Results

Summary

We apply the unsplit convolutional perfectly matched layer (CPML) absorbing boundary condition (ABC) to the viscoacoustic wave, which is derived from Kjartansson’s constant-Q model, with second-order spatial derivatives and fractional time derivatives, to annihilate spurious reflections from near-grazing incidence waves in the time domain. Computationally expensive temporal convolution in the unsplit CPML formulation are resolved by an effective recursive convolution update strategy which calculates time integration with the trapezoidal approximation, while the fractional time derivatives are computed with the Grünwald- Letnikov (GL) approximations. We verify the results by comparison with the 2D analytical solution obtained from wave propagation in homogeneous Pierre Shale.

Introduction

Perfectly matched layer (PML) absorbing boundary condition was introduced by Bérenger (1994) for the numerical simulations of electromagnetic waves in an unbounded medium. It can theoretically absorb the incident waves at the interface with the elastic volume, regardless of their incidence angle or frequency. However, the performance degrades upon the finite-difference time domain (FDTD) discretization, especially in the case of grazing incidence (Roden and Gedney, 2000; Bérenger, 2002a, 2002b). To deal with this problem, Kuzuoglu and Mittra (1996) proposed a general complex frequency shifted (CFS) method, in which a Butterworth-type filter is implemented in the layer. This approach is also known as convolutional PML (CPML) or complex frequency shifted PML (CFS-PML) (Bérenger, 2002a, 2002b), which has been proved to be more effective in absorbing the propagating wave modes at grazing incidence than the classical PMLs (Roden and Gedney, 2000; Komatitsch and Martin, 2007; Martin and Komatitsch, 2009; Drossaert and Giannopoulos, 2007a, 2007b).

Generally, the unsplit CPML has been applied to the wave equation recast as a first-order system in velocity and stress (Komatitsch and Martin, 2007; Martin and Komatitsch, 2009; Chen et. al., 2014). However, it was rarely used in numerical schemes based on the wave equation written as a second-order system in displacement. This form of wave equation is commonly used in finite-element methods (FEM), the spectral-element method (SEM) and some finite-difference methods. Several unsplit CPMLs have already been applied to the second-order wave equations (Li and Matar, 2010; René Matzen, 2011). Li and Matar (2010) presented an unsplit CPML for the second-order wave equation that contains auxiliary memory variables to avoid the convolution operators. Matzen (2011) developed a novel CPML formulation based on the second-order wave equation with displacements as the only unknowns, which is implemented by slightly modifying the existing displacement-based finite element framework.

Summary

Recent advances in multiscale methods have shown great promise in modeling multiphase flow in highly detailed heterogeneous domains. Existing multiscale methods, however, solve for the flow field (pressure and total velocity) only. Once the fine-scale flow field is reconstructed, the saturation equations are solved on the fine scale. With the efficiency in dealing with the flow equations greatly improved by multiscale formulations, solving the saturation equations on the fine scale becomes the relatively more expensive part. In this paper, we describe an adaptive multiscale finite-volume (MSFV) formulation for nonlinear transport (saturation) equations. A general algebraic multiscale formulation consistent with the operator-based framework proposed by Zhou and Tchelepi (SPE Journal, June 2008, pages 267-273) is presented. Thus, the flow and transport equations are solved in a unified multiscale framework. Two types of multiscale operators--namely, restriction and prolongation--are used to construct the multiscale saturation solution. The restriction operator is defined as the sum of the fine-scale transport equations in a coarse gridblock. Three adaptive prolongation operators are defined according to the local saturation history at a particular coarse block. The three operators have different computational complexities, and they are used adaptively in the course of a simulation run. When properly used, they yield excellent computational efficiency while preserving accuracy. This adaptive multiscale formulation has been tested using several challenging problems with strong heterogeneity, large buoyancy effects, and changes in the well operating conditions (e.g., switching injectors and producers during simulation). The results demonstrate that adaptive multiscale transport calculations are in excellent agreement with fine-scale reference solutions, but at a much lower computational cost.