Chen, Hanming (China University of Petroleum, Beijing) | Zhou, Hui (China University of Petroleum, Beijing) | Zhang, Qingchen (China University of Petroleum, Beijing) | Zhang, Qi (China University of Petroleum, Beijing)
Two staggered-grid finite-difference (SGFD) methods with fourth- and sixth-order accuracy in time have been developed recently based on two new SGFD stencils. The SGFD coefficients are determined by Taylor-series expansion (TE), which is accurate only nearby zero wavenumber. We adopt the new two SGFD stencils and optimize the SGFD coefficients by minimizing the errors between the wavenumber responses of the SGFD operators and the first-order k (wavenumber)-space operator in a least-squares (LS) sense. We solve the LS problems by performing weighted pseudo-inverse of nonsquare matrices to obtain the SGFD coefficients, and to yield two LS based SGFD methods. Dispersion analysis and numerical examples demonstrate that our LS based SGFD methods can preserve the original fourth- and sixth-order temporal accuracy and achieve higher spatial accuracy than the existing TE based time-space domain SGFD methods.
The staggered-grid finite-difference (SGFD) (Virieux, 1984) method has been widely used in seismic wave propagation modeling. Most of the SGFD applications adopt the traditional (2M, 2) scheme, which uses 2M-order Taylorseries expansion (TE) based FD operator to discretize spatial derivatives, and 2nd-order TE based FD operator to discretize temporal derivative. Although high-order spatial accuracy can be achieved by using a long stencil length, the temporal accuracy is only second-order. When a coarse time step is used, the traditional scheme suffers from obvious temporal dispersion during long time wave propagation.
Recently, Tan and Huang (2014a) propose two new SGFD methods with fourth-order and sixth-order accuracy in time respectively by incorporating a few of off-axial grid points into the standard SGFD stencil. The two methods are denoted as (2M, 4) and (2M, 6). The FD coefficients are determined in the time-space domain using TE approach. Althouth high-order temporal accuracy has been achieved, the TE based (2M, 4) and (2M, 6) methods still suffer from obvious spatial disperion when a large grid size or a short stencil length is adopted. Tan and Huang (2014b) continue to improve the spatial accuracy by using a nonlinear optimization to seek the optimal FD coefficients. However, the optimization requires repeated iterations, and the procedure may be time-consuming.
Wang, Yufeng (State Key Laboratory of Petroleum Resources and Prospecting, CNPC Key Lab of Geophysical Exploration, China University of Petroleum) | Zhou, Hui (State Key Laboratory of Petroleum Resources and Prospecting, CNPC Key Lab of Geophysical Exploration, China University of Petroleum) | Li, Qingqing (State Key Laboratory of Petroleum Resources and Prospecting, CNPC Key Lab of Geophysical Exploration, China University of Petroleum) | Chen, Hanming (State Key Laboratory of Petroleum Resources and Prospecting, CNPC Key Lab of Geophysical Exploration, China University of Petroleum) | Gan, Shuwei (State Key Laboratory of Petroleum Resources and Prospecting, CNPC Key Lab of Geophysical Exploration, China University of Petroleum) | Chen, Yangkang (The Unversity of Texas at Austin)
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.
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.
In this abstract, a new simple approach is presented to eliminate the harmonic noise in correlated vibroseis data. This technique is based on decomposition of the ground force signal into fundamental and harmonic components. Subsequently, we calculate the inverse-correlation operator and obtain the inverse-correlated data. Then, all the harmonics can be filtered out one by one after the inverse-correlated data cross-correlating with corresponding harmonic components. This method has some advantages as follows: First, it is simpler and more time-saving. Second, it can process truncated and incomplete correlated vibroseis seismic data. Moreover, this method only requires the ground force signal and shot location for each shot. Especially, the significant contribution of this procedure is a considerable reduction in the harmonics without any alteration of the desired signals. Both synthetic and field data are used to test its validity.
An efficient two-stage algebraic multiscale solver (TAMS) that converges to the fine-scale solution is described. The first (global) stage is a multiscale solution obtained algebraically for the given fine-scale problem. In the second stage, a local preconditioner, such as the Block ILU (BILU) or the Additive Schwarz (AS) method, is used. Spectral analysis shows that the multiscale solution step captures the low-frequency parts of the error spectrum quite well, while the local preconditioner represents the high-frequency components accurately. Combining the two stages in an iterative scheme results in efficient treatment of all the error components associated with the fine-scale problem. TAMS is shown to converge to the reference fine-scale solution. Moreover, the eigenvalues of the TAMS iteration matrix show significant clustering, which is favorable for Krylov-based methods. Accurate solution of the nonlinear saturation equations (i.e., transport problem) requires having locally conservative velocity fields. TAMS guarantees local mass conservation by concluding the iterations with a multiscale finite-volume step. We demonstrate the performance of TAMS using several test cases with strong permeability heterogeneity and large-grid aspect ratios. Different choices in the TAMS algorithm are investigated, including the Galerkin and finite-volume restriction operators, as well as the BILU and AS preconditioners for the second stage. TAMS for the elliptic flow problem is comparable to state-of-the-art algebraic multigrid methods, which are in wide use. Moreover, the computational time of TAMS grows nearly linearly with problem size.
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.
Previous research on multiscale methods for subsurface flow aims to obtain an efficient multiscale solution to the fine-scale problems. Such multiscale solution is usually a good approximation to the fine-scale problem. However, it has been reported that the multiscale solution deteriorates for high aspect ratios and channelized structures of permeability. Moreover, the multiscale solution does not converge to the fine-scale solution unless some special techniques are used.
In this paper, we propose an efficient two-stage algebraic multiscale (TAMS) method that converges to the fine-scale solution. The TAMS method consists of two stages, namely global and local stages. In the global stage, a multiscale solution is obtained purely algebraically from the fine-scale matrix. In the local stage, a local solution is constructed from a local preconditioner such as Block ILU(0) (BILU) or Additive Schwarz (AS) method. Spectral analysis shows that the multiscale solution step captures the low-frequency spectra in the original matrix very well and when combined with a local preconditioner that represents the high-frequency spectra, the full spectra can be well approximated. Thus the TAMS is guaranteed to converge to the fine-scale solution. Moreover, the spectra of the TAMS method tend to cluster together, which is a favorable property for Krylov subspace methods to converge fast. We also show that local mass conservation can be preserved if a multiscale solution step with the finite-volume restriction operator is applied before the iterative procedure exits. This allows for the TAMS method to be used to build efficient approximate solutions for multiphase flow problems that need to solve transport equations.
We test the numerical performance of the TAMS method using several challenging large-scale problems (fine-scale grid in the magnitude of one million) with complex heterogeneous structures and high aspect ratios. Different choices in the TAMS algorithm are employed, including the Galerkin or finite-volume type of restriction operator, BILU or AS preconditioner for the second stage, and the size of blocks for BILU and AS. The performance of TAMS is comparable or superior to the state-of-the-art algebraic multigrid (AMG) preconditioner when some optimal choice in the TAMS method is adopted. Moreover, the convergence of the TAMS method is insensitive to problem sizes, and the CPU time is almost linear to problem sizes. These indicate the TAMS method is efficient and
robust for large-scale problems.