A linear constitutive model for unsaturated poroelasticity by micromechanical analysis

This paper extends the Biot theory of poroelasticity from the saturated to unsaturated case. The Biot phenomenological model uses parameters that are easily observable, such as the deformation of porous frame, total stress, pore pressure, and fluid specific discharge. Such model is preferred for engineering applications. At this macroscopic level, the extension of Biot theory from saturated to unsaturated is straightforward. The constitutive constants, however, are combined properties of solid, pore space, and fluids. In the unsaturated case, the constants are functions of the degree of saturation. Their measurements and tabulation over a range of saturation is generally not feasible for practical applications. In this work, a Biot-Willis type analysis is performed for the unsaturated case to provide a theory that the bulk material constants can be evaluated using laboratory measurable micromechanical constants under saturated condition, plus a capillary pressure curve (saturation versus suction pressure) typically available for unsaturated porous medium, without the need of measurement at each state of saturation. In particular, it is demonstrated that the surface energy contained in the meniscus interface manifests as a “capillary modulus,” given by the negative inverse slope of the capillary pressure curve. A rigorous analysis based on the thermodynamic variational energy approach is also conducted to lend theoretical support to the phenomenological approach. The presented model can bring a closure to the practical engineering modeling of the deformation of partially saturated porous medium that lacks the information of material constants over the range of saturation.     

复杂应力路径下流体饱和砂岩本构模型

高孔隙率砂岩在不同围压及被不同流体（水及油）饱和时表现出不同的应力-应变特性及不同的破坏过程。复杂应力路径下油和水饱和砂岩的力学试验表明,存在某一临界围压,随着围压的增加,饱和砂岩从以压剪破坏为主,逐渐转变为以孔隙坍塌破坏为主的破坏机制。通过力学试验结果的分析,以两种破坏理论对油和水饱和砂岩的破坏特征进行了力学解释,并应用经典塑性力学的盖帽模型建立了相应的本构模型,应用非线性有限元法进行了数值验证,研究结果表明,提出的模型与试验资料吻合良好,能较好地模拟复杂应力路径下油和水饱和砂岩的破坏机制及模拟注水采油引起的砂岩附加沉降。     

A three-phase XFEM model for hydraulic fracturing with cohesive crack propagation

A three-phase hydro-mechanical model for hydraulic fracturing is proposed. Three phases include: porous solid, fracturing fluid and host fluid. Discontinuity is handled using extended finite element method (XFEM) while cohesive crack model is used as fracturing criterion. Flow through fracture is defined as one-dimensional laminar flow, and flow through porous medium (host reservoir) is defined as two-dimensional Darcy flow. Coupling between two fluids in each space, fracture and pore, is captured through capillary pressure–saturation relationship, while the identical fluids in fracture and pore are coupled through a so-called leak-off mass transfer term. Coupling between fluids and deformation is captured through compatibility of volumetric strain of fluids within fracture and pore, and volumetric strain of the matrix. Spatial and temporal discretisation is achieved using the standard Galerkin method and the finite difference technique, respectively. The model is verified against analytical solutions available from literature. The leaking of fracturing fluid into the medium and suction of porous fluid into the fracture around the tip, are investigated. Sensitivity analyses are carried out for cases with slow and fast injection rates. It is shown that the results by single-phase flow may underestimate the leak-off.     

高压环境下流体非饱和渗流对含水合物沉积物的影响

沉积物与流体流动的性质是影响水合物形成和聚集的两个重要因素,为研究水合物在沉积地层中的赋存机制必须探明高压环境下含水合物沉积物在非饱和渗流条件下的相互影响关系。以逸度差为水合物反应驱动力,反应动力学常数为Arrhenius类型,建立了包括非饱和流体流动-沉积物特征-水合物形成动力学耦合的二维模型,从理论上研究了多孔介质内流体与沉积物参数如含水率、去饱和系数、水力分布和水合物饱和度等在孔隙内的相互影响规律。结果表明,在设定的条件下,随着反应的进行孔隙水压力随时间逐渐大,在相同条件下水合物饱和度与温度增加导致孔隙水压力变大,其中水合物饱和度的影响较小,而沉积物基质吸力、去饱和系数与本征动力学常数则与孔隙水压力成反向变化,其中本征动力学常数的影响较大。     

坦桑尼亚恩泽加绿岩带隐伏金矿床综合找矿模型

恩泽加绿岩带是坦桑尼亚重要的成矿带之一,产出了高登普莱德(Golden Pride)等多个大型金矿。通过对坦桑尼亚恩泽加绿岩带已发现的隐伏金矿床进行分析研究,结果显示恩泽加绿岩带隐伏金矿床构造控矿作用明显;高精度磁法和激电中梯测量结果显示含矿构造蚀变带具有高磁、低阻、相对高极化的特征;土壤地球化学测量结果显示成矿元素Au异常在空间分布上大于相应的含矿构造蚀变带的分布范围,其浓集方向与含矿构造蚀变带展布一致,并与指示元素As、Sb的套合性较好。根据该区域隐伏金矿床的地质、地球物理及地球化学特征,对该区隐伏金矿床的找矿标志进行了总结,明确了各找矿标志与隐伏金矿床之间的关系,并在此基础上构建了恩泽加绿岩带隐伏金矿床的综合找矿模型,该模型的应用显示了多种技术手段联合实施在寻找绿岩带型隐伏金矿床的有效性。     

鄂尔多斯盆地延长东区块本溪组砂砾岩储层孔隙及可动流体差异赋存特征

以鄂尔多斯盆地延长东区块本溪组砂砾岩储层为研究对象,通过铸体薄片、扫描电镜、高压压汞和核磁共振等方法分析储层孔隙与流体差异赋存特征,探讨流体差异赋存影响因素。结果表明：储层主要发育粒间溶孔和晶间孔,进一步分为裂缝-粒间溶孔型、晶间孔-溶孔型、溶孔型、微孔型储层,以裂缝-粒间溶孔型、晶间孔-溶孔型储层发育较好。储层可动流体饱和度为51.16%～86.82%,T₂谱为双峰态,呈左低右高型。孔隙发育差异对可动流体流体赋存呈现出不同特征,孔隙发育较差的溶孔型和微孔型储层仍具有较高的可动流体饱和度。粒间溶孔、最大连通孔喉半径、中值半径及最大进汞饱和度等影响储层可动流体饱和度,以裂缝-粒间溶孔型、晶间孔-溶孔型储层尤为显著。溶孔型、微孔型储层可动流体饱和度受铁白云石影响更为明显,铁白云石能够改善界面的润湿性,使流体在其纳米孔隙中仍具备较好的流动能力。     

Surface waves in a cylindrical borehole through partially-saturated porous solid

Propagation of surface waves is discussed in a cylindrical borehole through a liquid-saturated porous solid of infinite extent. The porous medium is assumed to be a continuum consisting of a solid skeletal with connected void space occupied by a mixture of two immiscible inviscid fluids. This model also represents the partial saturation when liquid fills only a part of the pore space and gas bubbles span the remaining void space. In this isotropic medium, potential functions identify the existence of three dilatational waves coupled with a shear wave. For propagation of plane harmonic waves along the axially-symmetric borehole, these potentials decay into the porous medium. Boundary conditions are chosen to disallow the discharge of liquid into the borehole through its impervious porous walls. A dispersion equation is derived for the propagation of surface waves along the curved walls of no-liquid (all gas) borehole. A numerical example is studied to explore the existence of cylindrical waves in a particular model of the porous sandstone. True surface waves do not propagate along the walls of borehole when the supporting medium is partially saturated. Such waves propagate only beyond a certain frequency when the medium is fully-saturated porous or an elastic one. Dispersion in the velocity of pseudo surface waves is analysed through the changes in consolidation, saturation degree, capillary pressure or porosity.     

Hydro‐mechanical modeling of cohesive crack propagation in multiphase porous media using the extended finite element method

In this paper, a numerical model is developed for the fully coupled hydro‐mechanical analysis of deformable, progressively fracturing porous media interacting with the flow of two immiscible, compressible wetting and non‐wetting pore fluids, in which the coupling between various processes is taken into account. The governing equations involving the coupled solid skeleton deformation and two‐phase fluid flow in partially saturated porous media including cohesive cracks are derived within the framework of the generalized Biot theory. The fluid flow within the crack is simulated using the Darcy law in which the permeability variation with porosity because of the cracking of the solid skeleton is accounted. The cohesive crack model is integrated into the numerical modeling by means of which the nonlinear fracture processes occurring along the fracture process zone are simulated. The solid phase displacement, the wetting phase pressure and the capillary pressure are taken as the primary variables of the three‐phase formulation. The other variables are incorporated into the model via the experimentally determined functions, which specify the relationship between the hydraulic properties of the fracturing porous medium, that is saturation, permeability and capillary pressure. The spatial discretization is implemented by employing the extended finite element method, and the time domain discretization is performed using the generalized Newmark scheme to derive the final system of fully coupled nonlinear equations of the hydro‐mechanical problem. It is illustrated that by allowing for the interaction between various processes, that is the solid skeleton deformation, the wetting and the non‐wetting pore fluid flow and the cohesive crack propagation, the effect of the presence of the geomechanical discontinuity can be completely captured. Copyright © 2012 John Wiley & Sons, Ltd.     

Numerical modeling of multiphase fluid flow in deforming porous media: A comparison between two- and three-phase models for seismic analysis of earth and rockfill dams

In this paper, a fully coupled numerical model is presented for the finite element analysis of the deforming porous medium interacting with the flow of two immiscible compressible wetting and non-wetting pore fluids. The governing equations involving coupled fluid flow and deformation processes in unsaturated soils are derived within the framework of the generalized Biot theory. The displacements of the solid phase, the pressure of the wetting phase and the capillary pressure are taken as the primary unknowns of the present formulation. The other variables are incorporated into the model using the experimentally determined functions that define the relationship between the hydraulic properties of the porous medium, i.e. saturation, relative permeability and capillary pressure. It is worth mentioning that the imposition of various boundary conditions is feasible notwithstanding the choice of the primary variables. The modified Pastor–Zienkiewicz generalized constitutive model is introduced into the mathematical formulation to simulate the mechanical behavior of the unsaturated soil. The accuracy of the proposed mathematical model for analyzing coupled fluid flows in porous media is verified by the resolution of several numerical examples for which previous solutions are known. Finally, the performance of the computational algorithm in modeling of large-scale porous media problems including the large elasto-plastic deformations is demonstrated through the fully coupled analysis of the failure of two earth and rockfill dams. Furthermore, the three-phase model is compared to its simplified one which simulates the unsaturated porous medium as a two-phase one with static air phase. The paper illustrates the shortcomings of the commonly used simplified approach in the context of seismic analysis of two earth and rockfill dams. It is shown that accounting the pore air as an independent phase significantly influences the unsaturated soil behavior.     

流固耦合问题及研究进展

传统的渗流理论一般假设流体流动的多孔介质骨架是完全刚性的,即在孔隙流体压力变化过程中,固体骨架不产生任何弹性或塑性变形,这时可将渗流作为非耦合问题来研究。这种简化虽然可以得到问题的近似解,但存在许多缺陷,而且也不切合生产实际。比如:在油田开采过程中,孔隙流体压力会逐渐降低,将导致储层内有效应力的变化,使储层产生变形。近年来,流固耦合问题越来越受到人们的重视,这方面的研究涉及许多领域。该文介绍了有关工程涉及到的流固耦合问题,重点针对油、气开采问题,介绍了储层流固耦合渗流的特点及研究方法和理论进展,包括单相、多相流体渗流的流固耦合数学模型及有限元数值模型。      

Frictionless contact formulation for dynamic analysis of nonlinear saturated porous media based on the mortar method

A finite element algorithm for frictionless contact problems in a two‐phase saturated porous medium, considering finite deformation and inertia effects, has been formulated and implemented in a finite element programme. The mechanical behaviour of the saturated porous medium is predicted using mixture theory, which models the dynamic advection of fluids through a fully saturated porous solid matrix. The resulting mixed formulation predicts all field variables including the solid displacement, pore fluid pressure and Darcy velocity of the pore fluid. The contact constraints arising from the requirement for continuity of the contact traction, as well as the fluid flow across the contact interface, are enforced using a penalty approach that is regularised with an augmented Lagrangian method. The contact formulation is based on a mortar segment‐to‐segment scheme that allows the interpolation functions of the contact elements to be of order N. The main thrust of this paper is therefore how to deal with contact interfaces in problems that involve both dynamics and consolidation and possibly large deformations of porous media. The numerical algorithm is first verified using several illustrative examples. This algorithm is then employed to solve a pipe‐seabed interaction problem, involving large deformations and dynamic effects, and the results of the analysis are also compared with those obtained using a node‐to‐segment contact algorithm. The results of this study indicate that the proposed method is able to solve the highly nonlinear problem of dynamic soil–structure interaction when coupled with pore water pressures and Darcy velocity. Copyright © 2015 John Wiley & Sons, Ltd.     

Revisiting the thermodynamics of hardening plasticity for unsaturated soils

A thermodynamically consistent extension of the constitutive equations of saturated soils to unsaturated conditions is often worked out through the use of a unique ‘effective’ interstitial pressure, accounting equivalently for the pressures of the saturating fluids acting separately on the internal solid walls of the pore network. The natural candidate for this effective interstitial pressure is the space averaged interstitial pressure. In contrast experimental observations have revealed that, at least, a pair of stress state variables was needed for a suitable framework to describe stress–strain–strength behaviour of unsaturated soils. The thermodynamics analysis presented here shows that the most general approach to the behaviour of unsaturated soils actually requires three stress state variables: the suction, which is required to describe the invasion of the soil by the liquid water phase through the retention curve; two effective stresses, which are required to describe the soil deformation at water saturation held constant. However a simple assumption related to the plastic flow rule leads to the final need of only a Bishop-like effective stress to formulate the stress–strain constitutive equation describing the soil deformation, while the retention properties still involve the suction and possibly the deformation. Commonly accepted models for unsaturated soils, that is the Barcelona Basic Model and any approach based on the use of an effective averaged interstitial pressure, appear as special extreme cases of the thermodynamic formulation proposed here.     

TWO-PHASE OIL MIGRATION IN COMPACTING SEDIMENTARY BASINS MODELLED BY THE FINITE ELEMENT METHOD

The upstream-weighted finite element method with lumped mass matrix is applied to the modelling of oil migration in compacting sedimentary basins. An implicit formulation is made in Lagrangian co-ordinates of a pressure, saturation and a temperature equation, which is based on immiscible two-phase flow of oil and water. The formulation accounts for the compaction of the sediments, the generation of oil from solid organic material (kerogen), the eventual pore space generated by kerogen breakdown, and the density variations of the fluids which may set up thermal convection. The model is validated by comparison with results from a one-dimensional (1D) fractional flow-based migration model. A 2D case example showing oil expulsion from source rocks, and the filling of a trap is presented. The mass balance of the model is easily checked because all oil in the basin originates from breakdown of kerogen. Compared with other alternatives, the simple upstream-weighted finite element method is suggested as a possible first choice for a numerical method for the modelling of oil migration in compacting sedimentary basins. It easily deals with the complex geometry of a basin, it yields reasonably good results, is simple to implement, and the same implementation applies to all spatial dimensions. © 1997 by John Wiley & Sons, Ltd.     

层状非饱和土地基的轴对称稳态动力响应

考虑土颗粒、孔隙流体的压缩性及各相物质间的黏性、惯性耦合作用,采用理论上更加严谨的饱和度本构关系式,建立了非饱和土的动力控制方程。该组方程与饱和土的经典Biot波动方程完全兼容,因此,具有更广泛的适用性。通过引入一组状态向量,在圆柱坐标系下将非饱和土满足的波动方程转化为状态方程组,利用Hankel变换,求解状态方程组,得到了传递矩阵。结合边界条件及层间接触连续条件,求解了层状非饱和地基的稳态动力响应问题。数值算例表明：土层对地表动位移的影响主要集中在临界深度范围内;软硬土层的相对次序对地表动位移幅值有显著影响;饱和度增大会引起土的物理参数发生相应的改变,尤其是动剪切模量通常降幅较大,而动剪切模量是决定位移幅值的关键因素,最终的结果是导致地表动位移幅值明显增大。     

Coupling of solid deformation and pore pressure for undrained deformation—a discrete element method approach

This paper presents a numerical scheme for fluid‐particle coupling that uses the discrete element method by taking into consideration solid deformation and pore pressure generation. A new water particle element is introduced to calculate pore water pressure due to porosity changes. The water particle element has the same size and shape as the solid element and experiences the same amount of deformation. On the basis of the effective stress principle at the element contact, the total force is equal to the sum of the force transmitted through the solid element contact and the water particle force due to pore water pressure. Analytical solutions of traditional soil mechanics problems, such as isotropic compression and consolidated triaxial undrained test, are used to quantitatively validate the proposed model. The numerical results show good agreement between the model and the analytical solutions. The model therefore provides an effective method to calculate pore pressure in a porous medium in discrete modeling.     

The effective stress concept and the evaluation of changes in pore air pressure under jacketed isotropic compression tests for dry sand based on the 2‐phase porous theory

The effective stress concept for solid‐fluid 2‐phase media was revisited in this work. In particular, the effects of the compressibility of both the pore fluid and the soil particles were studied under 3 different conditions, i.e., undrained, drained, and unjacketed conditions based on a Biot‐type theory for 2‐phase porous media. It was confirmed that Terzaghi effective stress holds at the moment when soil grains are assumed to be incompressible and when the compressibility of the pore fluid is small enough compared to that of the soil skeleton. Then, isotropic compression tests for dry sand under undrained conditions were conducted within the triaxial apparatus in which the changes in the pore air pressure could be measured. The ratio of the increment in the cell pressure to the increment in the pore air pressure, m, corresponds to the inverse of the B value by Bishop and was obtained during the step loading of the cell pressure. In addition, the m values were evaluated by comparing them with theoretically obtained values based on the solid‐fluid 2‐phase mixture theory. The experimental m values were close to the theoretical values, as they were in the range of approximately 40 to 185, depending on the cell pressure. Finally, it was found that the soil material with a highly compressible pore fluid, such as air, must be analyzed with the multi‐phase porous mixture theory. However, Terzaghi effective stress is practically applicable when the compressibilities of both the soil particles and the pore fluid are small enough compared to that of the soil skeleton.     

湿度应力场理论的证明

在某些工作环境下,多孔介质中流体同固体骨架之间可能发生复杂的物理化学相互作用,吸附就是各种相互作用的表观现象之一。各种相互作用都伴随能量的转化或转移。吸附引起的固体骨架的膨胀是由于固体和流体之间复杂的能量转化或转移引起的,在此认识基础上,从热力学第一定律和建立本构方程的一般方法出发,将湿度（吸附量）当作系统的状态变量,对缪协兴提出的湿度应力场理论作了严格证明,并详细分析了该理论模型的适用条件及其力学意义。认为湿度应力场本构模型同经典弹性本构模型一样,确定了一种理想物质模型,它也只是实际材料本构关系的一种理想化。     

孔隙流体对岩体变形局部化的影响及数值模拟研究

本文首先叙述了孔隙流体对固体变形局部化影响的若干研究进展,并利用三维连续体快速拉格朗日分析,模拟了孔隙压力对岩体变形局部化应变场(剪切带)的影响。数值模拟结果表明,孔隙压力对变形局部化应变场有较大影响,随着孔隙压力的增加剪切带趋于不明显,岩体试件倾向于发生拉伸破坏。      

Poro-viscoelasticity modelling based on upscaling quasistatic fluid-saturated solids

In this paper, quasistatic models are developed for the slow flow of compressible fluids through porous solids, where the solid exhibits fading memory viscoelasticity. Problems of this type are important in practical geomechanics contexts, for example, in the context of fluid flow through unconsolidated reservoir sands and of wellbore deformation behaviour in gas and oil shale reservoirs, all of which have been studied extensively. For slow viscous fluid flow in the poro-viscoelastic media we are able to neglect the dynamic effects related to inertia forces, as well as the dissipation associated with the viscous flows. This is in contrast to the vast body of work in the poro-elastic context, where much faster flow of the viscous fluids may give rise to memory effects associated with microflows in pores of the solid media. Such problems have been treated extensively in both the dynamic and quasistatic cases. We are taking a specific type of the porous medium subject to slow deformation processes possibly inducing moderate pressure gradients and flow rates characterised by negligible inertia effects. As the result of homogenisation of such a two-phase medium, we observe the fading memory behaviour in the Biot modulus which controls the pressure increase due to skeleton macroscopic deformation and pore fluid content. Although our derivation leads to a result which is consistent with the formal phenomenological approach proposed by Biot (J Appl Phys 23:1482–1498, 1962), we offer the reader more insight into the structure of the poro-viscoelastic constitutive relations obtained; in particular, we can show that the Biot compressibility evolves in time according to the creep function while the skeleton stiffness is driven by the relaxation function.     

Influence of gas generation in electro‐osmosis consolidation

This paper presents a numerical model for the elasto‐plastic electro‐osmosis consolidation of unsaturated clays experiencing large strains, by considering electro‐osmosis and hydro‐mechanical flows in a deformable multiphase porous medium. The coupled governing equations involving the pore water flow, pore gas flow, electric flow and mechanical deformation in unsaturated clays are derived within the framework of averaging theory and solved numerically using finite elements. The displacements of the solid phase, the pressure of the water phase, the pressure of the gas phase and the electric potential are taken as the primary unknowns in the proposed model. The nonlinear variation of transport parameters during electro‐osmosis consolidation are incorporated into the model using empirical expressions that strongly depend on the degree of water saturation, whereas the Barcelona Basic Model is employed to simulate the elasto‐plastic mechanical behaviour of unsaturated clays. The accuracy of the proposed model is evaluated by validating it against two well‐known numerical examples, involving electro‐osmosis and unsaturated soil behaviour respectively. Two further examples are then investigated to study the capability of the computational algorithm in modelling multiphase flow in electro‐osmosis consolidation. Finally, the effects of gas generation at the anode, the deformation characteristics, the degree of saturation and the time dependent evolution of the excess pore pressure are discussed. Copyright © 2016 John Wiley & Sons, Ltd.