Poroelastic Media Research Articles

Modeling of seismic wave scattering in poroelastic media

Understanding wave scattering in the earth is considered fundamental in describing seismic wave propagation and providing information on structural features of the earth’s interior. Petrophysical parameters (especially porosity and permeability) affect the reflection coefficients of subsurface interfaces, which can better explain the field data and infer the subsurface structure. However, the numerical solutions to the scattering problem for efficient modeling of wave propagation in poroelastic earth structures have limitations. We develop a numerical algorithm for solving the poroelastic scattering integral equations. Specifically, applying perturbation theory to Biot’s equations, the solutions are expressed by the Lippman-Schwinger integral equations, which can express the displacement and pressure fields. We derive the contrast-source integral equations of the decoupled poroelastic wave equations. We apply a conjugate-gradient fast Fourier transform method for fast solutions of the integral equations. We show that despite the complexity of the geologic structure, the numerical method enables the modeling of the displacement and pressure fields in the frequency and time domains. We determine that the wave scattering problem for the Biot model provides a good description to understand the earth’s interior.

Numerical Simulations of the Hydraulic Fracture Propagation in Poroelastic Media Using the Coupled Hydro-Mechanical Field-Enriched Finite Element Method

Numerical Simulations of the Hydraulic Fracture Propagation in Poroelastic Media Using the Coupled Hydro-Mechanical Field-Enriched Finite Element Method

Comment on permeability conditions in finite element simulation of bone fracture healing

The most popular model of the bone healing considers the fracture callus as poroelastic medium. As such it requires an assumption of the callus’ external permeability. In this work a systematic study of the influence of the permeability of the callus boundary on the simulated bone healing progress is performed. The results show, that these conditions starts to play significant role with the decrease of the callus size. Typically enforced impermeability inhibits the progress of healing during simulation. A remedy for this effect is imposing drainage conditions at the callus’ boundary.

Finite domain solution of a hydraulic fracture in a permeable rock

In this work, we present a domain-based algorithm to simulate the propagation of a plane-strain hydraulic fracture in a zero-toughness permeable elastic medium. The algorithm utilizes a domain-based method to solve the elasticity equations and integrates a multi-scale tip asymptote, which is particular to hydraulic fractures, into this framework. This integration is key to accurately model the energy dissipation and the fluid leak-off in the fracture tip region. The algorithm combines a 2D finite volume method (FVM) for solving the elasticity equation with a 1D FVM for solving the nonlinear lubrication equation. Incorporating the far-field asymptotics and using a moving-mesh scheme reduces the computational burden while improving the accuracy of the scheme. The paper concludes with an analysis of the numerical results. This study demonstrates the potential of this domain-based approach for modeling hydraulic fractures in poroelastic media.

Master Curves for Poroelastic Spherical Indentation With Step Displacement Loading

Abstract Theoretical and numerical analyses are conducted to rigorously construct master curves that can be used for interpretation of displacement-controlled poroelastic spherical indentation test. A fully coupled poroelastic solution is first derived within the framework of Biot’s theory using the McNamee–Gibson displacement function method. The fully saturated porous medium is assumed to consist of slightly compressible solid and fluid phases and the surface is assumed to be impermeable over the contact area and permeable everywhere else. In contrast to the cases in our previous studies with an either fully permeable or impermeable surface, the mixed drainage condition yields two coupled sets of dual integral equations instead of one in the Laplace transform domain. The theoretical solutions show that for this class of poroelastic spherical indentation problems, relaxation of the normalized indentation force is affected by material properties through weak dependence on a single-derived material constant only. Finite element analysis is then performed in order to examine the differences between the theoretical solution, obtained by imposing the normal displacement over the contact area, and the numerical results where frictionless contact between a rigid sphere and the poroelastic medium is explicitly modeled. A four-parameter elementary function, an approximation of the theoretical solution with its validity supported by the numerical analysis, is proposed as the master curve that can be conveniently used to aid the interpretation of the poroelastic spherical indentation test. Application of the master curve for the ramp-hold loading scenario is also discussed.

Coupled finite element analysis of the dynamics of poroelastic media considering the relative fluid acceleration

AbstractThis paper mainly discusses the dynamics of poroelastic media using the finite element analysis based on the u‐v‐p full formulation, where , , and p denote the solid displacement, relative fluid velocity with respect to solid velocity, and pore fluid pressure. It incorporates the effect of relative fluid acceleration with respect to solid acceleration on soil dynamic response. The ‐‐p formulation is first verified through the comparison with the analytical solution. After that, the response of one‐dimensional saturated soil column and two‐dimensional partially saturated soil layer subjected to vertical loading with different combinations of soil permeability and loading frequency are investigated using the analyses with the ‐‐p and ‐p formulations, based on which the effect of relative pore fluid motion is discussed and the differences in the soil response between two kinds of analysis approaches are examined. Results reveal that the rapid fluid flow has a pronounced effect on soil dynamics and the soil with a greater degree of saturation is more sensitive to the increase of loading frequency and soil permeability. The soil dynamic response can basically be divided into two main categories that represent insignificant and significant flow motion depending on loading frequency and soil permeability. Furthermore, despite vertical loading, horizontal soil response can also be affected by the large relative pore fluid flow when loading frequency and soil permeability are large. In addition, the simplified formulation can still be applicable to predict dynamics of poroelastic media in cases with high loading frequency and low soil permeability.

Kinematic response of floating single piles in saturated and nearly-saturated soil to P-waves

This paper presents closed-form expressions for the analysis of floating single piles subjected to vertically propagating harmonic P-waves. The foundation soil is treated as poroelastic medium, and can be either fully-saturated or nearly-saturated. We show that the existence of non-continuous air phase in the form of air bubbles dissolved in the pore water will have noteworthy effect on the vertical kinematic response of piles. Through parametric analyses we demonstrate that hydromechanical analysis techniques that consider soil as fully saturated may significantly over-estimate pile head displacements for frequencies relevant to seismic P-waves. In addition, we quantify differences in the response between end-bearing and floating piles, and show that these differences are more prominent when soil is not fully saturated. Although the proposed expressions have been derived while considering elastic soil and pile response, findings relevant to the effect of the degree of soil saturation to the vertical kinematic response of piles are generally applicable to any coupled stress-flow analysis method.

Fluid-poroviscoelastic structure interaction problem with nonlinear geometric coupling

We investigate weak solutions to a fluid-structure interaction (FSI) problem between the flow of an incompressible, viscous fluid modeled by the Navier-Stokes equations, and a poroviscoelastic medium modeled by the Biot equations. These systems are coupled nonlinearly across an interface with mass and elastic energy, modeled by a reticular plate equation, which is transparent to fluid flow. We provide a constructive proof of the existence of a weak solution to a regularized problem. Next, a weak-classical consistency result is obtained, showing that the weak solution to the regularized problem converges, as the regularization parameter approaches zero, to a classical solution to the original problem, when such a classical solution exists. While the assumptions in the first step only require the Biot medium to be poroelastic, the second step requires additional regularity, namely, that the Biot medium is poroviscoelastic. This is the first weak solution existence result for an FSI problem with nonlinear coupling involving a Biot model for poro(visco)elastic media.

Transient computational homogenization of heterogeneous poroelastic media with local resonances

AbstractA computational homogenization framework is proposed for solving transient wave propagation in the linear regime in heterogeneous poroelastic media that may exhibit local resonances due to microstructural heterogeneities. The microscale fluid‐structure interaction problem and the macroscale are coupled through an extended version of the Hill‐Mandel principle, leading to a variationally consistent averaging scheme of the microscale fields. The effective macroscopic constitutive relations are obtained by expressing the microscale problem with a reduced‐order model that contains the longwave basis and the so‐called local resonance basis, yielding the closed‐form expressions for the homogenized material properties. The resulting macroscopic model is an enriched porous continuum with internal variables that represent the microscale dynamics at the macroscale, whereby the Biot model is recovered as a special case. Numerical examples demonstrate the framework's validity in modeling wave transmission through a porous layer.

Modelling of railway ballast as a poro-elastic medium and its effects on sleeper sound radiation

Conventional railway track is laid in a layer of coarse stones known as ballast. Due to the gaps between the stones, the railway ballast behaves as a porous acoustic material, with absorptive properties that are important for the noise produced by the railway system. Sound radiated by the bottom of the sleepers may also be transmitted through the ballast. Moreover, during a train pass-by the ballast can vibrate, which may contribute to the radiated noise. To consider all three effects simultaneously, the ballast is treated here as a poro-elastic medium. To obtain the properties of the ballast as a porous medium, transfer function measurements are performed in a vertical impedance tube to determine the surface impedance and absorption coefficient of different depths of reduced scale ballast. The complex wavenumber is also determined through transfer function measurements within the ballast in the vertical tube. Porosity and flow resistivity are determined by non-acoustic measurements, and the Johnson-Champoux-Allard model for porous materials is then fitted to the measured wavenumbers, allowing the remaining parameters to be determined. Comparison is made with the measured impedance and normal incidence absorption coefficient. The model is then used to predict the diffuse field absorption coefficients of both the reduced scale ballast and full-size ballast, which are compared with measurements, showing acceptable agreement. Finally, the effects on the sleeper radiation of introducing the poro-elastic ballast model are estimated and the sensitivity of the results to a practical range of ballast properties is assessed. The ballast vibration increases the sleeper radiation at low frequency, and especially between 120 and 280 Hz. This is caused by the structural vibration of the ballast driven by the sleepers, which has a resonance around 250 Hz. The acoustic velocity within the ballast is 180° out of phase with the structural velocity, and this leads to a small reduction in the sound radiation around 100 Hz. These effects could not be predicted using a surface impedance model for the ballast. Compared with the effect of the ballast vibration and sound transmission, the absorption of the ballast has a much smaller influence on the sleeper radiation.

An energy stable and mass-conservative numerical method for gas flow in poroelastic media

An energy stable and mass-conservative numerical method for gas flow in poroelastic media

Thermoelastic interactions in a micropolar poroelastic medium

The present problem investigates the two-dimensional disturbances in a homogeneous, isotropic, micropolar thermoelastic saturated porous medium under the purview of Green-Lindsay (GL) theory of thermoelasticity. The medium is assumed to be unstrained and unstressed initially and has uniform temperature. The formulation is subjected to a moving thermal shock. Normal mode technique is employed to obtain the exact expressions for the displacement components, stresses, pore pressure and temperature. Numerical computations have been carried out with the help of MATLAB software and the results are illustrated graphically. Comparisons are made to show the effects of time, velocity of the load, micropolar and porous parameters on the resulting quantities. The outcomes point out a strong impact of these parameters on the physical quantities and agree with the boundary conditions. Some particular cases of interest have been deducted from the current investigation to validate the results. The present work is useful and valuable for analysis of problems involving thermal shock, micropolar and poroelastic deformations.

Fluid identification in fractured media with genetic algorithm

The fluid identification of fracture media is susceptible to factors such as observation data, fluid sensitivity parameters, and inversion algorithms. However, appropriate nonlinear algorithms can be applied to reduce errors generated during the inversion process, thus helping to improve the reliability of fluid identification. In this paper, analysis and comparison are made to various reflection coefficient approximate formulas, of which the applicability and accuracy are demonstrated in fluid-saturated poroelastic media, and the objective function is constructed further from Russell's reflection coefficient formula. In this sense, the genetic algorithm (GA) can achieve the pre-stack inversion of seismic waves. Based on the inversion results such as P-, S-wave velocity, and density, the Russell fluid indicator, Poisson's ratio, and Poisson impedance are combined to identify the water-, gas-, and oil-bearing fracture. Finally, the applicability and reliability of the method are demonstrated through numerical calculations in some regions of Marmousi2 model and the field data. The results show that the seismic data can be preliminarily analyzed by the inversion method, followed by combining the Russell fluid indicator, Poisson's ratio, and Poisson impedance, effectively reducing the multiplicity of the inversion results and improving the reliability of fluid identification in fractured media.

Semi-analytical Solutions for Wellbores with Graded Skin Zones in Poroelastic Media

Semi-analytical Solutions for Wellbores with Graded Skin Zones in Poroelastic Media

New unconditionally stable and high-accuracy finite element procedures for high-frequency solid-fluid coupled dynamic systems

To achieve high-precision computation of the dynamic response of saturated poroelastic media under high-frequency loads, this paper develops new, unconditionally stable and high-accuracy finite element procedures based on the u-w and u-U solid-fluid coupled dynamic systems. In the procedures the solution domain is spatially discretized using the standard finite element method, while a precise time step integration method is introduced for temporal domain computations. A matrix-based spectral analysis is conducted to validate the stability of the proposed procedures. Subsequently, an error estimation and convergence analysis of the procedures are performed. Four classical examples involving different load types with varying time dependences are used to validate the numerical performance of the proposed procedures. Through rigorous comparison with analytical/reference solutions, the proposed procedures have been convincingly demonstrated to accurately capture the relative motion between the solid skeleton and the pore fluid under high-frequency loads. Furthermore, the procedures have remarkable advantages in both accuracy and efficiency compared to the commonly employed Newmark schemes.

Numerical analysis of the indentation of a poro-elastoplastic half space by a rigid cylinder

Poro-elasto-plastic cylindrical indentation as a means for material characterization is explored numerically in this work. When a rigid cylinder is being pressed into a semi-infinite, fully saturated poroelastic medium via step displacement loading, undrained and drained elastic constants are reflected in the responses at the early and late times, while hydraulic diffusivity is manifested in the transient behavior. Such an indentation test can therefore be used in principle to identify material parameters. For geomaterials, plastic deformation may occur due to the indentation action. How the hydromechanically coupled indentation response is affected by plastic deformation is investigated here using finite element analysis. We consider the problem to be plane strain with frictionless contact. Drucker-Prager failure criteria with and without a cap are considered. We show that if there is only limited plastic deformation, the concept of using the transient response for determining hydraulic diffusivity may be extended from poroelasticity to poro-elasto-plasticity.

Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media

Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media

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.

Lagrangian coherent structures control solute dispersion in heterogeneous poroelastic media

Lagrangian coherent structures control solute dispersion in heterogeneous poroelastic media

Willis coupling in one-dimensional poroelastic laminates

We employ the Baker–Campbell–Hausdorff formula to derive closed-form expressions for the effective properties, including emergent Willis coupling, of a one-dimensional heterogeneous poroelastic medium consisting of a periodically repeating two-layer unit-cell. In contrast to the elastic and fluidic analogs, the Willis coupling of this periodic poroelastic medium does not vanish in the low-frequency limit. However, the effective wavenumber and impedance of this asymmetric lamellar material demonstrate symmetric reflection and absorption behavior that is indicative of symmetric structures in the low-frequency limit due to the effect of Darcy’s law on the dynamic effective density, which is inversely proportional to frequency. These closed-form expressions are validated against results obtained by direct numerical evaluation. The scattering coefficients, particularly the two reflection coefficients for incidence from either side of a finite length asymmetric structure, are different at non-zero frequencies but still in the metamaterial limit and are correct when the Willis coupling is included. The results show that asymmetry in poroelastic layers results in direction-dependent absorption coefficients, one of which could be optimized based on layer properties and asymmetry factors. As a consequence, the frequency range of validity of these scattering coefficients is wider when the Willis coupling matrix is accounted for than in its absence. This work paves the way for better control of elastic and acoustic waves in multiphase materials by considering poroelastic behavior.