Soil Science Society of America Journal 65:648-654 (2001)
© 2001 Soil Science Society of America
DIVISION S-1 - SOIL PHYSICS
Correcting Laboratory Retention Curves for Hydrostatic Fluid Distributions
Marc Jalbert and
Jacob H. Dane
Dep. of Agronomy and Soils, Auburn Univ., Auburn AL 36849-5412
Corresponding author (jdane{at}acesag.auburn.edu)
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ABSTRACT
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Many experimental methods for obtaining capillary pressure–volumetric fluid content relations in porous media are affected by the occurrence of hydrostatic pressures that create nonuniform fluid content distributions throughout the sample of interest. Such conditions exist, for example, in suction apparatuses and pressure cells, which are widely used in vadose zone hydrology, agronomy and environmental engineering, and for Hg intrusion porosimetry routinely applied in the petroleum industry. We show how to correct experimental data for nonuniform pressure and fluid content distributions, which leads to retention or pore-size distribution curves applicable to physical points. This is necessary to make the retention information consistent with the differential equations modeling fluid flow in porous media. The advantage of the proposed method is that it does not require an a priori assumption of any given model describing the retention relation. The proposed correction formula is validated both numerically and experimentally and is compared with an existing correction procedure. By deconvoluting retention relations from the averaging taking place in the sample, the correction method presented should enhance the description of porous media–immiscible fluids systems at low capillary pressure values.
Abbreviations: DNAPL, denser-than-water nonaqueous phase liquid • hc, capillary pressure head • 
, volumetric fluid content
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INTRODUCTION
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MODELING THE FLOW of immiscible fluids in a porous medium requires knowledge of the capillary pressure head (hc)–volumetric fluid content () relation with maximal precision. Many experimental methods have been developed for obtaining this relation. Most methods, however, use samples of a certain height, zs, and neglect changes in fluid pressures and contents with elevation. Ignoring these changes can lead to substantial errors. This is especially true for coarse porous media, relatively high samples, fluids with low interfacial tensions, and dense fluids, such as Hg. In this paper we present the development and discuss the application of a mathematical formula that accounts for nonuniform fluid distributions due to hydrostatic fluid pressures, and thus enables the correction of experimentally obtained hc– relationships. The advantage of the proposed method over previously published correction procedures (Liu and Dane, 1995; Schroth et al., 1996) is that it does not assume any model for the retention relation. Consequently, it is applicable to porous media that are not well described by the commonly used retention models of Brooks and Corey (1964) and van Genuchten (1980). Furthermore, it guides one towards a correct modeling of the porous medium, if a closed-form expression for the retention relation is needed.
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THEORY
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Consider a homogeneous porous medium sample of constant horizontal cross-sectional area and height zs for which we would like to determine the drainage relation in a two-phase fluid system. The sample is initially saturated with the wetting fluid, for example, water in an air–water system, Hg vapor in a liquid Hg–Hg vapor system, or water in a tetrachloroethylene–water system. The sample is in contact with the nonwetting fluid, whose pressure is gradually stepwise increased. Static equilibrium is allowed to occur after each pressure increment and the in- or outflow volumes of the nonwetting or wetting fluid, respectively, are measured as they are needed to calculate the average volumetric nonwetting fluid content, 
n, and/or the average volumetric wetting fluid content, w, upon the completion of the experiment. The pressures in the wetting and nonwetting fluids are measured at elevations zw and zn, respectively. We will refer to these pressures as 
w and n because they are generally considered as average values of the pressures over the extent of the sample. At a physical point, the volumetric nonwetting and wetting fluid content are referred to as n and w, respectively. They are related by
 | (1) |
where 
is the porosity of the sample. Similarly, the capillary pressure, Pc, at a physical point is defined as
 | (2) |
where Pn (M L-1 T-2) is the pressure in the nonwetting fluid and Pw (M L-1 T-2) is the pressure in the wetting fluid. Consequently, the measured relation is n as a function of c = n - w. However, we are interested in the physical point relation between n and Pc of the soil sample, which we will assume to be homogeneous. We define the physical point and measured capillary pressure heads (L), respectively, as
 | (3) |
and
 | (4) |
where 
w is the density of the wetting fluid (M L-3), and g is the acceleration due to the earth's gravitational field (L T-2). We will denote n as the density of the nonwetting fluid (M L-3).
At equilibrium, the fluid pressures are hydrostatically distributed. Consequently, the pressures in the wetting and nonwetting fluids at elevation z above the bottom of the sample (z = 0) are, respectively
 | (5) |
 | (6) |
Combining Eq. [3], [4], [5], and [6], we obtain, following the notation presented by Liu and Dane (1995)
 | (7) |
where
 | (8) |
and
 | (9) |
As n is the average of n over the height of the sample, the two are related by
 | (10) |
We want to invert Eq. [10] to transform the experimentally obtained relation, n 
, into the relation of interest, n (hc). Making the change of variables stated in Eq. [7], Eq. [10] is transformed to
 | (11) |
Introducing 
n(x) = 
0xn(hc)dhc, we obtain
 | (12) |
Since n is the first derivative of n, and the system parameters A, B, and zs are independent of 
c, differentiating the above expression with respect to the measured capillary pressure head c results in
 | (13) |
Using Eq. [7], we obtain the following relationship, which is valid at the point z = zs, where the capillary pressure hc(zs) is denoted as hc,s
 | (14) |
and hence
 | (15) |
First, let us consider B > 0; that is, the wetting fluid is denser than the nonwetting fluid. We can shift Eq. [15] to obtain n(hc,s - Bzs), which is the nonwetting fluid content at the bottom of the sample
 | (16) |
Substituting Eq. [16] into Eq. [15] gives
 | (17) |
Iterating the recurrence process initiated in Eq. [16] and [17] and noting that, for any hc,s value, an integer M exists such that for any integer m > M, n(hc,s - mBzs) = 0 and 
= 0, we obtain
 | (18) |
where the infinity sign replaces M for the sake of notation convenience. The series in Eq. [18] is a finite sum, and all terms for i > M are equal to zero. Now, writing hc,s = hc, we obtain the relationship transforming the experimental curve n into the curve of interest n(hc), namely,
 | (19) |
For B < 0—that is, the wetting fluid is lighter than the nonwetting fluid—the final relation between n and n(hc) is now obtained as
 | (20) |
The final formula applicable to every fluid configuration, for a sample initially saturated with the wetting fluid, is
 | (21) |
where
 | (22) |
It should be noted that I = 0 if B < 0 and I = 1 if B > 0. The function I is thus identical to the Heaviside unit function, except for B = 0. Notice that in the particular case of two fluids of equal density (B = 0), we simply have
 | (23) |
Equation [21] expresses the nonwetting fluid content for a physical point at a capillary pressure head value hc as the sum of the terms
for i > I. Geometrically speaking, each of these terms can be represented by the surface area of a rectangle having |B|zs as one side length, and the derivative of the experimental relationship at hc - A - i|B|zs as the other side length. Consequently, n can be represented on a plot of the dn/dc curve as a cumulative area of adjacent rectangles having horizontal side length |B|zs. Figure 1
shows this geometric interpretation for the case A > 0 and B > 0. When the height of the sample tends to 0 (Bzs 
0), generally causing zn and zw to also have values approaching 0 (A 0), then the total area of the rectangles tends to the actual area located under the derivative of the experimental relationship. This is consistent with the fact that under such conditions the experimental relation n approaches the physical point relationship n(hc), as the sample becomes smaller and the experiment less sensitive to nonuniform pressure and fluid distributions.
It should be noted that Eq. [21] is equivalent to Eq. [11] in the case of a uniform nonwetting fluid content at some low value of the capillary pressure head. Consequently, it can be seen that at large capillary pressure values, as n becomes independent of c, Eq. [21] mathematically results in a periodic function n(hc) with period Bzs. This is because any such function having the value of its integral over a period equal to Bzsn
satisfies Eq. [11]. However, if the derivative of the experimental relation is correctly estimated, n(hc) should be representative of the physical behavior of the porous medium, that is, should be uniform at large capillary pressure values.
Equation [21] was obtained considering that dn/dc vanishes at low values of c. The condition of fluid content uniformity at low values of c may actually require measurements at negative values of the capillary pressure head if the porous medium exhibits mixed wettability (e.g., Lenhard and Oostrom, 1998). Even though Eq. [21] is still valid for these conditions, the experimenter may not have investigated the system at negative capillary pressure head values, thus prohibiting the use of the correction formula. This often occurs during the imbibition of water into initially air-dry, natural soils. In these cases, the retention curves have a non-zero slope at zero capillary pressure. Even if the experimenter did not collect enough data to meet the condition of uniform fluid contents at the lowest values of the capillary pressure head, the experimental curve can still be corrected by starting the recurrence process at the highest values if, at these values, homogeneous fluid contents exist throughout the sample. Using the same reasoning that led to Eq. [21], the physical point curve n(hc) is now obtained from the n data using the formula
 | (24) |
where
 | (25) |
It is important to note that the theory presented here is valid only for samples with initially homogeneous fluid contents. It is, therefore, valid on the primary drainage curve, main imbibition curve, and main drainage curve, which are the most important branches of the hysteretic capillary pressure–volumetric fluid content relationships. It should be noted that in case of entrapped nonwetting fluid, the nonwetting fluid content at low capillary pressure should be added to the right side of Eq. [21]. Our theory is not valid for any secondary curve because the porous medium–fluids system is not following the same secondary curve at the top and the bottom of the sample.
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MATERIALS AND METHODS
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Numerical Validation
To check the mathematical validity of Eq. [21], it was first tested on a hypothetical soil with a dual pore-size distribution and with water and air as the fluids. The Brooks and Corey (1964) function was used to describe drainage from the largest pores, while a lognormal distribution (Brutsaert, 1966; Kosugi, 1996) was considered to represent the smallest pores. Assuming a superposition principle for the retention relation (Zeiliguer, 1992), the general expression of the physical point retention curve for this hypothetical soil is
 | (26) |
where Sw is the water saturation (defined as the volumetric water content divided by the porosity), hd (L) is the displacement capillary pressure head, ß is the fraction of the pore volume occupied by pores of the Brooks–Corey type, 
is the so-called pore-size distribution index, erfc is the complementary error function, 
c (L) is the median capillary pressure head for the lognormal distribution, and 
is the variance of the lognormal distribution. We used the following parameter values: hd = 3 cm of water, ß = 0.2, = 3, c = 30 cm of water, = 0.25. We further assumed a 10-cm-high column with the fluids' pressures being recorded at the top. This configuration resulted in A = -10 cm and B = 1, which allowed us to calculate hypothetical pressure cell data according to Eq. [11]
 | (27) |
Figure 2
shows the graphs of the physical point curve (Eq. [26], solid line) and of the pressure cell curve (Eq. [27], dashed line). Using 1-cm intervals, from hc = 0 to hc = 60 cm, application of Eq. [21] to the hypothetical pressure cell curve resulted in the corrected points presented as open circles.
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Fig. 2. Numerical validation of Eq. [21] on hypothetical data. The solid line corresponds to the physical point retention curve (Eq. [26]), and the open circles resulted from the application of Eq. [21] to the hypothetical pressure cell curve (dashed line, Eq. [27])
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Experimental Validation
The theory was experimentally validated with a coarse sand (particle size between 0.84 and 2 mm), packed under water in a 10-cm-high glass column. Klute (1986) generally recommends suction apparatuses (Haines, 1930) over pressure cell systems (Richards, 1941) because they result in retention curves more representative of the field behavior of soils containing entrapped air. Consequently, a water-saturated ceramic plate at the bottom of the column was used to apply water pressures less than atmospheric to the fluid–porous medium system, while the top of the column was open to the atmosphere. The ceramic plate was hydraulically connected to a burette in which the air pressure, and hence the water pressure in the porous medium, were stepwise decreased using a regulated vacuum source. The amount of water drained from the porous medium into the burette was recorded each time the system had reached static equilibrium. Correspondingly, the capillary pressure head was calculated for the top of the sample. The simultaneous use of a gamma radiation system enabled us to obtain the retention curves at six physical points located along the height of the column, following the method presented by Dane et al. (1992). For its accuracy in two-fluid systems, a single radioactive source (241Am) procedure was chosen and calibrated according to Oostrom et al. (1998). The six physical point curves, determined during initial drainage, were very much alike (Fig. 3)
, indicating a nearly homogeneous porous medium. One of these curves (Fig. 3, solid circles, measured at z = 3 cm) was selected for comparison purposes. At the end of the experiment, the rate of evaporation through the connecting tubing was determined. The average curve obtained from knowledge of the amounts of drained water was corrected according to this evaporation rate and smoothed using adaptive spline curves (procedure SSLII-BSC2, Fujitsu Ltd., Tokyo, Japan). The latter was necessary to allow derivatives to be taken over the complete range of capillary pressure head values as needed in Eq. [21]. Although any kind of curve could have been used for smoothing the Haines apparatus data, adaptive spline curves were found to be flexible enough to fit the data well over the whole range of capillary pressure head values, and their derivatives were easily calculated. Smoothing of the experimental retention relation by spline curves, estimation of the derivatives, and computation of the corrected curve by application of Eq. [21] and [24], were incorporated into a Fortran 90 code (available upon request). For comparison purposes, the Haines apparatus data were also treated using the program TrueCell (available from the author at www.ag.auburn.edu/dept/ay/sp/, verified 4 Jan. 2001), which applies the correction procedure presented by Liu and Dane (1995).
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Fig. 3. Gamma radiation measurements of the physical point retention curve. The measurements at z = 3 cm are represented by solid circles. Open circles represent measurements at five other locations along the 10-cm-high column
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RESULTS
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Numerical Validation
Figure 2 shows the very good agreement between the physical point capillary pressure head–saturation relationship (solid line) and the corrected curve (open circles) resulting from the application of Eq. [21] to the hypothetical pressure cell data. Consequently, Eq. [21] appears to be mathematically correct. Furthermore, its application yields important information about the porous medium that was obscured by the averaging process taking place in the pressure cell. The hypothetical, experimental, 
w curve (Fig. 2) barely exhibits the Sw(hc) bump, representative of the dual pore-size distribution. Although the corrected curve and the physical point curve are almost on top of each other, the reader should notice a very slight scattering of the corrected points (open circles) around the physical point relationship (solid line) towards the high capillary pressure head values. This scattering is due to inexact estimations of the first derivatives of the pressure cell data. Because of the form of Eq. [21], the scattering appears periodic with period Bzs. This oscillating behavior will be more pronounced on actual experimental retention curves, where estimation of the slopes can become challenging.
Experimental Validation
Figure 4
shows the Haines apparatus data for the volumetric air content and the fitted, adaptive spline curve. The latter was used to eliminate noise in the experimental data to more easily obtain estimates of the first derivative of the experimental relation. However, application of Eq. [21] resulted in a physical point curve (Fig. 4) with a bulge (negative air content) in the region of initial water drainage. This bulge is an indication of the inability of the spline curve fitting procedure to correctly estimate the slope of the experimental data in this region, which was attributed to a discontinuity in the experimental data at the air entry value. Discontinuities cannot be taken into account by spline curves, which assume slope continuity at every point. Propagation of the initial error to the rest of the curve resulted in an oscillating behavior at higher capillary pressure head values. To eliminate the errors at this end of the curve we decided to use Eq. [24], which starts correcting the experimental data at the highest measured capillary pressure head value. The curve now obtained shows a regular behavior at the higher capillary pressure head values, but still bulges at the lower capillary pressure head values (Fig. 4). For further use, we adopted Eq. [21] to obtain the physical point curve at the lower and Eq. [24] at the higher capillary pressure head values. The negative air content values were eliminated, because they are physically impossible. The subsequently obtained curve is in good agreement with the physical point data obtained by gamma radiation scanning (Fig. 5)
. It is obvious that the lack of agreement between the Haines apparatus data and the actual retention curve shows the necessity of the correction.
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Fig. 4. Application of Eq. [21] (dashed line) and [24] (dash-dotted line) to the Haines' apparatus data (open circles and solid line). The obtained curves are compared with physical point data (solid circles)
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DISCUSSION
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Figure 5 includes the results of the correction procedure of Liu and Dane (1995) applied to the Haines apparatus data. The corrected curve, which was assumed to follow the Brooks and Corey model (1964), is in good agreement with the gamma radiation data representative of the relationship at a physical point. Notably, the displacement pressure head is very accurately determined. Our experiment thus confirms the applicability of the method of Liu and Dane (1995) to sandy porous media often used in laboratory research. However, the good agreement between gamma radiation data, the results of the correction procedure of Liu and Dane (1995), and the results of the correction procedure presented in this paper shows that the physical point capillary pressure head–fluid content relationship can also be estimated without assuming any a priori model. If a parametric form is desired for the porous medium capillary pressure head–fluid content relation, our correction procedure permits one to decide on the desired model to be used in the fitting procedures of Liu and Dane (1995) and Schroth et al. (1996), who pointed out that their correction methods can be extended to other relationships than the ones proposed by Brooks and Corey (1964) and van Genuchten (1980). For instance, a series of weighted van Genuchten relations, as proposed by Durner (1992), could be used if the corrected retention curve indicates a multimodal pore-size distribution.
As stated in Eq. [10], the Haines' and pressure cell methods result in average values of the fluids' contents over the height of the sample. At the region where the physical point curve is initially rather steep, indicative of a distinct displacement pressure (Brooks and Corey, 1964), the effect of averaging is to obscure this distinct value and to smooth the curve. Consequently, the physical point curve, representative of the porous medium, will generally be steeper and less regular than the curve experimentally obtained on a high sample, as shown in Fig. 2 and 5. The averaging effect in our correction procedure is represented by the product Bzs. Besides the smoothing effect, the position of the point where the average capillary pressure is determined creates a shift in the capillary pressure head values. This shift is taken into account in our correction procedure by incorporating the parameter A. The consideration of this shift is essential in order to obtain accurate scale parameters for the porous medium of interest, such as the displacement capillary pressure head, which represents the size of the largest interconnected pores. If the pressures are measured in the sample, the shifting cannot exceed certain limits determined by the height of the sample. The height of the sample is thus the most important parameter, and the importance of correcting experimental retention curves can generally be assessed by comparing this height with a capillary pressure head value representative of the porous medium, such as an estimate of the displacement capillary pressure head (Liu and Dane, 1995). If the representative capillary pressure head value is large compared with the height of the sample, then the averaging and shifting effects, even though their absolute influence is independent of the porous medium, will have a relatively small impact on the curve. In that case, the correction procedure presented in this paper would not be necessary, depending on the degree of accuracy wanted and the level of precision of the experiment. However, even if the investigator decides not to correct the average data, attention should be given to the position of the pressure ports. In their sensitivity analysis, Liu and Dane (1995) showed that pressure measurements should be obtained at the middle of the sample height to minimize the overall error due to the shifting effect.
A priori, our correction procedure permits one to obtain information about the retention curve at low capillary pressure head values in case the porous medium contains a wide range of pore sizes. As retention curves are, in many instances, used to estimate unsaturated hydraulic conductivity values of soils, accurate knowledge at low suctions, where the highest hydraulic conductivity values occur, is sought. However, it has been pointed out by several authors (e.g., White et al., 1972; Corey, 1992) that the wet end of drainage retention curves, obtained on samples in the laboratory, may not be representative of the behavior of the porous medium in the field and may not be a good indicator of the size of the largest pores present in the porous medium. This is due to the fact that during a drainage experiment, the sample first desaturates without occurrence of a fully connected nonwetting fluid network. The assumption of hydrostatic nonwetting fluid pressure may not hold under such conditions. This is especially critical when the sample contains some residual, gaseous nonwetting fluid, because changes in pressure induce extensive changes in the nonwetting fluid content. Consequently, the theory presented in this paper may not be valid for capillary pressures lower than the capillary pressure at which a non-zero nonwetting phase permeability is observed, which has been referred to as the bubbling capillary pressure. For that reason, Eq. [24] should be preferred over Eq. [21] for correcting experimentally obtained retention curves. To estimate the curve at the wet end, which could not be rigorously measured, the best solution may be to extrapolate the results of the correcting procedure using Eq. [24] to capillary pressures lower than the bubbling capillary pressure, following an approach presented by Corey (1992). White et al. (1972) showed that the relative importance of the first, noninterconnected desaturation stage was related to the exposed surface/volume ratio of the sample. It should be noted that the correction technique presented here, by allowing the use of relatively high, undisturbed field samples presenting low values for this ratio, may render measured retention curves more representative of in situ field conditions.
Finally, even though the correction procedure outlined in this study enhances retention relations obtained with commonly used devices such as suction apparatuses and pressure cells, we think that direct, pointwise measurements by gamma radiation scanning (e.g., Dane et al., 1992) provide the most robust method to obtain retention curves unaffected by hydrostatic fluid distributions and representative of the physical behavior of the porous medium of interest.
Additional Applications
Denser-than-Water Nonaqueous Phase Liquids–Water Systems
Pressure cells are also used in studies concerning the retention of nonaqueous phase liquids in soils. Denser-than-water nonaqueous phase liquids (DNAPLs) are of special interest because they form long-lasting and difficult to remediate sources of groundwater contamination. Modeling the flow of these liquids in water-saturated aquifers requires an accurate determination of the DNAPL content vs. capillary pressure head relation. Accurate knowledge of this relation at low capillary pressure head values, including the displacement pressure, is essential because the largest pores provide the initial flow paths for the contaminant, which is the nonwetting fluid. A correction procedure as we have outlined improves the information as used with pressure cells.
Mercury Intrusion Porosimetry
The Hg intrusion technique (Ritter and Drake, 1945) first requires the displacement of air by Hg vapor. It is noted that Hg vapor constitutes the wetting fluid, while liquid Hg is the nonwetting fluid. The sample is subsequently placed in a liquid Hg bath and in response to liquid Hg pressure increases, the volumes of liquid Hg invading the porous medium are recorded. Considering the very low density of the wetting fluid, we should not use capillary pressure head, but capillary pressure instead. Assuming the vapor density, and hence the pressure in the vapor phase, to be negligible, the expression for the capillary pressure at elevation z is (Eq. [6])
 | (28) |
where zP is the elevation at which the liquid Hg pressure is measured. This equation has the same form as Eq. [7], and the final formula correcting Hg intrusion data for nonuniform liquid Hg distributions is
 | (29) |
where Hg = 13600 kg m-3, and Hg is often called the cumulative Hg intrusion.
We can now estimate the need of the correction procedure, assuming the Young and Laplace equation gives the radius of the smallest invaded pore at the applied Hg pressure, that is,
 | (30) |
where r (L) is the pore radius, (M T-2) is the Hg vapor–liquid Hg interfacial tension, and 
is the angle of contact between the fluids' interface and the solid phase. Following the argument presented above, we consider the hydrostatic pressure corresponding to the height of the sample. The critical pore radius for a necessary correction of the experimental data is thus given as
 | (31) |
If the sample has a height zs = 2.5 cm, and assuming = 0.473 N m-1, = 130°, we can estimate rc = 0.1 mm. Consequently, the correction procedure is not necessary if the 2.5-cm-high sample only has pores with radii significantly smaller than 0.1 mm. Although most of the reservoir rocks studied in the petroleum industry satisfy this condition, allowing the experiment to be relatively unaffected by nonuniform Hg pressure and fluid distributions, the method presented in this paper may be of interest for studies concerning fractured reservoir rocks. We would like to emphasize that our theory permits one to treat Hg intrusion data in cases in which the sample would only be partially immersed in the Hg bath, thus extending the range of measurements to large pore sizes. The correction method also allows high cores to be used.
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CONCLUSIONS
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We presented a general method for correcting laboratory retention curves affected by nonuniform hydrostatic fluid pressure and fluid content distributions throughout the sample of interest. The approach is intended to assist in the important but difficult task of determining accurate porous medium retention properties at low capillary pressure values.
Received for publication May 19, 2000.
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REFERENCES
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- Brooks, R.H., and A.T. Corey. 1964. Hydraulic properties of porous media. Hydrol. Pap. no 3. Colorado State Univ., Fort Collins, CO.
- Brutsaert, W. 1966. Probability laws for pore-size distributions. Soil Sci. 101:85–92.
- Corey, A.T. 1992. Pore-size distribution. p. 37–44. In M.Th. van Genuchten et al. (ed.) Indirect methods for estimating the hydraulic properties of unsaturated soils. Univ. of California, Riverside.
- Dane, J.H., M. Oostrom, and B.C. Missildine. 1992. An improved method for the determination of capillary pressure-saturation curves involving TCE, water and air. J. Contam. Hydrol. 11:69–81.
- Durner, W. 1992. Predicting the unsaturated hydraulic conductivity using multi porosity retention curves. p. 185–202. In M.Th. van Genuchten et al. (ed.) Indirect methods for estimating the hydraulic properties of unsaturated soils. Univ. of California, Riverside.
- Haines, W.B. 1930. The hysteresis effect in capillary properties and the modes of moisture distribution associated therewith. J. Agric. Sci. 20:96–105.
- Klute, A. 1986. Water retention: Laboratory methods. p. 635–662. In A. Klute (ed.) Methods of soil analysis. Part 1. Agron. Monogr. 9. ASA and SSSA, Madison, WI.
- Kosugi, K. 1996. Lognormal distribution model for unsaturated soil hydraulic properties. Water Resour. Res. 32:2697–2703.
- Lenhard, R.J., and M. Oostrom. 1998. A general parametric model for predicting relative permeability–saturation–capillary pressure relationships of oil–water systems in mixed-wet porous media. Transp. Porous Media 31:109–131.
- Liu, H.H., and J.H. Dane. 1995. Improved computational procedure for retention relations of immiscible fluids using pressure cells. Soil Sci. Soc. Am. J. 59:1520–24.[Abstract/Free Full Text]
- Oostrom, M., C. Hofstee, J.H. Dane, and R.J. Lenhard. 1998. Single-source gamma radiation procedures for improved calibration and measurements in porous media. Soil Sci. 163:646–656.
- Richards, L.A. 1941. A pressure-membrane extraction apparatus for soil solution. Soil Sci. 51:377–386.
- Ritter, H.L., and L.C. Drake. 1945. Pore-size distribution in porous materials: Pressure porosimeter and determination of complete macropore-size distributions. Ind. Eng. Chem. Anal. Ed. 17:782–786.
- Schroth, M.H., S.J. Ahearn, J.S. Selker, and J.D. Istok. 1996. Characterization of Miller-similar silica sands for laboratory subsurface hydrologic studies. Soil Sci. Soc. Am. J. 60:1331–1339.[Abstract/Free Full Text]
- van Genuchten, M.Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44:892–898.[Abstract/Free Full Text]
- White, N.F., D.K. Sunada, H.R. Duke, and A.T. Corey. 1972. Boundary effects in desaturation of porous media. Soil Sci. 113:7–12.
- Zeiliguer, A.M. 1992. A hierarchical system to model the pore structure of soils. p. 499–514. In M.Th. van Genuchten et al. (ed.) Indirect methods for estimating the hydraulic properties of unsaturated soils. Univ. of California, Riverside.
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TOP
ABSTRACT
INTRODUCTION
