136 Steady-State Water Flow and Hydraulic Conductivity
for the various textural classes. There is a general trend towards lower conductivity with smaller mean particle diameter, but the variation within each texture class is large, so one must be careful in using average data like those in Table 6.1.
Correlation of saturated hydraulic conductivity with air entry potentials given in Table 6.1 gives
Ks= 0.0013|ψe|–3 (6.32)
which is similar to eqn (6.31), but with a power of 3 instead of 2 on the air-entry potential. Saturated hydraulic conductivity calculated using eqn (6.32) with represen- tative values for silt and clay contents are also shown in Table 6.1. Equation (6.32) was derived by integrating from the smallest to the largest pores, and considers all pores in the soil, even though the small pores have almost nothing to do with saturated hydraulic conductivity. Rawlset al.(1992) suggest a different approach that focuses on the largest pores. They suggest computing saturated hydraulic conductivity from the equation
Ks=Bφe4 (6.33)
where φe is the porosity between saturation and –33 J kg–1 water potential (the largest soil pores). If we assume a power-law water retention curve, this equation becomes
Ks= 0.07
θs
"
1 – ψe
–33 1/b#4
(6.34)
whereBhas been replaced with 0.07, the value that best fits the data in Table 6.1. The values predicted by eqn (6.34) are shown in Table 6.1. Even though this equation has only one free parameter (a value ofθs= 0.5 was assumed for all textures), it fits the data with only about half the error of eqn (6.32). It also has the advantage that it shows an explicit bulk density dependence (throughθs) ofKs.
Unsaturated Hydraulic Conductivity 137 much more tortuous as the soil desaturates. These factors have been successfully incorp- orated in models that give reasonable descriptions of unsaturated hydraulic conductivity as a function of water content (Campbell, 1974).
Campbell model
Based on the theory presented above, Campbell (1974) proposed a formulation to com- pute the unsaturated hydraulic conductivity. The advantage of this model is that the soil water retention curve proposed by Campbell (1985) is analytically integrable. The unsaturated hydraulic conductivity function is given by
K=
⎧⎪
⎨
⎪⎩ Ks
ψe
ψm
(2+3/b)
ifψm< ψe
Ks ifψm≥ψe
(6.35)
where K [kg s m–3] is the unsaturated conductivity and Ks [kg s m–3] is the saturated conductivity.
Mualem–van Genuchten model
Based on the same theory, Mualem (1976) later proposed a model in which the inte- gral was squared and normalized over the saturated water content using the degree of saturationSe:
Kr(Se) =Sel
⎡
⎢⎢
⎣ r
0
rF(r)dr rmax
0
rF(r)dr
⎤
⎥⎥
⎦
2
(6.36)
Note that the square in eqn (6.36) comes from eqn (6.27). As described above, by employing the Young–Laplace law, Mualem derived the final formula
Kr(Se) =Sle
⎡
⎢⎢
⎣ Se
0
1 ψ(Se) 1
0
1 ψ(Se)
⎤
⎥⎥
⎦
2
(6.37)
whereψis the water potential andl is a parameter related to the tortuosity of the pore space. The Mualem model was later combined with a model for the water retention curve (van Genuchten, 1980), to obtain the so-called Mualem–van Genuchten model.
The van Genuchten (1980) model was described in Chapter 5 and is given by Se(ψ) = θ–θr
θs–θr
= 1
[1 + (αψ)n]m (6.38)
138 Steady-State Water Flow and Hydraulic Conductivity
whereSeis the degree of saturation, lying in the range [0, 1], andα,n,m,θs andθr are fitting parameters.
The major limitation of this model is that when the parametersmandnare independ- ent in the van Genuchten (1980) equation, the integral has no analytical solution, and therefore the integration must be performed numerically using ‘special’ functions. Here we present an algorithm to perform the numerical integration of eqn (6.37). The integra- tion is performed using the incomplete beta function (Presset al., 1992). Equation (6.37) can be written as
K(Se) =KsSel[Iζ(p,q)]2 (6.39) wherep=m+ 1/n,q= 1 – 1/n, assuming independent parametersnandmandIζ is the incomplete beta function. A detailed descriptions of the derivation from eqn (6.37) to eqn (6.39) is provided by van Genuchtenet al.(1991) (pp. 13–17).
The following is a Python code, namedPSP_unsaturatedConductivity.py, where integration is performed to obtain the unsaturated hydraulic conductivity curve.
The parameters of the soil water retention curve (obtained from the nonlinear fitting presented in Chapter 5),α,n,m,θr,θsandKs, are read from a text file calledsoil.txt, where parameters for a silt loam soil are written.
#PSP_unsaturatedConductivity
from PSP_readDataFile import readDataFile import matplotlib.pyplot as plt
import math import numpy NODATA = -9999 def betacf(a, b, x):
maxNrIterations = 50 maxEpsilon = 0.0000003 am = 1
bm = 1 az = 1
bz = 1 - (a+b) * x / (a+1) myEpsilon = 1
m = 1
while (myEpsilon > (maxEpsilon * abs(az))):
if (m > maxNrIterations): return (NODATA)
d = (m * (b - m) * x) / ((a + 2*m -1) * (a + 2*m)) ap = az + d * am
bp = bz + d * bm
d = -((a + m) * (a + b + m) * x) / ((a + 2*m) * (a + 2*m + 1)) app = ap + d * az
bpp = bp + d * bz
Unsaturated Hydraulic Conductivity 139 am = ap / bpp
bm = bp / bpp old_az = az az = app / bpp bz = 1
m += 1
myEpsilon = abs(az - old_az) return (az)
def incompleteBetaFunction(a, b, x):
if ((x < 0.) or (x > 1.)):
return (NODATA)
if ((x == 0.) or (x == 1.)):
bt = 0.
else:
bt = (math.exp(math.lgamma(a + b) - math.lgamma(a) - math.lgamma(b) + a * math.log10(x) + b * math.log10(1. - x)))
if (x < ((a + 1.) / (a + b + 2.))):
return(bt * betacf(a, b, x) / a) else:
return(1. - bt * betacf(b, a, 1. - x) / b) def computeConductivity(currentSe, n, m, Ks):
p = m + 1. / n q = 1. - 1. / n
z = currentSe ** (1. / m)
myBeta = incompleteBetaFunction(p, q, z) if (myBeta == NODATA):
return (NODATA) else:
return(Ks * currentSe * (myBeta ** 2.)) def main():
A, isFileOk = readDataFile("soil.txt",1,’,’, True) if ((not isFileOk) or (len(A[0]) != 6)):
print(’warning: wrong soil file.’) return (False)
VG_alpha = A[0,0]
VG_n = A[0,1]
VG_m = A[0,2]
VG_thetaR = A[0,3]
140 Steady-State Water Flow and Hydraulic Conductivity thetaS = A[0,4]
Ks = A[0,5]
A, isFileOk = readDataFile("SWC.txt",1,’\t’, False) if (not isFileOk):
print(’warning: wrong SWC file in row nr.’, A+1) return (False)
waterPotential = A[:,0]
waterContent = A[:,1]
conductivity = numpy.zeros(len(waterContent)) for i in range(0, len(waterContent)):
currentSe = (waterContent[i] - VG_thetaR) / (thetaS - VG_thetaR) conductivity[i] = computeConductivity(currentSe, VG_n, VG_m, Ks) if (conductivity[i] == NODATA):
print (’Error in compute conductivity’) return (False)
plt.loglog (waterPotential, conductivity, ’ko’)
plt.xlabel(’Water Potential [J kg$^{-1}$]’,fontsize=14,labelpad=2) plt.ylabel(’Hydraulic Conductivity [kg s$^{-1}$ m$^{-3}$]’,
fontsize=14,labelpad=2) plt.show()
main()
10–2 10–3 10–4 10–5 10–6 10–7 10–8 10–9 10–10 10–11 10–12 10–13 10–14 Hydraulic conductivity [kg s–1 m–3]
100 101 102 103 104 105 106 Water potential [J kg–1]
Figure 6.3 Estimated unsaturated hydraulic conductivity curve as function of water potential obtained from numerical integration for a silt loam soil.
Unsaturated Hydraulic Conductivity 141 The numerical integration presented above requires evaluation of special functions such as the beta function. Convergence of these functions is not assured—in particular when the data set is incomplete or scattered. The restriction ofm= 1 – 1/nin the van Genuchten (1980) function allows for analytical integration of eqn (6.38):
K(h) = Ks{1 – (αψ)mn[1 + (αψ)n]–m}2
[1 + (αψ)n]ml (6.40)
A feature of this derivation is that because of the relationship between the hydraulic conductivity and the radius, the estimation of hydraulic conductivity is very sensitive to large pores. Since the curve is continuous around the air-entry potential, when the soil water potential h → 0, according to the Young–Laplace law, the radius becomes infinitely large, which is clearly not realistic. Ippisch et al. (2006) demonstrated that for values ofn < 2 and αha > 1, an air-entry value must be introduced in the van Genuchten (1980) formulation. A detailed theoretical description of these limitations was presented by Ippischet al.(2006), who described and specified the conditions for which the classical van Genuchten–Mualem model leads to wrong predictions of the hydraulic conductivity curve and presented a modified formulation that included an air-entry value. The modified formulation is presented below.
Ippisch–Mualem–van Genuchten model
The modified formulation as proposed by Ippischet al.(2006) has the form
Se=
⎧⎪
⎨
⎪⎩ 1
Sc[1 + (αψ)n]–mifψ≤ψe
1 ifψ > ψe
(6.41)
where Se is the degree of saturation, α, m and n are fitting parameters, and Sc= [1 + (αψe)n]–m is the water saturation at the air-entry potentialψe(the water poten- tial is expressed as a negative number). The resulting hydraulic conductivity using the Mualem model is
K=
⎧⎪
⎨
⎪⎩ KsSel
1 – (1 – (SeSc)1/m)m 1 – (1 –Sc1/m)m
2
ifSe<1
Ks ifSe≥1
(6.42)
wherel is the same parameter as in the original Mualem equation. Ippischet al.(2006) suggested thatψe can be obtained from knowledge of the largest pore size, from the water retention curve or by inverse modelling. The air-entry potential could be obtained by lettingψebe a fitting parameter during the fitting procedure.
142 Steady-State Water Flow and Hydraulic Conductivity
...
6.5 E X E R C I S E S
6.1. Plot ‘typical’ hydraulic conductivity for sand, silt loam and clay soil as a function of water potential using data from Table 6.1 and eqn (6.14). Use a log–log scale and plot both saturated and unsaturated conductivity. Use the graphs to determine the water potential at which water would flow from silt loam into sand in a layered profile with infiltration.
6.2. Write a Python function to compute saturated hydraulic conductivity from bulk density, silt fraction and clay fraction. Use your program to show how tillage affects hydraulic conductivity (assuming that tillage decreases the bulk density).
7
Variation in Soil Properties
In each of the chapters so far, physical properties of soil have been discussed. In sev- eral instances, relationships have been derived between measured properties and other variables. Implicit in all of this is the assumption that a particular measurement or esti- mate of a property contains information about the value of that property or of a related property in the entire system represented by that sample at that specific scale. In Chap- ter 2, we presented the concept of a representative elementary volume, a soil volume representing an average value of a given property, such that the differences among the values of the properties at increasing scales are smaller than a given number. This ap- proach was used to describe soil physical properties, from the pore scale of observations to the soil sample scale of observations. The concept of bulk density is an example of this ‘upscaling’ process, where the different densities of the different phases (solid, liquid and gas) are averaged into a value representing the ‘bulk’ density of the sample. As we pointed out in Chapter 2, this process is not reversible, meaning that when information is combined into an ‘average’ value, the information about that variable at smaller scales is lost. The dimension of the sample of a representative elementary volume is usually that of the cylindrical rings used for soil sampling, with height and diameter of a few centimetres. Clearly, this concept is limited in its applicability to a range of scales, since at increasing spatial scales, variation of the property in question may again be larger than a given number, because of soil layering, topographical or geological features. For in- stance, the average density of a soil volume of 2 m3, representing a soil profile, may be substantially different from the density of the chosen representative elementary volume, sampled at a given depth.
Moreover, experience shows that even measurements on several ‘identical’ soil sam- ples of the same size never produce identical results. A single sample or estimate is therefore unlikely to contain complete information about the value of that property for other possible samples, for instance over a given area. How much information then is or can be contained in a given observation or set of observations? The theory of stochastic processes provides a tool for properly describing systems with uncertainty or variation.
A stochastic description of a property can be used to determine the information content of data relating to that property. In this chapter, appropriate statistical descriptions of soil physical properties will be given and models will be extended to include uncertainty in input variables.
Soil Physics with Python. First Edition. Marco Bittelli, Gaylon S. Campbell and Fausto Tomei.
© Marco Bittelli, Gaylon S. Campbell and Fausto Tomei 2015. Published in 2015 by Oxford University Press.
144 Variation in Soil Properties