Conditional Stochastic Analysis for Multiphase Trsnsport in Randomly Heterogeneous,variably Saturated Media

Groundwater contamination of organics has recently become a problem of growing concern over the resulting health and environmental problems. In general, the multiphase system of nonaqueous phase liquid (NAPL), water and air has to be studied in order to realistically describe the movement of such materials in the subsurface. Numerous models have been developed to study multiphase flow and/or multispecies transport in porous media. However, using models to study the influence of medium heterogeneity on such flow and transport is only a recent event. It has been demonstrated for single-phase flow and transport in saturated and unsaturated media that the study of medium heterogeneity is amenable to stochastic analysis. In this paper, we extend our Eulerian–Lagrangian stochastic theory for single-phase transport to the problem of multiphase–multispecies transport in randomly heterogeneous media under the conditions that the flow is steady-state and the phases are in local chemical equilibrium. We present theoretical expressions to describe the first two conditional moments of the random concentration of any species in any phase. Though they reveal some of the fundamental properties and help gaining insight into the nature of the problem, these expressions cannot be evaluated without either high resolution Monte Carlo simulation or approximation (closure). Therefore, we propose two sets of workable approximations, one being a weak approximation and the other being a linearized pseudo-Fickian approximation. The former yields a nonlinear integro-differential equation for the first conditional moment and the latter yields a linear differential equation. Then the second moments can be computed from explicit expressions from either the weak or pseudo-Fickian approximation.     

Electrical Resistivity Index in Multiphase Flow through Porous Media

The simultaneous flow of two phases through a three-dimensional porous medium is calculated by means of a Lattice-Boltzmann algorithm. The time-dependent phase configurations can be derived and also macroscopic quantities such as the relative permeabilities. When one phase only is supposed to be conductive, the Laplace equation which governs electrical conduction can be solved in each phase configuration; an instantaneous value of the macroscopic conductivity is obtained and it is averaged over many configurations. The influence of saturation on the resistivity index is studied for six different samples and two viscosity ratios. The saturation exponent is systematically determined. The numerical results are also compared to other possible models and also to experimental results; finally, they are discussed and criticized.     

Steady-State Source Flow in Heterogeneous Porous Media

The flow of fluids in heterogeneous porous media is modelled by regarding the hydraulic conductivity as a stationary random space function. The flow variables, the pressure head and velocity field are random functions as well and we are interested primarily in calculating their mean values. The latter had been intensively studied in the past for flows uniform in the average. It has been shown that the average Darcy's law, which relates the mean pressure head gradient to the mean velocity, is given by a local linear relationship. As a result, the mean head and velocity satisfy the local flow equations in a fictitious homogeneous medium of effective conductivity. However, recent analysis has shown that for nonuniform flows the effective Darcy's law is determined by a nonlocal relationship of a convolution type. Hence, the average flow equations for the mean head are expressed as a linear integro-differential operator. Due to the linearity of the problem, it is useful to derive the mean head distribution for a flow by a source of unit discharge. This distribution represents a fundamental solution of the average flow equations and is called the mean Green function G _d (x). The mean head G _d(x) is derived here at first order in the logconductivity variance for an arbitrary correlation function (x) and for any dimensionality d of the flow. It is obtained as a product of the solution G _d ⁽⁰⁾(x) for source flow in unbounded domain of the mean conductivity K _A and the correction _d (x) which depends on the medium heterogeneous structure. The correction _d is evaluated for a few cases of interest.Simple one-quadrature expressions of _d are derived for isotropic two- and three-dimensional media. The quadratures can be calculated analytically after specifying (x) and closed form expressions are derived for exponential and Gaussian correlations. The flow toward a source in a three-dimensional heterogeneous medium of axisymmetric anisotropy is studied in detail by deriving ₃ as function of the distance from the source x and of the azimuthal angle . Its dependence on x, on the particular (x) and on the anisotropy ratio is illustrated in the plane of isotropy (=0) and along the anisotropy axis ( = /2).The head factor k ^* is defined as a ratio of the head in the homogeneous medium to the mean head, k ^*=G _d ⁽⁰⁾/G _d= _d ^–1. It is shown that for isotropic conductivity and for any dimensionality of the flow the medium behaves as a one-dimensional and as an effective one close and far from the source, respectively, that is, lim _x0 k ^*(x) = K _H/K _A and lim _x k ^*(x) = K ^efu/K _A, where K _A and K _H are the arithmetic and harmonic conductivity means and K ^efu is the effective conductivity for uniform flow. For axisymmetric heterogeneity the far-distance limit depends on the direction. Thus, in the coordinate system of (x) principal directions the limit values of k ^* are obtained as . These values differ from the corresponding components of the effective conductivities tensor for uniform flow for = 0 and /2, respectively. The results of the study are applied to solving the problem of the dipole well flow. The dependence of the mean head drop between the injection and production chambers on the anisotropy of the conductivity and the distance between the chambers is analyzed.     

Large Scale Mixing for Immiscible Displacement in Heterogeneous Porous Media

We derive a large scale mixing parameter for a displacement process of one fluid by another immiscible one in a two-dimensional heterogeneous porous medium. The mixing of the displacing fluid saturation due to the heterogeneities of the permeabilities is captured by a dispersive flux term in the large scale homogeneous flow equation. By making use of the stochastic approach we develop a definition of the dispersion coefficient and apply a Eulerian perturbation theory to determine explicit results to second order in the fluctuations of the total velocity. We apply this method to a uniform flow configuration as well as to a radial one. The dispersion coefficient is found to depend on the mean total velocity and can therefore be time varying. The results are compared to numerical multi-realization calculations. We found that the use of single phase flow stochastics cannot capture all phenomena observed in the numerical simulations.     

A Stochastic Model for Particulate Suspension Flow in Porous Media

A population balance model for a particulate suspension transport with size exclusion capture of particles by porous rock is derived. The model accounts for particle flux reduction and pore space accessibility due to restriction for large particles to move through smaller pores – a particle is captured by a smaller pore and passes through a larger pore. Analytical solutions are obtained for a uniform pore size medium, and also for a medium with small pore size variation. For both cases, the equations for averaged concentrations significantly differ from the classical deep bed filtration model.     

Analysis of Multiphase Non-Darcy Flow in Porous Media

Recent laboratory studies and analyses (Lai et al. Presented at the 2009 Rocky Mountain Petroleum Technology Conference, 14–16 April, Denver, CO, 2009) have shown that the Barree and Conway model is able to describe the entire range of relationships between flow rate and potential gradient from low- to high-flow rates through porous media. A Buckley and Leverett type analytical solution is derived for non-Darcy displacement of immiscible fluids in porous media, in which non-Darcy flow is described using the Barree and Conway model. The comparison between Forchheimer and Barree and Conway non-Darcy models is discussed. We also present a general mathematical and numerical model for incorporating the Barree and Conway model in a general reservoir simulator to simulate multiphase non-Darcy flow in porous media. As an application example, we use the analytical solution to verify the numerical solution for and to obtain some insight into one-dimensional non-Darcy displacement of two immiscible fluids with the Barree and Conway model. The results show how non-Darcy displacement is controlled not only by relative permeability, but also by non-Darcy coefficients, characteristic length, and injection rates. Overall, this study provides an analysis approach for modeling multiphase non-Darcy flow in reservoirs according to the Barree and Conway model.     

Exact Averaging of Stochastic Equations for Flow in Porous Media

It is well-known that at present, exact averaging of the equations for flow and transport in random porous media have been proposed for limited special fields. Moreover, approximate averaging methods—for example, the convergence behavior and the accuracy of truncated perturbation series—are not well-studied, and in addition, calculation of high-order perturbations is very complicated. These problems have for a long time stimulated attempts to find the answer to the question: Are there in existence some, exact, and sufficiently general forms of averaged equations? Here, we present an approach for finding the general exactly averaged system of basic equations for steady flow with sources in unbounded stochastically homogeneous fields. We do this by using (1) the existence and some general properties of Green’s functions for the appropriate stochastic problem, and (2) some information about the random field of conductivity. This approach enables us to find the form of the averaged equations without directly solving the stochastic equations or using the usual assumption regarding any small parameters. In the common case of a stochastically homogeneous conductivity field we present the exactly averaged new basic non-local equation with a unique kernel-vector. We show that in the case of some type of global symmetry (isotropy, transversal isotropy, or orthotropy), we can for three-dimensional and two-dimensional flow in the same way derive the exact averaged non-local equations with a unique kernel-tensor. When global symmetry does not exist, the non-local equation with a kernel-tensor involves complications and leads to an ill-posed problem.     

Solute Transport Analysis Through Heterogeneous Media in Nonuniform in the Average Flow by a Stochastic Approach

The stochastic approach has been shown to be an excellent tool for the characterisation and analysis of velocity fields and transport processes through heterogeneous porous formations. The main results (linear theory) have been obtained for problems with simplified flow conditions, usually in the assumption of uniform in the average flow, but a great effort is spent to reach theoretical results for more complex situations.This paper deals with 2D heterogeneous aquifers subject to uniform recharge; the stochastic approach is adopted to characterise, as ensemble behaviour, the velocity field and transport processes of a nonreactive solute. The impact of transmissivity conditioning on solute particles trajectories is analysed and an application is carried out. The analytical formulations, obtained by a first order analysis, are compared to the one resulting from constant in the average hydraulic gradient, and their reliability is investigated with numerical tests performed by a Monte Carlo method.The result of this study is that strong non-stationarities are present in the flow and transport process. A detailed analysis shows that the theoretical results cannot be extended to cases with high heterogeneity level, unlike the uniform in the average flow fields.     

流体饱和两相多孔介质拟静态问题的有限元解法

给出基于混合物理论的流体饱和两相多孔介质模型,该模型由一可变形固体 一流体相组成。采用Ｇａｌｅｒｋｉｎ加权残值法导出求解拟静态问题的有限元公式,并编制了二维有限元程序。用程序分析了一维和二维问题,得到合理的结果。     

On Mathematical Models of Average Flow in Heterogeneous Formations

We consider a general model of transient flow in media of random conductivity and storativity. The flow is driven by the spatially distributed source function (x, t) and the initial head distribution h ₀(x). The function models sources and wells and can be deterministic, random or a sum of both. The deterministic source function corresponds to singularities of deterministic strength, whereas the random models the head boundary condition. In the latter case, is shown to be proportional to the hydraulic conductivity. The aim of the study is to analyze the feasibility of averaging the flow equations and of developing the mathematical model of average flow (AFM) without solving problems in detail. It is shown that the problem of averaging is reduced to deriving two constitutive equations. The first equation, the effective Darcy's law (EDL) stems from averaging Darcy's law at local scale. The second one is related to the medium ability to store a fluid and expresses the correlation between the storativity and head in terms of the mean head. Both relationships are required to be completely determined by the medium structure (conductivity and storativity statistical properties) and independent of the flow configuration (functions and h ₀). We show that if one of the constitutive equations exists, the same is true respective to the second. This reduces the problem of averaging to the classic one of deriving the EDL. For steady flows the EDL is shown to exist for flows driven by sources (wells) of either deterministic flux or head boundary conditions. No EDL can be derived if both types of sources are present in the flow domain. For unsteady flows the EDL does not exist if the initial head correlates with the medium properties. For uncorrelated initial head distribution, its random residual (due to the measurement errors and scarcity of the data) has no impact on the EDL and is immaterial. For deterministic h ₀, the only case for which the EDL exists is the flow by sources of deterministic discharge. For sources of given head boundary condition the EDL can be derived only for uniform initial head distribution. For all other cases, the EDL does not exist. The results of the study are not limited by usually adopted assumptions of weak heterogeneity and of stationarity of the formation random properties.     

Inertia Effects in High-Rate Flow Through Heterogeneous Porous Media

The paper deals with the effects of large scale permeability–heterogeneity on flows at high velocities through porous media. The media is made of a large number of homogeneous blocks where the flow is assumed to be governed by the Forchheimer equation with a constant inertial coefficient. By assuming the validity of the Forchheimer equation at the large scale, an effective inertial coefficient is deduced from numerical simulations. Different media are investigated: serial-layers, parallel-layers and correlated media. The numerical results show that: (i) for the serial-layers, the effective inertial coefficient is independent of the Reynolds number and decreases when the variance and the mean permeability ratio increases; (ii) for the parallel-layers and the correlated media, the effective inertial coefficient is function of the Reynolds number and increases when the variance and the mean permeability ratio increases. Theoretical relationships are proposed for the inertial coefficient as function of the Reynolds number and the characteristics of the media.     

非饱和多孔介质非线性有限元分析的一致性算法

在文[1]工作的基础上，对非饱和多孔材料非线性问题进行分析，给出分析的本构模型，模型中考虑了毛吸压力的影响。给出问题分析的求解技术与算法策略，在此基础上，为保证迭代算法的收敛性，文中给出适合于广义塑性本构模型分析的一致性算法与一致性切线刚度矩阵。给出的数值算例证实了理论与算法的正确与有效性。     

明渠效应在多相计量中的应用研究

在多相流体力学和多相流测量方面大量的基础理论研究和实验研究的基础上,提出了全新的明渠流模型等多个流量计测模型用于多相流量计量。通过自主开发的智能型多相流量计测系统在室内多相流测试环道上,对模型的运用效果做了大量的实验研究。并且利用基于神经网络技术和模糊模式识别技术开发的计测软件对实验数据进行处理,结果表明:液相计测数据中 80%以上误差位于±5%的范围以内,除个别小流量外,所有误差位于±10%的范围以内;气相计测结果中 90%以上误差位于±10%的范围以内, 97%的误差位于±15%的范围以内。实验数据表明:该计测模型可以适用于多相流中不同粘度的液相流量计量;可适用于较宽的气、液相流量变化范围,模型计测误差稳定在可接受的水平。     

非饱和多孔介质有限元分析的基本控制方程与变分原理

本文在对问题研究现状进行阐述的基础上较系统地给出了骨架可变形非饱和多孔介质的全耦合分析模型，模型中考虑了孔隙气体，水（油）流动对介质力学性能的影响，多孔介质的饱和度，渗透系数与毛吸压力的关系，由实验给出，所导出的控制方程以固体骨架的位移与孔隙流体压力为基本未知量，由于问题的非自共轭特征，文中构造了非饱和介质动力问题的参数变分形式，并在此基础上给出有限元离散方程。     

Microvisual Study of Multiphase Gas Condensate Flow in Porous Media

Gas condensate reservoirs constitute a significant portion of hydrocarbon reserves worldwide. The liquid drop-out in these reservoirs may lead to recovery problems such as near wellbore permeability impairment and uncertainty in the actual location of the target condensate. Such technical issues can be addressed through improved understanding of the formation of condensate and the multiphase flow of gas/condensate/water in the reservoir as characterized by relative permeability curves. The appropriate relative permeability curves in turn can be used in reservoir simulators to assist in optimization of field development. This paper reports results of experiments conducted in micromodels, in support of possible core flow tests, using reservoir fluids under reservoir conditions. In particular, visualizations of condensate formation with and without connate water are presented and the differences between the two cases as well as the possible implications for the relative permeability measurements are discussed. Furthermore, the flow of gas and condensate at different force ratios (capillary and Bond numbers) are presented. It is postulated that a single dimensionless number may not be sufficient to characterize the multiphase flow in gas condensate reservoirs. The physical mechanisms occurring under various field conditions are examined in the light of these observations.     

The Discontinuous Solutions of the Transport Equations for Compositional Flow in Porous Media

In the present paper we consider a multicomponent multiphase isothermal flow in porous media with mass exchange between phases. The system of equations of multiphase multicomponent flow has discontinuous solutions, but is not hyperbolic, except some particular cases. For this general, non-hyperbolic system, we propose a free energy condition to select unique physically admissible discontinuous solutions. We also develop a geometrical procedure which provides a tool to analyze the free energy condition. For a two-component mixture, analytical formulae are obtained for the allowed discontinuities.     

Nonlinear Two-Phase Mixing in Heterogeneous Porous Media

For a two-phase immiscible flow through a heterogeneous porous medium in gravity field but with neglected capillary pressure, a macroscale model of first order is derived by a two-scale homogenization method while capturing the effect of fluid mixing. The mixing is manifested in the form of a nonlinear hydrodynamic dispersion and a transport velocity shift. The dispersion tensor is shown to be a nonlinear function of saturation. In the case offlow without gravity this function is proportional to the fractional flow derivative and depends on the viscosity ratio. For a flow which is one dimensional at the macroscale, the dispersion operator remains three dimensional and can be calculated in an analytical way. In the case of gravity induced flow, the longitudinal dispersion as the function of saturation is negative within some interval of saturation values. Numerical simulations of the microscale problemjustify the theoretical results of homogenization.