Xiuli Gai

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.

Models, methods and middleware for grid-enabled multiphysics oil reservoir management

Meeting the demands for energy entails a better understanding and characterization of the fundamental processes of reservoirs and of how human made objects affect these systems. The need to perform extensive reservoir studies for either uncertainty assessment or optimal exploitation plans brings up demands of computing power and data management in a more pervasive way. This work focuses on high performance numerical methods, tools and grid-enabled middleware systems for scalable and data-driven computations for multiphysics simulation and decision-making processes in integrated multiphase flow applications. The proposed suite of tools and systems consists of (1) a scalable reservoir simulator, (2) novel stochastic optimization algorithms, (3) decentralized autonomic grid middleware tools, and (4) middleware systems for large-scale data storage, querying, and retrieval. The aforementioned components offer enormous potential for performing data-driven studies and efficient execution of complex, large-scale reservoir models in a collaborative environment.

A Timestepping Scheme for Coupled Reservoir Flow and Geomechanics on Nonmatching Grids

This paper presents numerical techniques for coupled simulations with different time scales and space discretizations for reservoir flow and geomechanics. We use an explicitly coupled approach together with an iterative coupling to increase stability and reduce time discretization error. An error indicator is proposed to determine when displacement must be updated and whether the explicit or iterative coupling technique is required. Under this setting, one geomechanics calculation is performed for several reservoir flow steps. For time steps without geomechanics updates linear extrapolated pore volume is used for porous flow calculations. The resulting algorithm is computationally more efficient than the iterative coupling, and it is more stable and accurate than the loosely coupled technique. In the event that different meshes are used for the reservoir flow and geomechanics models, special treatments are required for the integration of the coupling terms over each element. To avoid complex 3D grid intersection calculations we propose to divide an element into a number of subelements and apply the midpoint integration rule over each subelement. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method for coupled simulations with different time and space discretizations.

Upscaling of Hydraulic Properties of Fractured Porous Media: Full Permeability Tensor and Continuum Scale Simulations

The simulation of flow and transport phenomena in fractured media is a challenging problem. Despite existing advances in computer capabilities, the fact that fractures can occur over a wide range of scales within porous media compromises the development of detailed flow simulations. Current discrete approaches are limited to systems that contain a small number of fractures. Alternatively, continuum approaches require the input of effective parameters that must be obtained as accurately as possible, based on the actual fracture network or its statistical description. In this work, a novel method based on the utilization of the Delta-Y transformation is introduced for obtaining the effective permeability tensor of a 2D fracture network. This approach entails a detailed description of the fracture network, where each fracture is represented as a segment with a given length, orientation and permeability value. A fine rectangular grid is then superimposed on the network, and the fractures are discretized so that each one of them is represented as a connected sequence of bonds on the grid with a hydraulic conductivity proportional to the ratio of effective permeability over fracture discretization length. The next step consists of the selection of a coarser rectangular grid on which the continuum simulation is performed. In order to obtain the permeability tensor for each one of the resulting blocks, the Delta-Y method is used. Finally, the resulting continuum permeability tensor is used to simulate the steady-state flow problem, and the results are compared with the actual flow pattern yielded by the fracture network simulation. The results obtained with both methods follow a similar flux pattern across the reservoir system. This shows that the proposed approach allows for efficient perform upscaling of hydraulic properties by honoring both the underlying physics and details of fracture network connectivity.

Streamline Tracing on Unstructured Grids

1. Order interior faces (or interior edges for 2D). 2. Compute the local conservation residual for each element. 3. For each face, compute velocity correction by using the local residual balancing process between the elements sharing the face; update velocity data and local conservation residuals. 4. Check convergence: if true, go to step 5; otherwise, go to step 3. 5. Extend the velocity into element interiors: a) If only the zeroth-order compatibility is needed, interpolate to obtain the velocity defined over the entire domain; b) If the rth-order compatibility is desired, apply a local high-order mixed finite element method for each individual element using the above corrected flux as boundary conditions. Single-phase flow with tracer injected on the left

An Analysis of Flow-simulation Scales And Seismic Response

Accurate integration of flow simulation and seismic modeling is one of the cornerstones of reliable time-lapse (4D) seismic monitoring. However, the question which scales flow simulations need to resolve to accurately capture reservoir changes during production and whether these scales are resolvable in seismic data is an open one. The answer impacts computational costs and our ability to predict reservoir changes from seismic observations. A sensitivity study is performed to determine the main seismic features due to pressure and saturation changes in an oil-gas reservoir during production. Numerical experiments show that saturation fronts can be effectively tracked at different flow and seismic resolution levels.

Weakly-Singular Integral Equations for Steady State Flow in Anisotropic Porous Media Containing Discontinuity Surfaces

In this paper, we present the development of the weakly-singular, weak-form fluid pressure and fluid flux integral equations for steady state Darcy’s flow in porous media. The integral equation for fluid flux is required for the treatment of flow in a domain which contains surfaces of discontinuities (e.g. cracks and impermeable surfaces), since the pressure integral equation contains insufficient information about the fluid flux on the surface of discontinuity. In this work, a systematic technique has been established to regularize the conventional fluid pressure and fluid flux integral equations in which the pressure equation contains a Cauchy singular kernel and the fluid flux equation contains both Cauchy and strongly-singular kernels. The key step in the regularization procedure is to construct a special decomposition for the fluid velocity fundamental solution and the strongly-singular kernel such that it is well-suited for performing an integration by parts via Stokes’ theorem. These decompositions involve weakly-singular kernels where their explicit form can be constructed, for general anisotropic permeability tensors, by the integral transform method. The resulting integral equations possess several features: they contain only weakly-singular kernels of order 1/r; their validity requires only that the pressure boundary data is continuous; and they are applicable for modeling fluid flow in porous media with a general anisotropic permeability tensor. A suitable combination of these weakly-singular, weak-form integral equations gives rise to a symmetric weak-form integral equation governing the boundary valued problem, thereby forming a basis for the weakly-singular, symmetric Galerkin boundary element method (SGBEM). As a consequence of that the integral equations are weakly-singular, the SGBEM allows standard C° elements to be employed everywhere in the discretization.Copyright