Dynamics of Wetting Fronts in Porous Media

We propose a new phenomenological approach for describing the dynamics of wetting front propagation in porous media. Unlike traditional models, the proposed approach is based on dynamic nature of the relation between capillary pressure and medium saturation. We choose a modified phase-field model of solidification as a particular case of such dynamic relation. We show that in the traveling wave regime the results obtained from our approach reproduce those derived from the standard model of flow in porous media. In more general case, the proposed approach reveals the dependence of front dynamics upon the flow regime.Comment: 4 pages, 2 figures, revte

ASYMPTOTIC PROPERTIES OF MASS TRANSPORT IN RANDOM POROUS MEDIA.

Suppose C(x,t) is the concentration at position x in Rᵈ and time t > 0 of a solute which is diffusing in some medium. If on a local scale the dispersion of the solute is governed by a constant dispersion matrix, 1/2(δ²), and a random velocity field, V(x), then C satisfies a convection-diffusion equation with random coefficients, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (1). Usually V(x) is taken to be μ + εU(x) where μ ε Rᵈ, U(x) is a given stationary random field with mean zero, and ε > 0 is a dimensionless parameter which measures the variability of V(x). Hydrological experiments suggest that on a regional scale the diffusion is classically Fickian with effective diffusion matrix D(ε) and drift velocity α(ε). Thus for large scales (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (2) is satisfied by the solute concentration. Here τ and χ are respectively time and space measured on large scales. It is natural to investigate the relation of the large scale coefficients D and α to the statistical properties of V(x). To relate (1) to (2)--and thus to approximate D(ε) and α(ε)--it is necessary to rescale t and x and average over the distribution of V. It can then be shown that the transition form (1) to (2) is equivalent to (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (3) where A = (∇•δ²∇)/2 + √nμ• ∇ and B(U) = √nU(√nx) • ∇. By expanding each side of (3) estimates of D(ε) and α(ε) can be obtained. The estimates have the form (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) (4). Both D₂ and α₂ depend on the power spectrum of U. Analysis shows that in at least the case of incompressible fluids D₂ is positive definite. In one dimensional transport α₂ < 0, hence α(k) < μ(k) through second order

Relationships among climate, tree-ring widths and grass production on the Santa Rita Experimental Range

I am indebted to many people for their advice and assistance in the completion of this thesis. At the Laboratory of Tree-Ring Research, Dr. Thomas P. Harlan aided and encouraged me in cross dating the sample cores; Linda G. Drew guided me through the data processing; Dr. Harold C. Fritts was kind enough to read an early copy of the thesis and to alert me to many of its weaknesses; Dr. Terence J. Biasing read and commented on the section on climate; James M. Burns measured the band-width series; Malcolm K. Cleaveland introduced me to the dendroclimatic potential of separate earlywood and latewood measurements and generously offered me his collection of literature on the subject. Dwight R. Cable of the Rocky Mountain Forest and Range Experiment Station, Santa Rita Experimental Range, provided the grass production data. His comments on grassclimate relationships in general, and on the response surfaces developed here in particular, were very valuable. I appreciate the patience and guidance given me by the members of my degree committee, Dr. Charles W. Stockton, Dr. Allen M. Solomon, and Marvin A. Stokes. I am especially grateful to Dr. Stockton, the director of this thesis, for his confidence in the study, for the degree of independence he allowed me, and for his thoughtful criticism. My sister, Susan W. Mills, labored mightily on the illustrations, and I find myself once again in her debt. Much appreciation is due Deborah R. Sliz for her careful review of the manuscript. None of these people are, of course, responsible for any errors of fact or interpretation which I may have committed.hydrology collectio

Statistical scaling of geometric characteristics in stochastically generated pore microstructures

We analyze the statistical scaling of structural attributes of virtual porous microstructures that are stochastically generated by thresholding Gaussian random fields. Characterization of the extent at which randomly generated pore spaces can be considered as representative of a particular rock sample depends on the metrics employed to compare the virtual sample against its physical counterpart. Typically, comparisons against features and/patterns of geometric observables, e.g., porosity and specific surface area, flow-related macroscopic parameters, e.g., permeability, or autocorrelation functions are used to assess the representativeness of a virtual sample, and thereby the quality of the generation method. Here, we rely on manifestations of statistical scaling of geometric observables which were recently observed in real millimeter scale rock samples as additional relevant metrics by which to characterize a virtual sample. We explore the statistical scaling of two geo-metric observables, namely porosity ($phi$) and specific surface area (SSA), of porous microstructures generated using the method of Smolarkiewicz and Winter [42] and Hyman and Winter [22]. Our results suggest that the method can produce virtual pore space samples displaying the symptoms of statistical scaling observed in real rock samples. Order q sample structure functions (statistical moments of absolute increments) of $phi$ and SSA scale as a power of the separation distance (lag) over a range of lags, and extended self-similarity (linear relationship between log structure functions of successive orders) appears to be an intrinsic property of the generated media. The width of the range of lags where power-law scaling is observed and the Hurst coefficient associated with the variables we consider can be controlled by the generation parameters of the method