Dynamic response to fluid extraction from a poroelastic half‐space

In this study, the dynamic response of a poroelastic half‐space to a point fluid sink is investigated using Biot's dynamic theory of poroelasticity. Based on Biot's theory, the governing field equations are re‐formulated in frequency domain with solid displacement and pore pressure. In a cylindrical coordinate system, a method of displacement potentials for axisymmetric displacement field is proposed to decouple the Biot's field equations to three scalar Helmholtz equations, and then the general solution to axisymmetric problems are obtained. The full‐space fundamental singular solution for a point sink is also derived using potential methods. The mirror‐image method is finally applied to construct the fundamental solution for a point sink buried in a poroelastic half‐space. Furthermore, a numerical study is conducted for a rock, that is, Berea sandstone, as a representative example. Copyright © 2013 John Wiley & Sons, Ltd.     

Steady‐state analytical solutions of flow and deformation coupling due to a point sink within a finite fluid‐saturated poroelastic layer

An exact steady‐state closed‐form solution is presented for coupled flow and deformation of an axisymmetric isotropic homogeneous fluid‐saturated poroelastic layer with a finite radius due to a point sink. The hydromechanical behavior of the poroelastic layer is governed by Biot's consolidation theory. Boundary conditions on the lateral surface are specifically chosen to match the appropriate finite Hankel transforms and simplify the transforms of the governing equations. Ordinary differential equations in the transformed domain are solved, and then the analytical solutions in the physical space for the pore pressure and the displacements are finally obtained by using finite Hankel inversions. The analytical solutions at some special locations such as the top and bottom surfaces, lateral surface, and the symmetrical axis are given and analyzed. And a case study for the consolidation of a water‐saturated soft clay layer due to pumping is conducted. The analytical solution is verified against the finite element solution. Meanwhile, an analysis of coupled hydromechanical behavior is carried out herein. The presented analytical solution is an exact solution to the practical poroelastic problem within an axisymmetric finite layer. It can provide us a better understanding of the poroelastic behavior of the finite layer due to fluid extraction. Besides, it can be applied to calibrate numerical schemes of axisymmetric poroelasticity within finite domains. Copyright © 2017 John Wiley & Sons, Ltd.     

Deformation of a poroelastic layer overlying an elastic half‐space due to dip‐slip faulting

An analytical solution of the plane strain problem of the deformation of a homogeneous, isotropic, poroelastic layer of uniform thickness overlying a homogeneous, isotropic, elastic half‐space due to two‐dimensional seismic sources buried in the elastic half‐space has been obtained. The integral expressions for the displacements, stresses and pore pressure have been obtained using the stress function approach by applying suitable boundary conditions at the free surface and the interface. The solution obtained is in the Laplace–Fourier transform domain. The case of a vertical dip‐slip line dislocation for the oceanic crust model of Earth is studied in detail. Schapery's formula is used for the Laplace inversion and the extended Simpson's formula for the Fourier inversion. Diffusion of pore pressure in the layer is studied numerically. Contour maps showing the pore pressure in the poroelastic layer have been plotted. The effect of the compressibility of the solid and fluid constituents on pore pressure has also been studied. Copyright © 2015 John Wiley & Sons, Ltd.     

Propagation of Rayleigh waves in fluid‐saturated non‐homogeneous soils with the graded solid skeleton distribution

The propagation characteristic of Rayleigh waves in a fluid‐saturated non‐homogeneous poroelastic half‐plane is addressed. Based on Biot's theory for fluid‐saturated media, which takes the inertia, fluid viscosity, mechanical coupling, compressibility of solid grains, and fluid into account, the dispersion equations of Rayleigh waves in fluid‐saturated non‐homogeneous soils/rocks are established. By considering the shear modulus of solid skeleton variation with depth exponentially, a small parameter, which reflects the relative change of shear modulus, is introduced. The asymptotic solution of the dispersion equation expressing the relationship between the phase velocity and wave number is obtained by using the perturbation method. In order to analyze the effects of non‐homogeneity on the propagation characteristic of Rayleigh waves, the variation of the phase velocity with the wave number is presented graphically and discussed through numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic response of road pavement resting on a layered poroelastic half‐space to a moving traffic load

This paper is dedicated to study the dynamic response of a thin‐plate resting on a layered poroelastic half‐space under a moving traffic load. Based on the dynamic poroelastic theory of Biot, the general solutions of the homogeneous poroelastic foundation are obtained by Fourier translation. By using the transmission and reflection matrices method in the frequency domain, the equivalent stiffness of the layered poroelastic half‐space is presented. Kirchhoff's hypotheses are applied to obtain the vertical displacement of the thin plate. By using the inverse Fourier transform, the time domain solution is obtained. As an example of three layers, the influences of the load velocity, the material properties of poroelastic layers, and the flexural rigidity of the plate on the response of the pavement system are examined. Analyses show that a soft intermediate layer results in the significant increase of vertical displacement of road pavement. Comparison with the existing work validates the present model. Copyright © 2013 John Wiley & Sons, Ltd.     

Analytical solutions of a finite two‐dimensional fluid‐saturated poroelastic medium with compressible constituents

An analytical solution is proposed for transient flow and deformation coupling of a fluid‐saturated poroelastic medium within a finite two‐dimensional (2‐D) rectangular domain. In this study, the porous medium is assumed to be isotropic, homogeneous, and compressible. In addition, the point sink can be located at an arbitrary position in the porous medium. The fluid–solid interaction in porous media is governed by the general Biot's consolidation theory. The method of integral transforms is applied in the analytical formulation of closed‐form solutions. The proposed analytical solution is then verified against both exact and numerical results. The analytical solution is first simplified and validated by comparison with an existing exact solution for the uncoupled problem. Then, a case study for pumping from a confined aquifer is performed. The consistency between the numerical solution and the analytical solution confirms the accuracy and reliability of the analytical solution presented in this paper. The proposed analytical solution can help us to obtain in‐depth insights into time‐dependent mechanical behavior due to fluid withdrawal within finite 2‐D porous media. Moreover, it can also be of great significance to calibrate numerical solutions in plane strain poroelasticity and to formulate relevant industry norms and standards. Copyright © 2014 John Wiley & Sons, Ltd.     

Generalized poroelastic wellbore problem

This paper presents a novel analytical solution to the transient, z‐dependent, and asymmetric problem of an infinite wellbore drilled into a fluid‐saturated porous medium. The formulations are based on Biot's linear theory of poroelasticity, in which the dependency of poroelastic field variables to spatial coordinates as well as time domain is considered in the most general form. This gives flexibility to the solution in cases that cannot be analyzed using the conventional plane strain or symmetric models. One such case is when calculating the stress variations around an inclined wellbore where the far‐field stresses are acting over a finite vertical section. The results of our solution to this case with a three‐dimensional state of far‐field stress are used to analyze the stability of inclined wellbores passing through abnormally stressed formations. The presented solution is capable of finding expressions for fundamental solutions with stress or flow boundary conditions at the wellbore. These solutions are here adopted to analyze the pressure disturbances generated by multiprobe formation tester, a standard wireline device that is designed for downhole fluid sampling as well as estimating the directional permeabilities of subsurface earth formations. A comparison with the conventional solution for the relevant pressure diffusion equation indicates that the poroelastic effect is relatively significant in relation to the transient response of the pore pressure. Further, it is shown that the finite dimensions of sink probe would, to a great extent, contribute to the formation's pore pressure variations at its immediate proximity. Copyright © 2013 John Wiley & Sons, Ltd.     

Displacement ring load Green's functions for saturated porous transversely isotropic tri‐material full‐space

Based on the Biot's poroelastic theory and using scalar potential functions both the ring load and point load displacement Green's functions for a transversely isotropic saturated porous full‐space composed of an upper half‐space, a finite thickness middle layer and a lower half‐space is analytically presented for the first time. It is assumed that each region consists of a different transversely isotropic material. The equations of poroelastodymanics in terms of the solid displacements and the pore fluid pressure are uncoupled with the help of two scalar potential functions, so that the governing equations for the potential functions are either a second order wave equation or a repeated wave‐heat transfer equation of sixth order. With the aid of Fourier expansion with respect to circumferential direction and Hankel integral transforms with respect to the radial direction in cylindrical coordinate system, the response is determined in the form of line integrals in the real space, followed by theorem of inverse Hankel integral transforms. The solutions degenerate to a single phase elastic material, and the results are compared with previous studies, where an excellent agreement may be observed with the results provided in the literature. Some examples of displacement Green's functions are finally given to illustrate the solution. Copyright © 2016 John Wiley & Sons, Ltd.     

A solution around a backfilled cavity in a low‐permeability poroelastic medium with application in in situ heating tests

Thermo‐hydro‐mechanical responses around a cylindrical cavity drilled or excavated in a low‐permeability formation are studied when the cavity is subjected to a time‐dependent thermal loading. The cavity is considered backfilled after it is supported by casing or lining. Solutions of temperature, pore water pressure, stress, and displacement responses are analytically formulated based on Biot's consolidation theory with the assumption that the backfilling material, supporting material, and surrounding low‐permeability formation are poroelastic media. The solution is expressed in Laplace space, and numerical inversion techniques are used to find field variables in the real‐time domain. After the solution is verified with the numerical results, it is applied in a large‐scale in situ heating test – PRACLAY heating test – for a predictive reference calculation and an extensive parametric study. Another medium‐scale in situ heating test – ATLAS III heating test – is also analyzed using the solution, which provides reasonable agreement with measurements. The new analytical solution proves to be a convenient tool for a good understanding of the resulting coupled thermo‐hydro‐mechanical behavior and is therefore valuable for the interpretation of measured data in engineering practices and for a rational design of potential radioactive waste repositories. Copyright © 2016 John Wiley & Sons, Ltd.     

Large‐scale poroelastic fractured reservoirs modeling using the fast multipole displacement discontinuity method

An effective approach to modeling the geomechanical behavior of the network and its permeability variation is to use a poroelastic displacement discontinuity method (DDM). However, the approach becomes rather computationally intensive for an extensive system of cracks, particularly when considering coupled diffusion/deformation processes. This is because of additional unknowns and the need for time‐marching schemes for the numerical integration. The Fast Multipole Method (FMM) is a technique that can accelerate the solution of large fracture problems with linear complexity with the number of unknowns both in memory and CPU time. Previous works combining DDM and FMM for large‐scale problems have accounted only for elastic rocks, neglecting the fluid leak‐off from the fractures into the matrix and its influence on pore pressure and stress field. In this work we develop an efficient geomechanical model for large‐scale natural fracture networks in poroelastic reservoirs with fracture flow in response to injection and production operations. Accuracy and computational performance of the proposed method with those of conventional poroelastic DDM are compared through several case studies involving up to several tens of thousands of boundary elements. The results show the effectiveness of the FMM approach to successfully evaluate field‐scale problems for the design of exploitation strategies in unconventional geothermal and petroleum reservoirs. An example considering faults reveals the impact of reservoir compartmentalization because of sealing faults for both geomechanical and flow variables under elastic and poroelastic rocks. Copyright © 2015 John Wiley & Sons, Ltd.     

Linear coupled analysis of desiccation shrinkage in a double‐layer partially saturated medium: semi‐explicit solutions

Solutions are presented for the problem of isothermal dessiccation shrinkage in a double‐layer porous partially saturated medium. The rheological model taken into account is linear poroelastic. Hence the analysis is mainly focused on hydromechanical coupling effects and contrasts of mechanical and hydraulic properties between two materials: a low thickness skin comprised between the outer boundary and the reference porous material. Three one‐dimensional ideal structures are taken into account: a wall of finite thickness (cartesian geometry), a thick cylinder and a thick sphere. The solution of the time‐dependent problem is arrived at by applying Laplace transforms to the field variables. Exact solutions are obtained in Laplace transform space using Mathematica^© to solve the field equations whilst taking into account the continuity equations at the interface and the boundary conditions. The Talbot's modified algorithm has been performed to invert the Laplace transform solutions. A bibliographical and numerical study shows that this method is remarkably precise, stable and close to the analytical inversion. Results are presented using poroelastic data representative of a concrete material and involve a strong coupling effect between hydraulical and mechanical behaviours. A first approach elastic modelling of degradation process have been presented using a thin outer layer. Apart from emphasising the semi‐explicit solution utility due to accurate speed calculation, this paper deals with more complex problems than those which can be solved using purely analytical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

The method of fundamental solution for 3‐D wave scattering in a fluid‐saturated poroelastic infinite domain

By using a complete set of poroelastodynamic spherical wave potentials (SWPs) representing a fast compressional wave P_I, a slow compressional wave P_II, and a shear wave S with 3 vectorial potentials (not all are independent), a solution scheme based on the method of fundamental solution (MFS) is devised to solve 3‐D wave scattering and dynamic stress concentration problems due to inhomogeneous inclusions and cavities embedded in an infinite poroelastic domain. The method is verified by comparing the result with the elastic analytical solution, which is a degenerated case, as well as with poroelastic solution obtained using other numerical methods. The accuracy and stability of the SWP‐MFS are also demonstrated. The displacement, hoop stress, and fluid pore pressure around spherical cavity and poroelastic inclusion with permeable and impermeable boundary are investigated for incident plane P_I and SV waves. The scattering characteristics are examined for a range of material properties, such as porosity and shear modulus contrast, over a range of frequency. Compared with other boundary‐based numerical strategy, such as the boundary element method and the indirect boundary integral equation method, the current SWP‐MFS is a meshless method that does not need elements to approximate the geometry and is free from the treatment of singularities. The SWP‐MFS is a highly accurate and efficient solution methodology for wave scattering problems of arbitrary geometry, particularly when a part of the domain extends to infinity.     

Rocking vibrations of a rigid embedded foundation in a poroelastic half‐space

This paper presents an analysis of the rocking vibrations of a rigid cylindrical foundation embedded in poroelastic soil. The foundation is subjected to time‐harmonic rocking excitation and is perfectly bonded to the surrounding soil. The soil underlying the foundation base is represented by a homogeneous poroelastic half‐space, whereas the soil along the side of the foundation is modeled as an independent poroelastic stratum composed of a series of infinitesimally thin layers. The behavior of the soil is governed by Biot's poroelastodynamic theory. The contact surface between the foundation base and the poroelastic soil is assumed to be smooth and either fully permeable or impermeable. The dynamic interaction problem is solved by employing a simplified analytical method. Some numerical results for the nondimensional rocking dynamic impedance and nondimensional angular displacement amplitude of the foundation are presented to show the effect of nondimensional frequency of excitation, poroelastic material parameters, hydraulic boundary condition, depth ratio and mass ratio of the foundation. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamic responses of a pile embedded in a layered poroelastic half‐space to harmonic lateral loads

A single pile embedded in a layered poroelastic half‐space subjected to a harmonic lateral load is investigated in this study. Based on Biot's theory, the frequency domain fundamental solution for a horizontal circular patch load applied in the layered poroelastic half‐space is derived via the transmission and reflection matrices method. Utilizing Muki and Sternberg's method, the second kind of Fredholm integral equation describing the dynamic interaction between the layered half‐space and the pile subjected to a top harmonic lateral load is constructed. The proposed methodology is validated by comparing results of this paper with some existing results. Numerical results show that for a two‐layered half‐space, the thickness of the upper softer layer has pronounced influences on the dynamic response of the pile and the half‐space. For a three‐layered half‐space, the presence of a softer middle layer in the layered half‐space will enhance the compliance for the pile significantly, while a stiffer middle layer will diminish the dynamic compliance of the pile considerably. Copyright © 2009 John Wiley & Sons, Ltd.     

Stochastic computational modelling of highly heterogeneous poroelastic media with long‐range correlations

The compaction of highly heterogeneous poroelastic reservoirs with the geology characterized by long‐range correlations displaying fractal character is investigated within the framework of the stochastic computational modelling. The influence of reservoir heterogeneity upon the magnitude of the stresses induced in the porous matrix during fluid withdrawal and rock consolidation is analysed by performing ensemble averages over realizations of a log‐normally distributed stationary random hydraulic conductivity field. Considering the statistical distribution of this parameter characterized by a coefficient of variation governing the magnitude of heterogeneity and a correlation function which decays with a power‐law scaling behaviour we show that the combination of these two effects result in an increase in the magnitude of effective stresses of the rock during reservoir depletion. Further, within the framework of a perturbation analysis we show that the randomness in the hydraulic conductivity gives rise to non‐linear corrections in the upscaled poroelastic equations. These corrections are illustrated by a self‐consistent recursive hierarchy of solutions of the stochastic poroelastic equations parametrized by a scale parameter representing the fluctuating log‐conductivity standard deviation. A classical example of land subsidence caused by fluid extraction of a weak reservoir is numerically simulated by performing Monte Carlo simulations in conjunction with finite elements discretizations of the poroelastic equations associated with an ensemble of geologies. Numerical results illustrate the effects of the spatial variability and fractal character of the permeability distribution upon the evolution of the Mohr–Coulomb function of the rock. Copyright © 2004 John Wiley & Sons, Ltd.     

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

Nonphysical 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.     

Transient dynamic response of multilayered saturated media subjected to impulsive loadings

This paper presents a stable and efficient method for calculating the transient solution of layered saturated media subjected to impulsive loadings by means of the analytical layer element method. Starting with the field equations based on Biot's linear theory for porous, fluid‐saturated media, and the seepage continuity equation, an analytical layer element for a single layer is established by applying Laplace‐Hankel integral transform. The global stiffness matrix in the transform domain for a layered saturated half‐space subjected to a transient circular patch loading is obtained by assembling the layer elements of each layer. The displacements in the time domain are derived by Laplace‐Hankel inverse transform of the global stiffness matrix. Numerical examples are conducted to verify the accuracy of the method and to demonstrate the influences of type of transient loading, buried depth of loading, permeability, and stratification of materials on the transient response of the multilayered saturated poroelastic media.     

Modelling of steady‐state fluid flow in 3D fractured isotropic porous media: application to effective permeability calculation

In this paper, 3D steady‐state fluid flow in a porous medium with a large number of intersecting fractures is derived numerically by using collocation method. Fluid flow in the matrix and fractures is described by Darcy's law and Poiseuille's law, respectively. The recent theoretical development presented a general potential solution to model the steady‐state flow in fractured porous media under a far‐field condition. This solution is a hypersingular integral equation with pressure field in the fracture surfaces as the main unknown. The numerical procedure can resolve the problem for any form of fractures and also takes into account the interactions and the intersection between fractures. Once the pressure field and then the flux field in fractures have been determined, the pressure field in the porous matrix is computed completely. The basic problem of a single fracture is investigated, and a semi‐analytical solution is presented. Using the solution obtained for a single fracture, Mori‐Tanaka and self‐consistent schemes are employed for upscaling the effective permeability of 3D fractured porous media. Copyright © 2012 John Wiley & Sons, Ltd.     

Dynamic responses of a poroelastic half‐space from moving trains caused by vertical track irregularities

The vibrations of railway tracks on a poroelastic half‐space generated by moving trains are investigated through a vehicle–track–ground coupling model. The theoretical model incorporates a vehicle, a track, and a fully saturated poroelastic half‐space soil medium. The source of vibration excitation is divided into two components: the quasi‐static loads and the dynamic loads. The quasi‐static loads are related to the static component of the axle loads, whereas the dynamic loads are due to the dynamic wheel–rail interaction. A linear Hertizian contact spring is introduced between each wheelset and the rail to consider the dynamic loads. Biot's dynamic theory is used to characterize the poroelastic half‐space soil medium. Using the Fourier transform, the governing equations for the track–ground system are solved and the numerical results are presented for a single axle vehicle model. The different dynamic characteristics of the elastic soil medium and the saturated poroelastic medium are investigated. In addition, the different roles of the moving axle loads and the roughness‐induced dynamic loads are identified. It is concluded that the vibration level of the free field off the track predicted by the poroelastic soil medium is smaller than that predicted by the elastic soil medium for vehicle speed below the Rayleigh wave speed of the poroelastic half‐space, whereas it is larger for vehicle speed above the Rayleigh wave speed. The dynamic loads play an important role in the dynamic responses of the track–ground system. Copyright © 2010 John Wiley & Sons, Ltd.     

Extension of the Griffith's fracture criteria to saturated clays

Inglis [1] has solved the problem of distribution of stress in an elastic plate around an elliptical hole. His works clarify the role of cracks in the failure of an elastic material. However, his solution cannot be applied to saturated clay because he considers only total stresses, while, in saturated clay, the criterion of rupture should be expressed in terms of effective and not total stresses. The solution of Atkinson and Craster [2] using Biot's poroelasticity theory, shows that there is no high pore pressure in the vicinity of the crack tips for saturated clay. The major difference between this approach and the Biot's theory of is that, in saturated clay, strain is a function of the variation of the effective stress [3], while, in poroelastic media, strain is only a function of the variation of the total stress [4, Equation 2.2]. Also in their solution there is continuity between the pore fluid and the inner fluid in the crack. Their solution is valid for poroelastic media involving a movement of the pore fluid. In our solution there is no movement of the pore fluid (Undrained condition). In this paper we have solved the same problem as Inglis [1], but for the particular case of saturated clay obeying elastic law. By solving this problem we obtained the expressions for pore pressure, effective stress, total stress and displacements. The results show that not only the total stress but also the pore pressure and the effective stress are also high in the vicinity of the crack tips. A new failure criterion, based on Griffith's strain energy principle [5] and maximum tensile stress [6], valid for saturated clay is developed in this paper. Copyright © 2003 John Wiley & Sons, Ltd.