Published online 20 November 2006
Published in Vadose Zone J 5:1278-1280 (2006)
DOI: 10.2136/vzj2006.0034
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
TECHNICAL NOTES
Equivalent Soil Pore Geometry to Determine Effective Water Permeability
E. Unsala,* and
J. H. Daneb
a Dep. of Chemical Eng., Loughborough University, Loughborough LE11 3TU UK
b Dep. of Agronomy and Soils, 202 Funchess Hall, Auburn University, Auburn, AL 36849
* Corresponding author (E.Unsal{at}lboro.ac.uk)
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ABSTRACT
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Knowledge of hydraulic properties is essential for predicting flow and transport in porous media. Previously, Unsal et al. presented a procedure to determine an equivalent pore size distribution and pore geometry of a sandstone core using effective air permeability values as a function of volumetric water content in conjunction with a genetic algorithm for optimization purposes. They also showed how the obtained equivalent pore size distribution and geometry could be used to obtain the water retention curve. In this technical note, we extend their work to predict the effective water permeability as a function of volumetric water content from the same information. As a check for the theoretically obtained effective water permeability values, we compared the predicted value at saturation with the measured value at saturation and to the predicted and measured effective air permeability values for a dry core. The four values, referred to as intrinsic permeability, were sufficiently close to give confidence in the procedure.
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INTRODUCTION
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THEORETICALLY THE INTRINSIC PERMEABILITY (m2) includes only geometric properties, such as size, arrangement, and connectivity, of the pore space and excludes the effects of fluid(s) (Corey, 1994). However, because of the complexity of a porous medium, it is difficult to describe it geometrically. It is possible to overcome this difficulty, at least in part, by considering the problem from a statistical point of view. Hence, instead of considering an actual porous medium, one can consider an ensemble of pores equivalent to the porous medium in question so that it exhibits similar retention and permeability functions for the fluid(s) under consideration. Two types of models have received great interest, namely, the capillary bundle model and the pore network model (Scheidegger, 1953, 1974, p. 125–150; Ball, 1981). Unsal et al. (2005) presented a modification of the capillary bundle model. Rather then using straight capillaries, they used parallel capillaries containing sections with different diameters. A procedure was then developed to predict the pore size distribution of a sandstone core from effective air permeability values measured as a function of water content. A genetic algorithm was used to optimize the ensemble of pores. The advantage of using air permeability is that it is an easy-to-measure soil parameter, at least for consolidated cores (Corey, 1994, p. 117–120). Application of silicone rubber mixtures (resin and hardener) has made air permeability measurement on unconsolidated samples easier as well (Unsal et al., 2007). When set, the silicone rubber provides strength, yet remains resilient and maintains good contact with the soil. Once the optimum ensemble of pores was determined, by matching calculated effective air permeability values with the measured ones at corresponding water content values, Unsal et al. (2005) proceeded to determine the main drainage retention curve by applying the Young and Laplace equation.
The term intrinsic permeability was introduced as a quantitative term to describe permeability of a porous medium in terms of solid phase properties only. Theoretically, therefore, the intrinsic permeability should be independent of the fluid being used. Consequently, it should be the same if measured with air in a completely dry porous medium as for water if measured in a completely saturated porous medium. In other words, we are assuming no interactions between the fluid and the solid phase and assume similar laminar flow behavior of the different fluids (i.e., for air, gas slippage (Klinkenberg effect), gas compression, and inertial forces were ignored). Although for unconsolidated soil samples, Jalbert and Dane (2003) showed a ratio of 1.43 for the intrinsic permeability values determined by air and water, respectively, we assumed that for a consolidated sandstone core this ratio should be close to one.
The objective of this note was to develop a procedure to predict the effective water permeability as a function of water content based on the optimum ensemble of pores determined from effective air permeability values. To lend support to the method, we compared the predicted intrinsic water permeability value with the predicted value for air and the measured values for water and air.
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THEORY
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As a starting point, we assume that we have determined the "best" pore size distribution based on the effective air permeability–water content data as outlined by Unsal et al. (2005). At this point we also know the water retention curve (main drainage curve).
Since we now deal with water flow rather than air, we need to modify Eq. [7] (Poiseuille's Law applied to air flow) of Unsal et al. (2005) to include gravity. We will also make a sign change to be more consistent with flow equations (water flows in the direction of decreasing hydraulic potential values); that is,
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where Qj is the volumetric flow rate (m3 s–1) through pore sequence j consisting of i sections, ri is the radius (m) of pore section i in pore sequence j, P is pressure (Pa) with Pi and Pi–1 being the entrance and exit pressure of section i, respectively, 
l is the density (kg m–3) of water, g is the gravitational acceleration (N kg–1), L is the height (m) of the pore sections, and 
w is the viscosity of water (Pa s). We assume vertical downward flow and all pore sequences to be parallel. When the porous medium is saturated, all pore sequences will conduct water. However, once the top section of a given pore sequence has drained, due to an increase in capillary pressure, that pore sequence will no longer conduct water and it will, therefore, no longer contribute to the hydraulic conductivity. The given sequence is, however, still subject to additional drainage at the same value of the capillary pressure head if an underlying section has a greater radius than the overlying section or at an increased value of the capillary pressure if the underlying section has a smaller radius than the overlying one. Keeping track of the drained water volumes at increasing values of the capillary pressure head provides information to determine the water retention curve (Unsal et al., 2005). To do the effective water permeability calculations, we assumed a hypothetical experiment in which an infinitesimal thin layer of water was maintained on top of those pore sequences that had all sections filled with water. Partially or completely drained sequences no longer contributed to the flow. Hence for i = 10, P10 = 0 (top of pore sequence), and assuming free drainage, P0 = 0 (bottom of pore sequence). For any pore sequence j that has all 10 sections filled with water, we then obtained 10 equations with 10 unknowns (one equation for each pore section) with the unknowns being Qj, which is the same in each pore section of a given pore sequence but differs between pore sequences, and P1 through P9. Hence, we can obtain Qj for any given pore sequence that is completely saturated and, therefore, conducts water. Next, we applied Eq. [10] of Unsal et al. (2005) to obtain Qtotal; that is,
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where 
is the number of conducting (saturated) pore sequences and 
a variable tortuosity factor that has been introduced to better represent flow through a real porous medium. For details on how to calculate the water content dependent tortuosity factor, the reader is referred to Corey (1994, p. 92–93). The Darcy flux density, q (m s–1), can then be calculated from
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where A is the total cross-sectional area (m2) of the porous medium. Using potentials on a volume basis, the flux density equation can be written as
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where K
is the hydraulic conductivity (m2 Pa–1 s–1) when using the potential on a volume basis, ph is the hydraulic pressure (Pa), that is
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where pt is referred to as the tensiometer pressure (Pa). Substituting Eq. [5] into [4] yields
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Because pt = 0 at both the top and bottom of the hypothetical core sample, we have
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and therefore
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Because we know q from Eq. [3], we can solve Eq. [8] for K. Values for the effective water permeability can then be calculated from
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MATERIALS AND METHODS
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A complete description of effective air permeability measurements was given by Unsal et al. (2005). The same paper provided values for the measured and predicted intrinsic air permeability values of the sandstone core (diameter = 2.5 cm, height = 6 cm). The experimental setup used to measure the effective air permeability was slightly modified to measure the saturated hydraulic conductivity of the same sandstone core. To assure complete saturation of the core while placed in the sample holder, it was first flushed with CO2 and subsequently saturated with a deaerated 0.005 M CaCl2 solution. A unit hydraulic head gradient was then established across the sandstone core and based on the measured flow rate, the hydraulic conductivity (m s–1) was determined and converted to K (m2 Pa–1 s–1) and kw (m2).
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RESULTS AND DISCUSSION
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Using a genetic algorithm, Unsal et al. (2005) determined an optimum pore ensemble (sizes and sequences), representing the pore structure of a sandstone core, by matching measured and predicted effective air permeability values. The predicted effective air permeability curve is presented in Fig. 1
and is based on 10 sections (i = 10) per pore sequence and 500 sequences (j = 500). Unsal et al. (2005) also calculated the water retention curve from the optimum pore ensemble. We extended these calculations to include effective water permeability values (Fig. 1). It should be noted that the selection of 500 pore sequences and 10 sections per pore sequence was based on an analysis in which first many more sequences and sections were used. By stepwise increasing these values, it was found that numbers greater than 500 and 10, for the number of sequences and sections per sequence, respectively, provided no additional advantage in the prediction of effective air permeability values. One could say that for the selected values we are dealing with a representative elementary volume. Comparison of calculated and measured intrinsic permeability values for both air (ka, measured = 1.4 µm2; ka, calculated = 1.52 µm2) and water (kw, measured = 1.25 µm2; kw, calculated = 1.55 µm2) showed these values to be very close. The calculated intrinsic permeability values for both air and water are slightly higher than the corresponding measured ones. This is because the model assumes all the pores are completely empty or filled with water for calculating the maximum air or water permeability (intrinsic), respectively. However, these conditions are almost impossible to obtain during experiments. There is likely to be some trapped air and water not contributing to flow (i.e., immobile water) in the pores. In addition, we would like to point out that the maximum air and water permeability (intrinsic) values are very close. The slight differences in the calculated intrinsic air and water permeability values were attributed to the use of two different applications of the Poiseuille equation, one for air and one for water. The results also show, however, that the presented method of first predicting an equivalent pore ensemble (Unsal et al., 2005), representative of the porous medium, has merit in the prediction of water retention curves (Unsal et al., 2005) and effective water permeability functions, and, consequently, hydraulic conductivity functions.
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Fig. 1. Model results for effective air and water permeability values (µm2) as a function of volumetric water content.
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SUMMARY AND CONCLUSIONS
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We presented a model that predicts effective water permeability and, therefore, hydraulic conductivity functions. The model is an extension of the Unsal et al. (2005) model, in which measured effective air permeability values of a sandstone core were used to determine an optimum equivalent pore ensemble to represent the real pore structure. A genetic algorithm, based on crossings and mutations of a number of initial realizations, was used as part of the optimization process. The physical part of the model was developed based on Poiseuille's Law of flow in capillaries consisting of sections with different diameters. The optimum equivalent pore ensemble was used to predict the main drainage curve using the Young and Laplace equation. The best results, as determined from comparison with the experimentally determined water retention curve, were obtained using a bimodal lognormal pore size distribution from which the pore sequences were estimated.
We extended the model to include calculations of effective water permeability values as a function of volumetric water content. As seen in Fig. 1, the maximum (intrinsic) values for air and water permeabilities are very close, as they should be. More importantly, however, the calculated and measured values for the water intrinsic permeability were very close as well, lending credence to the proposed model.
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REFERENCES
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- Ball, B.C. 1981. Modeling of soil pores as tubes using gas permeabilities, gas diffusivities and water release. J. Soil Sci. 32:465–481.
- Corey, A.T. 1994. Mechanics of immiscible fluids in porous media. Water Resources Publications, Highlands Ranch, CO.
- Jalbert, M., and J.H. Dane. 2003. A handheld device for intrusive and nonintrusive field measurements of air permeability. Available at www.vadosezonejournal.org. Vadose Zone J. 2:611–617.[Abstract/Free Full Text]
- Scheidegger, A.E. 1953. Theoretical models of porous matter. Producers Monthly 17:17–23.
- Scheidegger, A.E. 1974. The physics of flow through porous media. University of Toronto Press, Toronto, Canada.
- Unsal, E., J.H. Dane, and G.V. Dozier. 2005. A genetic algorithm for predicting pore geometry based on air permeability measurements. Available at www.vadosezonejournal.org. Vadose Zone J. 4:389–397.[Abstract/Free Full Text]
- Unsal, E., G.V. Dozier, and J.H. Dane. 2007. A novel optimization process for prediction of complex pore geometry via genetic search. Intelligent Automation and Soft Computing. (In press).
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ABSTRACT
INTRODUCTION
