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An explicit derivation of an exponential dependence of the hydraulic conductivity on relative saturation
Advances in Water Resources 27 (2004) 197–201 www.elsevier.com/locate/advwatres
Short communication
An explicit derivation of an exponential dependence of the hydraulic conductivity on relative saturation A.G. Hunt Cooperative Institute for Research in the Environmental Sciences, University of Colorado, Boulder, CO 80309, USA Received 2 November 2002; received in revised form 5 July 2003; accepted 4 November 2003
Abstract It is shown how to derive explicitly an exponential dependence of the unsaturated hydraulic conductivity, K, on relative saturation, S, and under what conditions such a relationship holds. An earlier derivation of the same result was so restrictive, that it appeared that realistic realizations were not possible, but the present derivation demonstrates the contrary. From existing results, the implication is that the steady-state unsaturated hydraulic conductivity will be log-normally distributed. The log-normal distribution, as well as an exponential dependence on moisture content, have been reported from the same experiment on the same field. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Continuum percolation; Critical path analysis; Fractal porous media; Constitutive relations; Exponential functions
1. Introduction Recently I used percolation theory [5], to derive an expression for the unsaturated hydraulic conductivity, K, of a ‘‘random fractal’’ porous media (such as described by Tyler and Wheatcraft [18], or Rieu and Sposito [14] in terms of the saturated hydraulic conductivity, KS and the relative saturation, S, K ¼ f ðKS ; SÞ. The most important difference between the Hunt [5], treatment of the porous medium itself and the earlier treatments, was that I allowed a continuous range of pore sizes, rather than a discrete distribution. The most important difference between the Hunt [5], derivation of the hydraulic conductivity and earlier results is in the application of critical path analysis [1,13], from percolation theory. For cases where explicit expressions for a fractal dimensionality, Dr 6¼ 3, of the pore space could be inferred from the particle-size distribution, the derived power-law relationships for K ¼ f ðKS ; SÞ agreed reasonably well with experiment [15], and with simulation [12], and were also quite similar in form to the result of [7] (who used a result of Burdine [3], in their derivation). As is known [18], the fractal formulation leads to a power-law water retention curve, which is in accord with the Brooks and Corey phenomenology [2].
E-mail address: [email protected] (A.G. Hunt).
0309-1708/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2003.11.005
In contrast, in the limit, Dr ! 3, an exponential relationship for K=KS ¼ f ðSÞ was derived [5]. An exponential dependence of K on S leads immediately to a log-normal distribution of the hydraulic conductivity [4,5], both results known from field studies [11]. It was noted, however [5], that a direct evaluation of the 3D expression for the porosity, / ¼ 1 ½r0 =rm r in the limit Dr ! 3 yielded zero pore space, and strictly speaking, zero KS . In this expression r0 and rm denote the smallest and largest pore radii, for which the soil geometry may be represented as fractal. While the limitation did hold in the previous derivation, I wish to show now that it is easy to overcome this difficulty. If the general notion that a fractal dimensionality equal to three implies that the partial porosity of each pore class between r0 and rm is equal, then it is not difficult to formulate a general expression for the pore volume probability density, which in turn yields the exponential relationship for K ¼ f ðKS ; SÞ in addition to a finite value of the saturated hydraulic conductivity, KS . This is the main point of the present note. It is necessary, however, to point out, that the water retention curve in the limit Dr ! 3 is no longer functionally compatible with the Brooks and Corey [2], phenomenology (it also turns out to be exponential). Two applications of critical path analysis are required to obtain the results for the log-normal distribution of the hydraulic conductivity. In the first application,
198
A.G. Hunt / Advances in Water Resources 27 (2004) 197–201
critical path analysis chooses a particular conductance out of a distribution of pore-scale conductance values with a uniform relative saturation. These pore scale conductances are given in terms of Poiseuille’s law, and are proportional to the fourth power of the pore radius, and inversely proportional to its length. In the case of uniform aspect ratios considered here, the net result is a local conductance proportional to the third power of the pore radius. Under the conditions assumed, and for Dr ! 3, the dependence of K on S turns out to be exponential. As pointed out in [5], a second application of critical path analysis is then required at larger scales, because it is necessary that the input (small scale) hydraulic resistances are exponential functions of a (random) moisture content. The relative saturation can equally be considered to be the quantity with the relevant spatial variability. These details have already been dealt with in [5]; what is missing was to address the problem of the last paragraph, namely that KS appeared to be identically zero, in the limit, Dr ! 3.
is proportional to an integral over an integrand, which is the product of Eq. (1) and the factor r3 , using the appropriate normalization constant, Z rm 3D /¼ r3 r1D dr ð3Þ rm3D r0 Eq. (3) is easily evaluated [5], and gives / ¼ 1 3D ðr0 =rm Þ , as is already known from the results of Rieu and Sposito [14]. There is no difficulty with extending the continuum pdf for pore sizes to the case Dr ¼ 3, however the normalization constant is different from its value when Dr 6¼ 3. To normalize Eq. (1) for Dr ¼ 3, set, Z rm Z rm Z rm 3 2Dr W ðrÞr dr ¼ A r dr ¼ A r1 dr ¼ / ð4Þ r0
r0
where the next to last equality follows from choosing Dr ¼ 3. Then it is easy to find that A¼ ln
2. Soil structure background The soil structure is not discussed in detail here, but the reader is referred to the work of Rieu and Sposito [14], denoted subsequently by RS. I use a power-law distribution of pore sizes over a finite range of pore sizes, from a minimum of r0 to a maximum of rm . This is supposed to be compatible with a random fractal structure, in which, in the mean, the geometrical relationships between successive pore classes are satisfied, allowing an identical representation of the statistics of pore classes to that satisfied by a deterministic fractal. It is shown in [5], that the power-law probability density function, pdf, for pore radii, W ðrÞ / r1D
ð1Þ
(equivalent to the discrete distribution, W ðrÞ / rDr Þ leads to the expression 3D r0 ¼/ ð2Þ 1 rm for the porosity, /. This relationship is identical to that of RS, and is thus generally compatible with the Brooks and Corey relationship for water retention characteristics, as shown in Hunt and Gee [6], and known from Tyler and Wheatcraft [18]. Eq. (2) may be shown consistent with Eq. (1) through the following. Assume a volume r3 associated with each pore of radius r (numerical factors relevant to specific geometries are suppressed here, as they will not be relevant for ratios of the unsaturated to the saturated conductivity; if the geometries are the same at all length scales, then these factors will ‘‘cancel’’). Then the total pore space volume
r0
/ h i rm r0
ð5Þ
3. Hydraulic conductivity and water retention curves The system is envisioned to consist of a network, or perhaps more accurately, of a net of connections between pores. The net consists of the connections, denoted as hydraulic conductances, with values of the conductance consistent with Poiseuille’s Law for flow through the narrowest constrictions, or throats. These conductances can also, in principle, be determined from numerical solution of the Navier–Stokes equation at the pore scale. The hydraulic conductivity is found through the application of critical path analysis. Critical path analysis yields directly the smallest value of the conductance on the most highly conducting path of infinite extent. Critical path analysis may be applied in the form of continuum percolation. As discussed in [5], it is generally possible to find a (critical) volume fraction, ac , consisting of individual pores, assembled at random, which percolates through the system. If this volume is chosen by including the largest pores down to some minimum pore radius, rc , then rc becomes the controlling radius for transport. KS must now be found for Dr ¼ 3. The condition reads simply, that Z rm / dr ¼ ac ð6Þ lnðrm =r0 Þ rc r where the integral is a volume fraction on account of the normalization above. The solution of Eq. (6) for the smallest, or critical, pore radius, rc , is
A.G. Hunt / Advances in Water Resources 27 (2004) 197–201
rc ¼ rm
r0 rm
ac =/ ð7Þ
Note that if ac > /, rc < r0 , and the hydraulic conductivity must vanish, there being no pores in the medium with r < r0 . Otherwise, the saturated hydraulic conductivity, KS , is proportional to rc3 . The process for S < 1 is to find the largest pore that remains filled, and apply the continuum percolation approach over pores restricted to be smaller than that limit. First find r> , the largest pore still filled with water. During drainage, it is assumed that, through film flow, the largest pores can be emptied first. Then r> can be found from the relative saturation, S, as follows: 1 Z r> rm dr ¼S ð8Þ ln r r0 r0 The factor / does not appear because the maximum value of S is not /, but 1. The value r> as a function of pressure can be obtained theoretically by the same general procedure as is used to develop the waterretention curve, below. Solution of Eq. (8) for r> yields, S rm ð9Þ r> ¼ r0 r0 In this case, where pores larger than r> can not contribute to the hydraulic conductivity, because they are empty, the critical volume must be constructed out of pores with smaller radii than r> , forcing the lower limit, rc , of the integration to smaller values as well: Z r> / dr ¼ ac ð10Þ lnðrm =r0 Þ rc r Solution of Eq. (9) for rc yields, ac =/ ac =/ 1S 1S r0 r0 r0 r0 rc ¼ r> ¼ rm ¼ r cS rm rm rm rm ð11Þ where rcS is the corresponding expression in the saturated case. The last expression can be put into an exponential form by using the identity x ¼ exp½lnðxÞ, and r0 rc ¼ rcS exp ð1 SÞ ln ð12Þ rm Since, in [5], it was argued that with uniform pore aspect ratio the critical hydraulic conductance, gc , is proportional to the cube of the critical radius, one can find, rm gc ¼ gcS exp ð1 SÞ3 ln ð13Þ r0 Eq. (13), with gcS the saturated value of the critical conductance, is nearly identical to that derived in [5], with the slight difference that the denominator of the exponential in Eq. (13) does not contain the factor
199
1 ac present in [5]. Since typical values of ac are about 0.15, this difference is not large, but the present derivation has no inconsistencies. In order to convert Eq. (13) into an equation for K, additional length scales are required, as discussed in [5]. Without these length scales, use of Eq. (13) to derive K as a function of KS is only approximate, rm Kc ¼ KS exp 3ð1 SÞ ln ð14Þ r0 Note that for practical applications to a porous medium, in which the relative saturation can vary over scales large relative to a pore scale, but small relative to the scale of measurement to the field scale, an additional application of critical path analysis, as described in Section 3.3.5 of Hunt [5] is required. The important modification here, compared with [5], however, is that the saturated conductivity is not zero, but is proportional to rc3 given in Eq. (7). The concern regarding the water retention curve must now be addressed. First, the problem is posed for arbitrary fractal dimensionality, in order to make a connection with [5]. Eq. (3), in the case Dr 6¼ 3, reads, Z rm 3 Dr dr 3 Dr ¼/ ð15Þ rr 3D r r rm r0 If it is assumed that the matric head, h, required to remove the water from a pore of radius r, is h ¼ A=r, then the air entry pressure is, hA ¼ A=rm , and, Z A=h 3 Dr r2D dr ð16Þ S¼ rm3Dr r0 When S ¼ 1, the upper limit is thus replaced by A=hA . Solution of Eq. (16) for the saturation as a function of pressure leads to, " D3 # D3 1 h h S ¼1 ð17Þ 1
/ hA hA where to obtain the approximate equality, identical to the Brooks and Corey [2], relationship, would require the porosity to equal 1. Note that substitution of Eq. (17) into [5] 3=3D 1S K ¼ KS 1 / ð18Þ 1 ac for K=KS as a function of relative saturation yields, "
3D !#3=ð3DÞ 1 hhA K ¼ KS 1 ð19Þ 1 ac Eq. (19) has some relationship with the Mualem [10], result. The solution for the matric potential as a function of saturation leads to,
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A.G. Hunt / Advances in Water Resources 27 (2004) 197–201
h ¼ hA
1 1 /ð1 SÞ
1 3D
4. Discussion ð20Þ
In the case D ¼ 3, however, the problem must be set up slightly differently. Now, Z A=h 1 dr ð21Þ S¼ lnðrm =r0 Þ r r0 From Eq. (21) it is possible to derive, ln hhA S ¼1 ln rrm0 Alternatively, rm h ¼ hA exp ð1 SÞ ln r0
ð22Þ
ð23Þ
As advised, this relationship is no longer a power-law between S and h. Nevertheless, representation of K in terms of h is still a power-law, in particular, K ¼ KS
hA h
3 ð24Þ
Note that the power n ¼ 3 here arises from the choice of a uniform aspect ratio at all pore sizes. Consider the power on ðh=hA Þ predicted from Eq. (19) for D 6¼ 3, d ln n¼
K KS
1 ¼3 3D d ln hhA 1 ac hhA
ð25Þ
It is easy to see that Eq. (25) yields a value of n which is slightly larger than 3 at large h (near hA ), but which increases slowly with diminishing h. However, it is also apparent that Eq. (19) predicts that K=KS should vanish 1=ð3DÞ at a very small, but finite, value of h ¼ hA ðac Þ . Miller–Miller similitude, Miller and Miller [8], implies n ¼ 2, although Snyder [16], as well as Hunt [5], note statistical complications that can invalidate such simple assumptions (and also the ideal result of n ¼ 3 here). Experiment typically yields values of n of from 2.6 to 2.8 (Sposito, personal communication). But an experimental fit to a Brooks and Corey formulation would constrain K=KS to be finite at all finite h, altering the fitted n to smaller values. It is not possible to state whether those values will or will not ultimately be in accord with the present derivation. What it is possible to state, however, is that if an exponential dependence of K=KS on S arises from fractal soils with D ¼ 3, then such soils will also have a power law relationship of K=KS on h, and will generate, according to the present treatment, n ¼ 3.
In this note, an explicit relationship between a powerlaw water retention curve developed for a fractal porous medium and the corresponding relationship between the unsaturated and saturated values of the hydraulic conductivity is given. Clearly it becomes possible to extract a fractal dimensionality from the former, apply to the latter, and predict the constitutive relationship, as introduced in [5]. For critical path analysis to have predictive value, the critical volume fraction, ac , for percolation must be known. In important recent work [9], consider a threshold water content for solute diffusion, which they call ht . In this reference a predictive phenomenological relationship for ht was given in terms of the relative surface area to volume ratio of the soil, SAvol . This method to calculate ht was shown in Hunt and Gee [6], to generate the same value for ac as is obtained from the minimum moisture content reached in unsteady drainage experiments. In fact, it turns out that fairly typical values of ht for loamy soils are about 0.15, a commonly cited value for the critical volume fraction in continuum percolation [17]. ac can, however, take on much larger values in certain geometries unfavorable for percolation. The most obvious such case is when pores are highly elongated, with similar orientations, but with spatial positions much more highly correlated in the direction perpendicular to elongation than in the direction parallel to the elongation. It may be interesting to note that for very large values of SAvol from clay soils, ca. 800 m2 / g, the Moldrup relationship ht ¼ 0:039ðSAvol Þ0:52 , predicts ht 1.
5. Conclusions In a previous paper it was shown possible to derive from Poiseuille’s law for saturated flow, and a fractal geometry of a porous medium an exponential dependence of K=KS on the relative saturation, S. The limitation that the relationship was strictly valid only in the limit D ! 3, was problematic, since according to the results of Hunt [5], the saturated hydraulic conductivity evaluated in that limit was identically zero. Now, however, an explicit accounting for the case D ¼ 3 has been made, and it has been shown that essentially the same exponential relationship holds, but the saturated hydraulic conductivity is non-zero. This allows explicit derivation of a log-normal distribution of the hydraulic conductivity as well [5]. The conditions derived, that the unsaturated hydraulic conductivity is an exponential function of the relative saturation, were reported to exist in a classic paper of Nielsen et al. [11], in which the steady-state unsaturated hydraulic conductivity was also reported to have been log-normally distributed. A po-
A.G. Hunt / Advances in Water Resources 27 (2004) 197–201
tential negative aspect, however, of the derived exponential relationship for K on S, is that the corresponding Brooks and Corey type relationship for the water retention curve also becomes exponential for the case Dr ¼ 3, while the dependence of K=KS on matric potential, h, remains a power-law.
Acknowledgements This study was supported by the US Department of Energy Climate Change Prediction Program, which is part of the DOE Biological and Environmental Research Program. The Pacific Northwest National Laboratory is operated for the DOE by Battelle Memorial Institute under contract DE-AC06-76RLO 1830. References [1] Ambegaokar VN, Halperin BI, Langer JS. Hopping conductivity in disordered systems. Phys Rev B 1971;4:2612. [2] Brooks RH, Corey AT. Hydraulic properties of porous media. Colorado State University Hydrology Paper 3, 1964. p. 27. [3] Burdine NT. Relative permeability calculations from pore-size distribution data. Petrol Trans Am Inst Min Eng 1953;198:71–7. [4] Hunt AG. Upscaling in subsurface transport using cluster statistics of percolation theory. Transp Porous Media 1998; 30:177. [5] Hunt AG. Applications of percolation theory to porous media with distributed local conductances. Adv Water Res 2001;24:279–307.
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[6] Hunt AG, Gee GW. Application of critical path analysis to fractal porous media: comparison with examples from the Hanford site. Adv Water Res 2001;25:129–46. [7] Kravchenko A, Zhang R. Estimating soil hydraulic conductivity from soil particle-size distribution. In: van Genuchten MTh, Leij FJ, Wu L, editors. Characterization and measurement of the hydraulic properties of unsaturated porous media. University of California, Riverside, Riverside, CA, 1999. p. 959–66. [8] Miller EE, Miller RW. Physical theory for capillary flow phenomena. J Appl Phys 1956;27:324–32. [9] Moldrup P, Oleson T, Komatsu T, Schjoning P, Rolston DE. Tortuosity, diffusivity, and permeability in the soil liquid and gaseous phases. Soil Sci Soc Am J 2001:613–23. [10] Mualem Y. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour Res 1976;12:513–22. [11] Nielsen DR, Biggar JW, Erh KT. Spatial variability of fieldmeasured soil–water properties. Hilgardia 1973;42:215–59. [12] Perrier E, Mullon C, Rieu M. Computer construction of fractal soil structures: simulation of their hydraulic and shrinkage properties. Water Resour Res 1995;31:2927–43. [13] Pollak M. A percolation treatment of dc hopping conduction. J Non-Cryst Solids 1972;11:1–24. [14] Rieu M, Sposito G. Fractal fragmentation, soil porosity, and soil water properties. I. Theory. Soil Sci Soc Am J 1991;55:1231. [15] Rockhold ML, Fayer MJ, Gee GW. Characterization of unsaturated hydraulic conductivity at the Hanford site. PNL 6488 Pacific Northwest National Laboratories, Richland, WA 99352, 1988. [16] Snyder V. Statistical hydraulic conductivity models and scaling of capillary phenomena in porous media. Soil Sci Soc Am J 1996;60:771–4. [17] Stauffer D, Aharony A. Introduction to percolation theory. 2nd ed. London: Taylor & Francis; 1994. [18] Tyler SW, Wheatcraft SW. Fractal processes in soil water retention. Water Resour Res 1990;26:1045–54.
Short communication
An explicit derivation of an exponential dependence of the hydraulic conductivity on relative saturation A.G. Hunt Cooperative Institute for Research in the Environmental Sciences, University of Colorado, Boulder, CO 80309, USA Received 2 November 2002; received in revised form 5 July 2003; accepted 4 November 2003
Abstract It is shown how to derive explicitly an exponential dependence of the unsaturated hydraulic conductivity, K, on relative saturation, S, and under what conditions such a relationship holds. An earlier derivation of the same result was so restrictive, that it appeared that realistic realizations were not possible, but the present derivation demonstrates the contrary. From existing results, the implication is that the steady-state unsaturated hydraulic conductivity will be log-normally distributed. The log-normal distribution, as well as an exponential dependence on moisture content, have been reported from the same experiment on the same field. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Continuum percolation; Critical path analysis; Fractal porous media; Constitutive relations; Exponential functions
1. Introduction Recently I used percolation theory [5], to derive an expression for the unsaturated hydraulic conductivity, K, of a ‘‘random fractal’’ porous media (such as described by Tyler and Wheatcraft [18], or Rieu and Sposito [14] in terms of the saturated hydraulic conductivity, KS and the relative saturation, S, K ¼ f ðKS ; SÞ. The most important difference between the Hunt [5], treatment of the porous medium itself and the earlier treatments, was that I allowed a continuous range of pore sizes, rather than a discrete distribution. The most important difference between the Hunt [5], derivation of the hydraulic conductivity and earlier results is in the application of critical path analysis [1,13], from percolation theory. For cases where explicit expressions for a fractal dimensionality, Dr 6¼ 3, of the pore space could be inferred from the particle-size distribution, the derived power-law relationships for K ¼ f ðKS ; SÞ agreed reasonably well with experiment [15], and with simulation [12], and were also quite similar in form to the result of [7] (who used a result of Burdine [3], in their derivation). As is known [18], the fractal formulation leads to a power-law water retention curve, which is in accord with the Brooks and Corey phenomenology [2].
E-mail address: [email protected] (A.G. Hunt).
0309-1708/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2003.11.005
In contrast, in the limit, Dr ! 3, an exponential relationship for K=KS ¼ f ðSÞ was derived [5]. An exponential dependence of K on S leads immediately to a log-normal distribution of the hydraulic conductivity [4,5], both results known from field studies [11]. It was noted, however [5], that a direct evaluation of the 3D expression for the porosity, / ¼ 1 ½r0 =rm r in the limit Dr ! 3 yielded zero pore space, and strictly speaking, zero KS . In this expression r0 and rm denote the smallest and largest pore radii, for which the soil geometry may be represented as fractal. While the limitation did hold in the previous derivation, I wish to show now that it is easy to overcome this difficulty. If the general notion that a fractal dimensionality equal to three implies that the partial porosity of each pore class between r0 and rm is equal, then it is not difficult to formulate a general expression for the pore volume probability density, which in turn yields the exponential relationship for K ¼ f ðKS ; SÞ in addition to a finite value of the saturated hydraulic conductivity, KS . This is the main point of the present note. It is necessary, however, to point out, that the water retention curve in the limit Dr ! 3 is no longer functionally compatible with the Brooks and Corey [2], phenomenology (it also turns out to be exponential). Two applications of critical path analysis are required to obtain the results for the log-normal distribution of the hydraulic conductivity. In the first application,
198
A.G. Hunt / Advances in Water Resources 27 (2004) 197–201
critical path analysis chooses a particular conductance out of a distribution of pore-scale conductance values with a uniform relative saturation. These pore scale conductances are given in terms of Poiseuille’s law, and are proportional to the fourth power of the pore radius, and inversely proportional to its length. In the case of uniform aspect ratios considered here, the net result is a local conductance proportional to the third power of the pore radius. Under the conditions assumed, and for Dr ! 3, the dependence of K on S turns out to be exponential. As pointed out in [5], a second application of critical path analysis is then required at larger scales, because it is necessary that the input (small scale) hydraulic resistances are exponential functions of a (random) moisture content. The relative saturation can equally be considered to be the quantity with the relevant spatial variability. These details have already been dealt with in [5]; what is missing was to address the problem of the last paragraph, namely that KS appeared to be identically zero, in the limit, Dr ! 3.
is proportional to an integral over an integrand, which is the product of Eq. (1) and the factor r3 , using the appropriate normalization constant, Z rm 3D /¼ r3 r1D dr ð3Þ rm3D r0 Eq. (3) is easily evaluated [5], and gives / ¼ 1 3D ðr0 =rm Þ , as is already known from the results of Rieu and Sposito [14]. There is no difficulty with extending the continuum pdf for pore sizes to the case Dr ¼ 3, however the normalization constant is different from its value when Dr 6¼ 3. To normalize Eq. (1) for Dr ¼ 3, set, Z rm Z rm Z rm 3 2Dr W ðrÞr dr ¼ A r dr ¼ A r1 dr ¼ / ð4Þ r0
r0
where the next to last equality follows from choosing Dr ¼ 3. Then it is easy to find that A¼ ln
2. Soil structure background The soil structure is not discussed in detail here, but the reader is referred to the work of Rieu and Sposito [14], denoted subsequently by RS. I use a power-law distribution of pore sizes over a finite range of pore sizes, from a minimum of r0 to a maximum of rm . This is supposed to be compatible with a random fractal structure, in which, in the mean, the geometrical relationships between successive pore classes are satisfied, allowing an identical representation of the statistics of pore classes to that satisfied by a deterministic fractal. It is shown in [5], that the power-law probability density function, pdf, for pore radii, W ðrÞ / r1D
ð1Þ
(equivalent to the discrete distribution, W ðrÞ / rDr Þ leads to the expression 3D r0 ¼/ ð2Þ 1 rm for the porosity, /. This relationship is identical to that of RS, and is thus generally compatible with the Brooks and Corey relationship for water retention characteristics, as shown in Hunt and Gee [6], and known from Tyler and Wheatcraft [18]. Eq. (2) may be shown consistent with Eq. (1) through the following. Assume a volume r3 associated with each pore of radius r (numerical factors relevant to specific geometries are suppressed here, as they will not be relevant for ratios of the unsaturated to the saturated conductivity; if the geometries are the same at all length scales, then these factors will ‘‘cancel’’). Then the total pore space volume
r0
/ h i rm r0
ð5Þ
3. Hydraulic conductivity and water retention curves The system is envisioned to consist of a network, or perhaps more accurately, of a net of connections between pores. The net consists of the connections, denoted as hydraulic conductances, with values of the conductance consistent with Poiseuille’s Law for flow through the narrowest constrictions, or throats. These conductances can also, in principle, be determined from numerical solution of the Navier–Stokes equation at the pore scale. The hydraulic conductivity is found through the application of critical path analysis. Critical path analysis yields directly the smallest value of the conductance on the most highly conducting path of infinite extent. Critical path analysis may be applied in the form of continuum percolation. As discussed in [5], it is generally possible to find a (critical) volume fraction, ac , consisting of individual pores, assembled at random, which percolates through the system. If this volume is chosen by including the largest pores down to some minimum pore radius, rc , then rc becomes the controlling radius for transport. KS must now be found for Dr ¼ 3. The condition reads simply, that Z rm / dr ¼ ac ð6Þ lnðrm =r0 Þ rc r where the integral is a volume fraction on account of the normalization above. The solution of Eq. (6) for the smallest, or critical, pore radius, rc , is
A.G. Hunt / Advances in Water Resources 27 (2004) 197–201
rc ¼ rm
r0 rm
ac =/ ð7Þ
Note that if ac > /, rc < r0 , and the hydraulic conductivity must vanish, there being no pores in the medium with r < r0 . Otherwise, the saturated hydraulic conductivity, KS , is proportional to rc3 . The process for S < 1 is to find the largest pore that remains filled, and apply the continuum percolation approach over pores restricted to be smaller than that limit. First find r> , the largest pore still filled with water. During drainage, it is assumed that, through film flow, the largest pores can be emptied first. Then r> can be found from the relative saturation, S, as follows: 1 Z r> rm dr ¼S ð8Þ ln r r0 r0 The factor / does not appear because the maximum value of S is not /, but 1. The value r> as a function of pressure can be obtained theoretically by the same general procedure as is used to develop the waterretention curve, below. Solution of Eq. (8) for r> yields, S rm ð9Þ r> ¼ r0 r0 In this case, where pores larger than r> can not contribute to the hydraulic conductivity, because they are empty, the critical volume must be constructed out of pores with smaller radii than r> , forcing the lower limit, rc , of the integration to smaller values as well: Z r> / dr ¼ ac ð10Þ lnðrm =r0 Þ rc r Solution of Eq. (9) for rc yields, ac =/ ac =/ 1S 1S r0 r0 r0 r0 rc ¼ r> ¼ rm ¼ r cS rm rm rm rm ð11Þ where rcS is the corresponding expression in the saturated case. The last expression can be put into an exponential form by using the identity x ¼ exp½lnðxÞ, and r0 rc ¼ rcS exp ð1 SÞ ln ð12Þ rm Since, in [5], it was argued that with uniform pore aspect ratio the critical hydraulic conductance, gc , is proportional to the cube of the critical radius, one can find, rm gc ¼ gcS exp ð1 SÞ3 ln ð13Þ r0 Eq. (13), with gcS the saturated value of the critical conductance, is nearly identical to that derived in [5], with the slight difference that the denominator of the exponential in Eq. (13) does not contain the factor
199
1 ac present in [5]. Since typical values of ac are about 0.15, this difference is not large, but the present derivation has no inconsistencies. In order to convert Eq. (13) into an equation for K, additional length scales are required, as discussed in [5]. Without these length scales, use of Eq. (13) to derive K as a function of KS is only approximate, rm Kc ¼ KS exp 3ð1 SÞ ln ð14Þ r0 Note that for practical applications to a porous medium, in which the relative saturation can vary over scales large relative to a pore scale, but small relative to the scale of measurement to the field scale, an additional application of critical path analysis, as described in Section 3.3.5 of Hunt [5] is required. The important modification here, compared with [5], however, is that the saturated conductivity is not zero, but is proportional to rc3 given in Eq. (7). The concern regarding the water retention curve must now be addressed. First, the problem is posed for arbitrary fractal dimensionality, in order to make a connection with [5]. Eq. (3), in the case Dr 6¼ 3, reads, Z rm 3 Dr dr 3 Dr ¼/ ð15Þ rr 3D r r rm r0 If it is assumed that the matric head, h, required to remove the water from a pore of radius r, is h ¼ A=r, then the air entry pressure is, hA ¼ A=rm , and, Z A=h 3 Dr r2D dr ð16Þ S¼ rm3Dr r0 When S ¼ 1, the upper limit is thus replaced by A=hA . Solution of Eq. (16) for the saturation as a function of pressure leads to, " D3 # D3 1 h h S ¼1 ð17Þ 1
/ hA hA where to obtain the approximate equality, identical to the Brooks and Corey [2], relationship, would require the porosity to equal 1. Note that substitution of Eq. (17) into [5] 3=3D 1S K ¼ KS 1 / ð18Þ 1 ac for K=KS as a function of relative saturation yields, "
3D !#3=ð3DÞ 1 hhA K ¼ KS 1 ð19Þ 1 ac Eq. (19) has some relationship with the Mualem [10], result. The solution for the matric potential as a function of saturation leads to,
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A.G. Hunt / Advances in Water Resources 27 (2004) 197–201
h ¼ hA
1 1 /ð1 SÞ
1 3D
4. Discussion ð20Þ
In the case D ¼ 3, however, the problem must be set up slightly differently. Now, Z A=h 1 dr ð21Þ S¼ lnðrm =r0 Þ r r0 From Eq. (21) it is possible to derive, ln hhA S ¼1 ln rrm0 Alternatively, rm h ¼ hA exp ð1 SÞ ln r0
ð22Þ
ð23Þ
As advised, this relationship is no longer a power-law between S and h. Nevertheless, representation of K in terms of h is still a power-law, in particular, K ¼ KS
hA h
3 ð24Þ
Note that the power n ¼ 3 here arises from the choice of a uniform aspect ratio at all pore sizes. Consider the power on ðh=hA Þ predicted from Eq. (19) for D 6¼ 3, d ln n¼
K KS
1 ¼3 3D d ln hhA 1 ac hhA
ð25Þ
It is easy to see that Eq. (25) yields a value of n which is slightly larger than 3 at large h (near hA ), but which increases slowly with diminishing h. However, it is also apparent that Eq. (19) predicts that K=KS should vanish 1=ð3DÞ at a very small, but finite, value of h ¼ hA ðac Þ . Miller–Miller similitude, Miller and Miller [8], implies n ¼ 2, although Snyder [16], as well as Hunt [5], note statistical complications that can invalidate such simple assumptions (and also the ideal result of n ¼ 3 here). Experiment typically yields values of n of from 2.6 to 2.8 (Sposito, personal communication). But an experimental fit to a Brooks and Corey formulation would constrain K=KS to be finite at all finite h, altering the fitted n to smaller values. It is not possible to state whether those values will or will not ultimately be in accord with the present derivation. What it is possible to state, however, is that if an exponential dependence of K=KS on S arises from fractal soils with D ¼ 3, then such soils will also have a power law relationship of K=KS on h, and will generate, according to the present treatment, n ¼ 3.
In this note, an explicit relationship between a powerlaw water retention curve developed for a fractal porous medium and the corresponding relationship between the unsaturated and saturated values of the hydraulic conductivity is given. Clearly it becomes possible to extract a fractal dimensionality from the former, apply to the latter, and predict the constitutive relationship, as introduced in [5]. For critical path analysis to have predictive value, the critical volume fraction, ac , for percolation must be known. In important recent work [9], consider a threshold water content for solute diffusion, which they call ht . In this reference a predictive phenomenological relationship for ht was given in terms of the relative surface area to volume ratio of the soil, SAvol . This method to calculate ht was shown in Hunt and Gee [6], to generate the same value for ac as is obtained from the minimum moisture content reached in unsteady drainage experiments. In fact, it turns out that fairly typical values of ht for loamy soils are about 0.15, a commonly cited value for the critical volume fraction in continuum percolation [17]. ac can, however, take on much larger values in certain geometries unfavorable for percolation. The most obvious such case is when pores are highly elongated, with similar orientations, but with spatial positions much more highly correlated in the direction perpendicular to elongation than in the direction parallel to the elongation. It may be interesting to note that for very large values of SAvol from clay soils, ca. 800 m2 / g, the Moldrup relationship ht ¼ 0:039ðSAvol Þ0:52 , predicts ht 1.
5. Conclusions In a previous paper it was shown possible to derive from Poiseuille’s law for saturated flow, and a fractal geometry of a porous medium an exponential dependence of K=KS on the relative saturation, S. The limitation that the relationship was strictly valid only in the limit D ! 3, was problematic, since according to the results of Hunt [5], the saturated hydraulic conductivity evaluated in that limit was identically zero. Now, however, an explicit accounting for the case D ¼ 3 has been made, and it has been shown that essentially the same exponential relationship holds, but the saturated hydraulic conductivity is non-zero. This allows explicit derivation of a log-normal distribution of the hydraulic conductivity as well [5]. The conditions derived, that the unsaturated hydraulic conductivity is an exponential function of the relative saturation, were reported to exist in a classic paper of Nielsen et al. [11], in which the steady-state unsaturated hydraulic conductivity was also reported to have been log-normally distributed. A po-
A.G. Hunt / Advances in Water Resources 27 (2004) 197–201
tential negative aspect, however, of the derived exponential relationship for K on S, is that the corresponding Brooks and Corey type relationship for the water retention curve also becomes exponential for the case Dr ¼ 3, while the dependence of K=KS on matric potential, h, remains a power-law.
Acknowledgements This study was supported by the US Department of Energy Climate Change Prediction Program, which is part of the DOE Biological and Environmental Research Program. The Pacific Northwest National Laboratory is operated for the DOE by Battelle Memorial Institute under contract DE-AC06-76RLO 1830. References [1] Ambegaokar VN, Halperin BI, Langer JS. Hopping conductivity in disordered systems. Phys Rev B 1971;4:2612. [2] Brooks RH, Corey AT. Hydraulic properties of porous media. Colorado State University Hydrology Paper 3, 1964. p. 27. [3] Burdine NT. Relative permeability calculations from pore-size distribution data. Petrol Trans Am Inst Min Eng 1953;198:71–7. [4] Hunt AG. Upscaling in subsurface transport using cluster statistics of percolation theory. Transp Porous Media 1998; 30:177. [5] Hunt AG. Applications of percolation theory to porous media with distributed local conductances. Adv Water Res 2001;24:279–307.
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