C. L. Winter

Moment differential equations for flow in highly heterogeneous porous media

Quantitative descriptions of flow and transport in subsurface environmentsare often hampered by uncertainty in the input parameters. Treatingsuch parameters as random fields represents a useful tool for dealingwith uncertainty. We review the state of the art of stochasticdescription of hydrogeology with an emphasis on statisticallyinhomogeneous (nonstationary) models. Our focus is on composite mediamodels that allow one to estimate uncertainties both in geometricalstructure of geological media consisting of various materials and inphysical properties of these materials.

Mean Flow in composite porous media

We develop probabilities and statistics for the parameters of Darcy flows through saturated porous media composed of units of different materials. Our probability model has two levels. On the local level, a porous medium is composed of disjoint, statistically homogeneous volumes (or blocks) each of which consists of a single type of material. On a larger scale, a porous medium is an arrangement of blocks whose extent and location are uncertain. Using this two-scaled model, we derive general formulae for the probability distribution of hydraulic conductivity and its mean; then we develop general perturbation expansions for mean head. We express distributions and parameters in terms of mixtures of locally homogeneous block densities weighted by large-scale block membership probabilities.

Stochastic analysis of steady‐state unsaturated flow in heterogeneous media: Comparison of the Brooks‐Corey and Gardner‐Russo Models

Existing stochastic models of unsaturated flow and transport are usually developed using the simple Gardner-Russo constitutive relationship though it is generally accepted that the more complex van Genuchten and Brooks-Corey relationships may perform better in describing experimental data. In this paper, we develop first-order stochastic models for gravity-dominated flow in second-order stationary media with both the Brooks-Corey and the Gardner-Russo constitutive relationships. These models also account for the spatial variability in effective water content, while the spatial variability is generally neglected in most existing stochastic models. Analytical solutions are obtained for the case of one-dimensional gravity-dominated flow. On the basis of the solutions, we illustrate the differences between results from these two constitutive models through some one-dimensional examples. It is found that the impacts of the constitutive models on the statistical moments of suction head, effective water content, unsaturated hydraulic conductivity, and velocity depend on the saturation ranges. For example, the mean head and the mean effective water content for the Brooks-Corey model differ in a great manner with their counterparts for the Gardner-Russo model near the dry and wet limits while the differences are small at the intermediate range of saturation. This finding is confirmed with some two-dimensional examples. It is also found that the Brook-Corey model has certain advantages over the Gardner-Russo model in analyzing unsaturated flow in heterogeneous media. For example, the stochastic model developed based on the Brooks-Corey function requires the coefficient of variation of head and soil parameter “αBC” to be small (≪1), whereas that based on the Gardner-Russo function assumes the one-point cross covariance of head and OLGR to be small (≪1). Illustrative examples reveal that the latter condition may be violated because the one-point covariance is found to increase rapidly to beyond unity as the soil becomes dry, whereas the former may be readily satisfied.

Nonstationary stochastic analysis of steady state flow through variably saturated, heterogeneous media

In this study we develop a first-order, nonstationary stochastic model for steady state, unsaturated flow in randomly heterogeneous media. The model is applicable to the entire domain of a bounded vadose zone, unlike most of the existing stochastic models. Because of its nonstationarity, we solve it by the numerical technique of finite differences, which renders the flexibility in handling different boundary conditions, input covariance structures, and soil constitutive relationships. We illustrate the model results in one and two dimensions for soils described by the Brooks-Corey constitutive model. It is found that the flow quantities such as suction head, effective water content, unsaturated hydraulic conductivity, and velocity are nonstationary near the water table and approach stationarity as the vertical distance from the water table increases. The stationary limits and the critical vertical distance at which stationarity is attained depend on soil types and recharge rates. The smaller the recharge rate is, the larger the critical distance; and the coarser the soil texture is, the smaller the distance. One important implication of this is that the existing simpler, gravity-dominated flow models may provide good approximations for flow in vadose zones of large thickness and/or coarse-textured soils although they may not be valid for vadose zones of fine-textured soils with a shallow water table. It is also found that the vertical extent of a domain where nonstationarity is important may be estimated by solving the one-dimensional Richards equation for mean head with average soil properties and appropriate boundary conditions. On the basis of the mean head, one may then determine whether the full, nonstationary model must be solved or whether a simpler, gavity-dominated model will suffice. The flow quantities are also nonstationary in the horizontal direction near the lateral boundaries, as found for flow in saturated zones.

Numerical solutions of moment equations for flow in heterogeneous composite aquifers

[1] We analyze flow in heterogeneous media composed of multiple materials whose hydraulic properties and geometries are uncertain. Our analysis relies on the composite media theory of Winter and Tartakovksy [2000, 2002], which allows one to derive and solve moment equations even when the medium is highly heterogeneous. We use numerical solutions of Darcy flows through a representative composite medium to investigate the robustness of perturbation approximations in porous medium with total log conductivity variances as high as 20. We also investigate the relative importance of the two sources of uncertainty in composite media, material properties, and geometry. In our examples the uncertain geometry by itself captures the main features of the mean head estimated by the full composite model even when the within-material conductivities are deterministic. However, neglecting randomness within materials leads to head variance estimates that are qualitatively and quantitatively wrong. We compare the composite media approach to approximations that replace statistically inhomogeneous conductivity fields with pseudohomogeneous random fields with deterministic trends. We demonstrate that models with a deterministic trend can be expected to give a poor estimate of the statistics of head.

Fractal nature and scaling of normal faults in the Española Basin, Rio Grande rift, New Mexico: implications for fault growth and brittle strain

Abstract Quaternary faults in the western Espanola Basin of the Rio Grande rift show a power-law size (displacement) distribution suggesting that faulting in this region is scale invariant, and that faults are self similar. The power law, or fractal, distribution is characterized by fractal dimension of 0.66 to 0.79 and represents a young, immature, active fault population in a continental extensional regime. Based on this distribution, it is estimated that unobserved faults with very small displacements account for up to 6% of the total strain. Since 1.2 Ma. total extension in this part of the basin has been at least 5%. A direct correlation exists between maximum displacement and length of faults in this area suggesting that they obey a scaling relationship in which the ratio of log d max /log L is 5 × 10 -3 . This ratio is nearly constant for faults whose lengths span three-orders of magnitude, indicating that there is no difference in the scaling relationship of displacement and length between faults of all sizes. Considering previous models, these fault characteristics suggest that, in the western Espanola Basin: (1) host rock shear strengths are low; (2) remote shear stresses were probably high: and (3) most faults do not extend throughout the brittle crust. Finally, displacement profiles on five of the largest faults arc asymmetric and show a rapid decrease in displacement from the point of maximum displacement toward the fault tip. The fractal nature, scaling relationship and distribution of displacement on faults are used to suggest that faults grew by nearly proportional increases in displacement and length, perhaps by mechanisms dominated by propagating shear fractures rather than by linkage of pre-existing joints or faults.

Dynamics of Free Surfaces in Random Porous Media

We consider free surface flow in random porous media by treating hydraulic conductivity of a medium as a random field with known statistics. We start by recasting the boundary-value problem in the form of an integral equation where the parameters and domain of integration are random. Our analysis of this equation consists of expanding the random integrals in Taylors series about the mean position of the free boundary and taking the ensemble mean. To quantify the uncertainty associated with such predictions, we also develop a set of integro-differential equations satisfied by the corresponding second ensemble moments. The resulting moment equations require closure approximations to be workable. We derive such closures by means of perturbation expansions in powers of the variance of the logarithm of hydraulic conductivity. Though this formally limits our solutions to mildly heterogeneous porous media, our analytical solutions for one-dimensional flows demonstrate that such perturbation expansions may remai...

Virtual watersheds: simulating the water balance of the Rio Grande Basin

Managers of water resources in arid and semi-arid regions must allocate increasingly variable surface water supplies and limited groundwater resources. This challenge is leading to a new generation of detailed computational models that can link multiple sources to a wide range of demands. Detailed computational models of complex natural-human systems can help decision makers allocate scarce natural resources such as water. This article describes a virtual watershed model, the Los Alamos Distributed Hydrologic System (LADHS), which contains the essential physics of all elements of a regional hydrosphere and allows feedback between them. Unlike real watersheds, researchers can perform experiments on virtual watersheds, produce them relatively cheaply (once a modeling framework is established), and run them faster than real time. Furthermore, physics-based virtual watersheds do not require extensive tuning and are flexible enough to accommodate novel boundary conditions such as land-use change or increased climate variability. Essentially, virtual watersheds help resource managers evaluate the risks of alternatives once uncertainties have been quantified.

Bayesian inference-based fusion of radar imagery, military forces and tactical terrain models in the image exploitation system/balanced technology initiative

Abstract The Imagery Exploitation System/Balanced Technology Initiative (IES/BTI) inputs synthetic aperture radar (SAR) imagery and outputs probabilistically ranked interpretations of the presence and location of military force membership, organization, and expected ground formations. There are also probabilistic models of underlying terrain types from a tactical perspective that provide evidence supporting or denying the presence of forces at a location. The system compares sets of detected military vehicles extracted from imagery against the models of military units and their formations to create evidence of force type and location. Based on this evidence, the system dynamically forms hypotheses of the presence, location and formations of military forces on the ground, which it represents in a dynamically modified Bayesian network. The IES/BTI functional design is based on a decision theoretic model in which processing choices are determined as a utility function of the current state of interpretation of imagery and a top-level goal to exploit imagery as accurately and rapidly as possible, given the available data, current state of the interpretation of force hypotheses and the system processing suite. In order to obtain sufficient throughput in processing multi-megabyte SAR imagery, and also to take advantage of natural parallelism in 2D-spatial reasoning, the system is hosted on a heterogeneous network of multiple parallel computers including a SIMD Connection Machine 2 and a MIMD Encore Multimax. Independent testing by the US Army using imagery of Iraqi forces taken during Desert Storm, indicated an average 260% improvement in the performance of expert SAR imagery analysts using IES/BTI as a front end to their image exploitation.

ANALYTICAL APPROXIMATION FOR THE GENERALIZED LAPLACE EQUATION WITH STEP FUNCTION COEFFICIENT

Many problems in science and engineering require the solution of the steady-state diffusion equation with a highly oscillatory coefficient. In this paper, we propose an analytical approximation