Robert P. Ewing

Percolation theory for flow in porous media

Percolation Theory: Topology and Structure.- Properties Relevant for Transport and Transport Applications.- Porous Media Primer for Physicists.- Specific Examples of Critical Path Analysis.- Hydraulic and Electrical Conductivity: Conductivity Exponents and Critical Path Analysis.- Other Transport Properties of Porous Media.- Pressure–Saturation Curves and the Critical Volume Fraction for Percolation: Accessibility Function of Percolation Theory.- Applications of the Correlation Length: Scale Effects on Flow.- Applications of the Cluster Statistics.- Properties based on Tortuosity.- Effects of Multi-Scale Heterogeneity.

Percolation theory and network modeling applications in soil physics

The application of percolation theory to porous media is closely tied to network models. A network model is a detailed model of a porous medium, generally incorporating pore-scale descriptions of the medium and the physics of pore-scale events. Network models and percolation theory are complementary: while network models have yielded insight into behavior at the pore scale, percolation theory has shed light, at the larger scale, on the nature and effects of randomness in porous media. This review discusses some basic aspects of percolation theory and its applications, and explores work that explicitly links percolation theory to porous media using network models. We then examine assumptions behind percolation theory and discuss how network models can be adapted to capture the physics of water, air and solute movement in soils. Finally, we look at some current work relating percolation theory and network models to soils.

Low pore connectivity in natural rock.

As repositories for CO(2) and radioactive waste, as oil and gas reservoirs, and as contaminated sites needing remediation, rock formations play a central role in energy and environmental management. The connectivity of the rocks porespace strongly affects fluid flow and solute transport. This work examines pore connectivity and its implications for fluid flow and chemical transport. Three experimental approaches (imbibition, tracer concentration profiles, and imaging) were used in combination with network modeling. In the imbibition results, three types of imbibition slope [log (cumulative imbibition) vs. log (imbibition time)] were found: the classical 0.5, plus 0.26, and 0.26 transitioning to 0.5. The imbibition slope of 0.26 seen in Indiana sandstone, metagraywacke, and Barnett shale indicates low pore connectivity, in contrast to the slope of 0.5 seen in the well-connected Berea sandstone. In the tracer profile work, rocks exhibited different distances to the plateau porosity, consistent with the pore connectivity from the imbibition tests. Injection of a molten metal into connected pore spaces, followed by 2-D imaging of the solidified alloy in polished thin sections, allowed direct assessment of pore structure and lateral connection in the rock samples. Pore-scale network modeling gave results consistent with measurements, confirming pore connectivity as the underlying cause of both anomalous behaviors: imbibition slope not having the classical value of 0.5, and accessible porosity being a function of distance from the edge. A poorly connected porespace will exhibit anomalous behavior in fluid flow and chemical transport, such as a lower imbibition slope (in air-water system) and diffusion rate than expected from classical behavior.

A generalized growth model for simulating initial migration of dense non‐aqueous phase liquids

Modeling the migration of spilled non–aqueous phase liquids (NAPLs) is currently difficult because the physics governing their movement is complex and knowledge of the local geology is always incomplete. NAPL movement is subject to buoyancy, capillary, and viscous forces in addition to being directed by the particular structural and hydraulic properties of the porous medium. Consideration of buoyancy forces suggests that the flow regime diagram of Lenormand et al. [1988] can be expanded to a third dimension. We develop a generalized growth model, based on invasion percolation, that captures the essential physics of initial NAPL migration but is simple enough computationally that simulations can be conducted much faster than by using continuum simulation models that attempt to capture all details of the physics. In comparison with available experimental data, our model realistically simulates movement of a NAPL for a wide range of values of Bond and capillary numbers, in any kind of porous medium. Because this approach allows much faster and simpler simulation than other approaches, a higher spatial resolution and/or the use of Monte Carlo methods to reduce the effect of geological uncertainty becomes a realistic possibility.

Percolation and permeability in partially structured networks

Structure, meaning a nonrandom arrangement of pores or pore domains, is present in many geologic porous media. We examined the effects of different nonrandom arrangements of the pore domains containing the largest pores on the percolation and flow properties of a simulated porous medium. Increasing the length of clusters (structural elements), or the fraction of the pore space occupied by them, decreases the percolation threshold (air entry value) and increases the permeability. Decreasing the internal homogeneity of clusters decreases the extent of their effects on percolation and permeability. Percolation threshold is not affected by the ratio between cluster length and network size as long as the cluster length is less than two thirds the network size. All cluster shapes display a similar relationship between percolation threshold and permeability, seen also in geologic porous media. The air entry value is a parameter that could potentially quantify both the degree of structure of a medium and its saturated permeability.

Universal scaling of the formation factor in porous media derived by combining percolation and effective medium theories

The porosity dependence of the formation factor for geologic media is examined from the perspective of universal scaling laws from percolation and effective medium theories. Over much of the range of observed porosity, the expected percolation scaling is observed, but the values of the numerical prefactor do not conform to the simple predictions from percolation theory. Combining effective medium and percolation theories produces a numerical prefactor whose value depends on both the threshold porosity and the porosity above which the formation factor crosses from percolation to effective medium scaling. This change allows extraction of a numerical value of the prefactor, which is reasonably close to experimental values. Subsequent evaluation of the porosity dependence of the formation factor shows that difficulties in prior comparisons of theory and experiment are largely removed when percolation scaling is allowed to transition to effective medium scaling far above the percolation threshold.

Stochastic pore-scale growth models of DNAPL migration in porous media ☆

Stochastic models that account for a wide range of pore-scale effects are discussed in the context of two-phase, immiscible displacement problems in natural porous media. We focus on migration of dense, nonaqueous phase liquids (DNAPLs) through water-saturated geological materials. DNAPL movement is governed by buoyancy, capillary, and viscous forces, as well as by the details of the porous medium. We examine key issues relevant to development of stochastic growth models. We then present a particular stochastic growth model, based on a generalization of invasion percolation and Eden growth approaches, which realistically simulates two-phase flows in a computationally efficient manner. Fingering patterns are shown to depend critically on the competing buoyancy, capillary, and viscous forces between the DNAPL and the water, and their interactions with the porous medium at the local scale. We conclude with recommendations for future research.

Modeling percolation properties of random media using a domain network

Many models for calculating hydraulic conductivity assume that the pore size distribution of a medium can be uniquely derived from the water retention curve. We examine this assumption by comparing water retention curves from simulations of drainage and imbibition of a domain network model to curves derived solely from the input pore size distribution. We use domain networks rather than the more usual oil reservoir pore scale networks to account for features of soils not commonly found in oil reservoirs. Simulated drainage and imbibition curves are substantially different from the input curve due to (1) the presence of a threshold in both the simulated drainage and imbibition curves and (2) the presence of entrapped air during imbibition. Differences between the simulated and input curves increase or decrease depending upon the network size, the type of percolation mechanism considered for drainage and imbibition, the aspect ratio of the pores, and the coordination number of the medium.

Quantitative 3-D elemental mapping by LA-ICP-MS of a basaltic clast from the hanford 300 area, Washington, USA

Laser ablation with inductively coupled plasma-mass spectrometry (LA-ICP-MS) was used to measure elemental concentrations at the 100-μm scale in a 3-dimensional manner within a basaltic clast sample collected from the Hanford 300 Area in south-central Washington State, United States. A calibration method was developed to quantify the LA-ICP-MS signal response using a constant-sum mass fraction of eight major elements; the method produced reasonable concentration measurements for both major and trace elements when compared to a standard basalt sample with known concentrations. 3-Dimensional maps (stacked 2-D contour layers, each representing 2100 μm × 2100 μm) show relatively uniform concentration with depth for intrinsic elements such as Si, Na, and Sr. However, U and Cu accumulation were observed near the sample surface, consistent with the sites release history of these contaminants. U and Cu show substantial heterogeneity in their concentration distributions within horizontal slices, while the intrinsic elements are essentially uniformly distributed. From these measured U concentrations and published grain size distributions, gravel and cobbles were estimated to contain about 1% of the contaminant U, implicating the coarse fraction as a long-term release source.

Internal Domains of Natural Porous Media Revealed: Critical Locations for Transport, Storage, and Chemical Reaction

Internal pore domains exist within rocks, lithic fragments, subsurface sediments, and soil aggregates. These domains, termed internal domains in porous media (IDPM), represent a subset of a materials porosity, contain a significant fraction of their porosity as nanopores, dominate the reactive surface area of diverse media types, and are important locations for chemical reactivity and fluid storage. IDPM are key features controlling hydrocarbon release from shales in hydraulic fracture systems, organic matter decomposition in soil, weathering and soil formation, and contaminant behavior in the vadose zone and groundwater. Traditionally difficult to interrogate, advances in instrumentation and imaging methods are providing new insights on the physical structures and chemical attributes of IDPM, and their contributions to system behaviors. Here we discuss analytical methods to characterize IDPM, evaluate information on their size distributions, connectivity, and extended structures; determine whether they exhibit unique chemical reactivity; and assess the potential for their inclusion in reactive transport models. Ongoing developments in measurement technologies and sensitivity, and computer-assisted interpretation will improve understanding of these critical features in the future. Impactful research opportunities exist to advance understanding of IDPM, and to incorporate their effects in reactive transport models for improved environmental simulation and prediction.