# Model help:TopoFlow-Soil Properties Page

## TopoFlow-Soil Properties Page

This page contains information on several different methods for modeling the properties of soil as a porous media in hydrologic models. It contains relationships and tables of values that are used for modeling the process of infiltration. A discussion of several soil characteristic relations is followed by two tables of standard values from the literature. This information is helpful for setting parameters in hydrologic models such as TopoFlow

## Main equations

Effective saturation or scaled water content

 $\displaystyle{ \Theta_{e} = \left ( \theta - \theta_{r} \right ) / \left ( \theta_{s} - \theta_{r} \right ) }$ (1)

Transitional Brooks-Corey (Smith, 1990)

(1) Hydraulic conductivity

 $\displaystyle{ K \left (\Theta_{e}\right ) = K_{s} * \Theta_{e}^ \left ({\frac{\eta}{\lambda}}\right ) }$ (2)

 $\displaystyle{ \Psi \left(\Theta_{e} \right ) = \Psi_{B} [\Theta_{e}^ \left ({\frac{-c}{\lambda}}\right ) - 1]^ \left ({\frac{1}{c}}\right ) - \Psi_{A} }$ (3)

(3) Hydraulic conductivity

 $\displaystyle{ K \left (\Psi \right ) = K_{s} * \{ 1 + [\left (\Psi + \Psi_{A} \right ) / \Psi_{B}]^c \} ^ \left ({\frac{-\eta}{c}} \right ) }$ (4)

Standard Brooks-Corey (1964)

(1) Hydraulic conductivity

 $\displaystyle{ K \left (\Theta_{e}\right ) = K_{s} * \Theta_{e}^ \left ({\frac{\eta}{\lambda}}\right ) }$ (5)

 $\displaystyle{ \Psi \left(\Theta_{e} \right ) = \Psi_{B} * \Theta_{e}^ \left ({\frac{-1}{\lambda}}\right ) }$ (6)

(3) Hydraulic conductivity

 $\displaystyle{ K \left (\Psi \right ) = K_{s} * \{ \Psi / \Psi_{B}\} ^ \left (-\eta \right ) }$ (7)

van Genuchten (1980)

(1) Hydraulic conductivity

 $\displaystyle{ K \left (\Theta_{e}\right ) = K_{s} * \Theta_{e}^ \left ({\frac{1}{2}}\right ) [1 - \left ( 1 - \Theta_{e}^ \left ({\frac{1}{m}}\right )\right )^m]^2 }$ (8)

 $\displaystyle{ \Psi \left(\Theta_{e} \right ) = \left ( 1/ \alpha_{g} \right ) [\Theta_{e}^ \left ({\frac{-1}{m}}\right ) - 1]^ \left ({\frac{1}{n}}\right ) }$ (9)

## Notes

Note on equations

1. The equations above are empirical and allow the hydraulic conductivity, K, and pressure head, ψ, to be computed as functions of the soil water content, θ. Such equations are called soil characteristic relations (or equations or functions). See Smith (2002, p. 13-23). Changing the parameters in these equations allows different soil types to be modeled. Typical values have been tabulated for many standard soil types.

2. There is a missing minus sign in front of λ in Smith(2002, p. 19, eqn. 2.14 and eqn. 2.16), for the standard Brooks-Corey relation. It has been fixed above.

3. The soil characteristic relations used by TopoFlow are those of the transitional Brooks-Corey (TBC) method. This method combines the advantages of the well-known Brooks-Corey (1964) and van Genuchten (1980) methods, as explained by Smith (2002). A key difference between the 3 methods is the value they give for ψ when the soil becomes saturated, that is, when θ = θs or Θ = 1. We usually want ψ < 0 (capillary-type suction) above the water table where θ < θs and ψ = 0 at the water table where θ = θs. By inserting Θ = 1 into the equations above, we find

• for TBC: ψ = -ψA and K = Ks (but often, ψA = 0)
• for BC: ψ = ψB and K = Ks
• for vG: ψ = 0 and K = Ks

For BC, the ψ curve can be made to rise vertically from ψB to 0 at Θ = 1. By design, the vG and TBC methods do not have this discontinuity at saturation.

4. Although K is computed quite differently for the transitional Brooks-Corey and van Genuchten methods, the equations for ψ are the same if we take: αg = 1/ψB, n = c and m = λ/c.

Note on parameters

1. Note that the permanent wilting point is associated with a pressure head of -15,000 cm. Water content associated with values less than this is considered to be unavailable to plants. The field capacity is the soil water content associated with a pressure head of about -340 cm and is an estimate of the water content that can be held against the force of gravity.

2. Soils in nature do not have water contents lower than that corresponding to hygroscopic water (Dingman, 2002, p. 236-238). At this extreme dryness water is absorbed directly from the air. The corresponding pressure head (tension head) is -31,000 cm.

3. Note that we need to specify θr and θs in order to compute a value of θ for a given value of ψ. Due to the way in which θr appears in the equations, we cannot solve for a value of θr given a value of ψ. However, in cases where θr << θ < θs, we can approximate Θ as (θ / θs).

4. The pore disconnectedness index, c, is a measure of the ratio of the length of the path followed by water in the soil to a straight-line path (Eagleson 1978; Bras 1990). It is unitless and can be approximated as:

c = 2 * b + 3

where b is again the unitless pore-size distribution index. This is consistent with the expression:

η = 2 + 3*λ

used by Smith (2002), because

c = η / λ = η * b

5. The capillary length scale, G, is a parameter that appears in both the Green-Ampt and Smith-Parlange 3-parameter methods of modeling infiltration (Smith, 2002, p. 69). For an initially dry soil (so we can neglect Ki), G can be computed by integrating K(ψ)/Ks over all values (-Infinity to 0) of the pressure head, ψ. As indicated by Smith (2002, p. 71), G can be computed in closed form for the Brooks-Corey (BC) relations to get:

G = -ψB * [η / (η - 1)]

For the transitional Brooks-Corey (TBC) relations with ψA = 0, the following appears to be a general expression for G:

G = -ψB * Γ[1 + 1/c] * Γ[(η - 1) / c] / Γ[η / c]

(Found by S. Peckham on 11/1/2010 by first finding expressions for c=1, c=3/2 and c=2 and then working out the pattern. Note also that the TBC expression for G approaches the BC expression in the limit as c becomes large. This is consistent with the comment by Smith (2002, p. 22) that TBC approaches BC in that limit. But for smaller values of c, they can be quite different. For example, if c=2 and λ=1, BC gives G = (-5/4) * ψB while TBC gives G = (-2/3) * ψB.) Recall that η = 2 + 3*λ. Some values of the general Gamma function are: Γ[1/2] = sqr(π), Γ[1]=1, Γ[3/2] = sqrt(π)/2, Γ[2]=1, Γ[n+1] = n * Γ[n], even if n is not an integer.

These expressions for G can be used when comparing results from the Green-Ampt, Smith-Parlange and Richards 1D methods of modeling infiltration. Note that the statement made by Smith(2002, p. 71) about G being in the interval [-ψB, -2 * ψB] only holds for the BC relation and relies on the fact that η > 2 when λ > 0.

6. If there is a depth of water at the surface, ds, then G should be replaced by (G + ds) in the Green-Ampt and Smith-Parlange equations (Smith, 2002, p. 84). Keep in mind, however, that this should be a water depth that is uniformly distributed over a grid cell and not the depth of water in a channel that is contained within the grid cell.

7. Let r0 be the rate at which water enters the top of the soil profile. If (r0 >= Ks), then the limiting value of the soil moisture content, θ, will be the saturated value, θs. However, if (r0 < Ks), then the limiting, maximum value of θ will be less than the saturated value and turns out to be the same for the Brooks-Corey and transitional Brooks-Corey relations. It is computed by setting K = r0 in the relations and solving for θ.

Note the usage of related equations in Topoflow model

1. TopoFlow 1.5 (IDL version) used: θi = θfc and

θr = θs (-ψB / 10000) λ

as defaults. This expression for θr uses a reference pressure head of -10,000 meters as the limit between air-dried and oven-dried soil. See Dingman (2002, p. 237). In addition, if a value of θi was entered that was less than the hygroscopic value, θH, then θH was used instead.

2. In TopoFlow 3.0 (Python version), values of θi and Ki are computed for TBC using the given values of θr and θs and assuming that the initial pressure head in the soil is that typically associated with the field capacity, i.e. ψ = -340 cm. In addition, the soil water content that corresponds to hygroscopic conditions (ψ = -31000 cm) is computed from the TBC parameters and compared to θr. When using the Richards 1D infiltration component, these values are printed in the CMT console window at the beginning of the run and can be compared to the user-entered values as a consistency check.

3. The unsaturated zone extends from the soil surface all the way down to the water table, where θ = θs. The water table is tracked by TopoFlow as a moving boundary, and may actually rise up to the land surface in some places. When this occurs, a subsurface seepage term is included in the water balance equations. Typically, ψA is set to zero so that the pressure head is zero at the water table, but it can also be used to model hysteresis effects, as explained by Smith (2002).

Notes on the Tables

Table 1 (below) is based on Table 6-1 in Dingman (2002, p. 235). It is based on the analysis of 1845 soils by Clapp and Hornberger (1978). Values in parentheses are standard deviations. Note that ψB is the bubbling pressure head, also known as the air entry pressure and then denoted by ψae. The parameter b is unitless and is known as the pore-size distribution index. Its inverse, λ, is also commonly used and is referred to in TopoFlow as the pore-size distribution parameter. For additional information on typical ranges of hydraulic conductivity, see Table 2.2 (p. 29) in Freeze and Cherry (1979). For typical ranges of porosity, see their Table 2.4 (p. 37).

Table 2 (below) is a reproduction of Table 8.1 (p. 136) from Smith (2002). It provides values for, G, known as the capillary length scale. G can be computed by integrating K(ψ)/Ks over all values (-Infinity to 0) of the pressure head, ψ. This table also provides alternate values for the saturated hydraulic conductivity, which when compared to Table 1 shows how much this parameter can vary within a soil texture group.

# Table 1

 Soil Texture Porosity, φ Ksat (cm/s) Ksat (hm/h) ψB b λ=1/b Sand 0.395 (0.056) 1.76 e-2 634 -12.1 (14.3) 4.05 (1.78) 0.247 Loamy sand 0.410 (0.068) 1.56 e-2 562 -9.0 (12.4) 4.38 (1.47) 0.228 Silty sand - - - - - - Sandy loam 0.435 (0.086) 3.47 e-3 125 -21.8 (31.0) 4.90 (1.75) 0.204 Loam 0.451 (0.078) 6.95 e-4 25.0 -47.8 (51.2) 5.39 (1.87) 0.186 Silt - - - - - - Loamy silt - - - - - - Silty loam 0.485 (0.056) 7.20 e-4 25.9 -78.6 (51.2) 5.30 (1.96) 0.189 Sandy clay loam 0.420 (0.059) 6.30 e-4 22.7 -29.9 (37.8) 7.12 (2.43) - Clay loam 0.476 (0.053) 2.45 e-4 8.82 -63.0 (51.0) 8.52 (3.44) 0.117 Silty clay loam 0.477 (0.057) 1.70 e-4 6.12 -35.6 (37.8) 7.75 (2.77) 0.129 Sandy clay 0.426 (0.057) 2.17 e-4 7.82 -15.3 (17.3) 10.4 (4.45) 0.096 Silty clay 0.492 (0.064) 1.03 e-4 3.71 -49.0(62.1) 10.4 (4.45) 0.096 Clay 0.482 (0.050) 1.28 e-4 4.61 -40.5 (39.7) 11.4 (3.70) 0.088

# Table 2

 Soil Texture Typical Ksat (mm/h) G (mm) Dry soil S (mm/h^0.5) Time scale, tc (h) Sand 30.0 82 38 0.08 Loamy sand 15.0 97 29 2.0 Silty sand - - - - Sandy loam 4.4 165 21 11.0 Loam 10.0 385 48 12.0 Silt 2.5 914 37 109 Loamy silt - - - - Silty loam 4.5 724 44 48 Sandy clay loam 13.0 240 43 5.5 Clay loam 2.6 804 35 92 Silty clay loam 0.7 1590 26 680 Sandy clay 1.2 589 21 73 Silty clay 0.4 3570 29 2600 Clay 4.0 2230 73 167

## Examples

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## References

• Beven, K.J. (2000) Rainfall-Runoff Modelling: The Primer, 360 pp., John Wiley and Sons, New York.
• Dingman, S.L (2002) Physical Hydrology, 2nd ed., Prentice Hall, New Jersey. (see Chapter 6)
• Eagleson, P.S. (1978) Climate, soil, and vegetation: 3. A simplified model of soil moisture movement in the liquid phase, Water Resources Research, 18, 722-730.
• Freeze, R.A. and J.A. Cherry ( 1979) Groundwater, 604 pp., Prentice-Hall, Inc., Englewood Cliffs, New Jersey.

Smith, R.E. (2002) Infiltration Theory for Hydrologic Applications, Water Resources Monograph 15, American Geophysical Union, Washington, DC.