Extension of a temperature effects model for capillary pressure saturation relations



[1] A series expression describing the effect of temperature on capillary pressure saturation relations was derived from the chemical thermodynamics of interfaces. Most systems studied could be described well by truncations of the series expression that retained one or two parameters. The one-parameter expression was appropriate for describing capillary pressures of soils experiencing modest changes in temperature. Thevan Genuchten [1980] equation modified with the one-parameter expression could be fitted precisely to 36 of 41 water-air capillary pressure saturation relations determined at more than one temperature. The residuals of the fitted equation were unaffected by degree of saturation. The one-parameter expression could describe equally well the effect of temperature on capillary pressures of homogeneous synthetic and natural heterogeneous porous media, for which a theoretical explanation was proposed. Asymptotic standard errors of the estimates for nonlinear regression analysis for the one-parameter expression fitted to water-oil and water-NAPL systems were of the order of the parameters themselves. This lack of fit may be more due to the wide temperature ranges studied for many of the water-oil and water-NAPL systems than the nature of the nonwetting phase. The two-parameter expression was appropriate for systems, such as petroleum reservoirs, subjected to pronounced changes in temperature. When compared to nonlinear regression fits to the one-parameter model, fits of van Genuchten's equation modified with the two-parameter expression generally reduced the errors of the parameter estimates but yielded only slight improvement in the mean square errors of the predicted values. A theoretical derivation indicates that the one-parameter expression should be appropriate also to describe the effect of temperature on the capillary pressures in very dry soils.

1. Introduction

[2] Capillary pressure of a wetting fluid in a porous medium at a constant degree of saturation is a linearly decreasing function of temperature. The phenomenon appears to be universal, having been observed in porous-media samples fashioned from soils, rocks, monosized glass beads, quartz sand, and even granular glass. At 298 K, the relative decrease in capillary pressure is typically around −1% K−1. The interfacial tension of water is also a decreasing linear function of temperature, but its relative decrease is much smaller, roughly −0.2% K−1. The differences between the relative changes between capillary pressure and water's surface tension are too large to ignore, but formulating a quantative explanation has proven to be elusive. The problem has initially entranced, eventually preoccupied, and ultimately frustrated a long line of geophysicists.

[3] Here it is reported that a one-parameter model presented earlier by Grant and Salehzadeh [1996] can be fitted to most of the available water-air capillary pressure saturation relation (CPSR) data collected at more than one temperature. Also reported here is a two-parameter model, also based on the chemical thermodynamics of interfaces, which describes the effect of temperature on CPSRs for water-oil systems over wide ranges of temperature. Both the one- and two-parameter models are shown to be members of a class of possible linear models of interfacial tensions. These underlying linear models would be expected to sum regularly in heterogeneous porous media (Defined here as porous media in which the solid matrices are composed of mixtures of mineral and organic solids having different surface chemical properties.) and may explain the apparent universality of the one- and two-parameter models and their applicability to both homogeneous and heterogeneous porous media.

[4] Although predictive, neither the one-parameter model nor the two-parameter model can be reconciled with well-documented independently measured changes with temperature of surface tension and contact angles. The goals of this paper are to reexamine the limits of applicability of the Young-Laplace equation and the nature of interfacial tensions in porous media, and to determine the effects of temperature on CPSRs for wider temperature ranges, heterogeneous porous media, nonwetting phases other than air, and lower degrees of saturation.

[5] This paper begins with a discussion of the Young-Laplace equation and how the individual interfacial tensions should be modeled to describe the influence of temperature. Two candidate models are derived and tested by nonlinear regression analysis. The applicability of the models is then evaluated for heterogeneous porous media and porous media in which the nonwetting phase is a fluid other than air. The article concludes with a discussion of the effect of temperature on the capillary pressures in very dry soils.

2. Derivation Based on Difference of Interfacial Tensions

[6] The so-called Young-Laplace equation is:

equation image

where pc is capillary pressure (Pa); pg, pressure in the gas phase (Pa); pl, pressure in the liquid phase (Pa); γlg is the interfacial tension between the liquid and gas phases (N m−1); and Θ, contact angle of the liquid gas interface with the solid (rad) [Evans et al., 1986]. It should be noted that by definition the product γlg cos Θ is equivalent to the following quantities:

equation image

where k is the wetting coefficient (1); γsg, interfacial tension of the solid-gas interface (N m−1); γls, interfacial tension of the liquid solid interface (N m−1) [Rowlinson and Widom, 1982].

[7] It is worthwhile here to discuss the relative changes in capillary pressure, liquid-gas interfacial tension, and, by inference, wetting coefficients. The temperature derivative of the interfacial tension for an arbitrary αβ interface is equal to the negative of corresponding entropy per unit area:

equation image

where sAαβ is the interfacial entropy per unit area (J K−1 m−2) and T is temperature (K).

[8] Without exception, liquid-gas interfacial tensions are almost perfectly linear decreasing functions of temperature [Jasper, 1972]. The entropy of the liquid-gas interface is therefore always a weak function of temperature and always positive. The heuristic explanation for the positive entropy is the increased dissorder in the liquid-phase as comes into contact with gas-phase at the interface. In contrast, molecules in a gas or liquid phase would be expected to become more ordered at the interface with a solid. I am aware of no determinations of this phenomenon for solid-gas interfacial tensions, but determinations of temperature changes in metal-melt and ice-melt interfacial tensions consistently find that these are linearly increasing functions of temperature-consistent with negative interfacial entropies [Spaepen, 1994; Huang and Bartell, 1995]. One can therefore assume reasonably that: dγlg/dT < 0, but dγ ls/dT > 0 and dγ sg/dT > 0. Figure 1 presents the capillary pressure during imbibition of an Elkmound sandy loam from 283 K to 313 K relative to its value at 298 K [Salehzadeh, 1990]. As is the norm, the wetting coefficient was not estimated, so that changes in the wetting coefficient with temperature must be expressed relative to its value at an arbitrary reference temperature Tr (K), as are pc and γlg. The relative value of liquid-gas interfacial tension of water is presented as the inferred wetting coefficient. The latter ratio is calculated by

equation image
Figure 1.

Water's liquid-gas interfacial tension γ, capillary pressure of an Elkmound sandy loam pc, and the soil's inferred wetting coefficient k, all relative to their values at 298 K.

[9] The values of both capillary pressure and liquid-gas interfacial tension decrease linearly with temperature with the former decreasing at a greater relative rate than liquid-gas interfacial tension. This implies that (1) the temperature sensitivity of capillary pressure is likely to be an interfacial phemonenon, (2) something other than liquid-gas interfacial tension, apparently the wetting coefficient, is responsible for the large relative decrease in capillary pressure with temperature, and (3) since k appears to be a linear function of temperature, it is reasonable to assume that γls and γsg are approximately linear functions of temperature.

[10] The choice of the functional form with which to represent the wetting coefficients or contact angles of the wetting liquids in porous media is a leitmotiv of theoretical studies of the transport of nonaqueous phase liquids in the subsurface [e.g., Gray, 1999]. Perhaps the lessons learned in the investigation of the effect of temperature on CPSR suggest fruitful approaches to solving the problem in more general situations. In deriving an expression for the effect of temperature on CPSRs, one can choose how to represent the interfacial effects in the Young-Laplace equation. In representing the product γlg cos Θ, one can (1) assume that k = cos Θ = 1, (2) assume that k = cos Θ ≠ 1 and attempt to model temperature changes in terms of the product γlg cos Θ (this has the powerful attraction that both γlg and Θ can be measured independently), or concede that neither γsg nor γls, let alone their temperature derivatives, are measurable, but assume that the responses of their difference to changes in temperature control the physics of the phenomenon and that by posing the problem in terms of their difference makes the algebra of the problem more tractable.

[11] Though it is not conceptually superior to the other two choices, experience indicates that the last path is the most fruitful one for many situations and will be followed in the present paper. The Young-Laplace equation becomes

equation image

The challenge then becomes to arrive at a responsible formulation for γsg − γls as a function of temperature.

3. Models for Interfacial Tensions

[12] Since there is very little experimental information on the changes in liquid-solid and solid-gas interfacial tensions with temperature, it was assumed that the selection of a functional form to model their changes with temperature would be guided by the behavior of the liquid-gas interfacial tension. Since the temperature derivative of interfacial tension can be derived from interfacial energy per unit area (us,αβ J m−2) via

equation image

[Everett, 1972, p. 600] it was decided that it was appropriate to define the liquid-solid and solid-gas interfacial phenomena initially in terms of interfacial energy and transform this formulation to the appropriate mathematical forms for γsg and γls.

[13] I have not noticed any evidence that the linear decrease in capillary pressure with temperature is affected by the number of components in the porous matrix. Apparently, the phenomenon is unaffected by the number of components in the colloidal mixture that makes up the porous matrices that have been studied. Any model selected to describe this phenomenon must be able to describe the decrease in capillary pressure with temperature for simple and mixed porous matrices. Accordingly, a power series expression was chosen, because pretty much any behavior can be described by the function and because a weighted sum of serveral such functions can be calculated easily. Accordingly, interfacial energy is defined here in terms of temperature by:

equation image
equation image

where us,ls and us, sg are the liquid-solid and solid-gas interfacial energies per unit area (J m−2); T, temperature (K); and ajls and ajsg coefficients.

[14] If liquid-solid and solid-gas interfacial energies can be described by equations (7) and (8) then equation (6) has the solutions:

equation image


equation image

where csg and cls are constants.

[15] The general forms for the wetting coefficient and capillary pressure, respectively, as functions of temperature, are therefore:

equation image
equation image

In practice, it is likely that equations (12) and (13) can be truncated. To guide this truncation it is useful to look carefully at the changes with temperature of water's interfacial tension and interfacial energy. The International Association for the Properties of Water and Steam (IAPWS) has released a recommended formulation for the surface tension of water from the melting temperature to the critical temperature [Haar et al., 1984, p. 316]. A plot of this relationship is presented in Figure A1.1

[16] The corresponding liquid-gas interfacial energy can be calculated via equation (6), yielding the relation presented in Figure 2 for water-air interfacial energy per unit area from the melting temperature to the critical temperature.

Figure 2.

Liquid-gas interfacial energy per unit area of water from its melting to its critical temperature.

[17] From this figure it appears that us,lg (and by extension us,ls and us,sg) can be modeled as a linear function of temperature from the melting point to roughly 450 K. A plot of ∂us,ls/∂T presented in Figure A2 (available electronically) indicates that the slope of the us,ls versus T curve is approximately contstant from the melting point to 450 K. Over this temperature range the following relation is therefore appropriate:

equation image

Solving equation (6), for which us,lg is described by a function with the form of equation (13), yields:

equation image

[18] To verify the suitability of equation (14) four candidate functions were fitted by regression to the water's liquid-gas interfacial tension values from 273.16 to 448 K provided by Vargaftik et al. [1983]:

equation image
equation image
equation image
equation image

[19] The results of this regression analysis are presented in Table 1. Although the linear model (equation (15)) gave a good description of the data, the square root mean square error (RMSE) of the fit can be reduced substantially by the inclusion of either a T ln(T) (equation (16)) or T2 (equation (17) term with the the former yielding the better fit of the two terms. The RMSE of the fit for equation (18), which includes both T ln(T) and T2 terms, improves the RMSE only slightly over equation (16). In most instances proper fitting of a four-parameter model of interfacial tension would require measurement at four temperatures. In the absence of few if any experimental measurements, it must be accepted tentatively that equation (16) would be an acceptable guide for description of interfacial tensions in water wet systems. It seems reasonable to assume that, below about 448 K, liquid-solid and solid-gas interfacial energies of water-wet systems are similarly well behaved functions of temperature. Therefore, the two most likely particular forms for equation (13) are for ajsg = ajls = 0 for j > 0 or j > 1.

Table 1. Estimated Parameter Values ( ± Their Standard Errors) and Square Root Mean Square Errors for Four Models of Liquid-Gas Interfacial Tension of Water Fitted by Nonlinear Regression to Its Value From 273.16 to 448 K
Equationa0lg, mN m−1a1lg, mN m−1 K−1a2lg, mN m−1 K−2clg, mN m−1 K−1RMSE, mN m−1
(15)124.48 ± 0.66  −0.1853 ± 0.00180.58386
(16)67.72 ± 0.14−0.16889 ± 0.00039 0.9766 ± 0.00270.00792
(17)97.53 ± 0.27 −0.000236 ± 0.0000022−0.01535 ± 0.00160.03141
(18)68.57 ± 1.07−0.2095 ± 0.00610.0000569 ± 0.00000851.215 ± 0.0360.00524

[20] Accordingly, equation (12) can be truncated and rearranged to yield:

equation image

By collecting the following terms

equation image
equation image
equation image

the following expressions are obtained,

equation image

where Tf is the observational temperature. After rearrangement these truncated expressions yield the following models of CPSR temperature sensitivity:

equation image

in which β2 = 0 and

equation image

Equation (24) is referred to here as the one-parameter model and equation (25) as the two-parameter model. (For each model β1 ≡ 1. An alternate derivation of equation (24) from interfacial thermodynamics was presented by Grant and Salehzadeh [1996].)

[21] It should be noted that equation (25) can be written

equation image

For modest changes in temperature, on the order of 20 K, ln Tf ≈ ln Tr and equation (26) can be approximated by

equation image


equation image

It can be assumed that us,sg and us,ls are linear functions of temperature, as will be their difference. Therefore, equation (25) reflects the thermophysical changes of the pertinent interfacial tensions. As a practical matter because of the very small changes in ln T over small temperature changes, equation (24) will in all probability describe experimental data in situations where the changes in temperature are modest (e.g., on the order of 20 K).

4. Regression Analysis of Data From Water-Air Systems

4.1. One-Parameter Model

[22] Grant and Salehzadeh [1996] incorporated equation (24) into the CPSR model of van Genuchten [1980] to yield

equation image


equation image


equation image

where pc° is a normalizing constant (e.g., pc° =1 Pa). Grant and Salehzadeh [1996] found that equation (28) could be fitted to some, but not all, of the data sets considered in their paper. Subsequently it was found that this difficulty was largely eliminated by pulling the temperature-correction term equation image from the exponential in equation (28) to yield

equation image

[23] Equation (31) was fitted by nonlinear regression to all of the available data sets of water-air CPSRs measured at more than one temperature. The results are presented in Table A1 (available electronically). Based on the results of the regression analysis, the following was concluded. (1) The one-parameter model describes most data well with low RMSE. (2) With few exceptions, model-parameter estimates have low asymptotic standard errors. (3) Based on plot of the residuals from the nonlinear regression analyses of the available data, goodness-of-fit appears to be unaffected by degree of saturation. (4) The one-parameter describes equally well capillary relations in porous media formed from homogeneous and heterogeneous porous media. (5) For those data sets in which the CPSRs were determined for both drainage and imbibition, the estimated β0 values for drainage were less (more negative) than those for imbibition.

4.2. Two-Parameter Model

[24] The van Genuchten equation modified by the two-parameter model

equation image

was also fitted by nonlinear regression to the water-air capillary pressure-saturation-temperature data sets of which I am aware. The results are presented in Table A2 (available electronically).

[25] When compared to the results of the nonlinear regression analysis with the one-parameter model, in all instances there was a modest decrease in the square root means square error of the fit, though in many instances the regression Jacobian was singular or nearly singular, resulting in parameter estimates with zero standard errors.

5. Applicability of the Model to Both Homogeneous and Heterogeneous Porous Media

[26] The results of nonlinear regression analysis of the one-parameter model presented in Table A1 indicate that equation (31) could be fitted to water-air capillary-saturation data from samples of soils (which are very heterogeneous porous media) and uniformly composed glass beads. This prompted an exploration for the nature of interfacial properties of porous materials that might explains this apparent universality.

[27] One could assume reasonably that there must be a rule by which the “gross” or “average” effect of component wetting coefficients are expressed in the capillary behavior of a porous medium sample. One could propose, for example, a relation such as

equation image

where kM and ki are the wetting coefficients of the medium and its i-th component, respectively; and ϕi, the fractional surface area of the i-th component.

[28] Equations (9) and (10) can be extended to each of the solid components in a heterogeneous porous media. For the i-th mixture component, the liquid-solid and solid-liquid interfacial tensions become

equation image
equation image

[29] The temperature sensitivity of wetting coefficient of a heterogeneous porous media could be described by

equation image

[30] If, as before, it assumed that ai,jsg = ai,jls = 0 for all j > 0 equations (2) and (24) may be manipulated to lead immediately to

equation image

with the corresponding forms a mixed porous medium and its components:

equation image


equation image

[31] Equation (39) may be rearranged to yield the following relation for component “i”:

equation image

[32] Similarly equation (39) can be rearranged to yield

equation image

where equation image and equation image.

[33] Equation (40), which is appropriate for a homogeneous porous medium, has the same mathematical form as equation (41), which is derived for a heterogeneous porous medium. This result, coupled with the general applicability of the model, provides additional evidence that the effect of temperature on capillary pressure is due primarily, if not exclusively, to capillary forces.

6. Regression Analysis of Data From Water-Oil Systems

6.1. One-Parameter Model

[34] Nonlinear regression analysis fits of the van Genuchten [1980] equation modified with equation (24) were found to describe well the change in capillary pressure with temperature for water-air systems. Inasmuch as the Young-Laplace equation is thought to apply equally as well to water-oil systems as water-air systems, it was thought initially that the application of this approach to water-oil systems would be a straightforward, but valuable, validation of this approach.

[35] In the 1970s, a research group in the Petroleum Engineering Department at Stanford University measured carefully the water-oil CPSR in studies of steam extraction of petroleum reservoirs. These studies have been augmented by more recent studies of water-NAPL systems. The results of nonlinear regression analysis of fits of equation (24) to these data are presented in Table A3 (available electronically).

[36] Twenty-four data sets were analyzed by nonlinear regression. An estimate of β0 with an absolute value less than 1000 could be obtained in only twelve of these cases. Of these twelve cases, five had asymptotic standard errors of the estimate of the same order as the estimate itself. Despite the ungainly estimates, the fitted parameters generally described the data well: the highest square root mean square error was 0.084, the lowest was 0.0047.

6.2. Two-Parameter Model

[37] The results of nonlinear regression analysis of water-oil data fitted to the two-parameter model are presented in Table A4 (available electronically). The results of these nonlinear regression analysis of the water-oil data sets were similar to those of the water-air systems. In comparision with the one-parameter model, the precision of the fit, as measured by residual sums of squares, was not improved by using the two-parameter model, though the asymptotic standard errors of many parameter estimates were reduced markedly.

7. Describing Temperature Effects on Capillary Pressure at Low Degrees of Saturation


[38] She and Sleep [1998] reported recently experimental studies of the effect of temperature on CPSRs for water-air and water-tetrachloroethylene systems. They concluded that (1) the one-parameter model of Grant and Salehzadeh [1996] could represent their CPSR data well, (2) the temperature derivative of contact angles for both water-air and water-tetrachloroethylene systems predicted by the fitted model were of opposite sign from those consistently reported in the literature, and (3) the apparent temperature derivative of the contact angle increased dramatically as the degree of saturation of the sample was lowered, a situation not addressed by the model of Grant and Salehzadeh [1996], which assumes that interfacial properties are unaffected by the degree of saturation.

[39] Their third conclusion raises important questions for the modeling of heat and matter transport in nonisothermal systems: Is the temperature sensitivity of capillary pressure affected by the degree of saturation? Below what degree of saturation does the one-parameter model no longer apply? Are the parameters in the one-parameter and two-parameter models functions of degree of saturation?

7.2. Analysis of Residuals

[40] It is well known that the vapor pressure in a porous medium decreases at low degrees of saturation and that this may have an effect on the responses of interfacial tensions to changes in temperature. To investigate this I plotted, in Figure 3, the residuals for the fit of equation (31) to three data sets reported by Wilkinson and Klute [1962], who measured CPSR for porous samples fashioned from silt-size and two sand-size fractions. In general, the residual error, as indicated by the scatter of the residual plot, was unaffected by degree of saturation.

Figure 3.

Residuals of three nonlinear regression fits of equation (24) to the three data sets of Wilkinson and Klute [1962].

[41] It is reasonable to expect that the behavior of the wetting coefficient would change at lower degrees of saturation, when the physical state of soil water becomes less a liquid held by capillarity to more like a gas adsorbed on a solid. The data of Campbell and Shiozawa [1992], who measured the CPSR on six diverse soils to very high capillary pressures are useful in illustrating this transition. A plot of the data for all soils in which the water content is expressed as nominal adsorbed water thickness is presented in Figure 4.

Figure 4.

Nominal adsorbed water thickness as a function of water-vapor pressure at 298 K for six soils having specific surfaces from 25 m2 g−1 to 200 m2 g−1: L soil, open circles; Palouse soil A horizon, squares; Palouse soil B horizon, triangles; Royal soil, diamonds; Salkum soil, inverted triangles; and Walla Walla soil, solid circles. The transition of water from the capillary to the adsorbed domain appears to occur at nominal adsorbed water thicknesses of 10–150 nm. Data from Campbell and Shiozawa [1992].

[42] The nominal adsorbed water thickness (δ, m) is calculated from the water content of the soil divided by the surface area via

equation image

where ρs and ρwater are the particle and water densities (Mg m−1), respectively, and rs is the nominal particle radius (m) calculated by the mean diameters of a particular particle-size class:

equation image

This indicates that the transition from capillary to adsorbed water occurs at a water thickness of approximately 10–150 nm. Figure 5 presents again the residuals from the nonlinear regression fit of equation (31) to the data of Wilkinson and Klute [1962] plotted against nominal adsorbed water thickness, rather than degree of saturation. The transition from capillary water to adsorbed water as deduced from the data of Campbell and Shiozawa [1992] is indicated by an arrow. This graphic indicates that the data of Wilkinson and Klute [1962] were collected at nominal adsorbed water thicknesses approximately three orders of magnitude greater than what is being designated here the transition from the capillary to the adsorbed zone.

Figure 5.

A representation of Figure 3 with apparent water thickness as the ordinate. All the data, and presumably most capillary pressure saturation data, are in the capillary domain, in which the temperature sensitivity of pc is not affected by degree of saturation.

[43] She and Sleep [1998] found that the residual water content differed from temperature to temperature. They fitted a linear equation of the form

equation image

I fitted the data of She and Sleep [1998] to van Genuchten's equation and also found that the estimated θr differed from temperature to temperature, though the differences were typically only slightly larger than the differences found in replicates at the same temperature. Table A5 (available electronically) presents an extension of this regression analysis to all of the water-air capillary pressure saturation data available in the published literature. The results presented in Table A5 did not support the trend proposed by She and Sleep [1998]. The linear increase in temperature could be either found, not found, or contradicted by different data sets. Interestingly, nonlinear regression analysis of the data of Nimmo and Miller [1986], who collected numerous values, yielding accordingly very precise parameter estimates, suggested no change in θr with temperature.

[44] She and Sleep [1998] also calculated the apparent ratio cos Θ (T = 353 K)/cos Θ (T = 293 K). They found that this ratio increased dramatically as the water content approached residual water content. One should note that there are slight differences in residual water contents from experimental run to experimental run, as presented in Table A6 (available electronically). The increase in the cos Θ (T = 353 K)/cos Θ (T = 293 K) ratio at lower degrees of saturation presented by She and Sleep [1998] may very well be due to combination of slight differences in residual saturation in experimental runs conducted at different temperatures and the enormous changes in capillary pressure with degree of saturation at lower degrees of saturation. In other words, the metric chosen by She and Sleep [1998] mapped modest changes in S into monumental changes in cos Θ (T = 353 K)/cos Θ (T = 293 K) at lower degrees of saturation, due more to the rapid changes in pc with S approaching residual saturation than changes in interfacial properties as the degree of saturation is lowered. This point is shown most clearly in Figure 6, which shows ∂pc/∂S (as calculated by the van Genuchten [1980] equation) against S.

Figure 6.

The partial derivative of capillary pressure with respect to degree of saturation for drainage by the sand sample studied by She and Sleep [1998].

7.3. Extension to Lower Degrees of Saturation

[45] Chemical potential of the water in a partially saturated porous medium, relative to a water in its standard state, can be related to capillary pressure (if the reference state is assumed to have the same composition, elevation, and atmospheric pressure) by

equation image

where μm and μH2O* (l, T, pl + g) are the chemical potentials of water in the porous medium and pure water at equilibrium with its saturated vapor (J mol−1), respectively and Vm,H2O(l) is the molar volume of the liquid (m3 mol−1) [Taylor and Ashcroft, 1972, p. 163].

[46] The partial pressure of water vapor at equilibrium with the soil water can be calculated (neglecting deviations from ideal gas law) via

equation image

where p is the vapor pressure of water above the porous medium, pl + g is the vapor pressure of pure water, and R, the gas constant (J K−1 mol−1). Equations (44) and (45) can be combined to yield

equation image

Assuming ∂Vm,H2O(l)/∂T ≈ 0 this leads immediately to:

equation image

The change with temperature in the logarithm of the vapor pressure above adsobed water is related to the isoteric enthalpy of adsorption (ΔadsgHm*) via

equation image

[Defay et al., 1966]. Similarly, the change with temperature in the logarithm of vapor pressure of liquid water is approximately related to the enthalpy of vaporization via:

equation image

[McGlashan, 1979]. These can be combined to yield:

equation image

which has the same form, and solutions, as equation (5). Since ΔadsgHm is affected by the amount of water adsorbed on the surface, it is expected that the β0 parameter in the one-parameter equation would be a function of the amount of adsorbed water at very low degrees of saturation.

8. Concluding Remarks

[47] The one- and two-parameter models presented in this paper provide an accurate description of the effect of temperature on capillary pressure. The use of either can be justified on both theoretical and practical grounds, the two-parameter model may be preferred in systems with large changes in temperature, if only because the parameters of this equation are typically estimated more precisely.

[48] The models describe the behavior of homogeneous and heterogeneous porous media equally well. The data analyzed in this paper were obtained from static determinations of CPSRs. A recent study reported by Bachmann et al. [2002] found that β0 values could be estimated precisely from the results of transient imbibition and drainage experiments.

[49] Several authors have found that the changes in contact angles with temperature inferred from the β0 values estimated thusfar are inconsistent with those changes reported in the published literature [She and Sleep, 1998; Bachmann et al., 2002]. While not questioning those published values, I must observe that equation (24) describes well the effect of temperature on capillary pressure.

[50] The challenge is to reconcile equation (24) as a description of the effect of temperature on capillary pressure, the apparent effect of temperature on contact angles, and the quantitative description of capillarity currently expressed in the Young-Laplace equation. Further studies are required to determine whether equation (24) is consistent with measured changes with temperature of the contact angle, though currently available values indicate that they are not. The resolution of this conflict should provide an insight into the limitations of the applicability of the Young-Laplace equation to liquids in partially saturated porous media.


fitted parameter, N/m.


ln(αpc°), 1.


fitted parameter, N/(m K).


fitted parameter, J/m2.


a constant, J/(mol K).


a constant, J/(m2 K).


maximum particle diameter, m.


minimum particle diameter, m.


fitted parameter, J/(m2 K).


wetting coefficient, 1.


wetting coefficient of a mixed porous medium component, 1.


wetting coefficient of a mixed porous medium, 1.


a parameter, 1.


capillary pressure, Pa.


pressure in the gas phase, Pa.


pressure in the liquid phase, Pa.


vapor pressure of water at equilibrium with water in a porous medium, Pa.

pl + g

vapor pressure of water at equilibrium with pure bulk water, Pa.

ln(pc/pc°), 1.


gas constant, J/mol.


pore radius, m.


particle radius, m.


degree of saturation, 1.


interfacial entropy per unit area between α β phases, J/(K m2)


temperature, K.


observational temperature, K.


reference temperature, K.


interfacial energy per unit area, J/m2.


interfacial energy per unit area of the solid-gas interface, J/m2.


interfacial energy per unit area of the solid-liquid interface, J/m2.


molar volume of liquid water, m3/mol.


a parameter, 1/Pa.


parameter, K.


parameter, K1.


parameter, 1.


interfacial tension between the liquid gas phases, N/m.


interfacial tension between the liquid solid phases, N/m.


interfacial tension between the solid gas phases, N/m.


interfacial tension, N/m.


nominal water film thickness, m.


isoteric enthalpy of adsorption, J/mol.


enthalpy of vaporization, J/mol.


volumetric soil-water content, 1.


residual volumetric soil-water content, 1.


saturated volumetric soil-water content, 1.


contact angle of the liquid-gas interface with the solid, rad.


chemical potential of water in a porous medium, J/mol.

μH2O* (l, T, pl + g)

chemical potential of pure water at equilibrium with its saturated vapor, J/mol.


density of water, kg m−3.


particle density of a solid, kg m−3.


fractional surface area of a component of a mixed porous material, 1.


[51] B. Sleep and J. R. Nimmo are thanked warmly for their comments on the manuscript in draft form. This work was supported by Environmental Quality Basic Research Enhancement Program work unit AT25-EC-B08, entitled “Pore-scale modeling of multiphase contaminant transport in freezing/thawing soils,” the Strategic Environmental Research and Development Program (SERDP), Deutsche Forschungsgemeinschaft (DFG) project number Ba 1359/5-1 within the priority program SPP 1090 “Soils as source and sink for CO2 - mechanisms and regulation of organic matter stabilization in soils” and the U.S. Army Engineer Research and Development Center work unit 61102/AT24/129/EE005 entitled “Chemistry of Frozen Ground.”