Gaussian Closure of Transient Unsaturated Flow in Random Soils

The Gaussian closure approximation, previously used by the authors to solve steady state stochastic unsaturated flow problems in randomly heterogeneous soils, is extended here to transient flow. The method avoids linearizing the governing flow equations or the soil constitutive relations. It places no theoretical limit on the variance of constitutive parameters and applies to a broad class of soils with flow properties that scale according to a linearly separable model. Closure is obtained by treating the dimensionless pressure head as a multivariate Gaussian function. It yields a system of coupled nonlinear differential equations for the first and second moments of . We apply the Gaussian closure technique to the problem of transient infiltration into a randomly stratified soil. In each layer, hydraulic conductivity and water content vary exponentially with . Elsewhere we show that application of the technique to other constitutive relations is straightforward. Our solution for the mean and variance of in a one-dimensional layer with random conductivity compares well with Monte Carlo results over a wide range of parameters, provided that the spatial variability of the constitutive exponent is small. The solution provides considerable insight into the behavior of the transient unsaturated stochastic flow problem.     

Recursive Conditional Moment Equations for Advective Transport in Randomly Heterogeneous Velocity Fields

Flow and transport parameters such as hydraulic conductivity, seepage velocity, and dispersivity have been traditionally viewed as well-defined local quantities that can be assigned unique values at each point in space-time. Yet in practice these parameters can be deduced from measurements only at selected locations where their values depend on the scale (support volume) and mode (instruments and procedure) of measurement. Quite often, the support of the measurements is uncertain and the data are corrupted by experimental and interpretive errors. Estimating the parameters at points where measurements are not available entails an additional random error. These errors and uncertainties render the parameters random and the corresponding flow and transport equations stochastic. The stochastic flow and transport equations can be solved numerically by conditional Monte Carlo simulation. However, this procedure is computationally demanding and lacks well-established convergence criteria. An alternative to such simulation is provided by conditional moment equations, which yield corresponding predictions of flow and transport deterministically. These equations are typically integro-differential and include nonlocal parameters that depend on more than one point in space-time. The traditional concept of a REV (representative elementary volume) is neither necessary nor relevant for their validity or application. The parameters are nonunique in that they depend not only on local medium properties but also on the information one has about these properties (scale, location, quantity, and quality of data). Darcy's law and Fick's analogy are generally not obeyed by the flow and transport predictors except in special cases or as localized approximations. Such approximations yield familiar-looking differential equations which, however, acquire a non-traditional meaning in that their parameters (hydraulic conductivity, seepage velocity, dispersivity) and state variables (hydraulic head, concentration) are information-dependent and therefore, inherently nonunique. Nonlocal equations contain information about predictive uncertainty, localized equations do not. We have shown previously (Guadagnini and Neuman, 1997, 1998, 1999a, b) how to solve conditional moment equations of steady-state flow numerically on the basis of recursive approximations similar to those developed for transient flow by Tartakovsky and Neuman (1998, 1999). Our solution yields conditional moments of velocity, which are required for the numerical computation of conditional moments associated with transport. In this paper, we lay the theoretical groundwork for such computations by developing exact integro-differential expressions for second conditional moments, and recursive approximations for all conditional moments, of advective transport in a manner that complements earlier work along these lines by Neuman (1993).     

Radial Flow in a Bounded Randomly Heterogeneous Aquifer

Flow to wells in nonuniform geologic formations is of central interest to hydrogeologists and petroleum engineers. There are, however, very few mathematical analyses of such flow. We present analytical expressions for leading statistical moments of vertically averaged hydraulic head and flux under steady-state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer. Like in the widely used Thiem equation, we prescribe a constant pumping rate deterministically at the well and a constant head at a circular outer boundary of radius L. We model the natural logarithm Y = lnT of aquifer transmissivity T as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. Perturbation of these nonlocal equations leads to a system of local recursive moment equations that we solve analytically to second order in the standard deviation of Y. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. It yields expected values of head and flux, and the variance–covariance of these quantities, as functions of distance from the well. It also yields an apparent transmissivity, T _a, defined as the negative ratio between expected flux and head gradient at any radial distance. The solution is supported by numerical Monte Carlo simulations, which demonstrate that it is applicable to strongly heterogeneous aquifers, characterized by large values of log transmissivity variance. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness. Potential uses include the analysis of pumping tests and tracer test conducted in such wells, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells.     

Boundary Integral Procedures for Unsaturated Flow Problems

Abstract. A novel numerical scheme based on the singular integral theory of the boundary element method. (BEM) is presented for the solution of transient unsaturated flow in porous media. The effort in the present paper is directed in facilitating the application of the boundary integral theory to the solution of the highly non-linear equations that govern unsaturated flow. The resulting algorithm known as the Green element method (GEM) presents a robust attractive method in the state-of -the-art application of the boundary element methodology. Three GEM models based on their different methods of handling the non-linear diffusivity, illustrate the suitability and robustness of this approach for solving highly non-linear 1-D and 2-D flows which would have proved cumbersome or too difficult to implement with the classical BEM approach.     

Uncertainty Propagation in Layered Unsaturated Soils

This study analyzes the wetting front migration in layered unsaturated soils which have uncertain hydraulic properties. A Monte Carlo scheme was used to propagate the uncertainty of hydraulic parameters. RANUF, a computer program, was developed to solve the one-dimensional, pressure-based form of Richards' equation and to implement the Monte Carlo scheme.Uncertainty propagation was investigated for two-layered soils of various alternating fine over coarse or coarse over fine layer configurations and of various nonrandomized and/or randomized layer arrangements. The effects of changing initial and boundary conditions were also investigated. Randomness was introduced via the saturated hydraulic conductivity, K _s, which was assumed to be distributed lognormally with a coefficient of variation of about 10 percent.It was found that in layered soils the mean profiles (i.e., water content and pressure head) remained essentially unchanged regardless of which layer (or layers) was (or were) randomized; however, the variance profiles were affected. Also, higher uniform initial water content tended to inhibit uncertainty, but higher supply rates did not show any characteristic trend for uncertainty behavior.     

Unsaturated Transport with Linear Kinetic Sorption Under Unsteady Vertical Flow

We consider transport of a solute obeying linear kinetic sorption under unsteady flow conditions. The study relies on the vertical unsaturated flow model developed by Indelman et al. [J. Contam. Hydrol. 32 (1998), 77–97] to account for a cycle of infiltration and redistribution. One of the main features of this type of transport, as compared with the case of a continuous water infiltration, is the finite depth of solute penetration. In the infiltration stage an analytical solution that generalizes the previous results of Lassey [Water Resour. Res. 24 (1988), 343–350] and Severino and Indelman [J. Contam. Hydrol. 70 (2004), 89–115] is derived. This solution accounts for quite general initial solute distributions in both the mobile and immobile concentration. When the redistribution is also considered, two timescales become relevant, namely: (i) the desorption rate k⁻¹, and (ii) the water application time t_ap. In particular, we have assumed that the quantity ε =(k t_ap)⁻¹ can be regarded as a small parameter so that a perturbation analytical solution is obtained. At field-scale the concentration is calculated by means of the column model of Dagan and Bresler [Soil Sci. Soc. Am. J. 43 (1979), 461–467], i.e. as ensemble average over an infinite series of randomly distributed and uncorrelated soil columns. It is shown that the heterogeneity of hydraulic properties produces an additional spreading of the plume. An unusual phenomenon of plume contraction is observed at long times of solute propagation during the drying period. The mean solute penetration depth is studied with special emphasis on the impact of the variability of the saturated conductivity upon attaining the maximum solute penetration depth.     

Localization of Mean Flow and Apparent Transmissivity Tensor for Bounded Randomly Heterogeneous Aquifers

We explore the concept of apparent transmissivity for bounded randomly heterogeneous media under steady-state flow regime. The novelty of our study consists of investigating a tensorial nature of apparent transmissivity. We demonstrate that apparent transmissivity of bounded domains is anisotropic even though an underlying local transmissivity field is statistically isotropic. For rectangular flow domains, we derive an analytical expression for the apparent transmissivity tensor via localization and perturbation expansion of the nonlocal mean flow equations in the variance of log-transmissivity. In this expression, almost everywhere the off-diagonal terms are several orders of magnitude smaller than the diagonal terms. When the domain size relative to the log-transmissivity correlation scale is large, the longitudinal and transverse components of the apparent transmissivity tensor approach the geometric mean of local transmissivity. While rigorously valid for mean uniform flows only, our expression for the apparent transmissivity tensor leads to mean hydraulic head distributions that compare favorably with those obtained through Monte-Carlo simulations and the nonlocal mean flow equations even in the presence of pumping wells. This agreement deteriorates in the vicinity of wells and as pumping rates increase.     

A Quasilinear Based Procedure for Saturated/Unsaturated Water Movement in Soils

The quasilinear form of Richards equation for one-dimensional unsaturated flow in soils can be readily solved for a wide variety of conditions. However, it cannot explain saturated/unsaturated flow and the constant diffusivity assumption, used to linearise the transient quasilinear equation, can introduce significant error. This paper presents a quasi-analytical solution to transient saturated/unsaturated flow based on the quasilinear equation, with saturated flow explained by a transformed Darcy's equation. The procedure presented is based on the modified finite analytic method. With this approach, the problem domain is divided into elements, with the element equations being solutions to a constant coefficient form of the governing partial differential equation. While the element equations are based on a constant diffusivity assumption, transient diffusivity behaviour is incorporated by time stepping. Profile heterogeneity can be incorporated into the procedure by allowing flow properties to vary from element to element. Two procedures are presented for the temporal solution; a Laplace transform procedure and a finite difference scheme. An advantage of the Laplace transform procedure is the ability to incorporate transient boundary condition behaviour directly into the analytical solutions. The scheme is shown to work well for two different flow problems, for three soil types. The technique presented can yield results of high accuracy if the spatial discretisation is sufficient, or alternatively can produce approximate solutions with low computational overheads by using large sized elements. Error was shown to be stable, linearly related to element size.     

Modelling Unsaturated Moisture Transport in Heterogeneous Limestone

The influence of macro-scale heterogeneities on the imbibition process is investigated for Savonnières, a French layered limestone. Free uptake experiments are performed both parallel and perpendicular to the bedding. It is found that the position of the different layers, and the exact material properties inside each layer can significantly influence the imbibition process. The experimental results are compared with numerical simulations. For the flow simulations, moisture permeability of the different layers is obtained with the upscaling technique presented in Part 1. Good agreement between simulations and experiments validate the proposed upscaling from meso to macroscopic scale.     

Probabilistic Study of Rainfall-Triggered Instabilities in Randomly Heterogeneous Unsaturated Finite Slopes

Transport in Porous Media - Water infiltration destabilises unsaturated soil slopes by reducing matric suction, which produces a decrease of material cohesion. If the porosity of the soil is...     

Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media

This study is concerned with developing a two-dimensional multiphase model that simulates the movement of NAPL in heterogeneous aquifers. Heterogeneity is dealt with in a probabilistic sense by modeling the intrinsic permeability of the porous medium as a stochastic process. The deterministic finite element method is used to spatially discretize the multiphase flow equations. The intrinsic permeability is represented in the model via its Karhunen–Loeve expansion. This is a computationally expedient representation of stochastic processes by means of a discrete set of random variables. Further, the nodal unknowns, water phase saturations and water phase pressures, are represented by their stochastic spectral expansions. This representation involves an orthogonal basis in the space of random variables. The basis consists of orthogonal polynomial chaoses of consecutive orders. The relative permeabilities of water and oil phases, and the capillary pressure are expanded in the same manner, as well. For these variables, the set of deterministic coefficients multiplying the basis in their expansions is evaluated based on constitutive relationships expressing the relative permeabilities and the capillary pressure as functions of the water phase saturations. The implementation of the various expansions into the multiphase flow equations results in the formulation of discretized stochastic differential equations that can be solved for the deterministic coefficients appearing in the expansions representing the unknowns. This method allows the computation of the probability distribution functions of the unknowns for any point in the spatial domain of the problem at any instant in time. The spectral formulation of the stochastic finite element method used herein has received wide acceptance as a comprehensive framework for problems involving random media. This paper provides the application of this formalism to the problem of two-phase flow in a random porous medium.     

Simulations of Unsaturated Flow in Multiply Zoned Media by Green Element Models

Flows in variably saturated media that exhibit second-type heterogeneity, in which abrupt changes of medium parameters occur, are simulated by the Green element method (GEM). Such media are usually encountered where soil formations have arisen by different geological or geomorphological processes spread over different time scales. Two challenges are posed when simulating flows in multiply zoned unsaturated media: one is the highly nonlinear nature of the flow within each zone, and the other is dealing with sharp contrast in medium parameters at the interfaces of different zones. Both challenges are accommodated in this paper using a flux-based Green element formulation to simulate the flow and incorporating the Picard and Newton–Raphson (N-R) algorithms to simplify the nonlinear discrete equations. Calculations are carried out on three numerical examples of infiltration into unsaturated soils in two spatial dimensions. The convergence rate of the N-R algorithm is superior to the Picard algorithm only for the first example, while none of the algorithms has a clear advantage for the other two examples. The N-R algorithm suffers from repeated calculation of derivatives of the medium parameters with respect to the pressure head, thereby compromising the accuracy of the solution and increasing computational cost.     

考虑非饱和土基质吸力作用的土石坝渗流分析

传统的渗流分析主要考虑饱和区而忽略非饱和区内的渗流,本文首先对影响非饱和土渗流特性的重要物理量基质吸力进行了分析,阐述了其形成机理并推出其理论计算公式,然后在考虑基质吸力的作用下,基于饱和．非饱和渗流计算原理,采用有限元法考虑了渗流场非饱和区的影响,并以各向同性均质坝和心墙土石坝为算例,进行了计算分析。算例表明,由于基质吸力产生的虹吸作用,使得浸润线上部的非饱和区内也存在着连续的水流,否定了传统分析中浸润线是最上部流线的假设,通过分析这种水流的特点,定性的得出其对土石坝渗流稳定性的影响,对土石坝渗流分析具有一定的指导意义。     

Error estimation of Gaussian closure for quasi-linear systems

An error estimation of the Gaussian closure technique for quasi-linear systems is given analytically by FPK method, under the assumption that the transition probability density function can be expressed by a power series in small parameter, which measures the nonlinearity. The error resulted from the Gaussian closure approximation is found to be of order o( ²).     

Symmetry Solutions for Transient Solute Transport in Unsaturated Soils with Realistic Water Profile

We examine solutions for solute transport using the convection-dispersion equation (CDE) during steady evaporation from a water table. It is common, when solving the CDE, to first approximate the volumetric water content of the soil as a constant. Here, we assume a reasonable function for the water content profile and construct realistic nonlinear hydraulic transport properties. Both classical and nonclassical symmetry techniques are employed. Invariant solutions are obtained for the one dimensional CDE even with a nontrivial background profile for volumetric water content.     

Analytical Solutions for Macrodispersion in a 3D Heterogeneous Porous Medium with Random Hydraulic Conductivity and Dispersivity

A stochastic analysis of macrodispersion for conservative solute transport in three-dimension (3D) heterogeneous statistically isotropic and anisotropic porous media when both hydraulic conductivity and local dispersivity are random is presented. Analytical expressions of macrodispersivity are derived using Laplace and Fourier transforms. The effects of various parameters such as ratio of transverse to longitudinal local dispersivity, correlation length ratio, correlation coefficient and direction of flow on asymptotic macrodispersion are studied. The behaviour of growth of macrodispersivity in preasymptotic stage is also shown in this paper. The variation in local dispersion coefficient causes change in transverse macrodispersivity. The consideration of random dispersivity along with random hydraulic conductivity indicates that the total dispersion is affected and important in the case when the hydraulic conductivity and dispersivity are correlated. It is observed that the pre-asymptotic behavior of the macrodispersivity is not sensitive to the choice of spectral density functions.     

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.     

A Lattice Gas Automaton Simulation of the Nonlinear Diffusion Equation: A Model for Moisture Flow in Unsaturated Porous Media

We investigate a two-dimensional lattice gas automaton (LGA) for simulating the nonlinear diffusion equation in a random heterogeneous structure. The utilility of the LGA for computation of nonlinear diffusion arises from the fact that, the diffusion coefficient in the LGA depends on the local density of fluid particles which statistically determines the collision rate and thus, the mean free path of the particles at the microscopic scale. The LGA may therefore be used as a physical analogue to simulate moisture flow in unsaturated porous media. The capability of the LGA to account for unsaturated flow is tested through a set of numerical experiments simulating one-dimensional infiltration in a simplified semi-infinite homogenous isotropic porous material. Different mechanisms of interactions are used between the fluid and the solid phase to simulate various fluid–solid interfaces. The heterogeneous medium, initially at low density is submitted to a steep density gradient by continuously injecting fluid particles at high concentration and zero velocity along one face of the model. The propagation of the infiltration front is visualized at different time steps through concentration profiles parallel to the applied concentration gradient and the infiltration rate is measured continuously until steady-state flow is reached. The numerical results show close agreement with the classical theory of flow in unsaturated porous media. The cumulative absorption exhibits the expected t ^1/2 dependence. The evolution of the effective diffusion coefficient with the particle concentration is estimated from the measured density profiles for the various porous materials. Depending on the applied fluid–solid interactions, the macroscopic effective diffusivity may vary by more than two orders of magnitude with density.     

软夹硬力学不均匀焊接接头疲劳裂纹闭合行为的实验研究

力学性能不均匀是焊接接头的三大待征之一，本文采用柔度法研究了不同硬夹层宽度的软夹硬力学不均匀焊接接头疲劳裂纹的闭俣行为。研究结果表明，裂纹尖端附近软区的局部屈服在疲劳载荷卸载过程中促使裂纹闭合。随着硬夹层宽度的减小，这一影响越来越明显。     

Stochastic Optimal Control of Nonlinear Systems via Short-Time Gaussian Approximation and Cell Mapping

A novel strategy to obtain global solutions of stochasticoptimal control problems with fixed state terminal conditions and controlbounds is proposed in this paper. The solution is global in the sense that theoptimal control solutions for all the initial conditions in a region of thestate space are obtained. The method makes use of Bellman's principle ofoptimality, the cumulant neglect closure method and the short-time Gaussianapproximation. A Markov chain with a control dependent transition probabilitymatrix is built using the generalized cell mapping method. This allows toevaluate the transient and steady state response of the controlled system. Themethod is applied to several linear and nonlinear systems leading to excellentcontrol performances.