Darcy s law describes water flux determined by a water potential gradient: q w = k dψ dx

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1 6 Hydraulic Properties The soil hydraulic properties are of utost iportance for a large variety of soil processes and in particular for soil water flow. The two ost coon hydraulic properties are the soil water characteristic function SWC also called soil water retention curve and the hydraulic conductivity function kψ or kθ. The first describes the relationship between the soil water potential ψ and the corresponding value of water content θ, while the second describes the relationship between the hydraulic conductivity k and either the water potential kψ or the water content kθ. 6. Darcy s Law Darcy s law describes water flux deterined by a water potential gradient: q w = k dψ dx 6. where q w is the flux density [ 2 s ], dψ/dx is the water potential gradient and k is the hydraulic conductivity [ s/ ], ψ is the water potential [J/] and x is the space diension []. The soil water potential is defined as a energy per unit ass [J/] or energy per unit volue [J/ ]. Since a Joule is equal to a Newton per eter [J = N ], a Joule per unit volue [J/ ] is equal to a Newton per eter squared [N / ]=[N/ 2 ], which is equivalent to a pressure or Pascal[Pa]. For this reason often the water potential is expressed in pressure units, and since the density of water is equal to 000 /, one Joule is equal to kilopascal. It is recoended to use units of [J/] for the water potential, first because they are consistent with the SI units and second because the unit of energy per ass J/ do not change with teperature or pressure, while the units of energy per volue J/ do change with teperature and pressure. Units of energy per ass are expressed in the SI units as: [ J = N units into Darcy s law: 2 s = s 2 s 2 = 2 s 2 ]. Substituting the proper with hydraulic conductivity in units of [ s ]. To obtain units which can be ore easily visualized, hydraulic conductivity is often expressed in [ s ]or[c day ]. To convert units,the water

2 potential is expressed as water head [] and the flux in [ s ]. To convert [ J ]into[ = J N ], ultiply [ J ]by[ g ] where [g] is the gravitational constant, which results into water potential units of eters: [ 2 s 2 s = ]. To convert the flux density into [ s ], divide by the density of water ρ l = 000. Into Darcy s law this conversion is perfored by dividing the left hand side by the density of water and the right hand side by the gravitational constant: 2 = s s ρ l 2 s 2 with units of hydraulic conductivity in [ s ] becoes: 2 = 2 s 2 s 000 s s where the units of flux J w arein[ s ], hydraulic conductivityk in[ s ], water potential ψ in[] and space diension x in[]. Rearranging eq. 6.3 and eq. 6.4 and solving for the hydraulic conductivity shows that to convert units of hydraulic conductivity fro [ s ]in[s ] it is necessary to ultiply by the density of water 000 [ ] and divide by the gravitational constant 9.8. Assuing that the density of water is equal to 000 [ ]at4 s 2 C, the 3 conversion factor is Correction of the conversion factor is necessary to account for the variation of the density of water with teperature. On the other hand to convert fro [ s ]in [ s ] it is necessary to divide by the density of water and ultiply by the gravitational constant, with conversion factor of Following is a table showing saturated hydraulic conductivity classes in the ost coon units. g 6.3 Table 6.. Saturated hydraulic conductivity classes in equivalent units s s c day c hr.02 * * * Usually hydraulic conductivity is reported in one of the units shown in Table 6.. For consistency with the International Syste of Units, hydraulic conductivity is expressed in [ s ]. Conversions are easily perfored by using the conversion factors described above. The variation of the density of water with teperature is ore easily accounted for by using units of [ s ], and by expressing the water potential in units of energy per ass [ J ]. 6.2 Water Potential The soil water potential is characterized by different coponents, depending on different physical and cheical phenoena in soils. The total poteantial is given by the following coponents: ψ = ψ + ψ o + ψ p + ψ h + ψ ω + ψ g

3 where ψ refers to the atric, ψ o to the osotic, ψ h to the hydrostatic, ψ p to the pneuatic, ψ ω to the overburden and ψ g to the gravitational coponent. 6.3 Water Content The soil water content is usually expressed on a gravietric base θ g [/] or on a voluetric base θ v /. To convert fro voluetric to gravietric or viceversa, the following forula is used: ρ w θ v = ρ b θ g 6.6 where ρ w is the density of water, and ρ b is the bulk density [/ ]. 6.4 Soil water characteristic The two coponents of the total soil water potential that are affected by water content are the atric and the osotic coponents. As previously described the soil water characteristic describes the relationship between the soil atric water potential ψ and the water content θ. Capbell 985 describes the soil oisture characteristic curve by a power-law relation: θ = θ ψ /b s ψe if ψ <ψ e 6.7 θ s if ψ ψ e where ψ J is the water potential, ψ e J is the air entry potential, θ 3 is the voluetric water content, θ s 3 is the saturated voluetric water content, and b is a shape paraeter related to the particle size distribution of the porous edia. This equation is discontinuous at the air entry potential, and it is analytically integrable. An alternative equation coonly used to describe the SWR is the van Genuchten 980 equation, which has the following for: S e ψ = θ θ r = θ s θ r [ + αψ n ] 6.8 which solving for θ can be written as: θ = θ r +θ s θ r [ + αψ n ] 6.9 where S e is the degree of saturation 0,, α, n,, θ s θ r are fitting paraeters. Fig. 6. shows a SWR for Salku soil, and the fitted Capbell 985a and van Genuchten 980c equations with independent paraeters. A non-linear least squares fitting algorith was ipleented Marquardt, 963; Press et al., 992 to obtain the data showed in fig. 6.. Different restriction can be iposed on the paraeters n and van Genuchten et al. 99 depending on the shape of the SWR curve. In particular, when only a liited range of water retention values are available usually in the wet range of the curve it ight be necessary to restrict the paraeters n and. More stable results are generally obtained when the restriction = /n is ipleented for incoplete data sets van Genuchten et al.,

4 Water Content [3 / 3] Water Content [3 / 3] a Air Entry Potential Experiental Fitted Water Potential log0 [J / ] c Experiental Fitted Water Potential log0 [J / ] Capbell Conductivity log0 [ s / 3] van Genuchten/Muale Conductivity log0 [ s / 3] b d -3 Water Potential log0 [J / ] Water Potential log0 [J / ] Fig. 6.. Fitted Capbell s a and van Genuchten s c SWR curves for Salku soil. Estiated unsaturated hydraulic conductivity using eq. 6.9b and eq. 6.2d. The saple has saturated hydraulic conductivity, k s =0 3 [ s 3 ]. 6.5 Hydraulic Condictivity function Derivation of hydraulic conductivity fro the soil water retention is obtained for the Capbell 985 forulation which describes the unsaturated hydraulic conductivity curve based on the cobined probability of finding continuous pores within a cross section of the porous edia. The calculation of the cobined probability requires to integrate twicefor each of the two sections over the pore space to obtain the hydraulic conductivity Capbell, 974. Since the SWR is analytically integrable, the unsaturated hydraulic conductivity function is given by: K = K ψe 2+3/b s ψ if ψ <ψ e 6.0 K s ifψ ψ e where K s 3 is the unsaturated conductivity and K s s 3 is the saturated conductivity. Figure 6.b shows the unsaturated hydraulic conductivity function obtained by using 27

5 eq Because of the discontinuous nature of the Capbell 985 equation, the unsaturated hydraulic conductivity function is also discontinuous at the air entry potential point. The derivation of the hydraulic conductivity function for the van Genuchten 980 equation is given by the Maule 976 odel which is written in the for: where KS e =K s S l e fs e = [ ] 2 fse 6. f ψθ 6.2 where S e is the degree of saturation, k s is the saturated hydraulic conductivity and l is a pore space connectivity paraeter assued to be equal to 0.5 as average for any soils. Note that the integration is first noralized over the spore space and then it is squared, which is equivalent to integrate twice as in Capbell 985. Nevertheless this integration when the paraeters n and are independent requires the use of special functions such as the Incoplete Beta function Press et al., 992. Equation 6. is then rewritten as: KS e =K s S l e[i ζ p, q] where p = +/n, q = /n, assuing independent n and paraeters. The incoplete Beta function is ore difficult to evaluate and in soe cases convergence is not assured. For this reason for scattered and incoplete SWR data sets, the restriction of = /n allows integration of equation 6. without use of the Incoplete Beta function, which results: Kh = K s αψ n [ + αψ n ] 2 [ + αψ n ] l 6.4 Note that the integration was always perfored in ters of the water potential. Figure 6.d shows the unsaturated hydraulic conductivity function obtained by following Maule 976 and ipleentation of the Incoplete Beta function Press et al., Exercises. Using the soil water characteristics listed in the WorkSheet Retention Curves in the files Ex4a and Ex4b, fit both the Capbell 985 and the van Genuchten 980 equations. Produce a table showing the fitting paraeters for the two equations, for each soil saple. Discuss the results based on the different physical properties of the tested saples. 2. Using the algoriths in Ex4a and Ex4b, calculate the unsaturated hydraulic conductivity for the soil saple Salku using both the Capbell 985 and the van Genuchten 980 equations. Plot the results and discuss the differences. 28