The velocity-stress finite-difference method with a rotated staggered grid applied to seismic wave propagation in a fractured medium

To enable a mathematical description, geologic fractures are considered as infinitely thin planes embedded in a homogeneous medium. These fracture structures satisfy linear slip boundary conditions, namely, a discontinuous displacement and continuous stress. The general finite-difference (FD) method described by the elastic wave equations has challenges when attempting to simulate the propagation of waves at the fracture interface. The FD method expressed by velocity-stress variables with the explicit application of boundary conditions at the fracture interface facilitates the simulation of wave propagation in fractured discontinuous media that are described by elastic wave equations and linear slip interface conditions. We have developed a new FD scheme for horizontal and vertical fracture media. In this scheme, a fictitious grid is introduced to describe the discontinuous velocity at the fracture interface and a rotated staggered grid is used to accurately indicate the location of the fracture. The new FD scheme satisfies nonwelded contact boundary conditions, unlike traditional approaches. Numerical simulations in different fracture media indicate that our scheme is accurate. The results demonstrate that the reflection coefficient of the fractured interface varies with the incident angle, wavelet frequency, and normal and tangential fracture compliances. Our scheme and conclusions from this study will be useful in assessing the properties of fractures, enabling the proper delineation of fractured reservoirs.

Fracture-Cross-Flow Equilibrium in Compositional Two-Phase Reservoir Simulation

Summary Compositional two-phase flow in fractured media has wide applications, including carbon dioxide (CO2) injection in the subsurface for improved oil recovery and for CO2 sequestration. In a recent work, we used the fracture-crossflow-equilibrium (FCFE) approach in single-phase compressible flow to simulate fractured reservoirs. In this work, we apply the same concept in compositional two-phase flow and show that we can compute all details of two-phase flow in fractured media with a central-processing-unit (CPU) time comparable with that of homogeneous media. Such a high computational efficiency is dependent on the concept of FCFE, and the implicit solution of the transport equations in the fractures to avoid the Courant-Freidricks-Levy (CFL) condition in the small fracture elements. The implicit solution of two-phase compositional flow in fractures has some challenges that do not appear in single-phase flow. The complexities arise from the upstreaming of the derivatives of the molar concentration of component i in phase α(cα,i) with respect to the total molar concentration (ci) when several fractures intersect at one interface. In addition, because of gravity, countercurrent flow may develop, which adds complexity when using the FCFE concept. We overcome these complexities by providing an upstreaming technique at the fracture/fracture interface and the matrix/fracture interface. We calculate various derivatives at constant volume V and temperature T by performing flash calculations in the fracture elements and the matrix domain to capture the discontinuity at the matrix/fracture interface. We demonstrate in various examples the efficiency and accuracy of the proposed algorithm in problems of various degrees of complexity in eight-component mixtures. In one example with 4,300 elements (1,100 fracture elements), the CPU time to 1 pore volume injection (PVI) is approximately 3 hours. Without the fractures, the CPU time is 2 hours and 28 minutes. In another example with 7,200 elements (1,200 fracture elements), the CPU time is 4 hours and 8 minutes; without fractures in homogeneous media, the CPU time is 2 hours and 53 minutes.