Dongxiao Zhang

Pore scale study of flow in porous media: Scale dependency, REV, and statistical REV

Flow in porous media is studied at the pore-scale with lattice Boltzmann simulations on pore geometries reconstructed from computed microtomographic images. Pore scale results are analyzed to give quantities such as permeability, porosity and specific surface area at various scales and at various locations. With this, some fundamental issues such as scale dependency and medium variability can be assessed quantitatively. More specifically, the existence and size of the well known concept, representative elementary volume (REV), can be quantified. It is found that the size of an REV varies spatially and depends on the quantity being represented. For heterogeneous media, a better measure may be the so called “statistical REV”, which has weaker requirements than does the deterministic REV.

Displacement of a two-dimensional immiscible droplet in a channel

We used the lattice Boltzmann method to study the displacement of a two-dimensional immiscible droplet subject to gravitational forces in a channel. The dynamic behavior of the droplet is shown, and the effects of the contact angle, Bond number (the ratio of gravitational to surface forces), droplet size, and density and viscosity ratios of the droplet to the displacing fluid are investigated. For the case of a contact angle less than or equal to 90°, at a very small Bond number, the wet length between the droplet and the wall decreases with time until a steady shape is reached. When the Bond number is large enough, the droplet first spreads and then shrinks along the wall before it reaches steady state. Whether the steady-state value of the wet length is greater or less than the static value depends on the Bond number. When the Bond number exceeds a critical value, a small portion of the droplet pinches off from the rest of the droplet for a contact angle less than 90°; a larger portion of the droplet is entrained into the bulk for a contact angle equal to 90°. For the nonwetting case, however, for any Bond number less than a critical value, the droplet shrinks along the wall from its static state until reaching the steady state. For any Bond number above the critical value, the droplet completely detaches from the wall. Either increasing the contact angle or viscosity ratio or decreasing the density ratio decreases the critical Bond number. Increasing the droplet size increases the critical Bond number while it decreases the critical capillary number.

Higher-order effects on flow and transport in randomly heterogeneous porous media.

Advective solute transport in nonuniform geologic media is generally nonlocal and non-Fickian [Neuman, 1993; Cushman and Ginn, 1993]. In statistically homogeneous log conductivity fields under uniform mean flow, the transport is expected to become asymptotically local and Fickian at late time. During the earlier preasymptotic regime, macrodispersivity (a measure of the rate at which a plume spreads) is expected to vary with solute residence time. A first-order (linear in the natural log hydraulic conductivity variance, σ2) analysis of this variation has been performed by Dagan [1984, 1987, 1988]. He found that, when local dispersion is neglected, the longitudinal macrodispersivity increases monotonically from zero toward a constant asymptote. However, the transverse macrodispersivity first increases from zero to a peak value, then decreases monotonically toward zero. The first-order asymptotic analyses by Winter [1982], Gelhar and Axness [1983] and Winter et al. [1984] also yield zero transverse macrodispersivity when local dispersion is disregarded.

Non-modal growth of perturbations in density-driven convection in porous media

In the context of geologic sequestration of carbon dioxide in saline aquifers, much interest has been focused on the process of density-driven convection resulting from dissolution of CO 2 in brine in the underlying medium. Recent investigations have studied the time and length scales characteristic of the onset of convection based on the framework of linear stability theory. It is well known that the non-autonomous nature of the resulting matrix does not allow a normal mode analysis and previous researchers have either used a quasi-static approximation or solved the initial-value problem with arbitrary initial conditions. In this manuscript, we describe and use the recently developed non-modal stability theory to compute maximum amplifications possible, optimized over all possible initial perturbations. Non-modal stability theory also provides us with the structure of the most-amplified (or optimal) perturbations. We also present the details of three-dimensional spectral calculations of the governing equations. The results of the amplifications predicted by non-modal theory compare well to those obtained from the spectral calculations.

Numerical solutions to statistical moment equations of groundwater flow in nonstationary, bounded, heterogeneous media

In this paper we investigate the combined effect of nonstationarity in log hydraulic conductivity and the presence of boundaries on flow in heterogeneous aquifers. We derive general equations governing the statistical moments of hydraulic head for steady state flow by perturbation expansions. Due to their mathematical complexity, we solve the moment equations by the numerical technique of finite differences. The numerical approach has flexibility in handling (moderately) irregular geometry, different boundary conditions, various trends in the mean log hydraulic conductivity, spatial variabilities in the magnitude and direction of mean flow, and different covariance functions, all of which are important factors to consider for real-world applications. The effect of boundaries on the first two statistical moments involving head is strong and persistent. For example, in the case of stationary log hydraulic conductivity the head variance is always finite in a bounded domain while the head variance may be infinite in an unbounded domain. As in many other stochastic models, the statistical moment equations are derived under the assumption that the variance of log hydraulic conductivity is small. Accounting for a spatially varying, large-scale trend in the log hydraulic conductivity field reduces the variance of log hydraulic conductivity. Although this makes the conductivity field nonstationary and significantly increases the mathematical complexity in the problem, it justifies the small-variance assumption for many aquifers.

Comprehensive Review of Caprock-Sealing Mechanisms for Geologic Carbon Sequestration

CO(2) capture and geologic sequestration is one of the most promising options for reducing atmospheric emissions of CO(2). Its viability and long-term safety, which depends on the caprocks sealing capacity and integrity, is crucial for implementing CO(2) geologic storage on a commercial scale. In terms of risk, CO(2) leakage mechanisms are classified as follows: diffusive loss of dissolved gas through the caprock, leakage through the pore spaces after breakthrough pressure has been exceeded, leakage through faults or fractures, and well leakage. An overview is presented in which the problems relating to CO(2) leakage are defined, dominant factors are considered, and the main results are given for these mechanisms, with the exception of well leakage. The overview includes the properties of the CO(2)-water/brine system, and the hydromechanics, geophysics, and geochemistry of the caprock-fluid system. In regard to leakage processes, leakage through faults or fracture networks can be rapid and catastrophic, whereas diffusive loss is usually low. The review identifies major research gaps and areas in need of additional study in regard to the mechanisms for geologic carbon sequestration and the effects of complicated processes on sealing capacity of caprock under reservoir conditions.

A stochastic analysis of steady state two‐phase flow in heterogeneous media

[1] We present a novel approach to modeling stochastic multiphase flow problems, for example, nonaqueous phase liquid flow, in a heterogeneous subsurface medium with random soil properties, in particular, with randomly heterogeneous intrinsic permeability and soil pore size distribution. A stochastic numerical model for steady state water-oil flow in a random soil property field is developed using the Karhunen-Loeve moment equation (KLME) approach and is numerically implemented. An exponential model is adopted to define the constitutive relationship between phase relative permeability and capillary pressure. The log-transformed intrinsic permeability Y(x) and soil pore size distribution b(x) are assumed to be Gaussian random functions with a separable exponential covariance function. The perturbation part of these two log-transformed soil properties is then decomposed into an infinite series based on a set of orthogonal normal random variables {xn}. The phase pressure, capillary pressure, and phase mobility are decomposed by polynomial expansions and the perturbation method. Combining these expansions of Y(x), b(x) and dependent pressures, the steady state water-oil flow equations and corresponding boundary conditions are reformulated as a series of differential equations up to second order. These differential equations are solved numerically, and the solutions are directly used to construct moments of phase pressure and capillary pressure. We demonstrate the validity of the proposed KLME model by favorably comparing firstand second-order approximations to Monte Carlo simulations. The significant computational efficiency of the KLME approach over Monte Carlo simulation is also illustrated.

Nonstationary stochastic analysis of transient unsaturated flow in randomly heterogeneous media

In this study, a first-order, nonstationary stochastic model of transient flow is developed which is applicable to the entire domain of a bounded vadose zone in the presence of sink/source. We derive general equations governing the statistical moments of the flow quantities by perturbation expansions. Owing to the mathematical complexity of the equations, in general we need to solve them numerically. The numerical moment equation approach, however, has the flexibility in handling different boundary conditions, flow configurations, input covariance structures, and soil constitutive relationships. The moment equations presented in this study are simpler and easier to solve than those in the literature for transient unsaturated flow. We solve these moment equations by the method of finite differences and demonstrate the developed model through some one- and two- dimensional examples under various transient conditions.

Eulerian-Lagrangian Analysis of Transport Conditioned on Hydraulic Data: 1. Analytical-Numerical Approach

Recently, a unified Eulerian-Lagrangian theory has been developed by one of us for nonreactive solute transport in space-time nonstationary velocity fields. We describe a combined analytical-numerical method of solution based on this theory for the special case of steady state flow in a mildly fluctuating, statistically homogeneous, lognormal hydraulic conductivity field. We take the unconditional mean velocity to be uniform but allow conditioning on measurements of log hydraulic conductivity (or transmissivity) and/or hydraulic head. This renders the velocity field nonstationary. We solve the conditional transport problem analytically at early time and express it in pseudo-Fickian form at later time. The deterministic pseudo-Fickian equations involve a conditional, space-time dependent dispersion tensor which we evaluate numerically along mean “particle” trajectories. These equations lend themselves to accurate solution by standard Galerkin finite elements on a relatively coarse grid. The final step is an explicit numerical computation of lower bounds on conditional concentration prediction variance-covariance (and coefficient of variation), travel time distribution, cumulative mass release across a “compliance surface,” the associated error, and plume spatial moments. Our method also allows quantification of the uncertainty in the original source location of any solute “particle” located anywhere in the field, at any time. This paper describes the methodology and presents some unconditional results. Conditioning and more advanced computations are presented in the subsequent papers.

Displacement of a three-dimensional immiscible droplet in a duct

The displacement of a three-dimensional immiscible droplet subject to gravitational forces in a duct is studied with the lattice Boltzmann method. The effects of the contact angle and capillary number (the ratio of viscous to surface forces) on droplet dynamics are investigated. It is found that there exists a critical capillary number for a droplet with a given contact angle. When the actual capillary number is smaller than the critical value, the droplet moves along the wall and reaches a steady state. When the capillary number is greater than the critical value, one or more small droplets pinch off from the wall or from the rest of the droplet, depending on the contact angle and the specific value of the capillary number. As the downstream part of the droplet is pinching off, a bottleneck forms and its area continues decreasing until reaching zero. The general trend found in a previous two-dimensional study that the critical capillary number decreases as the contact angle increases is confirmed. It is shown that at a fixed capillary number above the critical value, increasing the contact angle results in a larger first-detached portion. At a fixed contact angle, increasing the capillary number results in an increase of the size of the first detached droplet for