Daniel M. Tartakovsky

Numerical Methods for Differential Equations in Random Domains

Physical phenomena in domains with rough boundaries play an important role in a variety of applications. Often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both. This topological uncertainty can be efficiently handled by treating rough boundaries as random fields, so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains. To deal with this class of problems, we propose a novel computational framework, which is based on using stochastic mappings to transform the original deterministic/stochastic problem in a random domain into a stochastic problem in a deterministic domain. The latter problem has been studied more extensively, and existing analytical/numerical techniques can be readily applied. In this paper, we employ both a stochastic Galerkin method and Monte Carlo simulations to solve the transformed stochastic problem. We demonstrate our approach by applying it to an elliptic problem in single- and double-connected random domains, and comment on the accuracy and convergence of the numerical methods.

Applicability regimes for macroscopic models of reactive transport in porous media

We consider transport of a solute that undergoes a nonlinear heterogeneous reaction: after reaching a threshold concentration value, it precipitates on the solid matrix to form a crystalline solid. The relative importance of three key pore-scale transport mechanisms (advection, molecular diffusion, and reaction) is quantified by the Péclet (Pe) and Damköhler (Da) numbers. We use multiple-scale expansions to upscale a pore-scale advection-diffusion equation with reactions entering through a boundary condition on the fluid-solid interface, and to establish sufficient conditions under which macroscopic advection-dispersion-reaction equations provide an accurate description of the pore-scale processes. These conditions are summarized by a phase diagram in the (Pe, Da)-space, parameterized with a scale-separation parameter that is defined as the ratio of characteristic lengths associated with the pore- and macro-scales.

Type curve interpretation of late-time pumping test data in randomly heterogeneous aquifers

[1] The properties of heterogeneous media vary spatially in a manner that can seldom be described with certainty. It may, however, be possible to describe the spatial variability of these properties in terms of geostatistical parameters such as mean, integral (spatial correlation) scale, and variance. Neuman et al. (2004) proposed a graphical method to estimate the geostatistical parameters of (natural) log transmissivity on the basis of quasi-steady state head data when a randomly heterogeneous confined aquifer is pumped at a constant rate from a fully penetrating well. They conjectured that a quasi-steady state, during which heads vary in space-time while gradients vary only in space, develops in a statistically homogeneous and horizontally isotropic aquifer as it does in a uniform aquifer. We confirm their conjecture numerically for Gaussian log transmissivities, show that time-drawdown data from randomly heterogeneous aquifers are difficult to interpret graphically, and demonstrate that quasi-steady state distance-drawdown data are amenable to such interpretation by the type curve method of Neuman et al. The method yields acceptable estimates of statistical log transmissivity parameters for fields having either an exponential or a Gaussian spatial correlation function. These estimates are more robust than those obtained using the graphical time-drawdown method of Copty and Findikakis (2003, 2004a). We apply the method of Neuman et al. (2004) simultaneously to data from a sequence of pumping tests conducted in four wells in an aquifer near Tubingen, Germany, and compare our transmissivity estimate with estimates obtained from 312 flowmeter measurements of hydraulic conductivity in these and eight additional wells at the site. We find that (1) four wells are enough to provide reasonable estimates of lead log transmissivity statistics for the Tubingen site using this method, and (2) the time-drawdown method of Cooper and Jacob (1946) underestimates the geometric mean transmissivity at the site by 30-40%.

Conditional stochastic averaging of steady state unsaturated flow by means of Kirchhoff Transformation

We consider the effect of measuring randomly varying soil hydraulic properties on ones ability to predict steady state unsaturated flow subject to random sources and/or initial and boundary conditions. Our aim is to allow optimum unbiased prediction of system states (pressure head, water content) and fluxes deterministically, without upscaling and without linearizing the constitutive characteristics of the soil. It has been shown by Neuman et al. (1999) that such prediction is possible by means of first ensemble moments of system states and fluxes, conditioned on measured values of soil properties, when the latter scale in a linearly separable fashion as proposed by Vogel et al. (1991); the uncertainty associated with such predictions can be quantified by means of the corresponding conditional second moments. The derivation of moment equations for soils whose properties do not scale in the above manner requires linearizing the corresponding constitutive relations, which may lead to major inaccuracies when these relations are highly nonlinear, as is often the case in nature. When the scaling parameter of pressure head is a random variable independent of location, the steady state unsaturated flow equation can be linearized by means of the Kirchhoff transformation for gravity-free flow. Linearization is also possible in the presence of gravity when hydraulic conductivity varies exponentially with pressure head. For the latter case we develop exact conditional first- and second-moment equations which are nonlocal and therefore non-Darcian. We solve these equations analytically by perturbation for unconditional vertical infiltration and compare our solution with the results of numerical Monte Carlo simulations. Our analytical solution demonstrates in a rigorous manner that the concept of effective hydraulic conductivity does not apply to ensemble-averaged unsaturated flow except when gravity is the sole driving force.

Effective hydraulic conductivity of bounded, strongly heterogeneous porous media

We develop analytical expressions for the effective hydraulic conductivity Ke of a three-dimensional, heterogeneous porous medium in the presence of randomly prescribed head and flux boundaries. The log hydraulic conductivity Y forms a Gaussian, statistically homogeneous and anisotropic random field with an exponential autocovariance. By effective hydraulic conductivity of a finite volume in such a field, we mean the ensemble mean (expected value) of all random equivalent conductivities that one could associate with a similar volume under uniform mean flow. We start by deriving a first-order approximation of an exact expression developed in 1993 by Neuman and Orr. We then generalize this to strongly heterogeneous media by invoking the Landau-Lifshitz conjecture. Upon evaluating our expressions, we find that Ke decreases rapidly from the arithmetic mean KA toward an asymptotic value as distance between the prescribed head boundaries increases from zero to about eight integral scales of Y. The more heterogeneous is the medium, the larger is Ke relative to its asymptote at any given separation distance. Our theory compares well with published results of spatially power-averaged expressions and with a first-order expression developed intuitively by Kitanidis in 1990.

Hybrid Simulations of Reaction-Diffusion Systems in Porous Media

Hybrid or multiphysics algorithms provide an efficient computational tool for combining micro- and macroscale descriptions of physical phenomena. Their use becomes imperative when microscale descriptions are too computationally expensive to be conducted in the whole domain, while macroscale descriptions fail in a small portion of the computation domain. We present a hybrid algorithm to model a general class of reaction-diffusion processes in granular porous media, which includes mixing-induced mineral precipitation on, or dissolution of, the porous matrix. These processes cannot be accurately described using continuum (Darcy-scale) models. The pore-scale/Darcy-scale hybrid is constructed by coupling solutions of the reaction-diffusion equations (RDE) at the pore scale with continuum Darcy-level solutions of the averaged RDEs. The resulting hybrid formulation is solved numerically by employing a multiresolution meshless discretization based on the smoothed particle hydrodynamics method. This ensures seamless noniterative coupling of the two components of the hybrid model. Computational examples illustrate the accuracy and efficiency of the hybrid algorithm.

Anisotropy, lacunarity, and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms

It has been shown by Di Federico and Neuman [1997, 1998a, b] that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework, which views hydraulic conductivity as a random fractal characterized by a power variogram. Such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments. Di Federico and Neuman [1997] have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity (an infinite hierarchy) of exponential or Gaussian variograms and spectra of mutually uncorrelated fields (modes) that are homogeneous and isotropic. We show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic, provided the ratios between principal integral (spatial correlation) scales are the same for all modes. We then analyze the effect of filtering out (truncating) modes of low, high, and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains. A low-frequency cutoff renders the truncated hierarchy homogeneous. The integral scales of the lowest- and highest-frequency cutoff modes are related to length scales of the sampling window (domain) and data support (sample volume), respectively. Taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale (in a manner previously shown to be consistent with observations in the isotropic case). It also allows (in principle) bridging across scales at a specific locale, as well as among locales, by adopting either site-specific or generalized variogram parameters. The introduction of intermediate cutoffs allows us to account, in a straightforward manner, for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales (for example, where textural and structural features are associated with distinct ranges of scale, such as fractures having discrete ranges of trace length and density, which dissect the rock into matrix blocks having corresponding ranges of sizes). We explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale, variance, covariance, and spectra of a truncated fractal field. We then develop an expression for the equivalent hydraulic conductivity of a box-shaped porous block, embedded within a multiscale log hydraulic conductivity field, under mean-uniform flow. The block is larger than the support scale of the field but is smaller than a surrounding sampling window. Consequently, its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function, conditioned on a known mean value of support-scale conductivity across the window, are given explicitly by our multiscale theory.

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.

TRANSIENT EFFECTIVE HYDRAULIC CONDUCTIVITIES UNDER SLOWLY AND RAPIDLY VARYING MEAN GRADIENTS IN BOUNDED THREE-DIMENSIONAL RANDOM MEDIA

We have shown elsewhere [Tartakovsky and Neuman, this issue (a)] that in randomly heterogeneous media, the ensemble mean transient flux is generally nonlocal in space-time and therefore non-Darcian. We have also shown [Tartakovsky and Neuman, this issue (b)] that there are special situations in which this flux can be localized so as to render it Darcian in real or transformed domains. Each such situation gives rise to an effective hydraulic conductivity which relates mean gradient to mean flux at any point in real or transformed space-time. In this paper we develop first-order analytical expressions for effective hydraulic conductivity under three-dimensional transient flow through a boxshaped domain due to a mean hydraulic gradient that varies slowly in space and time. When the mean gradient varies rapidly in time, the Laplace transform of the mean flux is local but its real-time equivalent includes a temporal convolution integral; we develop analytical expressions for the real-time kernel of this convolution integral. The box is embedded within a statistically homogeneous natural log hydraulic conductivity field that is Gaussian and exhibits an anisotropic exponential spatial correlation structure. By the effective hydraulic conductivity of a finite box in such a field we imply the ensemble mean (expected value) of all random equivalent conductivities that one could associate with the box under these conditions. We explore the influence of domain size, time, and statistical anisotropy on effective conductivity and include a simple new formula for its variation with statistical anisotropy ratio in an infinite domain under steady state. Appendixes C-E are available with entire article on microfiche.Order by mail from American Geophysical Union, 2000 Florida Avenue, N.W., Washington, DC 20009 or by phone at 800-966-2481;