown that simultaneous integration can be achieved by computing the contribution to the coupled Jacobian matrix thru finite differencing of the residual equations. Forming the Jacobian matrix can also be avoided thru a partitioned CG iterative solution procedure of the Schur complement matrix. Iterative sequential integration (fixed point iterations) of pressure and stress equations are then discussed, and detailed algorithms are provided. Assessment of the accuracy of such iterative procedures is made thru comparison with available exact analytical solutions for the half-space consolidation (McName and Gibson [1960]). The iterative sequential procedure (one-way coupling iteration) is shown to require an unreasonably large number of iterations (>50) to capture accurately coupling effects. Finally, the coupling effects in a field case are investigated: CO2 injection at In Salah, Algeria where CO2 is injected at a temperature (50 dgC) below the resident brine reservoir (90 dgC) temperature. Surface uplift displacements and induced tensile stresses in the caprock computed with both a fully two-way coupled scheme versus a one-way scheme are compared.

Accuracy and convergence properties of the fixed‐stress iterative solution of two‐way coupled poromechanics

“…The preconditioner operator is applied to the residual —that is, to the correct discrete residual form of the equilibrium and continuity equations. This fact is overlooked in . The algorithm for applying to the residual vector R k is which takes advantage of the block‐triangular nature of by first updating the pressure components ( Y p ) followed by the displacement components ( Y u ) as follows: It is well known that a necessary and sufficient condition for the convergence of stationary iterative methods is that the spectral radius, ρ , of the iteration matrix, , is smaller than unity in magnitude .…”

“…The preconditioner operator P is applied to the residual R k D .B AX k /-that is, to the correct discrete residual form of the equilibrium and continuity equations. This fact is overlooked in [35,45]. The algorithm for applying P 1 to the residual vector R k is…”

“…This partitioned approach allows for design flexibility-such as the use of separate software modules for the geomechanics and flow sub-problems-and does not require the implementation of dedicated solvers and preconditioners to tackle the hydromechanical coupling. Unconditional stability and convergence of the scheme have been mathematically proved [32,34], but the practical performance of the scheme is still the subject of investigation [35].The present paper examines the fixed-stress iterative solution of the linear system obtained by a mixed FE/FV discretization, combining the most attractive features of the FE and FV discretization methods for the equilibrium and continuity equations, respectively. Specifically, the mechanical problem employs a continuous Galerkin formulation [36] to track the deformation field.…”

Two‐way coupling in reservoir–geomechanical models: vertex‐centered Galerkin geomechanical model cell‐centered and vertex‐centered finite volume reservoir models

“…Further, it should be noted that the C m approximation used in conventional reservoir simulators provides a correct representation of the diffusion time, which is controlled by the following diffusion coefficient (e.g., ): However, this is not the case with the scheme proposed by Kim et al , which therefore introduces undesirable distortions of the diffusion time and processes. Various one‐way coupling and one‐way iterative coupling strategies are discussed in detail in , where it is shown that one‐way iterative (sequential) integration of reservoir–geomechanical equations can work but requires an unreasonable number of iterations for accurate integration of strongly coupled equations, such as pressure and stress equations. This is not a surprise because it is well known that fixed‐point iterations (if they converge at all) always require a large number of iterations to converge.…”

Full Pressure Coupling for Geo-mechanical Multi-phase Multi-component Flow Simulations

“…This strategy is in part motivated from the availability of established and robust commercial software for both multi-phase flow (Schlumberger, n.d.-a) as well as mechanical deformation (Schlumberger, n.d.-b). Although some of these coupled approaches can be shown to be stable (Kim, Tchelepi, & Juanes, 2011b) and convergent (Mikelić, Wang, & Wheeler, 2014), in general it is desirable to consider a tighter coupling between flow and mechanics (Prevost, 2013).…”