Generalized solution to the anisotropic Mandel's problem

Existing solutions to Mandel's problem focus on isotropic, transversely isotropic, and orthotropic materials, the last two of which have one of the material symmetry axes coincide with the vertical loading direction. The classical plane strain condition holds for all these cases. In this work, analytical solution to Mandel's problem with the most general matrix anisotropy is presented. This newly derived analytical solution for fully anisotropic materials has all the three nonzero shear strains. Warping occurs in the cross sections, and a generalized plane strain condition is fulfilled. This solution can be applied to transversely isotropic and orthotropic materials whose material symmetry axes are not aligned with the vertical loading direction. It is the first analytical poroelastic solution considering mechanical general anisotropy of elasticity. The solution captures the effects of material anisotropy and the deviation of the material symmetry axes from the vertical loading direction on the responses of pore pressure, stress, strain, and displacement. It can be used to match, calibrate, and simulate experimental results to estimate anisotropic poromechanical parameters. This generalized solution is capable of reproducing the existing solutions as special cases. As an application, the solution is used to study the responses of transversely isotropic and orthotropic materials whose symmetry axes are not aligned with the vertical loading direction. Examples on anisotropic shale rocks show that the effects of material anisotropy are significant. Mandel-Cryer's effects are highly impacted by the degree of material anisotropy and the deviation of the material symmetry axes from the vertical loading direction.     

High‐order ADE scheme for solving the fluid diffusion equation in non‐uniform grids and its application in coupled hydro‐mechanical simulation

To improve the stability and efficiency of explicit technique, one proposed method is to use an unconditionally stable alternating direction explicit (ADE) scheme. However, the standard ADE scheme is only moderately accurate and restricted to uniform grids. This paper derives a novel high‐order ADE scheme capable of solving the fluid diffusion equation in non‐uniform grids. The new scheme is derived by performing a fourth‐order finite difference approximation to the spatial derivatives of the diffusion equation in non‐uniform grid. The implicit Crank‐Nicolson technique is then applied to the resulting approximation, and the subsequent equation is split into two alternating direction sweeps, giving rise to a new high‐order ADE scheme. Because the new scheme can be potentially applied in coupled hydro‐mechanical (H‐M) simulation, the pore pressure solutions from the new scheme are then sequentially coupled with an existing geomechanical simulator in the computer program Fast Lagrangian Analysis of Continua. This coupling procedure is called the sequentially explicit coupling technique based on the fourth‐order ADE scheme (SEA‐4). Verifications of well‐known consolidation problems showed that the new ADE scheme and SEA‐4 can reduce computer runtime by 46% to 75% to that of Fast Lagrangian Analysis of Continua's basic scheme. At the same time, the techniques still maintained average percentage error of 1.6% to 3.5% for pore pressure and 0.2% to 1.5% for displacement solutions and were still accurate under typical grid non‐uniformities. This result suggests that the new high‐order ADE scheme can provide an efficient explicit technique for solving the flow equation of a coupled H‐M problem, which will be beneficial for large‐scale and long‐term H‐M problems in geoengineering.     

Semi‐analytical solution to one‐dimensional consolidation for unsaturated soils with semi‐permeable drainage boundary under time‐dependent loading

This paper presents semi‐analytical solutions to Fredlund and Hasan's one‐dimensional consolidation of unsaturated soils with semi‐permeable drainage boundary under time‐dependent loadings. Two variables are introduced to transform two coupled governing equations of pore‐water and pore‐air pressures into an equivalent set of partial differential equations, which are easily solved by the Laplace transform. The pore‐water pressure, pore‐air pressure and settlement are obtained in the Laplace domain. Crump's method is adopted to perform the inverse Laplace transform in order to obtain semi‐analytical solutions in time domain. It is shown that the present solutions are more general and have a good agreement with the existing solutions from literatures. Furthermore, the current solutions can also be degenerated into conventional solutions to one‐dimensional consolidation of unsaturated soils with homogeneous boundaries. Finally, several numerical examples are provided to illustrate consolidation behavior of unsaturated soils under four types of time‐dependent loadings, including instantaneous loading, ramp loading, exponential loading and sinusoidal loading. Parametric studies are illustrated by variations of pore‐air pressure, pore‐water pressure and settlement at different values of the ratio of air–water permeability coefficient, depth and loading parameters. Copyright © 2017 John Wiley & Sons, Ltd.     

A simple analytical solution to one‐dimensional consolidation for unsaturated soils

This paper presents a simple analytical solution to Fredlund and Hasan's one‐dimensional (1‐D) consolidation theory for unsaturated soils. The coefficients of permeability and volume change for unsaturated soils are assumed to remain constant throughout the consolidation process. The mathematical expression of the present solution is much simpler compared with the previous available solutions in the literature. Two new variables are introduced to transform the two coupled governing equations of pore‐water and pore‐air pressures into an equivalent set of partial differential equations, which are easily solved with standard mathematical formulas. It is shown that the present analytical solution can be degenerated into that of Terzaghi consolidation for fully saturated condition. The analytical solutions to 1‐D consolidation of an unsaturated soil subjected to instantaneous loading, ramp loading, and exponential loading, for different drainage conditions and initial pore pressure conditions, are summarized in tables for ease of use by practical engineers. In the case studies, the analytical results show good agreement with the available analytical solution in the literature. The consolidation behaviors of unsaturated soils are investigated. The average degree of consolidation at different loading patterns and drainage conditions is presented. The pore‐water pressure isochrones for two different drainage conditions and three initial pore pressure distributions are presented and discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

On finite volume method implementation of poro‐elasto‐plasticity soil model

Accurate prediction of the interactions between the nonlinear soil skeleton and the pore fluid under loading plays a vital role in many geotechnical applications. It is therefore important to develop a numerical method that can effectively capture this nonlinear soil‐pore fluid coupling effect. This paper presents the implementation of a new finite volume method code of poro‐elasto‐plasticity soil model. The model is formulated on the basis of Biot's consolidation theory and combined with a perfect plasticity Mohr‐Coulomb constitutive relation. The governing equation system is discretized in a segregated manner, namely, those conventional linear and uncoupled terms are treated implicitly, while those nonlinear and coupled terms are treated explicitly by using any available values from previous time or iteration step. The implicit–explicit discretization leads to a linearized and decoupled algebraic system, which is solved using the fixed‐point iteration method. Upon the convergence of the iterative method, fully nonlinear coupled solutions are obtained. Also explored in this paper is the special way of treating traction boundary in finite volume method compared with FEM. Finally, three numerical test cases are simulated to verify the implementation procedure. It is shown in the simulation results that the implemented solver is capable of and efficient at predicting reasonable soil responses with pore pressure coupling under different loading situations. Copyright © 2015 John Wiley & Sons, Ltd.     

Pore‐scale modeling of fluid‐particles interaction and emerging poromechanical effects

A micro‐hydromechanical model for granular materials is presented. It combines the discrete element method for the modeling of the solid phase and a pore‐scale finite volume formulation for the flow of an incompressible pore fluid. The coupling equations are derived and contrasted against the equations of conventional poroelasticity. An analogy is found between the discrete element method pore‐scale finite volume coupling and Biot's theory in the limit case of incompressible phases. The simulation of an oedometer test validates the coupling scheme and demonstrates the ability of the model to capture strong poromechanical effects. A detailed analysis of microscale strain and stress confirms the analogy with poroelasticity. An immersed deposition problem is finally simulated and shows the potential of the method to handle phase transitions. Copyright © 2013 John Wiley & Sons, Ltd.     

The axisymmetric consolidation of a semi‐infinite transversely isotropic saturated soil

This paper presents an exact analytical solution to fully coupled axisymmetric consolidation of a semi‐infinite, transversely isotropic saturated soil subjected to a uniform circular loading at the ground surface. The analysis is under the framework of Biot's general theory of consolidation. First, the governing equations of consolidation are transformed into a set of equivalent partial differential equations with the introduction of two auxiliary variables. These partial differential equations are then solved using Hankel–Laplace integral transforms. Once solutions in the transformed domain have been obtained, the actual solutions in the physical domain for displacements and stress components of the solid matrix, pore‐water pressure and fluid discharge can be finally obtained by direct numerical inversion. The accuracy of the numerical solutions developed is confirmed by comparison with an existing exact solution for an isotropic and saturated soil that is a special case of the more general problem addressed. Numerical analyses are also presented to investigate the influence of the degree of material anisotropy on the consolidation settlement. Copyright © 2005 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.     

A direct boundary element method for plane strain poroelasticity

This paper presents a direct boundary element method (BEM), formulated in the Laplace transform space, for plane strain poroelasticity. The paper expands on work by Cheng and Liggett by recasting the theoretical foundation of BEM within the framework of Rice and Cleary's formulation of the Biot theory of poroelasticity. Furthermore, the numerical algorithm is generalized to deal with both interior and exterior domain problems, and a method for indirectly calculating the Cauchy principal value of the singular integrals is presented. Formulae for the stress and flux inside the domain are also derived. Finally, the algorithm is validated by comparing the numerical results with the analytical solution of a borehole subject to a far-field deviatoric stress (exterior domain) and with the solution of Mandel's problem (interior domain). These two examples provide a critical test of the algorithm.     

Two formulations for dynamic response of a cylindrical cavity in cross‐anisotropic porous media

Two formulations for calculating dynamic response of a cylindrical cavity in cross‐anisotropic porous media based on complex functions theory are presented. The basis of the method is the solution of Biot's consolidation equations in the complex plane. Employing two groups of potential functions for solid skeleton and pore fluid (each group includes three functions), the u–w formulation of Biot's equations are solved. Difference of these two solutions refers to use of two various potential functions. Equations for calculating stress, displacement and pore pressure fields of the medium are mentioned based on each two formulations. Copyright © 2009 John Wiley & Sons, Ltd.     

Controlling Factors of Organic Nanopore Development: A Case Study on Marine Shale in the Middle and Upper Yangtze Region,South China

The Upper Ordovician Wufeng-Lower Silurian Longmaxi and the Lower Cambrian Qiongzhusi shales are the major targets for shale gas exploration and development in China. Although the two organic-rich shales share similar distribution ranges and thicknesses, they exhibit substantially different exploration and development results. This work analyzed the nanopore structures of the shale reservoirs in this region. Pore development of 51 shale samples collected from various formations and locations was compared using the petromineralogical, geochemical, structural geological and reservoir geological methods. The results indicate that the reservoir space in these shales is dominated by organic pores and the total pore volume of micropores, mesopores, macropores in different tectonic areas and formations show different trends with the increase of TOC. It is suggested that organic pores of shale can be well preserved in areas with simple structure and suitable preservation conditions, and the shale with smaller maximum ancient burial depth and later hydrocarbon-generation-end-time is also more conducive to pore preservation. Organic pore evolution models are established, and they are as follows: ① Organic matter pore development stage, ② Early stage of organic matter pore destruction, and ③ late stage of organic matter pore destruction. The areas conducive to pore development are favorable for shale gas development. Research results can effectively guide the optimization and evaluation of favorable areas of shale gas.     

Dynamic Green's functions for an anisotropic poroelastic half-space

The dynamic responses of an anisotropic poroelastic half-space under an internal point load and fluid source are investigated in the frequency domain in this paper. By virtue of Fourier transform and Stroh formalism, the three-dimensional (3D) general solutions of the anisotropic Biot's coupling dynamics equations are derived in the frequency domain. Considering the two surface conditions, permeable and impermeable, the analytical solutions for displacement fields and pore pressure in half-space under a point source (point load or a fluid source) are obtained. When the material properties are isotropic, the numerical results of the poroelastic half-space are in excellent agreement with the existing analytical solutions. For anisotropic half-space cases, numerical results show the strong dependence of the dynamic Green's functions on the material properties.     

Estimation of the Shale Oil/Gas Potential of a Paleocene–Eocene Succession: A Case Study from the Meyal Area, Potwar Basin, Pakistan

The successful exploration and production of shale-gas resources in the United States and Canada sets a new possible solution towards the energy crisis presently affecting most countries of Asia. This study focuses on the use of well log and 2D seismic data for the characterization of the shale oil/gas potential of a Paleocene–Eocene succession — the Meyal area in the Potwar Basin of Pakistan. Two shaly plays are identified in Paleocene–Eocene strata in well logs using ΔLogR and modified ΔLogR cross-plot techniques. The results indicate that Paleocene shale(the Patala Formation) and the lower shaly part of Eocene limestone(Sakesar Formation) can be potentially mature source rocks. However, the thermal maturity modelling proves that only the Paleocene shale is mature. Our results also suggest that the maturity responses on ΔLogR models for the lower shaly part of the Eocene limestone are due to trapped hydrocarbons in the intra-formational fractures. Petroelastic/petrophysical analysis of the Patala Formation reveals two potential shale oil/gas zones on the basis of Young's modulus, Poisson's ratio, Brittleness index and Total Organic Content at an exploitation depth of 3980–3988 m. This work can provide valuable insight for estimating shale oil/gas potential in highly deformed basins not only in Asia but in other parts of the world.     

Analytical solution to axisymmetric consolidation of unsaturated soil stratum under equal strain condition incorporating smear effects

This paper proposes closed‐form analytical solutions to the axisymmetric consolidation of an unsaturated soil stratum using the equal strain hypothesis. Following the 1‐dimensional (1D) consolidation theory for unsaturated soil mechanics, polar governing equations describing the air and water flows are first presented on the basis of Fick's law and Darcy's law, respectively. The current study takes into account the peripheral smear caused by an installation of vertical drain. Separation of variables and Laplace transformation are mainly adopted in the analytical derivation to obtain final solutions. Then, the hydraulic conductivity ratio, the radius of influence zone and smear parameters influencing time‐dependent excess pore pressures, and the average degree of consolidation are graphically interpreted. In this study, a comparison made between the proposed equal strain results and the existing free strain results suggests that both hypotheses would deliver similar predictions. Moreover, it is found that the smear zone resulting from vertical drain installations would hinder the consolidation rate considerably.     

Transient response of a circular cavity in a poroelastic medium

This paper considers the transient response of a pressurized long cylindrical cavity in an infinite poroelastic medium. To obtain transient solutions, Biot's equations for poroelastodynamics are specialized for this problem. A set of exact general solutions for radial displacement, stresses, pore pressure and discharge are derived in the Laplace transform space by using analytical techniques. Solutions are presented for three different types of prescribed transient radial pressures acting on the surface of a permeable as well as an impermeable cavity surface. Time domain solutions are obtained by inverting Laplace domain solutions using a reliable numerical scheme. A detailed parametric study is presented to illustrate the influence of poroelastic material parameters and hydraulic boundary conditions on the response of the medium. Comparisons are also presented with the corresponding ideal elastic solutions to portray the poroelastic effects. It is noted that the maximum radial displacement and hoop stress at the cavity surface are substantially higher than the classical static solutions and differ considerably from the transient elastic solutions. Time histories and radial variations of displacement, hoop stress, pore pressure and fluid discharge corresponding to a cavity in two representative poroelastic materials are also presented.     

FINITE ELEMENT ANALYSES OF ANISOTROPIC POROELASTICITY: A GENERALIZED MANDEL'S PROBLEM AND AN INCLINED BOREHOLE PROBLEM

The finite element equations for non-linear, anisotropic poroelasticity are cast in the form of measurable engineering constants. Two problems of importance to the rock and petroleum industry are analysed by the FEM. First, the classical Mandel's problem with an extension to transversely isotropic case is investigated. Second, the problem of an inclined borehole is explored. In particular, the effect of material anisotropy on stress concentration near the wall with implication to borehole instability is examined in detail.     

Effect of wave non‐linearity on the standing‐wave‐induced seabed response

A standing wave in front of a seawall may reach a height more than twice of its incident component. When excess pore pressure occurs, it may even induce seabed instability, hence endangering the structure. This issue was studied previously using only linear wave theory. In this paper, standing‐wave theory to a second‐order approximation is applied, in order to demonstrate the differences between these two solutions. The spatial and temporal variations in the instantaneous pore pressure are first calculated, in addition to their vertical distributions. The effects of wave height, water depth and the degree of soil saturation on pore pressure distributions are then discussed, followed by the net pore pressure averaged over one wave cycle. The results suggest the existence of a residual pore pressure in the seabed and its net pore pressure can be used to estimate the wave‐induced liquefaction potential in a soil column. It also indicates that, in deep water, the second‐order solution predicts that a negative pore pressure at an antinode which may be greater than a positive pressure. Overall, the second‐order solution is found to agree better with the experimental results of the pore pressures available, compared to the linear solution. Copyright © 2000 John Wiley & Sons, Ltd.     

四川盆地下侏罗统陆相页岩气源储特征及耦合评价

四川盆地陆相页岩气勘探前景良好,目前已在川东北地区下侏罗统获得良好页岩气显示和工业气流.与海相页岩相比,陆相页岩具有相变快、岩相组合类型多(夹层发育)、有机质丰度低和源储配置关系复杂等特点.运用"源储耦合"研究思路,利用宏观-微观结合的储层表征技术对下侏罗统自流井组大安寨段页岩的烃源特征、储集特征及其耦合关系开展了分析...     

湘西北下古生界黑色页岩扫描电镜孔隙特征

利用扫描电镜对湘西北下古生界下寒武统牛蹄塘组和下志留统龙马溪组的黑色页岩样品进行了观察,进一步分析了黑色页岩中的微观孔隙类型、特征及其形成演化过程。区内两套黑色页岩内部发育多种储集空间类型,可概括分为3大类:矿物基质孔、有机质孔、微裂缝,其中矿物基质孔又可细分为粒间骨架孔、凝絮成因孔、溶蚀孔和基质晶间孔,有机质孔主要包括生物骨架孔和生烃残留孔。原始的沉积环境与成岩过程决定了泥页岩中的孔隙和微裂缝系统,不同的成岩阶段控制着页岩中孔隙结构的演化过程,进而影响着页岩气的赋存状态。泥页岩中这种复杂的微观孔隙网络系统为油气储集提供了有效空间,也为页岩气的渗流提供了主要通道,对页岩气的成藏和开发有重要意义。     

湘西北下古生界黑色页岩扫描电镜孔隙特征*

利用扫描电镜对湘西北下古生界下寒武统牛蹄塘组和下志留统龙马溪组的黑色页岩样品进行了观察,进一步分析了黑色页岩中的微观孔隙类型、特征及其形成演化过程。区内两套黑色页岩内部发育多种储集空间类型,可概括分为3大类:矿物基质孔、有机质孔、微裂缝,其中矿物基质孔又可细分为粒间骨架孔、凝絮成因孔、溶蚀孔和基质晶间孔,有机质孔主要包括生物骨架孔和生烃残留孔。原始的沉积环境与成岩过程决定了泥页岩中的孔隙和微裂缝系统,不同的成岩阶段控制着页岩中孔隙结构的演化过程,进而影响着页岩气的赋存状态。泥页岩中这种复杂的微观孔隙网络系统为油气储集提供了有效空间,也为页岩气的渗流提供了主要通道,对页岩气的成藏和开发有重要意义。