Solving the Buckley–Leverett equation with gravity in a heterogeneous porous medium

Immiscible two‐phase flow in porous media can be described by the fractional flow model. If capillary forces are neglected, then the saturation equation is a non‐linear hyperbolic conservation law, known as the Buckley–Leverett equation. This equation can be numerically solved by the method of Godunov, in which the saturation is computed from the solution of Riemann problems at cell interfaces. At a discontinuity of permeability this solution has to be constructed from two flux functions. In order to determine a unique solution an entropy inequality is needed. In this article an entropy inequality is derived from a regularisation procedure, where the physical capillary pressure term is added to the Buckley‐Leverett equation. This entropy inequality determines unique solutions of Riemann problems for all initial conditions. It leads to a simple recipe for the computation of interface fluxes for the method of Godunov. This revised version was published online in July 2006 with corrections to the Cover Date.     

Vanishing capillarity solutions of Buckley–Leverett equation with gravity in two-rocks’ medium

For the hyperbolic conservation laws with discontinuous-flux function, there may exist several consistent notions of entropy solutions; the difference between them lies in the choice of the coupling across the flux discontinuity interface. In the context of Buckley–Leverett equations, each notion of solution is uniquely determined by the choice of a “connection,” which is the unique stationary solution that takes the form of an under-compressive shock at the interface. To select the appropriate connection, following Kaasschieter (Comput Geosci 3(1):23–48, 1999), we use the parabolic model with small parameter that accounts for capillary effects. While it has been recognized in Cancès (Networks Het Media 5(3):635–647, 2010) that the “optimal” connection and the “barrier” connection may appear at the vanishing capillarity limit, we show that the intermediate connections can be relevant and the right notion of solution depends on the physical configuration. In particular, we stress the fact that the “optimal” entropy condition is not always the appropriate one (contrarily to the erroneous interpretation of Kaasschieter’s results which is sometimes encountered in the literature). We give a simple procedure that permits to determine the appropriate connection in terms of the flux profiles and capillary pressure profiles present in the model. This information is used to construct a finite volume numerical method for the Buckley–Leverett equation with interface coupling that retains information from the vanishing capillarity model. We support the theoretical result with numerical examples that illustrate the high efficiency of the algorithm.     

Analytic solutions to the advective contaminant transport equation with non-linear sorption

This paper considers advective transport of a soluble contaminant through saturated soil with non-linear sorption of the contaminant onto a stationary porous media. The non-linear sorption isotherms considered in the transport analysis are the Langmuir and Freundlich sorption isotherms. A special case of the Freundlich sorption isotherm is the linear sorption isotherm, and it is shown that in this case transport through a homogeneous soil results in the initial concentration profile simply being translated in the direction of the groundwater flow. However, when the sorption isotherm is non-linear the initial concentration profile distorts as it is translated with the groundwater flow, leading to the development of concentration shock fronts and rarefactions. Analytic solutions to the non-linear first-order hyperbolic equations are developed for a number of contaminant transport problems of practical significance. It is shown that in the case of the Langmuir sorption isotherms, shock fronts develop at the leading edge of the concentration profile while for the Freundlich sorption isotherm shock fronts may develop at either the leading or trailing edge of the concentration profile. Copyright © 1999 John Wiley & Sons Ltd.     

基于精确Riemann求解器的明满流过渡过程模拟

Preissmann窄缝法模拟明满流过渡过程方法简单,但存在明显的非物理振荡,抑制非物理振荡是该方法应用的关键。基于Godunov格式和精确Riemann求解器对明满流过渡过程进行模拟,针对Riemann问题代数恒等式在明满流交界处不光滑问题,提出了三阶收敛方法与二分法结合的迭代求解方法,保证迭代收敛至真实解;针对由于变量空间重构方法不能准确表达变量在空间中真实物理状态而导致的非物理振荡,提出了基于精确Riemann解的变量空间重构方法,准确表达激波间断在单元内的空间分布状态,从机理上抑制了非物理振荡。实例研究表明,数值计算结果与解析解或实测值吻合良好,研究成果为明满流过渡过程的高精度数值模拟提供了新的方法。     

基于精确Riemann求解器的明满流过渡过程模拟

Preissmann窄缝法模拟明满流过渡过程方法简单,但存在明显的非物理振荡,抑制非物理振荡是该方法应用的关键。基于Godunov格式和精确Riemann求解器对明满流过渡过程进行模拟,针对Riemann问题代数恒等式在明满流交界处不光滑问题,提出了三阶收敛方法与二分法结合的迭代求解方法,保证迭代收敛至真实解;针对由于变量空间重构方法不能准确表达变量在空间中真实物理状态而导致的非物理振荡,提出了基于精确Riemann解的变量空间重构方法,准确表达激波间断在单元内的空间分布状态,从机理上抑制了非物理振荡。实例研究表明,数值计算结果与解析解或实测值吻合良好,研究成果为明满流过渡过程的高精度数值模拟提供了新的方法。     

Uniqueness conditions in a hyperbolic model for oil recovery by steamdrive

In this paper we study a one-dimensional model for oil recovery by steamdrive. This model consists of two parts: a (global) interface model and a (local) steam condensation/capillary diffusion model. In the interface model a steam condensation front (SCF) is present as an internal boundary between the hot steam zone (containing water, oil and steam) and the cold liquid zone (containing only water and oil). Disregarding capillary pressure away from the SCF, a 2× 2 hyperbolic system arises for the water and steam saturation. This system cannot be solved uniquely without additional conditions at the SCF. To find such conditions we blow up the SCF and consider a parabolic transition model, including capillary diffusion. We study in detail the existence conditions for traveling wave solutions. These conditions provide the missing matching conditions at the SCF in the hyperbolic limit. We show that different transition models yield different matching conditions, and thus different solutions of the interface model. We also give a relatively straightforward approximation and investigate its validity for certain ranges of model parameters. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analytical and numerical solutions for carbonated waterflooding

We develop a Riemann solver for transport problems including geochemistry related to oil recovery. The example considered here concerns one-dimensional incompressible flow in porous media and the transport for several chemical components, namely H₂O, H⁺, OH^?, CO₂, \(\text {CO}_{3}^{2-}\), \(\text {HCO}_{3}^{-}\), and decane; they are in chemical equilibrium in the aqueous and oleic phases, leading to mass transfer of CO₂ between the oleic and aqueous phases. In our ionic model, we employ equations with zero diffusion coefficients. We do so because it is well known that for upscaled equations, the convection terms dominate the diffusion terms. The Riemann solution for this model can therefore be applied for upscaled transport processes in enhanced oil recovery involving geochemical aspects. In our example, we formulate the conservation equations of hydrogen, oxygen, hydrogen, and decane, in which we substitute regression expressions that are obtained by geochemical software. This can be readily done because Gibbs phase rule together with charge balance shows that all compositions can be rewritten in terms of a single composition, which we choose to be the hydrogen ion concentration (p H). In our example, we use the initial and boundary conditions for the carbonated aqueous phase injection in an oil reservoir containing connate water with some carbon dioxide. We compare the Riemann solution with a numerical solution, which includes capillary and diffusion effects. The significant new contribution is the effective Riemann solver we developed to obtain solutions for oil recovery problems including geochemistry and a variable total Darcy velocity, a situation in which fractional flow theory does not readily apply. We thus obtain an accurate solution for a carbonated waterflood, which elucidates some mechanisms of low salinity carbonated waterflooding.     

可压缩流体饱和孔隙介质中孔隙压力波传播数值分析

首次将高精度时空守恒元/解元方法推广到可压缩流体饱和孔隙介质中孔隙压力波传播的数值计算中。将孔隙度梯度从源（汇）项中分离,直接引入流通量,改进了理论模型。通过对孔隙介质激波问题的数值模拟,验证了方法的精度和有效性。在此基础上,提出了孔隙介质中二维黎曼问题,并揭示了孔隙压力波存在接触间断、激波、膨胀波、压缩波等复杂的结构特征。该成果对二氧化碳地质封存、二氧化碳提高石油采收率、页岩气压裂开采以及地震破裂过程的研究具有重要的理论与应用意义。     

Adjoint analysis of Buckley-Leverett and two-phase flow equations

This paper analyzes the adjoint equations and boundary conditions for porous media flow models, specifically the Buckley-Leverett equation, and the compressible two-phase flow equations in mass conservation form. An adjoint analysis of a general scalar hyperbolic conservation law whose primal solutions include a shock jump is initially presented, and the results are later specialized to the Buckley-Leverett equation. The non-convexity of the Buckley-Leverett flux function results in adjoint characteristics that are parallel to the shock front upstream of the shock and emerge from the shock front downstream of the shock. Thus, in contrast to the behavior of Burgers’ equation where the adjoint is continuous at a shock, the Buckley-Leverett adjoint, in general, contains a discontinuous jump across the shock. Discrete adjoint solutions from space-time discontinuous Galerkin finite element approximations of the Buckley-Leverett equation are shown to be consistent with the derived closed-form analytical solutions. Furthermore, a general result relating the adjoint equations for different (though equivalent) primal equations is used to relate the two-phase flow adjoints to the Buckley-Leverett adjoint. Adjoint solutions from space-time discontinuous Galerkin finite element approximations of the two-phase flow equations are observed to obey this relationship.     

A front-tracking method for the simulation of three-phase flow in porous media

Under certain physically reasonable assumptions, three-phase flow of immiscible, incompressible fluids can be described by a 2×2 nongenuinely nonlinear, hyperbolic system. We combine analytical solutions to the corresponding Riemann problem with an efficient front-tracking method to study Cauchy and initial-boundary value problems. Unlike finite difference methods, the front-tracking method treats all waves as discontinuities by evolving shocks exactly and approximating rarefactions by small entropy-violating discontinuities. This way, the method can track individual waves and give very accurate (or even exact) resolution of discontinuities. We demonstrate the applicability of the method through several numerical examples, including a streamline simulation of a water-alternating-gas (WAG) injection process in a three-dimensional, heterogeneous, shallow-marine formation.     

Analytical solutions to 1-D horizontal and vertical water infiltration in saturated/unsaturated soils considering time-varying rainfall

Analytical solutions to 1-D horizontal and vertical water infiltration in saturated/unsaturated soils are developed that can consider the variation of rainfall with time. In this model, water content and the permeability coefficient are assumed to be exponential functions of the pressure head, and diffusivity is constant. By means of Fourier integral transformation, the analytical solutions are expressed as infinite series. The steady-state solutions for horizontal and vertical infiltration in unsaturated soils are then derived. The solutions can consider both flux and pressure head boundaries. The solutions are easy to implement compared with numerical solutions, although the restricted assumptions may limit their applicability in some ways. The analytical solutions provide a reliable tool for checking the accuracy of various numerical methods under the condition of constant diffusivity. Finally, the analysis carried out in a case study indicates that the pressure head differences caused by the transient infiltration in both the horizontal and vertical directions can be estimated using the steady-state solutions, and the effect of gravity on water infiltration mainly depends on the boundary conditions.     

基于混合粒子群算法和能量法主动土压力计算

假定挡土墙后填土滑动面为通过墙踵的对数螺线滑动面,基于能量法,推导出了墙背倾斜、粗糙、墙后填土向上倾斜,适用于砂性土与黏性土的主动土压力上限解。以对数螺线通过斜坡的旋入角? 0和旋出角? h为变量,使用基于自然选择的混合粒子群优化算法对最危险滑动面进行全局搜索,从而获得主动土压力最优解。对于砂性土,将土压力系数与经典的极限分析上限解相比,发现在墙面倾角较小时两者基本一致,但当墙面倾角大于30°时,经典解明显偏小,而文中解与基于最优性原理的极限平衡解较接近。至于黏性土,对一工程实例进行计算,计算结果与实测值的相对误差为5.4%。     

A solution for the transition zone isosats in two-phase primary drainage in the presence of gravity

Primary drainage in a water-wet saturated medium in the absence of capillarity is typically a combination of shock (discontinuous) and rarefaction (continuous) waves. Using nonlinear relative permeability functions for the host fluid and the invading fluid leads to the existence of a shock wave front, and the degree of nonlinearity of the relative permeability functions has an inverse relationship with the size of the shock wave (i.e., difference of saturation between upstream and downstream of the shock wave), whereas for linear relative permeability functions, the shock wave size approaches 0. Injection of a lower-viscosity immiscible phase such as gas or solvent into a water-wet porous medium in the presence of large capillary pressure leads to development of an extended and growing saturation transition zone that follows the discontinuous shock wave front. In this article, a semianalytical solution for the position of equisaturation contours (isosats) in the transition zone in the presence of gravity is obtained for a set of linearized relative permeability functions. The capillary (diffusive) and buoyancy terms are neglected, and the generalized convective equation for mass conservation is obtained. The set of equations is then reduced to a one-dimensional steady-state differential equation through forcing the isosat formulation to obey mass conservation. This scheme allows the isosat distribution to be solved, and the case of injection into an axisymmetric geometry for a confined planar configuration is solved and presented. A finite element model was developed to demonstrate the reasonable agreement between analytical and numerical solutions.     

Analysis of DNAPL infiltration in a medium with a low-permeable lens

In this paper we study the infiltration of DNAPL in a porous medium containing a single low-permeable lens. Our aim is to determine whether or not DNAPL infiltrates into the lens. A key role is played by the capillary pressure: DNAPL cannot infiltrate into the lens unless the capillary pressure exceeds the entry pressure of the lens. In the model this is reflected by an interface condition, the extended capillary pressure condition. To derive analytical approximations we first consider a steady-state DNAPL plume in a homogeneous medium. This results in an estimate of the DNAPL plume width as a function of depth, and an asymptotic solution for small saturations. Assuming that the extent of the lens is much larger than the width of the unperturbed DNAPL plume in the homogeneous medium, we derive an explicit criterion for DNAPL infiltration into the lens in terms of a critical inflow rate. A numerical algorithm is presented in which the extended capillary pressure condition is incorporated. The numerical and analytical results show good qualitative agreement. This revised version was published online in August 2006 with corrections to the Cover Date.     

纳米硅对冻结黏性砂土强度影响试验研究

对加入1%纳米硅的黏性砂土进行温度-2℃、围压0.3~18 MPa的常规三轴压缩试验。试验结果表明：掺入纳米硅的冻结黏性砂土强度明显提高,在σ₃=3 MPa时强度提高甚至达到130%。将强度随围压的变化分成三个阶段：强化阶段,压融阶段,残余阶段。试验应力-应变曲线具有应变软化特性,修正的Duncan-Chang双曲线模型与其吻合良好。通过对修正的Duncan-Chang双曲线模型进行微分,分析得到初始切线模量随围压的变化可分成强化、压融和残余三个阶段。     

Variational inequalities for modeling flow in heterogeneous porous media with entry pressure

One of the driving forces in porous media flow is the capillary pressure. In standard models, it is given depending on the saturation. However, recent experiments have shown disagreement between measurements and numerical solutions using such simple models. Hence, we consider in this paper two extensions to standard capillary pressure relationships. Firstly, to correct the nonphysical behavior, we use a recently established saturation-dependent retardation term. Secondly, in the case of heterogeneous porous media, we apply a model with a capillary threshold pressure that controls the penetration process. Mathematically, we rewrite this model as inequality constraint at the interfaces, which allows discontinuities in the saturation and pressure. For the standard model, often finite-volume schemes resulting in a nonlinear system for the saturation are applied. To handle the enhanced model at the interfaces correctly, we apply a mortar discretization method on nonmatching meshes. Introducing the flux as a new variable allows us to solve the inequality constraint efficiently. This method can be applied to both the standard and the enhanced capillary model. As nonlinear solver, we use an active set strategy combined with a Newton method. Several numerical examples demonstrate the efficiency and flexibility of the new algorithm in 2D and 3D and show the influence of the retardation term. This work was supported in part by IRTG NUPUS.     

Uncoupling of coupled flows in soil—a finite element method

Coupled flow of water, chemicals, heat and electrical potential in soil are of significance in a variety of circumstances. The problem is characterized by the coupling between different flows, i.e. a flow of one type driven by gradients of other types, and by the dual nature of certain flows, i.e. combined convection and conduction. Effective numerical solutions to the problem are challenged due to the coupling and the dual nature. In this paper, we first present a general expression that can be used to represent various types of coupled flows in soil. A finite element method is then proposed to solve the generalized coupled flows of convection-conduction pattern. The unknown vector is first decomposed into two parts, a convective part forming a hyperbolic system and a conductive part forming a parabolic system. At each time step, the hyperbolic system is solved analytically to give an initial solution. To solve the multi-dimensional hyperbolic system, we assume that a common eigenspace exists for the coefficient matrices, so that the system can be uncoupled by transforming the unknown vector to the common eigenspace. The uncoupled system is solved by the method of characteristics. Using the solution of the hyperbolic system as the initial condition, we then solve the parabolic system by a Galerkin finite element method for space discretization and a finite difference scheme for time stepping. The proposed technique can be used for solving multi-dimensional, transient, coupled or simultaneous flows of convection-conduction type. Application to a flow example shows that the technique indeed exhibits optimality in convergence and in stability.     

Analysis of A New Model for Unsaturated Flow in Porous Media Including Hysteresis and Dynamic Effects

A new model for unsaturated flow in porous media, including capillary hysteresis and dynamic capillary effects, is analyzed. Existence and uniqueness of solutions are established and qualitative and quantitative properties of (particular) solutions are analyzed. Some results of numerical computations are given. The model under consideration incorporates simple ‘play’-type hysteresis and a dynamic term (time-derivative with respect to water content) in the capillary relation. Given an initial water content distribution, the model determines which parts of the flow domain are in drainage and which parts are in imbibition. The governing equations can be recast into an elliptic problem for fluid pressure and an evolution equation for water content. Standard methods are used to obtain numerical results. A comparison is given between J.R. Philip's semi-explicit similarity solution for horizontal redistribution in an infinite one-dimensional domain and solutions of the new model.     

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.