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.     

Large deformation elastic electro-osmosis consolidation of clays

This paper presents the theoretical background of an elastic electro-osmosis consolidation model for saturated soils experiencing large strains, which considers volumetric strains induced by changes in both the hydraulic and electric driven pore water flows. Three fully coupled governing equations, considering the soil mechanical behaviour, pore water transport and electrical field, and their numerical implementation within an updated Lagrangian finite element formulation, are presented. The proposed model is first verified against a classical one-dimensional analytical solution for electro-osmosis consolidation to demonstrate its accuracy and efficiency. Then, various numerical examples are investigated to study the deformation characteristics and time dependent evolution of excess pore pressure. Finally, the importance of considering large strains in a consistent and proper way is demonstrated, and differences compared to models based on small strain theory are highlighted.     

A large strain theory and its application in the analysis of the cone penetration mechanism

A computational method, based on an advanced elasto-plastic large strain formulation, well suited for the analysis of the cone penetration problem, is presented. A new approach of finite strain elasto-plastic analysis is employed.¹³ The basic (non-rate) constitutive relations are developed in a spatial reference space to preserve their physical significance. They are subsequently transformed in Lagrangian co-ordinates, and through simple time differentiation, their rate equations are introduced. The method is computationally implemented with the finite element method and special provisions are taken to allow for the moving boundary conditions of the problem.     

指数形式渗流定律下软土非线性大变形固结分析

考虑土中指数形式渗流定律以及土体的非线性固结特性,以超静孔隙水压力为变量在拉格朗日坐标系内建立了软土一维大变形固结问题的控制方程及其求解条件,并运用有限差分法获取其数值解答。在指数形式渗流定律退化为达西定律下,通过将差分解与已有的半解析解进行对比,验证了数值计算的可靠性。最后对指数形式渗流定律下软土一维非线性大变形固结性状进行计算分析,结果表明： 1时,软土的非线性大变形固结速率会随外载增大而减慢; 1时,软土的非线性大变形固结速率会随着外荷载的增加而加快;软土非线性大变形固结速率要比非线性小变形固结速率快,且差别会随荷载增大而加剧;此外,大变形固结理论的最终沉降值要小于小变形固结理论,且差别会随着荷载的增大而加剧。     

边坡大变形弹塑性有限元分析[Ⅱ]

本文应用Ｕｐｄａｔｅｄ Ｌａｇｒａｎｇｉａｎ有限元分析理论,分析了石龙庙滑坡的稳定性,其中包括滑坡的大变形,初始应力和超孔隙水压力。根据土的工程地质性质,滑坡体分为四层,土层被视为是弹塑性的,土的塑性屈服采用Ｄｒｕｃｋｅｒ－Ｐｒａｇｅｒ理想塑性屈服准则,挡土墙建成前后的滑坡应力和变形被分别分析和讨论,最后根据这些分析结果,提出了滑坡的整治方案。     

考虑渗透系数变化的非饱和土固结性状分析

基于Fredlund非饱和土一维固结理论,研究了有限厚度的表面透水透气、底面不透水不透气的线弹性和黏弹性非饱和土地基在加荷随时间指数性变化时的一维固结特性。分别得到了两类地基在固结过程中同时考虑液相、气相渗透系数非线性变化和仅考虑液相渗透系数变化两种情况下的半解析解答。利用典型算例进行计算,分析了不同情况下两类地基中超孔隙水、气压力消散以及地基固结度随时间的变化规律,并与不考虑渗透系数变化时的半解析解计算结果进行了对比。结果发现：固结过程中渗透系数呈非线性变化;只考虑液相渗透系数变化时,超孔隙气压力的消散变化不大,超孔隙水压力的消散加快;气相渗透系数变化对超孔隙气的消散产生明显影响,对超孔隙水压力消散影响不大。同时考虑液相和气相渗透系数变化时,土体中超孔隙水、气压力的消散均有明显变化,土体固结速度也相应加快;分析结果对非饱和土固结的进一步研究具有重要意义。     

Numerical and experimental studies of pressure-controlled cavity expansion in completely decomposed granite soils of Hong Kong

Compaction grouting is the injection of a viscous grout into a soil under high pressure, which then densifies the surrounding soil by reducing void space. Laboratory and field tests of compaction grouting have been carried out. In this paper, a numerical model is used to simulate the compaction grouting process with the primary purpose of investigating relationships among various control parameters, such as injection pressure, void ratio and excess pore water pressure at various radial distances from the injection point. The compaction process is treated as a cavity expansion process in the numerical simulation. The soil is modelled with an elasto-plastic Mohr–Coulomb model using the commercial finite element program ABAQUS. In addition to numerical simulations, pressure-controlled cavity expansion laboratory tests were carried out on completely decomposed granite (CDG) soil specimens. Data collected from laboratory tests are compared with the finite element simulation to validate the finite element analyses. Factors that control the compaction process, such as the coefficient of earth pressure (K), initial void ratio, number of loading cycles and effective confining pressure, are explored in the numerical simulations.     

非饱和土层一维固结特性分析

在Fredlund非饱和土的一维固结理论的基础上进行假设,由得出的液相及气相的控制方程、Darcy定律及Fick定律,采用Laplace 变换、逆变换等数学方法得到了大面积均布瞬时加载下表面为透水透气面、底面为不透水和不透气面的非饱和土层一维固结时间域内的超孔隙水压力、超孔隙气压力及土层沉降的解析解;应用典型算例,分析了不同气、水渗透系数比情况下土体超孔隙水压力、超孔隙气压力消散及土层沉降随时间的变化规律以及不同时间超孔隙水压力、超孔隙气压力消散随深度的变化规律。将得出的结果退化成相应的饱和土的解与太沙基饱和土固结理论结果比较,验证了其正确性。     

Numerical Analysis of Space–Time Finite Element Formulations for Miscible Displacements

A finite element formulation is proposed to approximate a nonlinear system of partial differential equations, composed by an elliptic subsystem for the pressure–velocity and a transport equation (convection–diffusion) for the concentration, which models the incompressible miscible displacement of one fluid by another in a rigid porous media. The pressure is approximated by the classical Galerkin method and the velocity is calculated by a post-processing technique. Then, the concentration is obtained by a Galerkin/least-squares space–time (GLS/ST) finite element method. A numerical analysis is developed for the concentration approximation. Then, stability, convergence and numerical results are presented confirming the a priori error estimates.     

软黏土层一维有限应变固结的超静孔压消散研究

根据土力学固结理论计算分析软黏土层固结过程的超静孔隙水压力值,确定软黏土体固结过程的强度增长,对排水固结法处理软土地基至关重要。软黏土层固结过程中土体变形较大时,有限应变固结理论和小应变固结理论计算分析软黏土固结所得结果差异较大。利用非线性有限元法及程序,通过对软黏土层固结工程算例的计算结果分析,研究了有限应变固结理论和小应变固结理论计算分析软黏土层一维固结超静孔压值消散的差异;探讨了软黏土体一维固结过程中,几何非线性、土体渗透性变化和压缩性变化对超静孔隙水压力消散的影响。研究结果表明,当土体的变形较大时,有限应变固结理论计算出的超静孔压要比小应变固结理论得到的值消散的更快。考虑土体固结过程中渗透性的变化时,超静孔压消散变慢;可用软黏土渗透性变化指数ck 反映渗透性变化对超静孔压消散的影响,渗透性变化指数ck值越小、超静孔压消散越慢。固结过程中软黏土压缩性的大小及变化也影响超静孔压的消散,可用软黏土的压缩指数cc反映固结过程中压缩性的大小及变化对超静孔压消散的影响,软黏土的压缩指数cc越小,固结过程软黏土层中的超静孔压消散越快。     

一维地下水非稳定流计算的配点法

传统的地下水数值计算方法(如有限元法和有限差分法)都需要网格或单元,网格生成需要占用大量的计算时间,求解所需数据量也较大。配点型无网格法摆脱了网格和单元的限制,只需节点信息,且节点布置灵活,易于实施。本文将配点型无网格法应用于解决一维地下水非稳定流计算问题,用MATLAB编制相应的程序,将结果与解析解和传统方法的计算结果相比较,计算得到的结果与解析解吻合很好,精度较高,计算过程简单,稳定性好。     

Finite deformation analysis of liquefaction-induced flow failure in soil embankments

Summary A finite element formulation is proposed for finite deformation dynamic analysis of saturated soil systems. The formulation is based on an updated Lagrangian approach and specifically considers the finite deformation effects on the flow of water through a soil element which undergoes a large deformation or rotation. A two-surface plasticity model is used to model the stress-strain behaviour of the soil skeleton. The proposed formulation has been implemented and is applied to simulate the response of a centrifuge model embankment. The calculated response is in good agreement with the observed behaviour of the soil embankment in the centrifuge test.     

Two‐dimensional finite element analysis of gravitational and lateral‐driven deformation in sedimentary basins

The paper deals with the modeling of some aspects, such as the formulation of constitutive equations for sediment material or finite element approach for basin analysis, related to mechanical compaction in sedimentary basins. In addition to compaction due to gravity forces and pore‐pressure dissipation, particular emphasis is given to the study of deformation induced by tectonic sequences. The numerical model relies upon the implementation of a comprehensive constitutive model for the sediment material formulated within the framework of finite poroplasticity. The theoretical model accounts for both hydromechanical and elasticity–plasticity coupling due to the effects of irreversible large strains. From the numerical viewpoint, a finite element procedure specifically devised for dealing with sedimentary basins as open systems allows to simulate within a two‐dimensional setting the process of sediment accretion or erosion. Several basin simulations are presented. The main objective is to analyze the behavior of a sedimentary basin during the different phases of its life cycle: accretion phase, pore‐pressure dissipation phase and compressive/extensional tectonic motions. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite element modeling of hydraulic fracturing in 3D

A procedure based on the finite element method is suggested for modeling of 3D hydraulic fracturing in the subsurface. The proposed formulation partitions the stress field into the initial stress state and an additional stress state caused by pressure buildup. The additional stress is obtained as a solution of the Biot equations for coupled fluid flow and deformations in the rock. The fluid flow in the fracture is represented on a regular finite element grid by means of “fracture” porosity, which is the volume fraction of the fracture. The use of the fracture porosity allows for a uniform finite element formulation for the fracture and the rock, both with respect to fluid pressure and displacement. It is demonstrated how the fracture aperture is obtained from the displacement field. The model has a fracture criterion by means of a strain limit in each element. It is shown how this criterion scales with the element size. Fracturing becomes an intermittent process, and each event is followed by a pressure drop. A procedure is suggested for the computation of the pressure drop. Two examples of hydraulic fracturing are given, when the pressure buildup is from fluid injection by a well. One case is of a homogeneous rock, and the other case is an inhomogeneous rock. The fracture geometry, well pressure, new fracture area, and elastic energy released in each event are computed. The fracture geometry is three orthogonal fracture planes in the homogeneous case, and it is a branched fracture in the inhomogeneous case.     

Electro-osmotic consolidation of soil with variable compressibility,hydraulic conductivity and electro-osmosis conductivity

In present study, the non-linear variations of soil compressibility, hydraulic and electro-osmosis conductivities were analyzed through laboratory experiments, and incorporated in a one-dimensional model. The analytical solutions for excess pore water pressure and degree of consolidation were derived, and numerical simulations were performed to verify its effectiveness. The results indicated that both the non-linear variations of hydraulic and electro-osmosis conductivities showed remarkable impacts on the excess pore water pressure and degree of consolidation, especially for soils with relative high compressibility. A further comparison with previous analytical solutions indicated that more accurate predictions could be obtained with the proposed analytical solutions.     

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.     

A semi-analytical solution to consolidation of unsaturated soils with the free drainage well

Based on Fredlund’s one-dimensional consolidation equation for unsaturated soil, Darcy’s law and Fick’s law, a semi-analytical solution was presented to the free drainage well with a finite thickness under application of uniform vertical loading and the boundary of the top and bottom surfaces impermeable to water and air. According to the polar governing equations of water and air phases and the boundary and initial conditions, the excess pore-air and pore-water pressures and the soil layer settlement in the Laplace transformed domain are obtained by performing the Laplace transform and utilizing the Bessel functions. Crump’s method is used to perform the inversion of Laplace transform in order to obtain numerical solutions in the real time domain. Finally, a typical example is given to illustrate the changes in the excess pore-air and pore-water pressures and soil layer settlement with time factor at different ratios of air–water permeability coefficient and/or different distances from the well.     

An operator‐split ALE model for large deformation analysis of geomaterials

Analysis of large deformation of geomaterials subjected to time‐varying load poses a very difficult problem for the geotechnical profession. Conventional finite element schemes using the updated Lagrangian formulation may suffer from serious numerical difficulties when the deformation of geomaterials is significantly large such that the discretized elements are severely distorted. In this paper, an operator‐split arbitrary Lagrangian–Eulerian (ALE) finite element model is proposed for large deformation analysis of a soil mass subjected to either static or dynamic loading, where the soil is modelled as a saturated porous material with solid–fluid coupling and strong material non‐linearity. Each time step of the operator‐split ALE algorithm consists of a Lagrangian step and an Eulerian step. In the Lagrangian step, the equilibrium equation and continuity equation of the saturated soil are solved by the updated Lagrangian method. In the Eulerian step, mesh smoothing is performed for the deformed body and the state variables obtained in the updated Lagrangian step are then transferred to the new mesh system. The accuracy and efficiency of the proposed ALE method are verified by comparison of its results with the results produced by an analytical solution for one‐dimensional finite elastic consolidation of a soil column and with the results from the small strain finite element analysis and the updated Lagrangian analysis. Its performance is further illustrated by simulation of a complex problem involving the transient response of an embankment subjected to earthquake loading. Copyright © 2007 John Wiley & Sons, Ltd.     

非均匀层状介质一维波动方程精确解的有限差分算法

平面波的传播问题通常可以归结为一维波动方程的定解问题。在非均匀介质中,即使简单的一维波动方程也需要借助于数值方法获得近似解。3层5点古典差分格式是计算偏微分方程一种常用算法,作为一种显式迭代格式,需要满足稳定性条件 ,其中 为波速, 为空间采样间隔, 为时间采样间隔。当 时, ,古典差分格式达到临界稳定状态。在这种情况下,平面波在 时间内的传播距离恰好等于空间采样间隔,差分格式真实地反映了平面波的传播原理,因而可以得到一维波动方程的精确解。但是,由于在非均匀介质中存在不连续的波阻抗界面,此方法不适于计算非均匀介质的波场。为了将临界稳定情况下的古典差分格式推广应用至非均匀层状介质,提出了一种能够处理波阻抗界面的有限差分格式,并应用傅里叶分析法得到其稳定性条件。模型算例验证了此算法的正确性。     

复杂电性结构大地电磁二维响应特征

为了改进计算区域离散化问题,本文利用自适应非结构化网格有限单元法求解二维地电结构下大地电磁场满足的加权余量表达式。在有限元求解电磁场的过程中,网格剖分越精细、计算精度越高,计算量也会越大。此外,结构化网格难以适应任意地形以及复杂地质构造。而自适应非结构化网格在电性变化剧烈的区域会自动加密,在电性缓变的区域则生成粗疏的网格,从而优化网格质量与数量。因此,文中引入COMSOL Multiphysics软件,以实现若干地电模型的构建及非结构化自由四边形单元网格化。将网格数据信息导入本文算法,计算大地电磁场响应,并与解析解及数值解对比。结果表明,基于非结构化网格的正演模拟精度高、适应性强,为计算区域网格化提供了新的方法。