Wave Propagation In Poroelastic Media Research Articles

Simulation of dynamic pulsing fracking in poroelastic media by a hydro-damage-mechanical coupled cohesive phase field model

This study develops a hydro-damage-mechanical fully coupled numerical method capable of modelling complex fluid-driven transient dynamic crack propagation in quasi-brittle poroelastic media. In this method, the fluid flow in both fractures and porous media is described by a fluid continuity equation with the modified Darcy-Poiseuille law based on the Biot's poroelastic theory. The fluid pressure and the inertial force of solids are coupled by governing equations of the mesh-insensitive phase-field regularized cohesive zone model that can simulate quasi-brittle multi-crack initiation, propagation, branching and merging without remeshing or crack tracking. The resultant displacement-pressure-damage coupled multiphysics system of equations is solved using an alternative minimization Newton-Raphson iterative algorithm with an implicit Newmark integration scheme within the finite element framework. The new method was first validated by a few 2D problems, including crack branching and deflecting in solids, fracking in a concrete cube, and consolidation and stress wave propagation in poroelastic media, subjected to various impulsive loadings. It was then applied to pressure pulsing fracking of a 40 m granite rock reservoir with extensive parametric studies of fluid viscosity and pulsing injection rate, mode and period. It was found that pulsing injection with higher fluid rates and lower fluid viscosities resulted in more developed crack patterns, and in particular, there existed an optimal pulsing injection period that could promote fracking under relatively low injection pressures. 3D horizontal well problems with multiple non-planar crack propagation and bifurcation were also successfully simulated to demonstrate the capacity and potential of the new method for engineering design and optimization of pressure pulsing fracking.

2D parallel simulation of seismic wave propagation in poroelastic media to monitor a CO2 geological sequestration process

Geological sequestration of carbon dioxide CO2 involves storing CO2 beneath geological formations to reduce atmospheric emissions. Typically, CO2 is pressurized into a liquid state and injected into porous rock formations. Monitoring this process relies on seismic data. For the effective monitoring of seismic wave propagation in extensive formations, a highly efficient and robust algorithm is essential. We developed an efficient approach to alleviate the computational workload associated with simulating CO2 sequestration using the staggered grid finite difference method. The method developed was applied to a moderately complex medium which represents pre- and post-CO2 injection of Utsira Formation, North Sea. The model was developed for two scenarios, representing the stages before and after the injection of CO2 into the Formation. The effect of CO2 was observed and picked efficiently in the reservoir by the algorithm developed for the scenarios before and after the injection of CO2. We could see the brightening of reflections with an increased CO2 saturation in the post-CO2 injection cases. The method developed greatly reduces the computational requirement of the simulation which makes it more efficient for large scale applications.

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.

Electromagnetic field generated by acoustic wave scattering at a poroelastic inclusion located in a fluid

The study of elastic wave propagation in poroelastic media is of considerable scientific interest because of some effects that do not appear in monophasic elastic or viscoelastic media. In the case of interconnected pores containing a moving fluid in a poroelastic medium, there exist two types of longitudinal waves. In heterogeneous porous media, the energy of elastic oscillations is redistributed due to mutual transformation of elastic waves of different types at the inclusion boundaries. The filtration flow that appears there results in the case of polar fluids in a displacement of electric double layers at the pore boundaries, and, as a consequence, in generation of an electromagnetic field.We consider the problem of the scattering of an elastic wave by a spherical poroelastic inclusion in a fluid. The solution was obtained for sufficiently low frequencies so that the dispersion of the elastic wave on individual grains of the porous media may be neglected and the averaged equations of the mechanics of continuous media (at the so-named mesoscale) may be used. In order to describe the elastic wave propagation, we have used the Pride system of electrokinetic equations. For the solution of our problem, we have applied the decomposition of the potentials of acoustic and electromagnetic waves into spherical harmonics. We have obtained the dependencies of the generated electromagnetic field on the inclusion porosity and the fluid properties in pores. The results were obtained for fully open pore interface at the inclusion boundary.

Time domain numerical modeling of wave propagation in poroelastic media with Chebyshev rational approximation of the fractional attenuation

In this work, we investigate the poroelastic waves by solving the time‐domain Biot‐JKD equation with an efficient numerical method. The viscous dissipation occurring in the pores depends on the square root of the frequency and is described by the Johnson‐Koplik‐Dashen (JKD) dynamic tortuosity/permeability model. The temporal convolutions of order 1/2 shifted fractional derivatives are involved in the time‐domain Biot‐JKD model, causing the problem to be stiff and challenging to be implemented numerically. Based on the best relative Chebyshev approximation of the square‐root function, we design an efficient algorithm to approximate and localize the convolution kernel by introducing a finite number of auxiliary variables that satisfy a local system of ordinary differential equations. The imperfect hydraulic contact condition is used to describe the interface boundary conditions and the Runge‐Kutta discontinuous Galerkin (RKDG) method together with the splitting method is applied to compute the numerical solutions. Several numerical examples are presented to show the accuracy and efficiency of our approach.

Computer Simulation of the Seismic Wave Propagation in Poroelastic Medium

This article presents an algorithm for the numerical solution of an initial-boundary value problem for a symmetric t-hyperbolic system of partial differential equations. This problem is based on continual filtration model, which describes the propagation of seismic waves in a poroelastic medium saturated with a fluid characterized by such physical parameters as the propagation velocities of longitudinal P- (fast and slow) and transverse S-waves, the density of the medium materials, and porosity. The system of linearized equations of saturated porous media is formulated in terms of physical variables of the velocity–stress tensor of the porous matrix and the velocity–pressure of the saturating fluid in the absence of energy dissipation. The solution is implemented numerically using an explicit finite difference upwind scheme built on a staggered grid to avoid the appearance of oscillations in the solution functions. The program code implementing parallel computing is developed in the high-performance Julia programming language. The possibility of using the approach is demonstrated by the example of solving the problem of propagation of seismic waves from a source located in the formation. Computational experiments based on real data from oil reservoirs have been implemented, and dynamic visualization of solutions consistent with the first waves arrival times has been obtained.

An optimized staggered-grid finite-difference operator for seismic wave simulation in poroelastic media

Most of the finite-difference methods for simulating seismic wave propagation in poroelastic media compute the spatial derivatives using fixed-length, which is the 2Mth-order operator. An optimized staggered-grid finite-difference (SGFD) method, which uses mixed operator length to approximate spatial derivatives, is derived from Biot’s equations to improve accuracy and stability. The numerical stability and dispersion of our optimized scheme are derived using the dispersion relation for the poroelastic equations. We have compared the dispersion curve plots for the developed and the conventional method. We introduce evidence that the developed scheme is comparatively more stable and accurate than the existing conventional method. The numerical solutions obtained using our optimized scheme closely approximate the analytical solution, and all wave types predicted by Biot’s theory are generated. Our scheme is used to simulate seismic wave propagation in a 2D homogeneous model, a two-layered model, a moderately complex model derived from the Utsira Formation, Sleipner North Sea, where a large amount of CO2 is being sequestrated, and a 3D homogeneous model. The developed scheme takes less computational time than the conventional SGFD method without compromising the accuracy or the stability.

Resolving Wave Propagation in Anisotropic Poroelastic Media Using Graphical Processing Units (GPUs)

AbstractBiot's equations describe the physics of hydromechanically coupled systems establishing the widely recognized theory of poroelasticity. This theory has a broad range of applications in Earth and biological sciences as well as in engineering. The numerical solution of Biot's equations is challenging because wave propagation and fluid pressure diffusion processes occur simultaneously but feature very different characteristic time scales. Analogous to geophysical data acquisition, high resolution and three dimensional numerical experiments lately redefined state of the art. Tackling high spatial and temporal resolution requires a high‐performance computing approach. We developed a multi‐ graphical processing units (GPU) numerical application to resolve the anisotropic elastodynamic Biot's equations that relies on a conservative numerical scheme to simulate, in a few seconds, wave fields for spatial domains involving more than 1.5 billion grid cells. We present a comprehensive dimensional analysis reducing the number of material parameters needed for the numerical experiments from ten to four. Furthermore, the dimensional analysis emphasizes the key material parameters governing the physics of wave propagation in poroelastic media. We perform a dispersion analysis as function of dimensionless parameters leading to simple and transparent dispersion relations. We then benchmark our numerical solution against an analytical plane wave solution. Finally, we present several numerical modeling experiments, including a three‐dimensional simulation of fluid injection into a poroelastic medium. We provide the Matlab, symbolic Maple, and GPU CUDA C routines to reproduce the main presented results. The high efficiency of our numerical implementation makes it readily usable to investigate three‐dimensional and high‐resolution scenarios of practical applications.

Simulation of seismic wave propagation in poroelastic media using vectorized Biot’s equations: an application to a CO$$_{2}$$ sequestration monitoring case

Wave propagation through porous media allows us to understand the response and interaction that occur between the elastic rock matrix and the fluid. This interaction has been described by Biot in his theory of poroelasticity. Seismic wave simulation using Biot’s formulations is computationally expensive when compared with the acoustic and elastic cases. This computational burden can be reduced by reformulating the numerical derivative operators to improve the efficiency. To achieve this, we used a staggered-grid finite difference operator to discretize 2D velocity stress equations as given by Biot’s theory. A vectorized derivative is applied on the staggered grid by shifting the coordinates. The reformulated equations were applied to compute the seismic response of a reservoir, where $$\hbox {CO}_2$$ is being injected and the effect of injected $$\hbox {CO}_2$$ in the formation is clearly seen in the synthetic data generated. The algorithm was coded in Python and to test its efficiency, the simulation run-time was compared for both serial and vectorized equations, and the speed-up ratio was calculated. Our results show a decrease in the simulation run-time for the vectorized execution with over a factor of a hundred percent (100%). We further observed that the amplitudes of the events increase with an increase in $$\hbox {CO}_2$$ saturation in the formation. This matches well with the real data.

Adaptive Discontinuous Galerkin Modeling of Intrinsic Attenuation Anisotropy for Fluid-Saturated Porous Media

An hp- and memory-adaptive discontinuous Galerkin time-domain algorithm is presented to efficiently model wave propagation in poroelastic media, with the incorporation of 3-D fully anisotropic intrinsic attenuation. From the perspective of physics, the attenuation from the triclinic rock frame, the loss in the pore fluid, and the friction for their interaction are all considered. From the perspective of implementation, a new frequency-domain constitutive equation is introduced, involving complex-valued poroelasticity matrix. A new $Q$ value is defined as the ratio between its real and imaginary parts for every entry. Then, a generalized Maxwell body is adopted to approximate this frequency-dependent $Q$ in the time domain. Mathematically speaking, the hyperbolicity is preserved for the new viscous poroelastic system. Our results corroborate that the intrinsic attenuation anisotropy makes tangible effects in fluid-saturated porous formations.

Analytical modelling of sound transmission loss across finite clamped triple-wall sandwich panels in the presence of external mean flow

• The triple-wall panels have superior sound performance to their double-wall counterparts due to an additional partition. • The finite dimensions and clamped boundary influence the sound transmission of triple-wall sandwich panels. • The external flow enhances sound transmission loss significantly in the low-frequency domain. This paper analytically addresses the sound transmission loss (STL) performance of finite triple-wall sandwich panels lined with poroelastic materials and clamp mounted on an infinite rigid baffle in the presence of an external mean flow. The wave propagation in poroelastic media is addressed using Biot's theory. The coupling methods between the poroelastic core and facing plates are associated with different boundary conditions and various panel configurations. The Galerkin method and the modal superposition theory are used to deal with the clamped boundary conditions and finite extensions as well as to transfer the integral function to a matrix equation. The analytical model is verified by comparing to previous results of infinite triple-wall sandwich panels and finite triple-wall panels without poroleastic materials. The numerical results show that the triple-wall panels have superior STL to their double-wall counterparts due to the existence of an additional partition. The finite dimensions of the triple-wall sandwich panels with clamped boundary strongly influence the STL at low frequencies, and this effect is coupled with that of the external mean flow. The air layer plays a critical role in the sound insulation properties that vary among different panel configurations, and the configurations with two air layers exhibit the optimal overall sound insulation performance.

Radial vibrations of an infinitely long poroelastic composite hollow circular cylinder

The problem of radial vibrations of an infinitely long poroelastic composite hollow circular cylinder is solved by employing Biot’s theory of wave propagation in poroelastic media. A poroelastic composite hollow cylinder consists of two concentric poroelastic cylindrical layers both of which are made of different poroelastic materials with each poroelastic material as homogeneous and isotropic. The boundaries of composite hollow poroelastic cylinder are free from stress. The frequency equations of radial vibrations of poroelastic composite hollow cylinder with rigid core, poroelastic composite solid cylinder, poroelastic composite solid cylinder with rigid casing and of rigid core and poroelastic composite bore are derived as particular cases. Non-dimensional frequency is computed as a function of ratio of thickness to inner radius of core. The results are presented graphically for two types of poroelastic composite cylinders and then discussed.Keywords:Radial vibrations, Poroelasticity, Composite cylinder, Frequency, Rigid.

Analytical modelling of sound transmission through finite clamped double-wall sandwich panels lined with poroelastic materials

This paper addresses analytically the vibroacoustic problem of sound transmission across a rectangular double-wall sandwich panel clamp mounted on an infinite rigid baffle and lined with poroelastic materials. The wave propagation in poroelastic media is described by Biot’s theory and the coupling methods between the poroelastic core and the panel determine the various configurations and associated boundary conditions. The modal superposition theory and the weighted residual (Galerkin) method are employed to account for the finite extension with the clamped boundary and to obtain a double-series solution of the problem through a matrix equation. The sound transmission loss (STL) of the structure is calculated for a single incident wave after validating the analytical model against previous theories and experiments. The numerical results show that the finite panel and clamped boundary influence the STL dominantly in the low-frequency range. The poroelastic materials exhibit strong damping effects on the resonances of STL spectra at high frequencies as well as panel vibrations and enhance significantly the sound insulation performance of the sandwich panel.

Analysis of flexural wave propagation in poroelastic composite hollow cylinder

Wave propagation in an infinitely long poroelastic composite hollow cylinder in is examined by employing Biot’s theory of wave propagation in poroelastic media. A poroelastic composite hollow cylinder consists of two concentric poroelastic cylindrical layers both of which are made of different poroelastic materials with each poroelastic material as homogeneous and isotropic. The inner and outer boundaries of composite hollow poroelastic cylinder are free from stress. The frequency equation of flexural vibrations of poroelastic composite hollow cylinder is obtained. In addition some particular cases such as poroelastic composite hollow cylinder with rigid casing, poroelastic composite bore and poroelastic bore are discussed. Non-dimensional phase velocity is computed as a function of non-dimensional wavenumber. The results are presented graphically for two types of poroelastic composite cylinders and then discussed.Keywords: Biot’s theory, poroelastic composite hollow cylinder, wavenumber, rigidity, phase velocity

On the Use of Transfer Approaches to Predict the Vibroacoustic Response of Poroelastic Media

The transfer matrix method (TMM) is a famous analytic method in the vibroacoustic community. It is classically considered as a high frequency approach, because of the hypothesis of acoustic plane waves impinging on a flat infinite panel. Thus, it cannot take into account directly finite-size effects or lateral boundary conditions (BCs), and it needs specific algorithms to correct its results in the low frequency range. Within the transfer matrix framework, the use of finite elements makes it possible to generalize the range of applications of transfer approaches. Thus, the study of wave propagation in poroelastic media, in presence of lateral BCs can be carried out. The links between theses waves and the acoustic response of a sample are investigated. Finally, it shows that transfer approaches are not limited in the low frequency range, as usually stated. In fact, the validity of analytic transfer approaches depends more on the material and on the geometry than on the frequency range.

Optimal implicit 2-D finite differences to model wave propagation in poroelastic media

Optimal implicit 2-D finite differences to model wave propagation in poroelastic media

Reflection and transmission coefficients of a single layer in poroelastic media

Wave propagation in poroelastic media is a subject that finds applications in many fields of research, from geophysics of the solid Earth to material science. In geophysics, seismic methods are based on the reflection and transmission of waves at interfaces or layers. It is a relevant canonical problem, which has not been solved in explicit form, i.e., the wave response of a single layer, involving three dissimilar media, where the properties of the media are described by Biot's theory. The displacement fields are recast in terms of potentials and the boundary conditions at the two interfaces impose continuity of the solid and fluid displacements, normal and shear stresses, and fluid pressure. The existence of critical angles is discussed. The results are verified by taking proper limits-zero and 100% porosity-by comparison to the canonical solutions corresponding to single-phase solid (elastic) media and fluid media, respectively, and the case where the layer thickness is zero, representing an interface separating two poroelastic half-spaces. As examples, it was calculated the reflection and transmission coefficients for plane wave incident at a highly permeable and compliant fluid-saturated porous layer, and the case where the media are saturated with the same fluid.

Model-based Assessment of Seismic Monitoring of CO2 in a CCS Project in Alberta, Canada, Including a Poroelastic Approach

The theoretical detectability of CO2 is investigated for the Quest Carbon Capture and Storage (CCS) project in Alberta, Canada. This is completed by using Gassmann fluid substitution and elastic seismic modeling. The study indicates that for the period of one year injection, the location and spatial distribution of the CO2 plume could be detected in the seismic data. This is providing the data have good bandwidth and a high signal to noise ratio. However, pore fluid properties such as density, velocity, viscosity, and saturation are neglected in elastic modeling in spite of the reservoir rocks being porous media saturated with fluids. Another phase of the study includes developing a more accurate forward modeling algorithm that takes the fluid properties into account. For this purpose, the Biot's equations of motion in poroelastic media are employed to develop a finite difference algorithm to simulate the wave propagation in poroelastic media. To examine the algorithm, a numerical example is defined based on the Quest project. The results reveal that the algorithm properly handles the layered models and thus can be used in the future to examine more complex models. This approach may also be better than elastic modeling for theoretical monitoring of CO2 since the reservoir fluid content is considered in the modeling process.

A hybrid finite difference/control volume method for the three dimensional poroelastic wave equations in the spherical coordinate system

In this paper, we consider the numerical approximation of the three-dimensional poroelastic wave equations in the spherical coordinate system. One difficulty in the design of an efficient numerical scheme is that the problem is singular in the center and the polar axes of the computational domain. Nevertheless, we develop a hybrid finite difference/control volume method for solving this problem. Our method is explicit and is second order accurate in both space and time. Numerical results are shown to confirm the convergence rate of our method and the effectiveness to simulate wave propagation in poroelastic media in the spherical coordinate system.

Vibrations in a Fluid-Loaded Poroelastic Hollow Cylinder Surrounded by a Fluid in Plane-Strain Form

Plane-strain vibrations in a fluid-loaded poroelastic hollow cylinder surrounded by a fluid are investigated employing Biot’s theory of wave propagation in poroelastic media. The poroelastic hollow cylinder is homogeneous and isotropic, while the inner and outer fluids are homogeneous, isotropic and inviscid. The frequency equation of the fluid-loaded poroelastic cylinder surrounded by a fluid is obtained along with several particular cases, namely, fluid-loaded poroelastic cylinder, fluid-loaded bore, poroelastic cylinder surrounded by a fluid and poroelastic solid cylinder submerged in a fluid. The frequency equations are obtained for axially symmetric, flexural and anti-symmetric vibrations each for a pervious and an impervious surface. Nondimensional frequency for propagating modes is computed as a function of the ratio of thickness to the inner radius of the core. The results are presented graphically for two types of poroelastic cylinders and then discussed.