An enhanced sequential fully implicit scheme for reservoir geomechanics

Abstract

In this paper, it is proposed an enhanced sequential fully implicit ESFI algorithm with a fixed stress split to approximate robustly poro-elastoplastic solutions related to reservoir geomechanics. The constitutive model considers the total strain effect on porosity/permeability variation and associative plasticity. The sequential fully implicit algorithm SFI is a popular solution to approximate solutions of a coupled system. Generally, the SFI consists of an outer loop to solve the coupled system, in which there are two inner iterative loops for each equation to implicitly solve the equations. The SFI algorithm occasionally suffers from slow convergence or even convergence failure. In order to improve the convergence (robustness) associated with SFI, a new nonlinear acceleration technique is proposed employing Shanks transformations in vector-valued variables to enhance the outer loop convergence, with a Quasi-Newton method considering the modified Thomas method for the internal loops. In this ESFI algorithm, the fluid flow formulation is defined by Darcy's law including nonlinear permeability based on Petunin model. The rock deformation includes a linear part being analyzed based on Biot's theory and a nonlinear part being established using Mohr-Coulomb associative plasticity for geomechanics. Temporal derivatives are approximated by an implicit Euler method and spatial discretizations are adopted using finite element in two different formulations. For the spatial discretization, two weak statements are obtained: the first one uses a continuous Galerkin for poro-elastoplastic and Darcy's flow; the second one uses a continuous Galerkin for poro-elastoplastic and a mixed finite element for Darcy's flow. Several numerical simulations are presented to evaluate the efficiency of ESFI algorithm in reducing the number of iterations. Distinct poromechanical problems in 1D, 2D, and 3D are approximated with linear and nonlinear settings. Where appropriate, the results were verified with analytic solutions and approximated solutions with an explicit Runge-Kutta solver for 2D axisymmetric poroelastoplastic problems.

Evaluation of the impact of strain-dependent permeability on reservoir productivity using iterative coupled reservoir geomechanical modeling

“…By inserting equation (9) into equation ( 7) and considering S e = S +φ • c f , an alternative expression for (φρ f ) is obtained as (Kim et al, 2012;Duŕan et al, 2019):…”

“…Subsequently, the development of poroelasticity theory for anisotropic materials and its extension to poro-elastoplasticity was developed by (Biot, 1955(Biot, , 1962Coussy, 1995). In recent years, several researchers have been published in porous media to find coupled geomechanics and reservoir flow such as, Phillips & Wheeler (2007); Wei & Zhang (2010); Sanei et al (2017); Jiang & Yang (2018); Duŕan et al (2019).…”

A multiscale mixed finite element method applied to the simulation of two-phase flows

“…Figure 4 on the right side shows that the cost of having small mass loss carries larger number of SFI iterations. However, this issue can be addressed by some nonlinear acceleration techniques [38,56,32]. Figure 5 shows the evolution of the water segregated by the effect of gravity at times values t = {0, 25, 50, 100} d.…”

“…For instance, they have been promoted by a research group at The University of Texas at Austin, USA, under Prof. M. Wheeler leadership, and by other groups (see e.g. [20,21,22,23,24,25,13,26,27,28,29,30,31,32]). The success of this approach is mainly due to some relevant properties valid for these methods.…”

Applying two-grid method on unstructured tetrahedra to study fault stability and surface deformation in multiphase geomechanics

“…As shown in Fig. 2b, the two-grid approach is built on top of a staggered solution algorithm in which the flow and geomechanics subproblems are solved sequentially and iteratively using a fixed stress split iterative scheme [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,21,46,47,48,49,50,51,52]. The accuracy of the two-grid method has been demonstrated for the classical Mandel's problem [53,54] in our earlier work [21].…”