Maximum-likelihood estimation of unsaturated hydraulic parameters

Maximum-likelihood estimators properly represent measurement error, thus provide a statistically sound basis for evaluating the adequacy of a model fit and for finding the multivariate parameter confidence region. We demonstrate the advantages of using maximum-likelihood estimators rather than simple least-squares estimators for the problem of finding unsaturated hydraulic parameters. Inversion of outflow data given independent retention data can be treated by an extension to a Bayesian estimator. As an example, we apply the methodology to retention and transient unsaturated outflow observations, both obtained on the same medium sand sample. We found the van Genuchten expression to be adequate for the retention data, as the best fit was within measurement error. The Cramer–Rao confidence bound described the true parameter uncertainty approximately. The Mualem–van Genuchten expression was, however, inadequate for our outflow observations, suggesting that the parameters (, n) may not always be equivalent in describing both retention and unsaturated conductivity.     

Estimating unsaturated soil hydraulic parameters using ant colony optimization

Although models are now routinely used for addressing environmental problems, both in research and management applications, the problem of obtaining the required parameters remains a major challenge. An attractive procedure for obtaining model parameters in recent years has been through inverse modeling. This approach involves obtaining easily measurable variables (model output), and using this information to estimate a set of unknown model parameters. Inverse procedures usually require optimization of an objective function. In this study we emulate the behavior of a colony of ants to achieve this optimization. The method uses the fact that ants are capable of finding the shortest path from a food source to their nest by depositing a trail of pheromone during their walk. Results obtained with the ant colony parameter optimization method are very promising; in eight different applications we were able to estimate the `true' parameters to within a few percent. One such study is reported in this paper plus an application to estimating hydraulic parameters in a lysimeter experiment. Despite the encouraging results obtained thus far, further improvements could still be made in the parameterization of the ant colony optimization for application to estimation of unsaturated flow and transport parameters.     

Artificial recharge through a thick, heterogeneous unsaturated zone

Thick, heterogeneous unsaturated zones away from large streams in desert areas have not previously been considered suitable for artificial recharge from ponds. To test the potential for recharge in these settings, 1.3 x 10(6) m(3) of water was infiltrated through a 0.36-ha pond along Oro Grande Wash near Victorville, California, between October 2002 and January 2006. The pond overlies a regional pumping depression 117 m below land surface and is located where thickness and permeability of unsaturated deposits allowed infiltration and saturated alluvial deposits were sufficiently permeable to allow recovery of water. Because large changes in water levels caused by nearby pumping would obscure arrival of water at the water table, downward movement of water was measured using sensors in the unsaturated zone. The downward rate of water movement was initially as high as 6 m/d and decreased with depth to 0.07 m/d; the initial time to reach the water table was 3 years. After the unsaturated zone was wetted, water reached the water table in 1 year. Soluble salts and nitrate moved readily with the infiltrated water, whereas arsenic and chromium were less mobile. Numerical simulations done using the computer program TOUGH2 duplicated the downward rate of water movement, accumulation of water on perched zones, and its arrival at the water table. Assuming 10 x 10(6) m(3) of recharge annually for 20 years, a regional ground water flow model predicted water level rises of 30 m beneath the ponds, and rises exceeding 3 m in most wells serving the nearby urban area.     

Effective parameters for flow in saturated heterogeneous porous media

Effective parameters for flow in saturated porous media are obtained via Taylor-Aris-Brenner moment analysis considering both periodic as well as stationary porous medium properties. It is assumed that a slug is instantaneously introduced into an unbounded, anisotropic porous medium having a compressible matrix, and that the correlation length of the local hydraulic conductivity and specific storage fluctuations is smaller than the correlation length of hydraulic head fluctuations (gradually varying flow). It is shown that the effective specific storage is equal to its volume average. The effective hydraulic conductivity is derived by a small-perturbation analysis and it is shown to consist of its volume average and of a second term which accounts for the ‘small’ local conductivity fluctuations.     

Upscaling of flow in heterogeneous porous formations: Critical examination and issues of principle

We consider heterogeneous media whose properties vary in space and particularly aquifers whose hydraulic conductivity K may change by orders of magnitude in the same formation. Upscaling of conductivity in models of aquifer flow is needed in order to reduce the numerical burden, especially when modeling flow in heterogeneous aquifers of 3D random structure. Also, in many applications the interest is in average values of the dependent variables over scales larger or comparable to the conductivity length scales. Assigning values of the conductivity K_b to averaging domains, or computational blocks, is the topic of a large body of literature, the problem being of wide interest in various branches of physics and engineering. It is clear that upscaling causes loss of information and at best it can render a good approximation of the fine scale solution after averaging it over the blocks.The present article focuses on upscaling approaches dealing with random media. It is not meant to be a review paper, its main scope being to elucidate a few issues of principle and to briefly discuss open questions. We show that upscaling can be usually achieved only approximately, and the result may depend on the particular upscaling scheme adopted. The typically scarce information on the statistical structure of the fine-scale conductivity imposes a strong limitation to the upscaling problem. Also, local upscaling is not possible in nonuniform mean flows, for which the upscaled conductivity tensor is generally nonlocal and it depends on the domain geometry and the boundary conditions. These and other limitations are discussed, as well as other open topics deserving further investigation.     

Generation of three dimensional flow fields for statistically anisotropic heterogeneous porous media

A methodology for generating three dimensional (3D) flow fields for statistically anisotropic heterogeneous porous media is presented and demonstrated. The simulated flow fields are shown to exhibit the input spatial correlation structure and observe mass continuity. Sample flow fields are presented in the form of cross sectional slices of the 3D formation. These cross sections demonstrate visually the characteristics of subsurface flow. The method was found to be faster than traditional techniques in terms of its computational requirements. Given this method, it is possible to generate the large number of realizations of a velocity field necessary to compute high order statistics in transport problems.     

Numerical investigation of the anisotropic hydraulic conductivity behavior in heterogeneous porous media

The behavior of the mean equivalent hydraulic conductivity normal and parallel to stratification (K₁, and K₂, respectively) is studied here through Monte Carlo simulations of three-dimensional, steady-state flow in statistically anisotropic, bounded, and heterogeneous media. For water flow normal to stratification in strongly heterogeneous porous media (²_Y=3) the value of K₁ is not unique; it ranges from an arithmetic to a geometric, and finally, to a harmonic mean behavior depending on field dimensions, and medium anisotropy. For a fixed anisotropy ratio and variance of Y = ln K, the larger the distance, in the direction perpendicular to stratification, over which water flow takes place, the faster the rate at which, K_H, behavior is approached. However, even for large anisotropy ratios, harmonic mean behavior appears to be a good approximation only for aquifer thickness L₁ that is large enough to allow stratified flow to occur. For small aquifer thickness (L₁/₁<8, where ₁ is the integral scale normal to stratification) the limiting behavior, for large anisotropy ratios, appears to be, instead, that of two-dimensional flow, i.e., water flows primarily parallel to the planes of stratification. When the aquifer thickness is very small compared to the horizontal dimensions (and with relative similar integral scales in the three directions) a behavior resembling arithmetic mean conditions is exhibited, i.e., water flow takes place through heterogeneous, vertical, soil volumes. The geostatistical expressions of Desbarats (1992a) for upscaling hydraulic conductivity values were utilized and closed form empirical relations were developed for the main components of the upscaled hydraulic conductivity tensor.     

A Neumann expansion approach to flow through heterogeneous formations

A stochastic approach is used for the study of flow through highly heterogeneous aquifers. The mathematical model is represented by a random partial differential equation in which the permeability and the porosity are considered to be random functions of position, defined by the average value, constant standard deviation and autocorrelation function characterized by the integral scale. The Laplace transform of the solution of the random partial differential equation is first written as a solution of a stochastic integral equation. This integral equation is solved using a Neumann series expansion. Conditions of convergence of this series are investigated and compared with the convergence of the perturbation series. For mean square convergence, the Neumann expansion method may converge for a larger range of variability in permeability and porosity than the classic perturbation method. Formal expressions for the average and for the correlation moments of the pressure are obtained. The influence of the variability of the permeability and porosity on pressure is analyzed for radial flow. The solutions presented for the pressure at the well, as function of the permeability coefficient of variation, may be of practical interest for evaluating the efficiency of well stimulation operations, such as hydraulic fracturing or acidizing methods, aimed at increasing the permeability around the well.     

Influence of heterogeneous air entry pressure on large scale unsaturated flow in porous media

The paper presents numerical simulations of water infiltration in unsaturated porous media containing coarse-textured inclusions embedded in fine-textured background material. The calculations are performed using the two-phase model for water and air flow and a simplified model known as the Richards equation. It is shown that the Richards equation cannot correctly describe flow in the presence of heterogeneities. However, its performance can be improved by introducing appropriately defined effective capillary and permeability functions, representing largescale behaviour of the heterogeneous medium.     

Stochastic analysis of bounded unsaturated flow in heterogeneous aquifers: Spectral/perturbation approach

This paper describes a stochastic analysis of steady state flow in a bounded, partially saturated heterogeneous porous medium subject to distributed infiltration. The presence of boundary conditions leads to non-uniformity in the mean unsaturated flow, which in turn causes non-stationarity in the statistics of velocity fields. Motivated by this, our aim is to investigate the impact of boundary conditions on the behavior of field-scale unsaturated flow. Within the framework of spectral theory based on Fourier–Stieltjes representations for the perturbed quantities, the general expressions for the pressure head variance, variance of log unsaturated hydraulic conductivity and variance of the specific discharge are presented in the wave number domain. Closed-form expressions are developed for the simplified case of statistical isotropy of the log hydraulic conductivity field with a constant soil pore-size distribution parameter. These expressions allow us to investigate the impact of the boundary conditions, namely the vertical infiltration from the soil surface and a prescribed pressure head at a certain depth below the soil surface. It is found that the boundary conditions are critical in predicting uncertainty in bounded unsaturated flow. Our analytical expression for the pressure head variance in a one-dimensional, heterogeneous flow domain, developed using a nonstationary spectral representation approach [Li S-G, McLaughlin D. A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis. Water Resour Res 1991;27(7):1589–605; Li S-G, McLaughlin D. Using the nonstationary spectral method to analyze flow through heterogeneous trending media. Water Resour Res 1995; 31(3):541–51], is precisely equivalent to the published result of Lu et al. [Lu Z, Zhang D. Analytical solutions to steady state unsaturated flow in layered, randomly heterogeneous soils via Kirchhoff transformation. Adv Water Resour 2004;27:775–84].     

A technique for evaluating flow parameters in water bodies with a highly heterogeneous depth

A method is proposed for calculating the hydrodynamic parameters of flows in water bodies with shallow and deep zones. Calculation of flow parameters in the Sea of Azov is given as an example.Translated from Vodnye Resursy, Vol. 32, No. 1, 2005, pp. 55–60.Original Russian Text Copyright © 2005 by Chikin.     

Analytical solutions to steady state unsaturated flow in layered,randomly heterogeneous soils via Kirchhoff transformation

In this study, we derive analytical solutions of the first two moments (mean and variance) of pressure head for one-dimensional steady state unsaturated flow in a randomly heterogeneous layered soil column under random boundary conditions. We first linearize the steady state unsaturated flow equations by Kirchhoff transformation and solve the moments of the transformed variable up to second order in terms of σ_Y and σ_β, the standard deviations of log hydraulic conductivity Y=ln(K_s) and of the log pore size distribution parameter β=ln(α). In addition, we also give solutions for the mean and variance of the unsaturated hydraulic conductivity. The analytical solutions of moment equations are validated via Monte Carlo simulations.     

Real gas flow through heterogeneous porous media: theoretical aspects of upscaling

We consider the problem of upscaling transient real gas flow through heterogeneous bounded reservoirs. One of the commonly used methods for deriving effective permeabilities is based on stochastic averaging of nonlinear flow equations. Such an approach, however, would require rather restrictive assumptions about pressure-dependent coefficients. Instead, we use Kirchhoff transformation to linearize the governing stochastic equations prior to their averaging. The linearized problem is similar to that used in stochastic analysis of groundwater flow. We discuss the effects of temporal localization of the nonlocal averaged Darcy's law, as well as boundary effects, on the upscaled gas permeability. Extension of the results obtained by means of small perturbation analysis to highly heterogeneous porous formations is also discussed.     

A Linking Test to reduce the number of hydraulic parameters necessary to simulate groundwater recharge in unsaturated soils

In environmental studies, numerical simulation models are valuable tools for testing hypothesis about systems functioning and to perform sensitivity studies under scenarios of land use or climate changes. The simulations depend upon parameters which are not always measurable quantities and must be calibrated against observations, using for instance inverse modelling. Due to the scarcity of these observations, it has been found that parameter sets allowing a good matching between simulated and measured quantities are often non-unique, leading to the problem of equifinality. This can lead to non-physical values, erroneous fluxes and misleading sensitivity analysis. Therefore, a simple but robust inverse method coined the Linking Test is presented to determine if the parameters are linked. Linked parameters are then sub-divided into classes according to their impact on water fluxes. The Linking Test establishes the causes of non-uniqueness of parameter sets and the feasibility of the inverse modelling.     

The mechanism of unsaturated flow through a volcanic ash layer under humid climatic conditions

It is very interesting and meaningful to investigate the rainfall-groundwater recharge process under the humid climatic condition of Japan, where mean annual precipitation is about 1600 mm. The present study has investigated soil water movement in the unsaturated zones of a volcanic ash layer, called the ‘Kanto Loam formation’, using environmental tritium as a tracer. The site selected is a flat ground surface on a terraced upland which has a deep unsaturated zone (about 20 m) with a relatively high water content (about 70 per cent) consisting of nearly uniform Kanto Loam formation. The tritium concentrations in groundwater, soil waters having different matric potentials, precipitation, and the seepage water moving through the formation into a man-made cave were measured to characterize the rainfall-groundwater recharge process and the effect of large pore spaces in the formation mentioned by previous studies. Because of the humid climate of Japan, there appears to be a unique soil water flow characteristic which may involve percolation through large pore spaces during heavy rainfall. However, in a fine grained and high water content soil like the Kanto Loam formation, the existence of this flow through large pore spaces does not have a significant effect upon the whole recharge process. The recharge model of displacement flow with dispersion is useful in estimating the tritium concentration profile of soil water. The calculated result shows a recharge rate of 2.5 mm/day. The value obtained reflects the hydrological characteristics of the uplands covered with volcanic ash.     

Similarity solutions to the second boundary value problem of unsaturated flow through porous media

Similarity solutions to the second boundary value problem of unsaturated flow are studied in one-dimensional, semi-infinite porous media with the soil-water diffusivity proportional to some power of the water content. The existence and uniqueness of two types of similarity solutions to the problem are investigated and the properties of these solutions are presented. It is shown that these two types of similarity solutions exist and that they may not be unique for every parameter range studied. The use of the similarity solutions is discussed for the experimental determination of soil-water diffusivity.     

变井源距垂直地震剖面各向异性参数反演

利用微扰理论推导了弱各向异性(WA)介质参数正反演计算的基本公式;给出了在已知慢度矢量的一个分量和偏振矢量情况下确定WA参数的方法.如果这一方法被用于单一的变井源距垂直地震剖面(walkawayVSP)资料,可以获得9个WA参数.这9个WA参数完全决定了qP波和两个qS波在由剖面和钻井所决定的平面内传播的特性.对单条walkawayVSP观测系统数据的产生和WA参数的反演进行了数值模拟计算,对所能确定的WA参数及其可靠性进行了详细的讨论.