Susan E. Minkoff

Coupled fluid flow and geomechanical deformation modeling

Accurate prediction of reservoir production in structurally weak geologic areas requires both mechanical deformation and fluid flow modeling. Loose staggered-in-time coupling of two independent flow and mechanics simulators captures much of the complex physics at a substantially reduced cost. Two 3-D finite element simulators—Integrated Parallel Accurate Reservoir Simulator (IPARS) for flow and JAS3D for mechanics—together model multiphase fluid flow in reservoir rocks undergoing deformation ranging from linear elasticity to large, nonlinear inelastic compaction. The loose coupling algorithm uses a highlevel driver to call the flow simulator for a set of time steps with fixed reservoir properties. Pore pressures from flow are used as loads for the geomechanics code in the determination of stresses, strains, and displacements. The mechanics-derived strain is used to calculate changes to the reservoir parameters (porosity and permeability) for the next set of flow time steps. Mass is conserved in the coupled code despite dynamically changing reservoir parameters via a modification to the Newton system for the flow equations, and an approximate rock compressibility becomes a useful preconditioner to help with convergence of the modified flow equations. Two numerical experiments illustrate the accuracy of the coupled code. The first example is a quarterfive-spot waterflood undergoing poroelastic deformation, which is validated against a fully coupled simulator. Vertical displacements at the well locations match to within 10%. Moreover, experimentation shows that 13 mechanics time steps (taken over the course of 5 years of simulation time) were sufficient to achieve this result (a substantial cost savings over full coupling in which both the mechanics and flow equations must be solved at each time step). The second numerical example is based on real data from the Belridge Field in California, which illustrates one of the complex plastic constitutive relationships available in the coupled code. The results mimic behavior which was observed in the field. The coupled code serves as a prototype for loosely coupling together any two preexisting simulators modeling diverse physics. This technique produces a coupled code relatively quickly and inexpensively and has the advantage of accurately modeling complex nonlinear phenomena often

A Comparison of Techniques for Coupling Porous Flow and Geomechanics

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 threephase 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.

Coupled geomechanics and flow simulation for time‐lapse seismic modeling

To accurately predict production in compactible reservoirs, we must use coupled models of fluid flow and mechanical deformation. Staggered‐in‐time loose coupling of flow and deformation via a high‐level numerical interface that repeatedly calls first flow and then mechanics allows us to leverage the decades of work put into individual flow and mechanics simulators while still capturing realistic coupled physics. These two processes are often naturally modeled using different time stepping schemes and different spatial grids—flow should only model the reservoir, whereas mechanics requires a grid that extends to the earths surface for overburden loading and may extend further than the reservoir in the lateral directions. Although spatial and temporal variability between flow and mechanics can be difficult to accommodate with full coupling, it is easily handled via loose coupling. We calculate the total stress by adding pore pressures to the effective rock stress. In turn, changes in volume strain induce up...

An Operator-based Approach To Upscaliiig ThePressure Equation

Permeability and porosity parameters of a porous medium are known only in a statistical sense. For risk assessment, one must perform multiple flow simulations of a single site, varying these input parameters. Because multiple simulations of large sites are computationally prohibitive, upscaling from fine to coarse scales is necessary. Traditional upscaling techniques determine a new effective or upscaled permeability field defined on a coarser scale, which is then used in a standard coarse grid discretization operator. We develop here a method of determining a new coarse grid discretization operator that provides an upscaled solution but bypasses the determination of effective permeability and porosity fields. The method has two steps. We first solve for fine scale flow information internal to each coarse grid cell. Because the problems are small, this step is relatively fast. Then we determine a modified coarse grid operator for solving the upscaled problem that includes the fine scale flow information from the first step. The method is developed for single-phase flow in the context of the mixed finite element method; therefore, the method is locally mass conservative. Unlike traditional upscaling methods (such as homogenization) we do not impose arbitrary boundary conditions on the coarse grid. We present comparisons of our method with the harmonic average permeability upscaling technique.

Operator upscaling for the acoustic wave equation

Modeling of wave propagation in a heterogeneous medium requires input data that varies on many different spatial and temporal scales. Operator-based upscaling allows us to capture the effect of the fine scales on a coarser domain without solving the full fine-scale problem. The method applied to the constant density, variable sound velocity acoustic wave equation consists of two stages. First, we solve small independent problems for approximate fine-scale information internal to each coarse block. Then we use these subgrid solutions to define an upscaled operator on the coarse grid. The fine-grid velocity field is used throughout the process (i.e.,no averaging of input fields is required). An equivalence between the variational form of the problem and a staggered finite-difference scheme allows us to use finite differences to solve the subgrid wave propagation problems. Due to the homogeneous Neumann boundary conditions imposed on each coarse block, the subgrid problems decouple, which leads to the natura...

A Matrix Analysis of Operator-Based Upscaling for the Wave Equation

Scientists and engineers who wish to understand the earths subsurface are faced with a daunting challenge. Features of interest range from the microscale (centimeters) to the macroscale (hundreds of kilometers). It is unlikely that computational power limitations will ever allow modeling of this level of detail. Numerical upscaling is one technique intended to reduce this computational burden. The operator-based algorithm (developed originally for elliptic flow problems) is modified for the acoustic wave equation. With the wave equation written as a first-order system in space, we solve for pressure and its gradient (acceleration). The upscaling technique relies on decomposing the solution space into coarse and fine components. Operator-based upscaling applied to the acoustic wave equation proceeds in two steps. Step one involves solving for fine-grid features internal to coarse blocks. This stage can be solved quickly via a well-chosen set of coarse-grid boundary conditions. Each coarse problem is solved independently of its neighbors. In step two we augment the coarse-scale problem via this internal subgrid information. Unfortunately, the complexity of the numerical upscaling algorithm has always obscured the physical meaning of the resulting solution. Via a detailed matrix analysis, the coarse-scale acceleration is shown to be the solution of the original constitutive equation with input density field corresponding to an averaged density along coarse block edges. The pressure equation corresponds to the standard acoustic wave equation at nodes internal to coarse blocks. However, along coarse cell boundaries, the upscaled solution solves a modified wave equation which includes a mixed second-derivative term.

Spatial Parallelism of a 3D Finite Difference Velocity-Stress Elastic Wave Propagation Code

In a three-dimensional isotropic elastic earth, the wave equation solution consists of three velocity components and six stresses. We discretize the partial derivatives using second order in time and fourth order in space staggered finite difference operators. The parallel implementation uses the message passing interface library for platform portability and a spatial decomposition for efficiency. Most of the communication in the code consists of passing subdomain face information to neighboring processors. When the parallel communication is balanced against computation by allocating subdomains of reasonable size, we observe excellent scaled speedup. Allocating subdomains of size

Full waveform inversion of marine reflection data in the plane‐wave domain

25 \times 25 \times 25

A Two-Scale Solution Algorithm for the Elastic Wave Equation

on each node, we achieve efficiencies of 94\% on 128 processors of an Intel\linebreak Paragon.

Staggered In Time Coupling of Reservoir Flow Simulation and Geomechanical Deformation: Step 1 - One-Way Coupling

Full waveform inversion of a p‐τ marine data set from the Gulf of Mexico provides estimates of the long‐wavelength P‐wave background velocity, anisotropic seismic source, and three high‐frequency elastic parameter reflectivities that explain 70% of the total seismic data and 90% of the data in an interval around the gas sand target. The forward simulator is based on a plane‐wave viscoelastic model for P‐wave propagation and primary reflections in a layered earth. Differential semblance optimization, a variant of output least‐squares inversion, successfully estimates the nonlinear P‐wave background velocity and linear reflectivities. Once an accurate velocity is estimated, output least‐squares inversion reestimates the reflectivities and an anisotropic seismic source simultaneously. The viscoelastic model predicts the amplitude‐versus‐angle trend in the data more accurately than does an elastic model. Simultaneous inversion for reflectivities and source explains substantially more of the actual data than d...