Flow In Heterogeneous Media Research Articles

Flow in Heterogeneous Media Displaying a Linear Trend in the Log Conductivity

The purpose of this study is to derive a general solution for the problem of flow in nonstationary geological formations where the nonstationarity manifests itself in the form of a spatial trend in the mean log conductivity. A stochastic frame of reference is adopted to account for the spatial variability of the hydraulic conductivity. For a complete stochastic description we derive the expected values and spatial covariances of the hydraulic head and the fluid flux vector, as well as a relation between the expected values of the head and the fluxes. These expressions are obtained using a perturbation expansion of the log conductivity about its nonstationary mean, and they are correct to the first order in the variance of the log conductivity. The expressions we derive are applicable for any space dimensionality and for arbitrary orientation of the trend in space. A general methodology is outlined for derivation of these expressions for any type of spatial covariance; and for demonstration, explicit results are obtained for a Gaussian isotropic covariance.

The code IVA 4: second law of thermodynamics for multi-phase multi-component flows in heterogeneous media

The code IVA 4: second law of thermodynamics for multi-phase multi-component flows in heterogeneous media

Nonlocal properties of nonuniform averaged flows in heterogeneous media

The properties of nonuniform average potential flows in media of stationary random conductivity are studied. The mathematical model of average flow is derived as a system of governing equations to be satisfied by mean velocity and mean head. The averaged Darcy's law determines the effective conductivity as an integral operator of the convolution type, relating the mean velocity to the mean head gradient in a nonlocal way. In Fourier domain the mean velocity is proportional to the mean head gradient. The coefficient of proportionality is referred to as the effective conductivity tensor and is derived by perturbation methods in terms of functionals of the conductivity spatial moments. It is shown that the effective conductivity cannot be defined uniquely for potential flows. However, this nonuniqueness does not affect the spatial distribution of the mean head and velocity. Analytical expressions of the effective conductivity tensor are derived for two‐ and three‐dimensional flows and for exponential and Gaussian correlations of isotropic conductivity. The fundamental solution of the governing equations in effective media (mean Green function) is calculated for the same cases. Two new asymptotic models of the averaged Darcy's law are developed to be applicable to large and small scales of heterogeneity. Several asymptotic expansions of tue two‐dimensional and three‐dimensional mean Green functions are derived for exponential and Gaussian correlations.

One- and Two-Dimensional Pulse Sequences for Diffusion Experiments in the Fringe Field of Superconducting Magnets

Abstract The fringe field of superconducting magnets provides extremely strong and stable field gradients which can be used for diffusion experiments. A number of corresponding supercon fringe-field variants of the pulsed-gradient spin-echo method are presented. These include procedures and pulse sequences for the record of relaxation-independent diffusion decays, multislice excitation, and one- or two-dimensional Fourier-transform evaluation from the k to the z domain. Several test experiments are described. It is demonstrated that reliable diffusion decays are obtained. The one-dimensional z -domain evaluation produces grid patterns analogous to the optical forced Rayleigh scattering method. The two-dimensional experiments are suitable for the detection of spatially varying probability densities of diffusion or incoherent flow in heterogeneous media. As the resolution is particularly high, fluids can be studied close to solid surfaces.

Stochastic modeling of unsaturated flow in heterogeneous media with water uptake by plant roots: Tests of the parallel columns model under two‐dimensional flow conditions

In the first paper of this series (Rubin and Or, 1993) we developed a stochastic model predicting spatial moments of soil water matric potential ψ and saturation S under conditions of unsaturated steady state flow as function of the spatial moments of log saturated hydraulic conductivity Y, capillarity index α, and plant rooting depth δ. In this study, we investigate the ability of the proposed parallel columns model to reconstruct two‐dimensional flow solutions, and apply the model to spatial estimation using the geostatistical formalism. A finite difference numerical scheme was developed to solve the two‐dimensional Richards equation in heterogeneous flow domain and obtain detailed images of the spatial distribution of ψ and S and their moments. Two‐dimensionally‐based images representing “true” distributions of ψ and S in a vertical cross section (x‐z plane) of laterally heterogeneous soil were used in a synthetic spatial estimation problem for comparison with the one‐dimensionally‐based solution and with kriging estimates. The results show that (1) the means of ψ and S were practically identical for the one‐ and two‐dimensional solutions; (2) ψ variance obtained from the two‐dimensional solution was smaller than the one‐dimensional, whereas S variance was larger in the two‐dimensional case; (3) a smoothing of ψ under two‐dimensional flow regime resulted in spatial covariance function of a Bessel type, whereas two‐dimensional spatial covariance of S exhibited a steep decay near its origin denoting spatial discontinuity; (4) the least smoothing of ψ was observed near the surface where nearly one‐dimensional conditions exist; (5) during transient flow, spatial covariances assume their steady state form quite rapidly; (6) spatial covariances predicted by the parallel columns model provided good estimates at unmeasured locations with conservative confidence regions for ψ estimation and less conservative for S estimation; and (7) analytically predicted spatial covariances provide a means for network design.

On spectral measure and Monte Carlo Approaches to flow in heterogeneous media

A pedagogical approach to the spectral integral method as applied to flow problems is presented. The deterministic Fourier analog to the random spectral integral is highlighted, thus illustrating the close ties between the two. The general solution to the deterministic flow problem is used to illustrate that a weakly stationary conductivity field does not in general lead to a weakly stationary head field. Several alternative Monte Carlo methods are presented and compared to illustrate the problems associated with dropping terms of the form ƒ′h′ from the flow equation, where ƒ and h are respectively perturbations in the conductivity and head fields.

Monte Carlo Simulation of Linear Two-Phase Flow in Heterogeneous Media

In this paper an algorithmic formulation is given for the synthesis of linear two-phase systems with inherent variability. It is assumed that uncertainties in the flow system arise due to the heterogeneities of the porous medium. Methods of stochastic mechanics are adopted to offer a novel approach, whereby the absolute permeability is taken as a spatial stochastic process with a log-normal distribution at each point of the flow domain. A correlation structure is built into the model so as to achieve a realistic representation. In addition, a constraint on the invariance of the harmonic means of the random permeabilities is attained by way of topological scaling. A computer-synthesized waterflooding experiment is conducted to explore the effect of heterogeneity on output variables such as the oil pressure head and cumulative recovery. It is observed that output uncertainties can become significant even when the heterogeneities are small. By employing a least-squares procedure, the sensitivity of relative permeabilities to induced stochastic variations is also qualitatively investigated.