Summary This paper compares three techniques for coupling multiphase porous flow and geomechanics. Sample simulations are presented to highlight the similarities and differences in the techniques. One technique uses an explicit algorithm to couple porous flow and displacements in which flow calculations are performed every timestep and displacements are calculated only during selected timesteps. A second technique uses an iteratively coupled algorithm in which flow calculations and displacement calculations are performed sequentially for the nonlinear iterations during each timestep. The third technique uses a fully coupled approach in which the program's linear solver must solve simultaneously for fluid flow variables and displacement variables. The techniques for coupling porous flow with displacements are described and comparison problems are presented for single-phase and three-phase flow problems involving poroelastic deformations. All problems in this paper are described in detail, so the results presented here may be used for comparison with other geomechanical/porous-flow simulators. Introduction Many applications in the petroleum industry require both an understanding of the porous flow of reservoir fluids and an understanding of reservoir stresses and displacements. Examples of such processes include subsidence, compaction drive, wellbore stability, sand production, cavity generation, high-pressure breakdown, well surging, thermal fracturing, fault activation, and reservoir failure involving pore collapse or solids disposal. It would be useful to compare porous flow/geomechanics techniques for all of these processes, because some of these processes involve a stronger coupling between porous flow and geomechanics than others. However, this paper looks at a subset of these processes and compares three coupling techniques for problems involving subsidence and compaction drive. All of the sample problems presented in this paper assume that the reservoir absolute permeabilities are constant during a run. Displacements influence fluid flow through the calculation of pore volumes, and fluid pressures enter the displacement calculations through the poroelastic constitutive equations. Several authors have presented formulations for modeling poroelastic, multiphase flow. Settari and Walters (1999) discuss the different methods that have been used to combine poroelastic calculations with porous flow calculations. They categorize these different methods of coupling poroelastic calculations with porous flow calculations as decoupled (Minkoff et al. 1999a), explicitly coupled, iteratively coupled, and fully coupled. The techniques discussed in this paper are explicitly coupled, iteratively coupled, and fully coupled.

In coupled geomechanics and reservoir modeling, the finite element discretization of the force balance equation leads to very large linear systems, whose solution is both time and memory consuming. ICCG (Incomplete Cholesky Factorized Conjugate Gradient) is a popular technique for solving for displacements, but the technique is limited to about 60,000 nodal points on desktop machines. Most large 3D field scale problems will have to be run on parallel machines. In this paper, we present a reduced-communication, super coarsening multigrid method that can be combined with other domain decomposition-based preconditioner to achieve faster convergence with high parallel scalability. A preliminary test case of 1.5 million grid blocks with up to 59 processors shows a parallel efficiency of above 90%. Introduction The modeling of fluid-structure interactions is of growing importance for both energy and environmental applications. Due to complexities, nonlinearities, phase behavior, and the number of partial differential equations required to describe a coupled system of poro-elasticity and/or poro-plasticity with multiphase flow, extending a conventional reservoir model to a coupled fluid-flow and geomechanical model is not trivial, though considerable success has been achieved in recent years. Several authors1,2,3,4,5 have presented mathematical and numerical formulations for the coupled equations. Different coupling techniques have been investigated6,7 that are applicable to existing reservoir flow models and one's choice normally depends on speed, accuracy, or ease of implementation. Comparisons have been done between coupled and uncoupled simulations, and between different coupling techniques for accuracy and efficiency. Recently, stability issues have also been discussed concerning oscillations in low permeability zones. Individual models are becoming more sophisticated, and coupling methods are becoming more accurate and more stable. However, the computational bottleneck of solving the large linear system still remains. The finite element formulation of the geomechanics model has a 27-point stencil compared to the standard 7-point stencil in the finite difference formulation of multiphase flow equations. More than 70 percent of the total CPU time for a coupled simulation is spent in the geomechanics model solving the system of linear equations. So it is very important to develop efficient linear solvers to reduce the computational cost without losing scalability on parallel computers, so that coupled analyses can be economically and numerically feasible for practical field applications. Thomas et al.8 describe a coupled procedure running on multiple processors, but the parallel efficiency with 16 processors is less than 50 percent. In this work, we present a parallel poroelastic model combined with a multiphase flow model using the iteratively coupled technique. The linear solver we employ is PCG preconditioned by Incomplete Cholesky Factorization (IC) or block Jacobi. Faster convergence and higher scalability is achieved by calling a super coarsening multigrid routine to make further global corrections after each preconditioning step. The rest of the paper is organized as follows. First the governing flow and deformation equations are described briefly. Instability issues are also briefly discussed. Then the super coarsening multigird preconditioner and its parallel implementation are presented. Results of numerical experiments conducted on a 64-node Beowulf PC cluster are shown to demonstrate the efficiency and parallel scalability of the combined IC (or block Jacobi) and multigrid preconditioned BiCGSTAB (Biconjugate Gradient Stablized) method. Geomechanics and Multiphase Flow Equations Terzaghi first analyzed the fluid-flow-stress coupling equations in 1925 as a 1D consolidation problem. Later, Biot9 extended the theory to a more generalized three-dimensional case, based on a linear stress-strain relation and a single-phase fluid flow. Here we present an extension of Biot's equations for three-phase immiscible and isothermal flow.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractThis paper compares three techniques for coupling multiphase porous flow and geomechanics. Sample simulations are presented to highlight the similarities and differences in the techniques. One technique uses an explicit algorithm to couple porous flow and displacements where flow calculations are performed every time step and displacements are calculated only during selected time steps. A second technique uses an iteratively coupled algorithm where flow calculations and displacement calculations are performed sequentially for nonlinear iterations during time steps. The third technique uses a fully coupled approach where the program's linear solver must solve simultaneously for fluid flow variables and displacement variables. The techniques for coupling porous flow with displacements are described, and comparison problems are presented for single-phase and three-phase flow problems involving poroelastic deformations. All problems in this paper are described in detail so the results presented here may be used for comparison with other geomechanical/porous flow simulators.

This paper presents fracture simulations of multiple, interacting, non-planar fractures in three dimensions. The paper provides a short description of the mathematical formulation of the model but the primary focus is on fracture propagation examples and illustrations of how multiple hydraulic fractures interact in three dimensions. The examples which are presented are intended to provide insight into how the number of growing fractures affects fracture shapes, how changes in fluid viscosity can cause fractures to grow together or grow apart, and how limited-entry at the perforations affects the propagation of interacting fractures. The fracture simulator discussed in this paper models the simultaneous growth of non-planar hydraulic fractures in a three-dimensional linear elastic media, and it can also incorporate stress shadows from hydraulic fractures created during earlier fracture stages. The program uses a symmetric Galerkin boundary element method to model fracture shapes and fracture growth, while flow in the fractures is modeled as power-law fluid flow in arbitrary curved channels. The program uses an effective mode-I stress intensity factor to determine which portions of each fracture will propagate and employs mixed-mode stress intensity factors, K I and K II , to determine propagation directions.

This paper describes a three-dimensional (3D), three-phase, black-oil model being used to simulate naturally fractured reservoirs. The program is fully implicit and can perform single-porosity, dual-porosity, or dual-permeability computations. Sample simulations are presented to illustrate the differences between the three computational techniques. Single-porosity recoveries are much larger than the recoveries predicted by the dual techniques. The dual-permeability, primary-depletion recoveries are very similar to the dual-porosity, primary-depletion recoveries, while the dual-permeability waterflood recoveries are significantly larger than the dual-porosity waterflood recoveries.Pseudocapillary pressures are generated from fine-grid, single-matrix block studies. The pseudocapillary pressures are then used in the dual-porosity simulations to account for fluid distributions in the matrix and fracture systems.