Heterogeneous Anisotropic Media Research Articles

Comparative analysis of double and single porosity effects on SH-wave induced vibrations in periodic porous lattices

The manuscript presents a novel study comparing the impact of single and double porosity on horizontally polarized shear wave (SH-wave) propagation in corrugated elastic void materials. The considered mathematical model comprises two cases; the first one depicts SH-wave propagation through a void porous layer having creased boundaries and resting over a heterogeneous anisotropic fluid-saturated fractured porous half-space, whereas according to the second case, heterogeneous anisotropic fluid-saturated porous semi-infinite medium without fractures has been considered. In both cases, rigidity and density of the half-space vary quadratically with depth. The separable variable method is used to attain the complex frequency equation for each case that leads to two different dispersion relations associated with two distinct wave fronts. The first wave front depends on the void parameters, whereas the second wave front defines the propagation of SH-waves in an elastic layer without void pores. In each case, the complex dispersion relation has been separated into two equations that illustrate the dispersion and attenuation properties of SH-waves. Using the variations in the inhomogeneity, position, fluctuation, flatness, total porosity, and anisotropy parameters, case I and case II have been compared in each graphical execution. In addition, the surface response of shear stress and displacement have been implemented graphically.

INTEGRAL TRANSFORM SOLUTION OF HEAT CONDUCTION IN ANISOTROPIC HETEROGENEOUS MEDIA

The generalized integral transform technique (GITT) is advanced to deal with conduction heat transfer in anisotropic heterogeneous media. A formal solution for exact integral transformation of conduction in anisotropic media is extended to account for heterogeneities expressed as space variable equation coefficients and source terms. The proposed eigenfunction expansion is based on biorthogonal eigenvalue problems, which results in an exact integral transformation for a class of linear problems and in a coupled transformed ordinary differential system for nonlinear situations. An algorithm is proposed for the associated eigenvalue problems, also handled through the GITT, by considering simpler biorthogonal eigenvalue problems of known analytical solution, leading to transformed algebraic eigenvalue problems. A single domain reformulation strategy is adopted to merge the information from multiple regions and materials, either isotropic or anisotropic, into one single diffusion equation. A two-dimensional transient test case is considered that presents an abrupt transition between isotropic and anisotropic materials yielding a marked change in thermal behavior in a defined region of interest formed by the anisotropic inclusion. Convergence behavior of the integral transform solution is illustrated, and the fully converged results are employed as a benchmark to inspect the accuracy of a commercial finite element code for automatically defined mesh refinement levels.

Wavefield simulation with the discontinuous Galerkin method for a poroelastic wave equation in triple-porosity media

In this paper, we derive a new poroelastic wave equation in triple-porosity media and develop a weighted Runge-Kutta (RK) discontinuous Galerkin method (DGM) for solving it. Based on Biot’s theory and Lagrangian formulas, we obtain 3D Biot’s equations in a heterogeneous anisotropic triple-porosity medium. We also summarize poroelastic wave equations in single- and double-porosity media. The traditional two-phase theory is a special case of the porous theory. Compared with single- or dual-porosity wave equations, our triple-porosity wave equation generates more accurate wavefield information. The isotropic and anisotropic cases are considered. Subsequently, we formulate the new poroelastic equation into a first-order hyperbolic conservation system, which is suitable to be solved by DGM. An optimized local Lax-Friedrichs flux and an implicit-weighted RK time discretization scheme are used for this computation. We use two types of mesh elements — quadrilateral and unstructured triangular elements. We find that there are two kinds of slow P waves, P1 and P2 waves, in double-porosity media, whereas three kinds of slow P waves, P1, P2, and P3 waves, exist in three-porosity media. We also study the analytical and numerical solutions of propagation velocities for different waves in isotropic media without dissipation using the Jacobian matrix of DGM and provide a comparison of field variables about three types of wave equations. Finally, we conduct a series of examples to quantitatively investigate the propagation properties of seismic waves in isotropic and anisotropic multiporosity media computed by DGM. The slow P wave in multiporosity media with dissipation decays rapidly, which also will lead to phase distortion. Numerical results verify the correctness and applicability of our proposed new equation and indicate that the weighted RK DGM is a stable and accurate algorithm to simulate wave propagation in poroelastic media.

Hydrogeological feasibility of mine water deep geological storage in Baotashan coarse sandstone: A case study in Ordos Basin

AbstractFor the sake of mine water drainage and sustainable groundwater protection, the new approach of mine water deep geological storage (MWDGS) is highly necessary to save water resources in the semi‐arid region of China. However, up to now, little academic research has been done on mine water geological storage. Given this situation, the hydrogeological feasibility of MWDGS was explored in Baotashan coarse sandstone (BCS) of Jurassic measure in Ordos Basin. The results show that the white‐gray BCS with a fragile skeleton of quartz (41.4%), feldspar (21.1%), and clay minerals (16.4%) provides the potential variable‐void for mine water; and its hydro‐chemical type of BCS aquifer is CO3‐Na and Cl‐Na. As the burial depth increases, the strong alkaline groundwater is in stagnant and poor recharge‐runoff‐discharge condition. The lab test shows that the pores whose diameter is over 10 μm could be treated as the main storage of mine water; and the effective porosity varies from 1.36% to 3.46%. When mine water is injected, the strong hydrodynamics of mine water storage would change the permeability significantly and about 0.201% soluble solids would be dissolved. Partial clay minerals obstruct the pores and induce the saturated phase of high permeability to evolve into steady phase of lower permeability. Under the condition of nonhydraulic fracturing during continuous storage, the heterogeneous anisotropic medium obtained by Transition PRObability GeoStatistics (TPROGS) shows that the capacity of BCS aquifer is 0.455 to 1.226 Mm3 for 1 km2 in the study area. The simulation shows that the groundwater mound in well‐scale and mine‐scale would be formed. The groundwater quality characteristics of “Three Zone” would occur around and gradually drop to approximate the original brine within 10 years. The hydrogeological feasibility reveals that this approach is useful for the well design and groundwater environment management during the mine water deep geological storage project in the Ordos basin.

Finite volume method for coupled subsurface flow problems, II: Poroelasticity

The article introduces a cell-centered finite-volume method for the Biot problem in heterogeneous anisotropic media, characterized with full-tensor properties. We derive the expression for the coupled flux and the interpolation method across the discontinuity of the properties. The obtained flux expression consists of a two-point part, a transversal part and an additional contribution due to gravity. The interpolation method is the generalization of the harmonic averaging point concept to coupled problems. The method is stable despite collocation of both pressure and displacement at cell centers due to eigensplitting of the matrix coefficients in the flux expression and upstream approximation. A general type of boundary condition is integrated without introduction of auxiliary degrees of freedom. Our flux discretization method is a realization of our more general concept of stable flux discretization for saddle-point systems with vector of several unknowns. We demonstrate the applicability of the method on a set of challenging numerical benchmarks.

General finite-volume framework for saddle-point problems of various physics

AbstractThis article is dedicated to the general finite-volume framework used to discretize and solve saddle-point problems of various physics. The framework applies the Ostrogradsky–Gauss theorem to transform a divergent part of the partial differential equation into a surface integral, approximated by the summation of vector fluxes over interfaces. The interface vector fluxes are reconstructed using the harmonic averaging point concept resulting in the unique vector flux even in a heterogeneous anisotropic medium. The vector flux is modified with the consideration of eigenvalues in matrix coefficients at vector unknowns to address both the hyperbolic and saddle-point problems, causing nonphysical oscillations and an inf-sup stability issue. We apply the framework to several problems of various physics, namely incompressible elasticity problem, incompressible Navier–Stokes, Brinkman–Hazen–Dupuit–Darcy, Biot, and Maxwell equations and explain several nuances of the application. Finally, we test the framework on simple analytical solutions.

A Fast Algorithm for Elastic Wave‐Mode Separation Using Deep Learning With Generative Adversarial Networks (GANs)

AbstractWave‐mode separation and wave‐vector decomposition aims to separate a full elastic wavefield into different wavefields corresponding to their respective wave mode. The wave mode separation process allows the handling of different wave modes independently, which is important to ensure the accuracy of migration. Most of the previously developed algorithms require the knowledge of polarization vectors that are calculated using the Christoffel equation in the wave‐number domain by using the anisotropic parameters of the mesh nodes. This process can be really expensive, especially for quasi‐differential operators. We propose to implement a framework using deep neural networks (DNNs) for elastic wave‐mode separation in a heterogeneous anisotropic medium. To avoid the necessity of solving the Christoffel equation at each spatial location of the medium, we develop an efficient algorithm using generative adversarial networks (GANs). In particular, we propose a wave‐mode decoupling workflow where we train the neural network using five to six time slices for a particular source location and test on all other time slices for that source location, as well as for all other source locations that are not a part of training. Numerical examples of increasing complexity show that the proposed approach provides an effective wave‐mode decomposition method for heterogeneous and strongly anisotropic media.

Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: transformation between Cartesian and ray-centred coordinates

SUMMARY Within the field of seismic modelling in anisotropic media, dynamic ray tracing is a powerful technique for computation of amplitude and phase properties of the high-frequency Green’s function. Dynamic ray tracing is based on solving a system of Hamilton–Jacobi perturbation equations, which may be expressed in different 3-D coordinate systems. We consider two particular coordinate systems; a Cartesian coordinate system with a fixed origin and a curvilinear ray-centred coordinate system associated with a reference ray. For each system we form the corresponding 6-D phase spaces, which encapsulate six degrees of freedom in the variation of position and momentum. The formulation of (conventional) dynamic ray tracing in ray-centred coordinates is based on specific knowledge of the first-order transformation between Cartesian and ray-centred phase-space perturbations. Such transformation can also be used for defining initial conditions for dynamic ray tracing in Cartesian coordinates and for obtaining the coefficients involved in two-point traveltime extrapolation. As a step towards extending dynamic ray tracing in ray-centred coordinates to higher orders we establish detailed information about the higher-order properties of the transformation between the Cartesian and ray-centred phase-space perturbations. By numerical examples, we (1) visualize the validity limits of the ray-centred coordinate system, (2) demonstrate the transformation of higher-order derivatives of traveltime from Cartesian to ray-centred coordinates and (3) address the stability of function value and derivatives of volumetric parameters in a higher-order representation of the subsurface model.

Locating microseismic events using multiplicative time reversal imaging based on decoupled wavefields in 2D VTI media: Theoretical and synthetic cases studies

Estimation of microseismic event location is of great significance in unconventional resources development and exploitation with its advantage of imaging the fractures induced by hydraulic fracturing stimulation. Time reversal imaging technique (TRI), which reversely propagates the recorded microseismic energy to refocus to its real location, has been verified as a satisfying method in noisy environment. However, the location images obtained from TRI techniques are frequently smeared with strong imaging noises and artifacts resulting from the cross talk of coupled wave modes coinciding in space and time with sparse measurements especially in heterogeneous anisotropic media, resulting in erroneous location estimations. To mitigate these disturbances, a multiplicative time reversal imaging method based on decoupled wavefields is proposed (DC-MTRI). Wave-mode separators constructed based on Christoffel equation are first utilized to fully separate quasi P wave (qP) and quasi vertically pozlarized shear wave (qSV) wavefields at each receiver in 2D vertical transverse isotropy (VTI) media. Multiplicative time reversal imaging condition is then applied to the auto and cross-correlations of decoupled wavefields of each receiver to construct the final location image. The synthetic examples show that the wave-mode separators applied in this paper can fully separate qP and qSV potentials in 2D VTI media compared to Helmholtz decomposition. As a consequence, the proposed method substantially improves the quality of the location image with a much sharper focus and less noisy imaging energy at shallow depth of the sophisticated Sigesbee 2A model. In addition, the numerical experiments show that DC-MTRI can give robust source location images with distinct radiation pattern of high resolution for sparse acquisition plan. Furthermore, DC-MTRI has same sensitivity on velocity perturbations compared with conventional TRI techniques. Therefore, the proposed technique can potentially be employed as the constraints to estimate source mechanism and update the velocity structure in the future.

Interaction between P, S1 and S2 waves in multilayered periodic monoclinic media at low frequencies

SUMMARYApplication of the Floquet theorem and the matrix propagator method reduces the problem of the plane wave propagation in a periodically layered anisotropic media, to analysis of the properties of stationary envelopes of different wave modes propagating up- and downwards. We analyse the interchanging of stop- and pass-bands and their structure at low frequencies for a periodically layered medium with monoclinic symmetry. The analysis shows the effect of interaction between P,S1 and S2 wave multipliers for stop- and pass-band structure and gives insight into the wave propagation in vertically heterogeneous anisotropic media which is important in modelling and interpretation of seismic data.

Source‐independent waveform inversion for attenuation estimation in anisotropic media

ABSTRACTIn previous publications, we presented a waveform‐inversion algorithm for attenuation analysis in heterogeneous anisotropic media. However, waveform inversion requires an accurate estimate of the source wavelet, which is often difficult to obtain from field data. To address this problem, here we adopt a source‐independent waveform‐inversion algorithm that obviates the need for joint estimation of the source signal and attenuation coefficients. The key operations in that algorithm are the convolutions (1) of the observed wavefield with a reference trace from the modelled data and (2) of the modelled wavefield with a reference trace from the observed data. The influence of the source signature on attenuation estimation is mitigated by defining the objective function as the ℓ2‐norm of the difference between the two convolved data sets. The inversion gradients for the medium parameters are similar to those for conventional waveform‐inversion techniques, with the exception of the adjoint sources computed by convolution and cross‐correlation operations. To make the source‐independent inversion methodology more stable in the presence of velocity errors, we combine it with the local‐similarity technique. The proposed algorithm is validated using transmission tests for a homogeneous transversely isotropic model with a vertical symmetry axis that contains a Gaussian anomaly in the shear‐wave vertical attenuation coefficient. Then the method is applied to the inversion of reflection data for a modified transversely isotropic model from Hess. It should be noted that due to the increased nonlinearity of the inverse problem, the source‐independent algorithm requires a more accurate initial model to obtain inversion results comparable to those produced by conventional waveform inversion with the actual wavelet.

Finite volume method for coupled subsurface flow problems, I: Darcy problem

The article introduces a finite-volume method for the Darcy problem in heterogeneous anisotropic media. The method is based on the mixed formulation for the pressure and its gradient. The method is stable despite collocation of both pressure and its gradient at cell centers and demonstrates the first order convergence on numerous benchmarks as well as good monotonicity property. The method produces quasi-definite matrix, which is numerically shown to have good asymptotics of the condition number. Our flux discretization method is a realization of our more general concept of stable flux discretization for saddle-point systems with vector of several unknowns. In this paper this vector is composed of pressure and its gradient and the saddle-point system is the mixed formulation of the Darcy problem.

Self-consistent and Maxwell approximations to describe the excess conductivity anisotropy in FeSe above superconducting transition temperature

Using the self-consistent approximation we calculate conductivity in an anisotropic heterogeneous media with superconducting inclusions and compare the results with those obtained previously using the Maxwell approximation and with available experimental data on excess conductivity above Tc = 8K in FeSe. Advantages and drawbacks of these two approximations are discussed. The obtained analytical formulas can be applied to various other anisotropic heterogeneous superconductors, including high-Tc layered superconductors.

Calculation of the amplitudes of elastic waves in anisotropic media in Cartesian or ray-centred coordinates

We derive various expressions for the amplitude of the ray-theory approximation of elastic waves in heterogeneous anisotropic media, and show their mutual relations. The amplitude of a wavefield with general initial conditions can be expressed in terms of two paraxial vectors of geometrical spreading in Cartesian coordinates, and in terms of the 2×2 matrix of geometrical spreading in ray-centred coordinates. The amplitude of the Green tensor can be expressed in six different ways: (a) in terms of the paraxial vectors corresponding to two ray parameters in Cartesian coordinates, (b) in terms of the 2×2 paraxial matrices corresponding to two ray parameters in ray-centred coordinates, (c) in terms of the 3×3 upper right submatrix of the 6×6 propagator matrix of geodesic deviation in Cartesian coordinates, (d) in terms of the 2×2 upper right submatrix of the 4×4 propagator matrix of geodesic deviation in ray-centred coordinates, (e) in terms of the 3×3 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to the source and receiver Cartesian coordinates, and (f) in terms of the 2×2 matrix of the mixed second-order spatial derivatives of the characteristic function with respect to the source and receiver ray-centred coordinates. The step-by-step derivation of various equivalent expressions, both known or novel, elucidates the mutual relations between these expressions.

The Onset of Convection in an Anisotropic Heterogeneous Porous Medium: A New Hydrodynamic Boundary Condition

The Horton–Rogers–Lapwood problem with strong heterogeneity and anisotropy is examined for a simple case, namely where the heterogeneity is provided by two layers, each of which is homogeneous and isotropic in a horizontal plane. We derived a new hydrodynamic boundary condition at the interface between two different porous media and then formulated and numerically solved the eigenvalue problem to determine the critical values of the wavenumber and Rayleigh number. We found that our approach works and gives the correct result for the homogeneous situation independent of the position of the interface. We also showed that for weak heterogeneity, by using modified anisotropy parameters weighted with the two layer depths, results obtained in Kvernvold and Tyvand (J Fluid Mech 90:609–624, 1979) for a single-layer problem can be used to approximate the critical Rayleigh number for the double-layer problem. We found that the agreement with the two-layer solution is best if a harmonic mean is used to define the mean permeability ratio and an arithmetic mean is utilized to define the mean conductivity ratio.

Superpositions of Gaussian beams and column Gaussian packets in heterogeneous anisotropic media

Superpositions of Gaussian beams and column Gaussian packets in heterogeneous anisotropic media

An accurate discontinuous Galerkin method for solving point-source Eikonal equation in 2-D heterogeneous anisotropic media

Accurate numerical computation of wave traveltimes in heterogeneous media is of major interest for a large range of applications in seismics, such as phase identification, data windowing, traveltime tomography and seismic imaging. A high level of precision is needed for traveltimes and their derivatives in applications which require quantities such as amplitude or take-off angle. Even more challenging is the anisotropic case, where the general Eikonal equation is a quartic in the derivatives of traveltimes. Despite their efficiency on Cartesian meshes, finite-difference solvers are inappropriate when dealing with unstructured meshes and irregular topographies. Moreover, reaching high orders of accuracy generally requires wide stencils and high additional computational load. To go beyond these limitations, we propose a discontinuous-finite-element-based strategy which has the following advantages: (1) the Hamiltonian formalism is general enough for handling the full anisotropic Eikonal equations; (2) the scheme is suitable for any desired high-order formulation or mixing of orders (p-adaptivity); (3) the solver is explicit whatever Hamiltonian is used (no need to find the roots of the quartic); (4) the use of unstructured meshes provides the flexibility for handling complex boundary geometries such as topographies (h-adaptivity) and radiation boundary conditions for mimicking an infinite medium. The point-source factorization principles are extended to this discontinuous Galerkin formulation. Extensive tests in smooth analytical media demonstrate the high accuracy of the method. Simulations in strongly heterogeneous media illustrate the solver robustness to realistic Earth-sciences-oriented applications.

Parallel numerical modeling of hybrid-dimensional compositional non-isothermal Darcy flows in fractured porous media

This paper introduces a new discrete fracture model accounting for non-isothermal compositional multiphase Darcy flows and complex networks of fractures with intersecting, immersed and non-immersed fractures. The so called hybrid-dimensional model using a 2D model in the fractures coupled with a 3D model in the matrix is first derived rigorously starting from the equi-dimensional matrix fracture model. Then, it is discretized using a fully implicit time integration combined with the Vertex Approximate Gradient (VAG) finite volume scheme which is adapted to polyhedral meshes and anisotropic heterogeneous media. The fully coupled systems are assembled and solved in parallel using the Single Program Multiple Data (SPMD) paradigm with one layer of ghost cells. This strategy allows for a local assembly of the discrete systems. An efficient preconditioner is implemented to solve the linear systems at each time step and each Newton type iteration of the simulation. The numerical efficiency of our approach is assessed on different meshes, fracture networks, and physical settings in terms of parallel scalability, nonlinear convergence and linear convergence.

A $k$ -Space Pseudospectral Method for Elastic Wave Propagation in Heterogeneous Anisotropic Media.

This paper presents the theory of the k -space method generalized to model elastic wave propagation in heterogeneous anisotropic media. The k -space methods are promising time integration techniques giving, in conjunction with collocation spectral methods, accurate and efficient numerical schemes for problems in heterogeneous media. In this paper, the k -space operator is derived in a spatially continuous form using the Fourier analysis of the displacement formalism of elastodynamics. An efficient numerical algorithm is then constructed by applying a Fourier collocation spectral method, leading to define the discrete k -space scheme. The proposed method is temporally exact for homogeneous media, unconditionally stable for heterogeneous media, and also allows larger time steps without loss of accuracy. Implementation of the method is discussed in detail. The method is validated through a set of numerical tests. The numerical results show the efficacy of the method compared with the conventional schemes.

Modeling anisotropic flow and heat transport by using mimetic finite differences

Modeling anisotropic flow in porous or fractured rock often assumes that the permeability tensor is diagonal, which means that its principle directions are always aligned with the coordinate axes. However, the permeability of a heterogeneous anisotropic medium usually is a full tensor. For overcoming this shortcoming, we use the mimetic finite difference method (mFD) for discretizing the flow equation in a hydrothermal reservoir simulation code, SHEMAT-Suite, which couples this equation with the heat transport equation. We verify SHEMAT-Suite-mFD against analytical solutions of pumping tests, using both diagonal and full permeability tensors. We compare results from three benchmarks for testing the capability of SHEMAT-Suite-mFD to handle anisotropic flow in porous and fractured media. The benchmarks include coupled flow and heat transport problems, three-dimensional problems and flow through a fractured porous medium with full equivalent permeability tensor. It shows firstly that the mimetic finite difference method can model anisotropic flow both in porous and in fractured media accurately and its results are better than those obtained by the multi-point flux approximation method in highly anisotropic models, secondly that the asymmetric permeability tensor can be included and leads to improved results compared the symmetric permeability tensor in the equivalent fracture models, and thirdly that the method can be easily implemented in existing finite volume or finite difference codes, which has been demonstrated successfully for SHEMAT-Suite.