Brine transport in porous media: On the use of Von Mises and similarity transformations

In this paper we use a Von Mises transformation to study brine transport in porous media. The model involves mass balance equations for fluid and salt, Darcy's law and an equation of state, relating the salt mass fraction to the fluid density. Application of the Von Mises transformation recasts the model equations into a single nonlinear diffusion equation. A further reduction is possible if the problem admits similarity. This yields a formulation in terms of a boundary value problem for an ordinary differential equation which can be treated by semi‐analytical means. Three specific similarity problems are considered in detail: (i) one‐dimensional, stable displacement of fresh water and brine in a porous column, (ii) flow of fresh water along the surface of a salt rock, (iii) mixing of parallel layers of brine and fresh water. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Some aspects of head‐variance evaluation

We compare two methods of evaluating head covariance for two‐dimensional steady‐state flow in mildly heterogeneous bounded rectangular aquifers. The quasi‐analytical approach, widely used in stochastic subsurface hydrology, is based on the Green's function representation, and involves numerical four‐fold integration. We compare this approach with a numerical solution of the two‐dimensional boundary‐value problem for head covariance. We show that the finite differences integration of this problem is computationally less expensive than numerical four‐fold integration of slowly‐convergent infinite series. This revised version was published online in July 2006 with corrections to the Cover Date.     

Upwind‐mixed methods for transport equations

In this paper, we continue our analysis of upwind‐mixed methods for advection–diffusion equations, which have been developed and analyzed by the first author over the past several years. In previous work, our analysis has been limited to low order approximating spaces, positive definite diffusion coefficients and Dirichlet boundary conditions. In this paper, we extend our results to higher order approximating spaces, possibly zero diffusion, and more physically realistic boundary conditions. Moreover, unlike previous papers, we avoid the use of Gronwall's Inequality, which can result in extremely large constants in the stability and error bounds. Numerical results are presented for constant, linear and quadratic approximating spaces. This revised version was published online in July 2006 with corrections to the Cover Date.     

A mixed finite element method for Hele-Shaw cell equations

A new numerical method to solve the system of equations describing two phase flow in a Hele-Shaw cell is presented. It combines a mixed finite element method, the method of subtraction of the singularity and a front tracking grid in a single computational strategy. This choice of discretization techniques is well motivated by the difficulties present in the system of equations and the physics of the problem. The new method was tested against analytical solutions and also by solving the Saffman–Taylor viscous fingering problem for finite and infinite mobility ratios. In both cases convergence under mesh refinement is achieved for the fingers developed from an initial sinusoidal interface. Finger splitting is observed for low values of the surface tension and high mobility ratio. Different explanations, based in our results, are provided for this phenomenon. This revised version was published online in July 2006 with corrections to the Cover Date.     

An upscaling method for one‐phase flow in heterogeneous reservoirs. A weighted output least squares (WOLS) approach

In this paper we study the problem of determining the effective permeability on a coarse scale level of problems with strongly varying and discontinuous coefficients defined on a fine scale. The upscaled permeability is defined as the solution of an optimization problem, where the difference between the fine scale and the coarse scale velocity field is minimized. We show that it is not necessary to solve the fine scale pressure equation in order to minimize the associated cost‐functional. Furthermore, we derive a simple technique for computing the derivatives of the cost‐functional needed in the fix‐point iteration used to compute the optimal permeability on the coarse mesh. Finally, the method is illustrated by several analytical examples and numerical experiments. This revised version was published online in July 2006 with corrections to the Cover Date.     

Traveling waves in a finite condensation rate model for steam injection

Steam drive recovery of oil is an economical way of producing oil even in times of low oil prices and is used worldwide. This paper focuses on the one-dimensional setting, where steam is injected into a core initially containing oil and connate water while oil and water are produced at the other end. A three-phase (oil, water, steam) hot zone develops, which is abruptly separated from the two-phase (oil + water) cold zone by the steam condensation front. The oil, water and energy balance equations (Rankine–Hugoniot conditions) cannot uniquely solve the system of equations at the steam condensation front. In a previous study, we showed that two additional constraints follow from an analysis of the traveling wave equation representing the shock; however, within the shock, we assumed local thermodynamic equilibrium. Here we extend the previous study and include finite condensation rates; using that appropriate scaling requires that the Peclet number and the Damkohler number are of the same order of magnitude. We give a numerical proof, using a color-coding technique, that, given the capillary diffusion behavior and the rate equation, a unique solution can be obtained. It is proven analytically that the solution for large condensation rates tends to the solution obtained assuming local thermodynamic equilibrium. Computations with realistic values to describe the viscous and capillary effects show that the condensation rate can have a significant effect on the global saturation profile, e.g. the oil saturation just upstream of the steam condensation front.     

A wavelet method for detecting S‐waves in seismic data

Seismic signals consist of several typically short energy bursts, called phases, exhibiting several patterns in terms of dominant frequency, amplitude and polarisation. We present a fast algorithm to detect the so‐called S‐phase in a three‐component seismic signal. This new approach combines traditional S‐phase detection methods and the discrete wavelet transform. This revised version was published online in July 2006 with corrections to the Cover Date.     

On simulating reactive chemical transport with dominating precipitated species controlling chemical equilibrium

We present an approach developed to compute chemical equilibrium and its corresponding reactive chemical transport when dominating precipitated species (DPS) exist. In computing chemical equilibrium, most models take the concentrations or activities of component species and precipitated species as the master variables. However, when the amount of a precipitated species is much larger than those of other species, small computational errors on this DPS concentration might introduce large errors on the concentrations of other species and would cause non‐mass‐conserved numerical results. To deal with the existence of DPS, we pick as master variables the concentration change, rather than the concentration, of DPS to compute chemical equilibrium. Since the concentration changes of DPS will no longer be much larger than the concentrations of other species in determining equilibrium, our approach is able to provide correct numerical results. We also employ the modified total analytical concentrations, rather than the total analytical concentrations, of aqueous components as the dependent variables in presenting and solving corresponding transport equations. Several examples are given to reveal the numerical problems associated with DPS and to demonstrate the success of our approach. This revised version was published online in July 2006 with corrections to the Cover Date.     

Inexact Newton methods and the method of lines for solving Richards' equation in two space dimensions

Richards' equation (RE) is often used to model flow in unsaturated porous media. This model captures physical effects, such as sharp fronts in fluid pressures and saturations, which are present in more complex models of multiphase flow. The numerical solution of RE is difficult not only because of these physical effects but also because of the mathematical problems that arise in dealing with the nonlinearities. The method of lines has been shown to be very effective for solving RE in one space dimension. When solving RE in two space dimensions, direct methods for solving the linearized problem for the Newton step are impractical. In this work, we show how the method of lines and Newton-iterative methods, which solve linear equations with iterative methods, can be applied to RE in two space dimensions. We present theoretical results on convergence and use that theory to design an adaptive method for computation of the linear tolerance. Numerical results show the method to be effective and robust compared with an existing approach.     

Calculating the internodal transmissibilities using finite analytic method and its application for multi‐phase flow in heterogeneous porous media

In the traditional numerical reservoir simulations, the internodal transmissibility is usually defined as the harmonic mean of the permeabilities of the adjacent grids. This definition underestimates the phase flux and the speed of the saturation front, especially for the strong heterogeneous case. In this article, the internodal transmissibility is recalculated according to the nodal analytic solution. The redefined internodal transmissibility can be used directly to calculate the multiphase flow in the numerical reservoir simulations. Numerical examples show that, compared to the traditional numerical methods, the proposed scheme makes the convergences much faster as the refinement parameter increases, and the accuracy is independent of the heterogeneity. Copyright © 2016 John Wiley & Sons, Ltd.     

分层沉积物河流-地下水系统水力连通演化

为揭示河流-地下水系统水力连通状态演化特征,以分层沉积物季节性失水河流-地下水系统为研究对象,解析水力连通状态判别标准,剖析系统内渗流变异机制,阐明水力连通状态变异类型与机制,完善演化理论框架。构建6个具有不同分层沉积物的河流-地下水系统渗流模型,利用数值模拟方法模拟水力连通状态演化及变异。结果表明：湿润锋曲线和饱和带前锋曲线是刻画水力连通状态演化的特征曲线;沉积物内渗透能力和毛细力作用差异是导致变异的原因。湿润锋曲线变异类型为陡坎型、缓坡型或交替型;饱和带前锋曲线变异类型为后移型、前突型或交替型。研究成果为进一步开展河流-地下水系统内水量评估、物质迁移和能量传输研究提供理论支撑。     

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.     

松散沉积物中水合物降压分解阵面演化实验及数值模拟

水合物分解阵面是水合物 开采现场监测关注重点之一,其传播速率与水合物开采效率密切相关,但是目前松散沉积物中水合物降压分解阵面演化数据积累明显不足。进行了松散沉积物中水合物降压分解阵面演化实验 ,采用时域反射技术测量了水合物饱和度随时间的变化,分析了水合物分解阵面的传播规律;提出了轴对称水合物降压分解数学模型并进行了适用性验证,通过敏感性分析探讨了影响因素对 水合物分解阵面演化过程的影响关系。基于室内实验和数值模拟认为：(1)水合物降压分解阵面传播距离与其传播时间平方根呈近似线性关系;(2)水合物降压分解阵面传播速率随其传播距离 的增加而迅速减小;(3)水合物降压分解阵面传播速率随着绝对渗透率基准值、气体饱和度初始值和环境温度的增加而增大,随着水合物饱和度初始值、下降指数和出口压力的增加而减小。     

A system-theoretical approach to selective grid coarsening of reservoir models

From a system-theoretical point of view and for a given configuration of wells, there are only a limited number of degrees of freedom in the input–output dynamics of a reservoir system. This means that a large number of combinations of the state variables (pressure and saturation values) are not actually controllable and observable from the wells, and accordingly, they are not affecting the input–output behavior of the system. In an earlier publication, we therefore proposed a control-relevant upscaling methodology that uniformly coarsens the reservoir. Here, we present a control-relevant selective (i.e. non-uniform) coarsening (CRSC) method, in which the criterion for grid size adaptation is based on ranking the grid block contributions to the controllability and observability of the reservoir system. This multi-level CRSC method is attractive for use in iterative procedures such as computer-assisted flooding optimization for a given configuration of wells. In contrast to conventional flow-based coarsening techniques our method is independent of the specific flow rates or pressures imposed at the wells. Moreover the system-theoretical norms employed in our method provide tight upper bounds to the ‘input–output energy’ of the fine and coarse systems. These can be used as an a priori error-estimate of the performance of the coarse model. We applied our algorithm to two numerical examples and found that it can accurately reproduce results from the corresponding fine-scale simulations, while significantly speeding up the simulation.     

A mathematical and numerical model for reactive fluid flow systems

We formulate mathematical and numerical models for multispecies, multi-phase and non-isothermal reactive fluid flow in porous media focusing on the chemical reactions and the transport of solutes. Mass conservation and stability in the time integration are emphasized. We use cell-centered finite volume differencing in space and an implicit Runge-Kutta method in time. Assuming two space dimensions, we introduce flux approximation for arbitrarily shaped convex quadrilaterals. On equidistant and variable sized rectangular grids we choose limited κ= related schemes to approximate the advective flux and the central difference scheme for the diffusive flux. On non-rectangular grids we recommend the VF9 scheme for the estimation of the diffusive flux. Our model exists as a code. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Comparison of Numerical Formulations for Two-phase Flow in Porous Media

Numerical approximation based on different forms of the governing partial differential equation can lead to significantly different results for two-phase flow in porous media. Selecting the proper primary variables is a critical step in efficiently modeling the highly nonlinear problem of multiphase subsurface flow. A comparison of various forms of numerical approximations for two-phase flow equations is performed in this work. Three forms of equations including the pressure-based, mixed pressure–saturation and modified pressure–saturation are examined. Each of these three highly nonlinear formulations is approximated using finite difference method and is linearized using both Picard and Newton–Raphson linearization approaches. Model simulations for several test cases demonstrate that pressure based form provides better results compared to the pressure–saturation approach in terms of CPU_time and the number of iterations. The modification of pressure–saturation approach improves accuracy of the results. Also it is shown that the Newton–Raphson linearization approach performed better in comparison to the Picard iteration linearization approach with the exception for in the pressure–saturation form.     

A dynamic network model for two‐phase immiscible flow

A dynamic pore‐scale network model is formulated for two‐phase immiscible flow. Interfaces are tracked through the pore throats using a modified Poiseuille equation, whereas special displacement rules are used at the pore bodies. The model allows interfaces to move over several pore‐lengths within a time step. Initial computational results are presented for a drainage experiment to demonstrate some of the features of the model. This revised version was published online in July 2006 with corrections to the Cover Date.     

Adaptive numerical simulation of the remediation of soil contamination by in‐situ gas venting

This paper describes a mathematical model and adaptive numerical simulation of the time‐dependent multiphase, multicomponent flow which occurs when a gas venting process is used to remove a volatile contaminant from a porous medium. The numerical simulation is adaptive in both space and time and involves the use of a finite element spatial discretization and the SPRINT2D software [6] for time integration. Results are presented which demonstrate the high quality of the simulation, both in terms of the length scales of the features that are resolved and the efficiency of the solutions relative to those obtained on fixed grids. This revised version was published online in July 2006 with corrections to the Cover Date.