Biot's Poroelastic Equations Research Articles

Acoustic response of patchy-saturated porous media: Coupling Biot's poroelasticity equations for mono- and biphasic pore fluids.

Patchy saturation is a term used in the seismic prospecting literature to describe the state of a geological formation in which two immiscible pore fluids prevail in mesoscopic-scale clusters. If the pore fluids have contrasting compressibilities, wave-induced fluid pressure diffusion (FPD) processes may induce significant attenuation and velocity dispersion on seismic waves. Biot's monophasic poroelasticity theory is widely used to model the seismic response of rocks containing binary patches of two immiscible pore fluids. Even though effective fluid approximations may help to represent more realistic partially saturated patches using Biot's monophasic equations, the so inferred dissipation may not be representative of actual biphasic FPD phenomena. In this work, Biot's equations for mono- and biphasic fluids are combined to model FPD processes in porous media, comprising fully and partially saturated patches. An analytical solution for one-dimensional layered patchy-saturated media is presented which permits to explain some, as of yet enigmatic, experimentally observed characteristics such as increased seismic attenuation and stiffening effects occurring at low saturations. The results show that the existence of an additional diffusive wave mode within partially saturated patches may render conventional binary and effective fluid approaches incorrect and error prone.

Evolutionary topology optimization approach to design multiphase soundproof systems with poroelastic media

SummaryWith the constant development of cities, noise sources have become increasingly present inside and outside living environments. Consequently, soundproof systems comprised of porous materials have been widely adopted as filling fabric of closed‐space structures, such as in the components of buildings, airplanes or automobiles. However, in many situations, simply filling spaces may not be the most effective approach. In that scenario, this work introduces a multiphase acoustic topology optimization methodology to design closed‐space structures for sound attenuation. Based on the Bi‐directional Evolutionary Structural Optimization (BESO) algorithm, the proposed approach combines Biot's poroelasticity equations, expressed in the mixed /p form, and the Unified Multiphase (UMP) technique to fully describe the multiphysics involved in the acoustic, poroelastic and elastic model relations. An objective function contemplating different combinations of structural, viscous and thermal dissipated powers is maximized over multiple frequencies. Volume constraints in each material phase and a novel material interpolation scheme are also considered. The resultant topologies present enhanced dissipated power levels and manufacturability, even when compared with various baseline configurations of similar volume fractions.

Analysis of a linearized poromechanics model for incompressible and nearly incompressible materials

In this work, we thoroughly analyze the linearized version of a poromechanics model developed to simulate soft tissues perfusion. This is a fully unsteady model in which the fluid and solid equations are strongly coupled through the interstitial pressure. As such, it generalizes Darcy, Brinkman and Biot equations of poroelasticity. The mathematical and numerical analysis of this model was initially performed for a compressible porous material. Here, we focus on the nearly incompressible case with a semigroup approach, which also allows us to prove the existence of weak solutions. We show the existence and uniqueness of strong and weak solutions in the incompressible limit case, for which a divergence constraint on the mixture velocity appears. Due to the special form of the coupling, the underlying problem is not coercive. Nevertheless, by using the notion of $ T $-coercivity, we obtain stability estimates and well-posedness results. Our study also provides guidelines to propose stable and robust approximations of the problem with mixed finite elements. In particular, we recover an inf-sup condition that is independent of the porosity. Finally, we numerically investigate the elliptic regularity of the associated steady-state problem and illustrate the sensitivity of the solution with respect to the different model parameters.

Dynamic SV‐Wave Signatures of Fluid‐Saturated Porous Rocks Containing Intersecting Fractures

AbstractDetecting fracture intersections is crucial for understanding rock hydraulic properties. For this purpose, we study the dynamic SV‐wave signatures of fluid‐saturated porous rocks containing intersecting fractures. A theoretical model is derived using the dynamic Biot's poroelasticity equations. Using this model, we analyze the features of SV‐waves and compare to those of previously studied P‐waves. The results show that the dispersion and attenuation of SV‐waves caused by elastic scattering and FF‐WIFF (Fracture‐Fracture Wave‐induced Fluid Flow) have a similar dependence on properties of intersecting fractures and fluid as those of P‐waves. However, the FB‐WIFF (Fracture‐Background Wave‐induced Fluid Flow) causes much smaller dispersion and attenuation for SV‐waves than for P‐waves. In particular, such dispersion and attenuation of SV‐waves are negligibly small for the orthogonally intersecting fractures regardless of wave incidence angles. In addition to the dispersion and attenuation, the WIFF and elastic scattering also greatly affect the anisotropy properties, which gives rise to frequency‐dependent velocity and attenuation anisotropies for both SV‐ and P‐ waves. Such anisotropy properties are sensitive to fracture intersection angles and are quite different between SV‐ and P‐ waves. These complementary features of SV‐ and P‐ waves provide the basis for fracture intersection detection using the combined features of these two waves. By comparing our model to the known results for the limiting cases, we validate our model.

Green's function of the Lord–Shulman thermo-poroelasticity theory

SUMMARY The Lord–Shulman thermoelasticity theory combined with Biot equations of poroelasticity, describes wave dissipation due to fluid and heat flow. This theory avoids an unphysical behaviour of the thermoelastic waves present in the classical theory based on a parabolic heat equation, that is infinite velocity. A plane-wave analysis predicts four propagation modes: the classical P and S waves and two slow waves, namely, the Biot and thermal modes. We obtain the frequency-domain Green's function in homogeneous media as the displacements-temperature solution of the thermo-poroelasticity equations. The numerical examples validate the presence of the wave modes predicted by the plane-wave analysis. The S wave is not affected by heat diffusion, whereas the P wave shows an anelastic behaviour, and the slow modes present a diffusive behaviour depending on the viscosity, frequency and thermoelasticity properties. In heterogeneous media, the P wave undergoes mesoscopic attenuation through energy conversion to the slow modes. The Green's function is useful to study the physics in thermoelastic media and test numerical algorithms.

Semi-analytical solution to the problem of frequency dependent anisotropy of porous media with an aligned set of slit cracks

Fractures control fluid flow in geological formations because their presence can greatly affect effective permeability. Existence of cracks in a porous rock can induce seismic dispersion attenuation and frequency-dependent anisotropy, due to wave-induced fluid flow (WIFF) between the fractures and pores. Thus, dispersion and attenuation can be valuable seismic signatures for fracture characterization. In previous studies these effects were modeled by considering cracks as low aspect-ratio inclusions embedded in a porous background medium modeled by Biot's poroelasticity equations. However, previous works are limited to normal wave incidence. In this research, we study the P-wave dispersion and attenuation for oblique incidence in a saturated porous medium with aligned slit (two-dimensional) fractures, through solving a mixed boundary value problem for Biot's dynamic poroelasticity equations. The semi-analytical solution of this problem gives dispersion and attenuation as functions of the incidence angle. The strongest dispersion and attenuation occur when the P-wave propagates along the fracture normal and decrease with increasing incident angle (as measured from the fracture normal). For the frequency-dependent anisotropy at low frequencies, the sample behaves as nearly elliptical, and at high frequencies the medium exhibits a strong anellipticity. The attenuation anisotropy parameters reach their maximum when the fluid diffusion length is comparable to the crack radius. Comparison of the heuristic solution, which is s based on an interpolation between low- and high frequencies using a relaxation function, to the semi-analytical solution of the rigorous equations shows excellent agreement, demonstrating the validity of the heuristic approach.

Numerically quantifying energy loss caused by squirt flow

ABSTRACTIn interconnected microcracks, or in microcracks connected to spherical pores, the deformation associated with the passage of mechanical waves can induce fluid flow parallel to the crack walls, which is known as squirt flow. This phenomenon can also occur at larger scales in hydraulically interconnected mesoscopic cracks or fractures. The associated viscous friction causes the waves to experience attenuation and velocity dispersion. We present a simple hydromechanical numerical scheme, based on the interface‐coupled Lamé–Navier and Navier–Stokes equations, to simulate squirt flow in the frequency domain. The linearized, quasi‐static Navier–Stokes equations describe the laminar flow of a compressible viscous fluid in conduits embedded in a linear elastic solid background described by the quasi‐static Lamé–Navier equations. Assuming that the heterogeneous model behaves effectively like a homogeneous viscoelastic medium at a larger spatial scale, the resulting attenuation and stiffness modulus dispersion are computed from spatial averages of the complex‐valued, frequency‐dependent stress and strain fields. An energy‐based approach is implemented to calculate the local contributions to attenuation that, when integrated over the entire model, yield results that are identical to those based on the viscoelastic assumption. In addition to thus validating this assumption, the energy‐based approach allows for analyses of the spatial dissipation patterns in squirt flow models. We perform simulations for a series of numerical models to illustrate the viability and versatility of the proposed method. For a 3D model consisting of a spherical crack embedded in a solid background, the characteristic frequency of the resulting P‐wave attenuation agrees with that of a corresponding analytical solution, indicating that the dissipative viscous flow problem is appropriately handled in our numerical solution of the linearized, quasi‐static Navier–Stokes equations. For 2D models containing either interconnected cracks or cracks connected to a circular pore, the results are compared with those based on Biot's poroelastic equations of consolidation, which are solved through an equivalent approach. Overall, our numerical simulations and the associated analyses demonstrate the suitability of the coupled Lamé–Navier and Navier–Stokes equations and of Biot's equations for quantifying attenuation and dispersion for a range of squirt flow scenarios. These analyses also allow for delineating numerical and physical limitations associated with each set of equations.

A generalized effective anisotropic poroelastic model for periodically layered media accounting for both Biot's global and interlayer flows

ABSTRACTWe present a generalized effective poroelastic model for periodically layered media in the mesoscopic scale range, which accounts for both Biot's global and interlayer wave‐induced fluid flow, as well as for the anisotropy associated with the layering. Correspondingly, it correctly predicts the existence of the fast and slow P‐waves as well as quasi and pure S‐waves. The proposed analytical model is validated through comparisons of the P‐wave and S‐wave phase velocity dispersion and attenuation characteristics with those inferred from a one‐dimensional numerical solution of Biot's poroelastic equations of motion. We also compare our model with the classical mesoscopic model of White for a range of scenarios. The results demonstrate that accounting for both wave‐induced fluid flow mechanisms is essential when Biot's global flow prevails at frequencies that are comparable or smaller with respect to those governing interlayer flow. This is likely to be the case in media of high permeability, such as, for example, unconsolidated sediments, clean sandstones, karstic carbonates, or fractured rocks. Conversely, when interlayer flow occurs at smaller frequencies with respect to Biot's global flow, the predictions of this model are in agreement with White's model, which is based on quasi‐static poroelasticity.

A Model for Interstitial Drainage Through a Sliding Lymphatic Valve.

This study investigates fluid flow and elastic deformation in tissues that are drained by the primary lymphatic system. A model is formulated based on the Rossi hypothesis that states that the primary lymphatic valves, which are formed by overlapping endothelial cells around the circumferential lining of lymphatic capillaries, open in response to swelling of the surrounding tissue. Tissue deformation and interstitial fluid flow through the tissue are treated using the Biot equations of poroelasticity and, the fluid flux (into the interstitium) across the walls of the blood capillaries, is assumed to be linearly related to the pressure difference across the walls via a constant of proportionality (the vascular permeability). The resulting model is solved in a periodic domain containing one blood capillary and one lymphatic capillary starting from a configuration in which the tissue is undeformed. On imposition of a constant pressure difference between blood and lymphatic capillaries, the solutions are found to settle to a steady state. Given that the magnitude of pressure fluctuations in the lymphatic system is much smaller than this pressure difference between blood and lymph, it is postulated that the resulting steady-state solution gives a good representation of the state of the tissue under physiological conditions. The effects of changes to the Young's modulus of the tissue, the blood-lymphatic pressure difference, vascular permeability and valve dimensions on the steady state are investigated and discussed in terms of their effects on oedema in the context of age- and pregnancy-related changes to the body.

Attenuation of sound in sand sediments due to porosity fluctuations.

At high frequencies, the attenuation measured in sand sediments is larger than that predicted by Biot theory. To account for this discrepancy, perturbation theory is used to incorporate losses due to scattering by porosity variations into both Biot's poroelastic equations and the effective density fluid model. While previous results showed that fluctuations in the bulk frame modulus were insufficient to produce significant attenuation in a sand sediment, modest levels of fluctuations in the porosity produce significant scattering loss. By using the sediment parameters and the heterogeneity power spectrum measured during the Sediment Acoustics Experiment in 2004, the perturbation theory result shows good agreement with the sound speed and attenuation data without any free parameters.

Finite difference modeling of ultrasonic propagation (coda waves) in digital porous cores with un-split convolutional PML and rotated staggered grid

Abstract Ultrasonic wave propagation in heterogeneous porous cores under laboratory studies is an extremely complex process involved with strong scattering by microscale heterogeneous structures. The resulting coda waves, as an index to measure scattering attenuation, are recorded as continuous waveforms in the tail portion of wavetrains. Because of the contamination of reflections from the side ends and reverberations between the sample surfaces, it is difficult to extract pure coda waves from ultrasonic measurements for the estimation of the P - and S -coda attenuation quality factors. Comparisons of numerical and experimental ultrasonic wave propagation in heterogeneous porous cores can give important insight into understanding the effect of boundary reflections on the P - and S -codas in the laboratory experiment. It challenges numerical modeling techniques by three major issues: the creation of a digital core model to map heterogeneous rock properties in detail, the perfect simulation with a controllable and accurate absorbing boundary, and overcoming the numerical dispersions resulting from high-frequency propagation and strong heterogeneity in material. A rotated staggered-grid finite-difference method of Biot's poroelastic equations is presented with an unsplit convolutional perfectly matched layer (CPML) absorbing boundary to simulate poroelastic wave propagation in isotropic and fluid-saturated porous media. The contamination of boundary reflections on coda waves is controlled by the CPML absorbing coefficients for the comparison between numerical and experimental ultrasonic waveforms. Numerical examples with a digital porous core demonstrate that the boundary reflections contaminate coda waves seriously, causing much larger coda quality factors and thus underestimating scattering attenuation.

A hybrid-dimensional approach for an efficient numerical modeling of the hydro-mechanics of fractures

Characterization of subsurface fluid flow requires accounting for hydro-mechanical coupling between fluid-pressure variations and rock deformation. Particularly, flow of a compressible fluid along compliant hydraulic conduits, such as joints, fractures, or faults, is strongly affected by the associated deformation of the surrounding rock. We investigated and compared two alternative numerical modeling approaches that describe the transient fluid-pressure distribution along a single deformable fracture embedded in a rock matrix. First, we analyzed the coupled hydro-mechanical problem within the framework of Biot's poroelastic equations. Second, in a hybrid-dimensional approach, deformation characteristics of the surrounding rock were combined with a one-dimensional approximation of the fluid-flow problem to account for the high aspect ratios of fractures and the associated numerical problems. A dimensional analysis of the governing equations reveals that the occurring physical phenomena strongly depend on the geometry of the hydraulic conduit and on the boundary conditions. For the analyzed geometries, hydro-mechanical coupling effects dominate and convection effects can be neglected. Numerical solutions for coupled hydro-mechanical phenomena were obtained and compared to field data to characterize the fractured rock in the vicinity of an injection borehole. Either approach captures convection, diffusion, and hydro-mechanical effects, yet the hybrid-dimensional approach is advantageous due to its applicability to problems involving high-aspect-ratio features. For such cases, the modeling of pumping tests by means of the hybrid-dimensional approach showed that the observed inverse-pressure responses are the result of the coupling between the fluid flow in the fracture and the rock deformation caused by fluid-pressure variations along the fracture. Storage capacity as a single parameter of a fracture is insufficient to address all aspects of the coupling.

An alternative Biot's formulation for dissipative porous media with skeleton deformation

This paper presents an alternative formulation of Biot's theory to account for the elastic frame effects in a porous medium in which the acoustical properties of the fluid phase are predicted with an equivalent fluid model. This approach was originally developed for a double porosity medium. In this paper, the alternative formulation is applied to predict the transmission loss and absorption coefficient in the case of a single layer fibrous material, a multi-layer system, vibrating perforated plates, and porous composite materials. In the proposed formulation the coupling coefficients in Biot's poroelasticity equations are expressed in terms of the dynamic volumic mass and dynamic bulk modulus. By doing so, the elastic properties of the material frame are considered independently from the properties of the fluid. This formulation is implemented in the form of a transfer matrix algorithm which is validated against experimental data on sound absorption and sound transmission which are obtained for a range of various sound excitations and material arrangements. It is shown that this approach is able to predict accurately the acoustical properties of vibrating perforated plates and porous composites. The proposed approach is sufficiently general to be implemented in a finite element method.

On the causes of pressure oscillations in low‐permeable and low‐compressible porous media

SUMMARYNonphysical pressure oscillations are observed in finite element calculations of Biot's poroelastic equations in low‐permeable media. These pressure oscillations may be understood as a failure of compatibility between the finite element spaces, rather than elastic locking. We present evidence to support this view by comparing and contrasting the pressure oscillations in low‐permeable porous media with those in low‐compressible porous media. As a consequence, it is possible to use established families of stable mixed elements as candidates for choosing finite element spaces for Biot's equations. Copyright © 2011 John Wiley & Sons, Ltd.

Dispersion and attenuation due to scattering from heterogeneities of the frame bulk modulus of a poroelastic medium

While Biot theory can successfully account for the dispersion observed in sand sediments, the attenuation at high frequencies has been observed to increase more rapidly than Biot theory would predict. In an effort to account for this additional loss, perturbation theory is applied to Biot's poroelastic equations to model the loss due to the scattering of energy from heterogeneities in the sediment. A general theory for propagation loss is developed and applied to a medium with a randomly varying frame bulk modulus. The theory predicts that these heterogeneities produce an overall softening of the medium as well as scattering of energy from the mean fast compressional wave into incoherent fast and slow compressional waves. This theory is applied to two poroelastic media: a weakly consolidated sand sediment and a consolidated sintered glass bead pack. The random variations in the frame modulus do not have significant effects on the propagation through the sand sediment but do play an important role in the propagation through the consolidated medium.

A method of solving the stiffness problem in Biot's poroelastic equations using a staggered high-order finite-difference

In modelling elastic wave propagation in a porous medium, when the ratio between the fluid viscosity and the medium permeability is comparatively large, the stiffness problem of Biot's poroelastic equations will be encountered. In the paper, a partition method is developed to solve the stiffness problem with a staggered high-order finite-difference. The method splits the Biot equations into two systems. One is stiff, and solved analytically, the other is nonstiff, and solved numerically by using a high-order staggered-grid finite-difference scheme. The time step is determined by the staggered finite-difference algorithm in solving the nonstiff equations, thus a coarse time step may be employed. Therefore, the computation efficiency and computational stability are improved greatly. Also a perfect by matched layer technology is used in the split method as absorbing boundary conditions. The numerical results are compared with the analytical results and those obtained from the conventional staggered-grid finite-difference method in a homogeneous model, respectively. They are in good agreement with each other. Finally, a slightly more complex model is investigated and compared with related equivalent model to illustrate the good performance of the staggered-grid finite-difference scheme in the partition method.

Finite difference modeling of Biot's poroelastic equations at seismic frequencies

Across the seismic band of frequencies (loosely defined as <10 kHz), a seismic wave propagating through a porous material will create flow in the pore space that is laminar; that is, in this low‐frequency “seismic limit,” the development of viscous boundary layers in the pores need not be modeled. An explicit time stepping staggered‐grid finite difference scheme is presented for solving Biot's equations of poroelasticity in this low‐frequency limit. A key part of this work is the establishment of rigorous stability conditions. It is demonstrated that over a wide range of porous material properties typical of sedimentary rock and despite the presence of fluid pressure diffusion (Biot slow waves), the usual Courant condition governs the stability as if the problem involved purely elastic waves. The accuracy of the method is demonstrated by comparing to exact analytical solutions for both fast compressional waves and slow waves. Additional numerical modeling examples are also presented.

Finite element frequency-domain acoustic analysis of open-cell plastic foams

A numerical method using an impulsive excitation to excite and collect the acoustic frequency response functions of corrugated open-cell plastic foams is presented in this study. In the study, the Biot's poroelasticity equations are first phrased in terms of solid and fluid displacements and then transformed into the Laplace domain. With the use of the general quadrilateral or triangular elements, the stiffness matrixes for the foams in the Laplace domain are then derived by the Galerkin-type finite element method. After solving and obtaining the dynamic stiffness transfer functions for the foams that are excited by an impulsive pressure on their upper surface, the Laplace transformed stiffness transfer functions are then transformed into frequency domain called dynamic stiffness functions, which can be further used in calculating the acoustic properties of foams. For validations, the proposed Laplace transformed finite element method (LTFEM) is first used to predict the acoustic properties of a planar and rigidly backed open-cell plastic foam with infinite width and permeable upper surface. Thereafter, the influences of the thickness, the width-to-thickness ratio, and the roller as well as fixed side edge restraints on the planar foams’ acoustic properties are examined. Furthermore, the use of LTFEM to predict the acoustic properties of corrugated foams are demonstrated and discussed. Without the use of the additional acoustic field that is required in the earlier studies, results predicted in the present study are in good agreement with either the exact solutions or the experimental data.

A mixed finite-element method for solving the poroelastic Biot equations with electrokinetic coupling

SUMMARY In this paper, a numerical method for solving the Biot poroelastic equations is developed. These equations comprise acoustic (typically water) and elastic (porous medium frame) equations, which are coupled mainly through fluid/solid drag terms. This wave solution is coupled to a simplified form of Maxwell’s equations, which is solved for the streaming potential resulting from electrokinesis. The ultimate aim is to use the generated electrical signals to provide porosity, permeability and other information about the formation surrounding a borehole. The electrical signals are generated through electrokinesis by seismic waves causing movement of the fluid through pores or fractures of a porous medium. The focus of this paper is the numerical solution of the Biot equations in displacement form, which is achieved using a mixed finite-element formulation with a different finite-element representation for displacements and stresses. The mixed formulation is used in order to reduce spurious displacement modes and fluid shear waves in the numerical solutions. These equations are solved in the time domain using an implicit unconditionally stable time-stepping method using iterative solution methods amenable to solving large systems of equations. The resulting model is embodied in the MODELLING OF ACOUSTICS, POROELASTICS AND ELECTROKINETICS (MAPEK) computer model for electroseismic analysis.

Investigation of the convergence of the mixed displacement-pressure formulation for three-dimensional poroelastic materials using hierarchical elements.

Recently, a mixed pressure displacement [u, P] formulation based on Biot's poroelasticity equations has been presented for porous materials. This model leads to a reduction of the number of degrees of freedom required for the modeling of three-dimensional porous media in comparison to classical displacement-displacement [u, U] formulations. In this paper, an extension of the [u, P] formulation based on hierarchical elements is presented. First, a variant of the weak integral form of the [u, P] formulation is presented and its numerical implementation using hierarchical elements is detailed, together with the application of boundary and loading conditions. Numerical results are presented to show the accuracy and performance of the present approach. In particular, the importance of correctly capturing the coupling effects between the two phases is highlighted.