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Immiscible displacement, involving two fluids in a porous medium, can be unstable and fingered under certain conditions. In this paper, the original linear instability criterion of Chuoke et al. (1959) is generalized, considering wettability of two immiscible fluids to the porous medium. This is then used to predict 24 specific flow and porous medium conditions for the onset of wetting front instability in the subsurface. Wetting front instability is shown to be a function of the driving fluid wettability to the medium, differences in density and viscosity of the fluids, the magnitude of the interfacial tension, and the direction of flow with respect to gravity. Scenarios of water and nonaqueous-phase liquid infiltration into the vadose zone are examined to predict preferential flow and contamination of groundwater. The mechanisms of finger formation, propagation, and persistence in the vadose zone are reviewed, and the existing equations for calculating the size, the number and velocity of fingers are simplified for field applications. The analyses indicate that fingers initiate and propagate according to spatial and temporal distribution of the dynamic breakthrough (water- or air-entry) pressures in the porous medium. The predicted finger size and velocity are in close agreement with the experimental results.

The in situ determination of the field-saturated hydraulic conductivity of low-permeability porous materials is a major concern for both geotechnics and soil physics with regards to environmental protection or water resources management. Recent early-time single-ring infiltration experiments, involving sequential constant head and falling head conditions, allow its efficient estimation. Nevertheless, the theory on which the interpretation was based was still strictly valid to nondeformable soils and implicity relied on a particular form of the hydraulic conductivity-soil water pressure head relationship. This theory is now extended to deformable materials, without any restrictive hypothesis. A new concept, bulk sorptivity, which characterizes the solid phase movement, is introduced. Field experiments, conducted on two liners of swelling and slowly permeable materials, revealed that neglecting the soil deformation induces an underestimation of the actual coefficient of permeability of the soil.

Spatial variability of hydraulic parameters is an important factor in prediction of water flow and solute transport in unsaturated soils. An analytical solution for one-dimensional constant rate rainfall infiltration in a homogeneous soil is incorporated into a model of a spatially variable field. The model is a composite of vertically homogeneous soil columns that differ in their hydraulic properties. In this study, horizontal variation in infiltration profiles is due to variation in the inverse macroscopic capillary length (α) of each column. The sill in travel time variance of wetting profiles with depth for the homogeneous column confirms the notion of a long-time traveling wave of fixed shape. Variation in α slightly enhances the spread of wetting profiles and delays formation of the traveling wave. Identification of the mean α value is more critical than inclusion of variability in α for wetting profile simulations under the examined conditions.

We used time domain reflectometry (TDR) probes installed vertically at the soil surface beneath a constant-rate rainfall simulator to measure cumulative water storage and the soil's unsaturated hydraulic conductivity. The slope from linear regression of water storage on time before any applied water infiltrates to the bottom of the TDR probe gives an estimate of the local infiltration rate. Local infiltration rates measured by TDR in the field were plotted against the corresponding local steady state water contents to give an estimate of the soil's unsaturated hydraulic conductivity over a range in water content of 20% using only two applied rainfall rates. The spatial variability in local infiltration rates may be the result of infrequent high-intensity pulses of rainfall leading to temporary ponding and redistribution of water at the soil surface. Nonlinear optimization was used to estimate the saturated hydraulic conductivity and inverse capillary length scale from TDR data.

The field-saturated hydraulic conductivity Kfs can be obtained from measurements of the long-time steady flow rates from ring infiltrometers by assuming either that the soil is a “Green and Ampt” soil that has a relationship between hydraulic conductivity K and soil water pressure ψ, such that K = Kfs, 0 > ψ > ψf; K → 0, ψƒ < ψf, or that the soil is a “Gardner” soil with an exponential relationship K = Kfs exp(αψ), where α is a constant. It is shown that infiltration into a Green and Ampt soil from a surface source with zero head is the same as that for a Gardner soil with α → ∞ (gravity dominant) and with a surface source at a head |ψfs|. The shape factor G calculated for Gardner soils using a numerical solution of Richards's equation is used to calculate Kfs and α from ring infiltrometer tests based on Q = [aH/G + a/(αG) + πa2]Kfs, where Q is the steady state flow rate, H is the constant ponded head, and a is the ring radius. The equivalent G factor for Green and Ampt soils calculated using electric analogue solutions of Laplace's equation is shown to agree very well with the numerical solution. The slope of the linear relationship between the shape factor and the depth of insertion of the ring divided by the radius of the ring is shown to have a slope approximately equal to 1/π.

Recently, two exact analytical solutions of Richards' flow equation in Fokker-Planck form using empirical nonlinear diffusivity and hydraulic conductivity functions were published. A direct application of the solutions is the modeling of cumulative water storage in a fixed depth under constant rainfall infiltration. Analytical solutions for storage require a change of variable of integration and a numerical inversion technique. The solutions contain three independent parameters including saturated hydraulic conductivity, the α parameter, and a parameter which defines the shape of the wetting front. Sensitivity analyses indicate that changes in parameter values give significant changes in cumulative storage curves. Cumulative storage can be measured by vertical parallel time domain reflectometry probes which are capable of measuring an evolving volumetric moisture content within a fixed depth. A nonlinear least squares fitting procedure may be used to evaluate the three parameters.

In the in situ determination of hydraulic conductivity most procedures are based on a close attainment of steady state infiltration. The accuracy of the measurement is therefore dependent on both the degree to which steady state flow is attained and the validity of the assumptions in the analysis procedure. Several approximate analytical solutions of one-dimensional and three-dimensional flow provide reasonable estimates of both transient and steady state infiltration compared to the corresponding numerical solutions. Three-dimensional infiltration initially approaches steady state flow much faster than one-dimensional infiltration, but at some point close to steady flow and dependent on soil properties, one-dimensional infiltration approaches steady state faster. Some practical answers are supplied on the time required to reach reasonable estimates of steady state flow, the amount of water used, and the distance to the wetting front for a range of soil properties.

[1] In the paper ‘‘Prediction of fingering in porous media’’ by Zhi Wang, Jan Feyen, and David E. Elrick (Water Resources Research 34(9), 2183–2190, 1998), the authors developed 24 specific criteria for predicting unstable flow in porous media, based on the Chuoke equation [Chuoke et al., 1959]. However, some of the results were inconsistent with predictions by others. For instance, de Rooij [2000] noted that the Wang criteria showed unstable flow for overly rapid upward flow in contrary to Philip’s [1975] theory that predicted an unconditionally stable flow. These discrepancies resulted from three mistakes in the theoretical development, namely, (1) an inconsistency in the transcription of the Chuoke equation (equation (1) in the original paper), (2) a discrepancy between the definition of s* compared to the Chuoke definition (leading to a wrong equation (3a)) and, (3) sign errors in the development of the criteria (Table 1). In the following, we give the correct development and revise the 24 stability criteria of Wang et al. [1998].

A series of miscible displacement experiments was conducted in an unsaturated glass bead medium maintained at a constant average water content during each displacement. The variation in the form of the breakthrough curve with decreasing water content was not large and was not related in a simple way to the water content; however, there was consistent shift of the breakthrough curve to the left of the relative concentration value of 0.5 and 1 pore volume and a lone tail or slow approach to the final relative concentration of 1.0. Some of these effects may be attributed to the presence of stagnant liquid in the pores. (11 refs.)

A new apparatus for studying miscible displacement processes in soils is described. Both the soil water content and the pore water velocity can be controlled within certain ranges. The usefulness of this technique is illustrated by a brief description of: (1) the mixing of Cl− during miscible displacement in a glass bead medium; (2) the movement and adsorption of Atrazine, a herbicide, in soil; and (3) the movement and microbiological nitrification of NH4+−N to NO3−−N during unstaturated flow conditions in soil.

A series of experiments have been conducted at a constant temperature on the intrusion of salt from a reservoir containing a fixed volume of solution into an underlying long vertical column of porous material, initially saturated with pure water and sealed at the bottom. The variation with time of the salt concentration in the reservoir was measured for different types of porous materials, different initial concentrations, and different mechanical treatments of the solution. The experimental results, substantiated by the theoretical analyses, indicate that the main contributor to the influx of salt into the porous material is the convective dispersion arising from the onset of horizontal variations of the salt concentration and of the vertical velocity component. The rate of this process is governed by a characteristic time where ρU and ρL are the initial densities in the reservoir and in the porous material, respectively; K is the hydraulic conductivity of the porous material with respect to the liquid initially saturating it; and H is a characteristic head of the solution. Approximate expressions have been derived for the rate of decrease of the salt concentration in the reservoir, for the influx of salt into the porous material, and for the advances of the salt front within it.