Geotechnical and Geological Engineering

, Volume 29, Issue 2, pp 161–169

# Influence of Unsaturated Soil Properties Uncertainty on Moisture Flow Modeling

Original paper

## Abstract

The quality of a numerical modeling solution of moisture flow through unsaturated soil, in part, depends on properly described unsaturated soil properties. The variability of the Soil Water Characteristic Curve, SWCC, is attributed to hysteresis and reproducibility of measurement. Because the unsaturated conductivity function is rarely directly measured, the variability of the unsaturated soil hydraulic conductivity function is attributed to the uncertainty associated with the estimation of this parameter with currently available fitting functions, and hence a range of reasonable variation was considered. One-dimensional modeling of expansive soil under dry initial conditions (suction of 1,500 kPa) was performed; both potential evaporation and infiltration boundary conditions were considered. It was found that small variations in the unsaturated soil hydraulic conductivity function result in significantly different modeling outputs, as expected, while substential variation in SWCC alone (assuming the same unsaturated soil hydraulic conductivity for all SWCCs) produced almost identical soil response in terms of soil suction when the slope of the SWCC is similiar. Thus, proper characterization of the slope of the SWCC is important to proper suction profile determination.

## Keywords

Unsaturated SWCC Hysteresis Uncertainty Modeling

## 1 Introduction

Numerical analysis of moisture flow through the vadose zone is commonly performed to aid engineers and soil scientists in the design of high impact environmental and geohazard projects. Currently, there is a large number of available commercial and public domain software aimed at solving a form of Richard’s equation describing moisture flow through unsaturated media. The pressure head h-based formulation
$${\frac{\partial }{\partial x}}\left( {k_{\left( h \right)x} {\frac{\partial h}{\partial x}}} \right) = \gamma_{w} m_{2w} {\frac{\partial h}{\partial t}}$$
(1)
is a nonlinear, parabolic, advection–diffusion partial differential equation involving: the elevation x, the time t, the total head h (m), the specific weight of water γ w (9.81 kN/m3), the slope of the Soil Water Characteristic Curve, SWCC, $$m_{2w} = \partial \theta /\partial u\, \ge \, 0$$ (where θ represents the volumetric water content), and the unsaturated soil hydraulic conductivity, $$k_{\left( h \right)x} > 0$$ (m/h). Both m 2w and k (h)x are functions of h, sometimes described in terms of matric suction, $$\psi = \max \left( {0, - u} \right)$$ (kPa), where the pore water pressure u is defined by
$$u = \gamma_{w} \left( {h - x} \right)\quad [{\text{kPa}}]$$
(2)
The solution of Richard’s equation requires that both Soil Water Characteristic Curve, SWCC, and unsaturated soil hydraulic conductivity are defined. Preliminary review of the equation suggests that both k(h) and m 2w are parameters consequential to the behavior of the numerical solution since they appear explicitly in the equation. It is postulated that additional SWCC characteristics might have a limited impact on the overall analysis.

## 2 Uncertainty of Unsaturated Soil Functions

The quality of the numerical solution, in part, depends on properly described unsaturated soil properties, such as SWCC and unsaturated hydraulic conductivity, k(h). In industry, laboratory testing is rarely used to determine these functions due to practical challenges associated with test procedures which include test duration, sophisticated test equipment, the procedure know-how and analysis of data, to name few. More commonly, the unsaturated soil properties are estimated with fitting functions such as van Genuchten, Brooks and Corey, and Fredlund and Xing equations for SWCC and van Genuchten and Mualem, Brooks and Corey, and Leong and Rahardjo equations for unsaturated soil coefficient of permeability; see Fredlund and Rahardjo (1993) for detailed descriptions. These functions are estimated either by fitting them through a few measured data points or by statistical analysis based on commonly quantified soil properties such as gradation and Atterberg Limits. As shown by van Genuchten and Nielsen (1985), Vogel et al. (2001) and Vogel and Cislerova (1988), the choice of the analytical model for SWCC estimation can significantly affect the predicted k(h) function. Vogel illustrated that small changes in SWCC near saturation result in large changes in k(h). “The differences are more pronounced in fine textured soils than in coarse textured soils” (Vogel et al. 2001).

### 2.1 SWCC

Soil water characteristic curve correlates soil moisture content to soil suction where the moisture can be expressed in terms of degree of saturation, volumetric or gravimetric water content. The research done by Zapata (1999) indicates that different saturation levels can be obtained for the same soil sample at the same suction. That phenomenon can be explained by the variation in dry density, hysteresis, different test methodologies, variability in test procedures and operator error. Fredlund and Rahardjo (1993) determined that when testing is done on the same soil material with different dry density, the soil behavior will follow a different path on the saturation versus suction curve. It is further explained that the change in dry density of soil results in a change of soil fabric and the soil, at new density, exhibits somewhat different soil behavior. Many of these factors are challenging to quantify. Hysteresis is one exception. It is well established that SWCC will follow one path during desaturation process and another one during saturation (Fredlund and Rahardjo 1993; Zapata 1999; Hillel 1980; Scott et al. 1983). An essentially closed hysteresis loop is formed when a backpressure saturated soil is desaturated and then saturated again, assuming bottom up wetting which allows for escape of air ahead of the wetted front. The wetting curve typically has the same slope as the drying curve and is shifted to the left of drying curve by 0.5–1.5 log of suction (Fredlund and Rahardjo 1993; Zapata 1999). When the sample is wetted by ponding water on top of the soil, the specimen reaches a lower value of saturation due to a high percentage of entrapped air, resulting in an open loop hysteresis with the drying curve obtained as described above. The ratio of θ(sat ponding)(sat Back Pressure) has been estimated by Hillel (1980) to be about 90% and Basile et al. (2003) providing range from 78 to 95%.

The uncertainty band in Fig. 1 reflects the influence of the above-mentioned factors. It shows experimental data for Fountain Hills, Arizona clay, the best fit function, and the 95% confidence band developed by Zapata (1999). The data points, for this example, vary over two orders of magnitude in suction per specific water content. Similar conclusion was presented by Gribb (2000) who reported data scatter over one order of magnitude for sandy soil. Figure 1 shows that the representation of the SWCC as a single unique curve is, in general, an approximation.

### 2.2 Unsaturated Soil Hydraulic Conductivity

The unsaturated hydraulic conductivity describes moisture flow characteristics of soil as a function of pore water pressure (suction). As the soil desaturates, the number of saturated pores decreases, decreasing the number of moisture flow passages, hence the unsaturated hydraulic conductivity decreases as suction increases. Field methods of k(h) estimation include the well-known instantaneous profile method which produces a wetting curve (Watson 1966). The evaporation method of the instantaneous profile method developed by (Wind 1968), with the modifications by Tamari et al. (1993) and Romano and Santini (1999), is typically adopted in a laboratory setting for estimation of the drying curve unsaturated hydraulic conductivity function.

Only limited empirical information is available about unsaturated hydraulic conductivity of clayey soils. The available data usually provides k(h) only up to a suction of 100 kPa. It’s been shown that within this low suction range the k(h) variation due to hysteresis is insignificant (Mualem 1986; Kool and Parker 1987 and Fredlund and Rahardjo 1993). It should be noted that the unsaturated hydraulic conductivity, presented in terms of volumetric water content essentially shows no hysteresis while in terms of suction, exhibits significant hysteresis, which usually does not exceed one log of suction (conclusion drawn based on data by Brooks and Corey 1964).

Normalized unsaturated hydraulic conductivity data from Soil Vision 4.0 were obtained to observe the data scatter for clayey material with similar index properties (SVS 2005a). This data was used in estimating the scatter in the property functions for the soil of the analyses presented in this paper. While similar slope of the unsaturated hydraulic conductivity function was obtained for the normalized plot, considering all 16 soils, the scatter of the data ranged over two orders of magnitude of suction for a given conductivity. As with SWCC estimates, the k(h) function is usually represented as a single unique curve, which, in general, is an approximation typically based on the drying SWCC estimate. Small variations in the SWCC near saturation cause large differences in k(h) near saturation, with significant consequences on numerical results, stability of solution and rate of convergence (Vogel et al. 2001).

## 3 Problem Set Up

The influence of uncertainty of unsaturated soil properties on numerical modeling of moisture flow through soil is examined on a simple 1-D problem with a 10-m deep cleyey profile. Both infiltration and potential evaporation are considered separately over the estimated ranges of SWCC and k(h).

### 3.1 Soil Properties

A clayey soil from Litchfield, Arizona with the following properties: LL = 85, PI = 53, Gs = 2.797, p d  = 1.36 g/cm3, θ s = 51.2% and k sat = 8.71e-6 m/h was used in this analysis. The drying SWCC up to 1,500 kPa suction was obtained experimentally on an undisturbed specimen using a pressure plate apparatus. One filter paper test was performed on an air dried specimen to estimate the SWCC in the high suction range. This experimentally-determined SWCC was used as a best estimate curve, and varabilty consistent with Sect. 2 above was imposed. The laboratory results are given in Fig. 2a. Initially, all fit functions listed in Sect. 2. where considered. It was found that a best-fit curve to the SWCC experimental data produced very similar k(h) estimates with all fitting equations; therefore, when experimental data is available, uncertainty associated with the choice of fitting model is considered to be insignificant and within the range of expected empirical data scatter. In this study, the Fredlund and Xing equation was used to estimate the SWCC
$$\theta = C_{\left( h \right)} {\frac{{\theta_{s} }}{{\left( {\ln \left( {e^{1} + \left( {{\frac{\psi }{a}}} \right)^{n} } \right)} \right)^{m} }}},\quad C_{\left( h \right)} = 1 - {\frac{{\ln \left( {1 + {\frac{\psi }{hr}}} \right)}}{{\ln \left( {1 + {\frac{{10^{6} }}{hr}}} \right)}}}$$
(3)
involving θ s , saturated volumetric water content, and soil coefficients a, n, m and hr. The Leong and Rahardjo equation was used to predict k(h) function
$$k\left( h \right) = {\frac{{k_{\text{sat}} }}{{\left( {\ln \left( {e(1) + \left( {{\frac{\psi }{a}}} \right)^{n} } \right)} \right)^{mp} }}}$$
(4)
where p is a pore size distribution coefficient.
Based on the literature review presented in Sect. 2.1 an open and closed loop hysteretic soil behavior, which roughly includes the expected data scatter, was considered. Table 1 summarizes the SWCC parameters while Fig. 2a provides a visual representation. F1 represents a drying curve for a back pressure saturated soil specimen, F2 describes the wetting curve obtained with soil saturated (submerged) from the bottom up, and F3 represents the expected wetting curve for surface-down wetting of the specimen (soaked) conditions, where a ratio of 0.8 for θ(sat surface-down)/θ(sat Back Pressure) was assumed. All three curves are used in the sensitivity analysis.
Table 1

SWCC parameters

Function #

Description

θ s

(%)

a

n

m

hr

F1

Drying

51.2

140

0.6

0.9

2,000

F2

Wetting from bottom up

51.2

10

0.7

0.9

200

F3

Wetting from top down

41.0

40

0.7

1.1

2,000

Based on the literature review for clayey soils presented in Sect. 2.2, the data scatter of k(h) is expected to range over two orders of magnitude of suction (at a given value of k(h)) for suction values smaller then 100 kPa, including the k(h) variation due to hysteresis. For suctions larger than 100 kPa the uncertainty is more challenging to quantify, therefore, different values of the fitting parameter p were considered to obtain estimates of variability in k(h) at higher values of suction. This resulted in different k(h) slopes for larger suctions because larger p values produce steeper k(h) slopes. The p parameters of 8 and 20 were considered in evaluation of k(h) variability values.

Considering the k(h) variability so obtained (depicted in Fig. 2b and obtained using fitting equation 4), it can be seen that the k(h) estimate using F1 SWCC and p = 12, F1 and p = 8, and F2 and p = 12, produces reasonable variation of the k(h) function for use in the sensitivity analyses. The selected curves for the sensitivity study capture slopes and variability consistent with k(h) curve fits to Soil Vision 4.0 data on clays.

Although both drying and wetting data have been used to develop laboratory k(h) functions, it is unclear which fitting curve models described in Sect. 2.2 were intended for use with the drying path and which ones for use with the wetting path. Most likely all laboratory data was combined to arrive at these fitting methodologies, though the vast majority of literature data represents drying curves. The identified potential range in k(h) discussed above and used in the sensitivity studies presented herein is used to model both infiltration and evaporation. The sensitivity studies were used to determine the potential consequence of improper k(h) selection on the assessment of depth of influence and magnitude of saturation (desaturation), as this variation in k(h) is believed to be sufficient to encompass variations due to hysteresis.

### 3.2 Initial and Boundary Conditions

In situ matric suction measurements at depths larger than what is commonly considered the active zone depth were estimated for Arizona climatic conditions (precipitation of 8′′ and potential evaporation of 92′′), and for clayey material, were determined to vary between 1,000 and 2,500 kPa (Dye 2008). Based on this data, a total head of −152.9 m was applied as the bottom boundary condition for these simulations. The initial profile condition was assumed to be constant with depth and equal to the bottom boundary condition (h = −152.9 m).

Two surface boundary conditions were considered. The first one consisted of constant irrigation of 0.001 m/h applied for 50 h. The second condition was a potential evaporation, PE, flux simulating average flux conditions in June in Arizona where PE is 0.0002 m/h, relative humidity is 18% and temperature is 32°C. The duration of the PE flux was 500 h.

### 3.3 Modelling Software, Mesh Size and Time Step

Numerical analyses were performed with commercial software, SVFlux 5.80, which is a finite element program based on a FlexPDE kernel, a general software for solving systems of PDEs in 1D, 2D or 3D (SVS 2005b). FlexPDE utilizes adaptive unstructured mesh generation and adaptive time stepping based on an implicit Backwards Difference formula (BDF) of low order (order 1 is implicit Euler, order 2 is “Gear’s method”). Infiltration flow is analysed with the h-form of Richards’ equation. More complex atmospheric conditions consisting of both infiltration and potential evaporation are solved with a form of Richards’ equation modified for vapour flow based on the work done by Wilson et al. (1994). In order to obtain a numerical solution that satisfies accuracy, stability and convergence criteria an appropriate mesh size, dx, and time step, dt, must be used to advance the solution. Inappropriate selection of dx and/or dt is characterized by poor water balance and numerical oscillations in actual flux, pore water pressures at the soil surface and with depth. For most of the scenarios presented in this paper, the mesh spacing was varied exponentially with depth, where a mesh spacing of 0.05-m was used at the bottom of the profile and 0.0005-m was applied at the soil surface. The adaptive time step increased from 1e-7-h at the beginning of the analysis to 0.1-h at the end with average time step of 0.1-h. Tighter mesh and time step discretization had to be used for the infiltration problem. Here discretization in space, dx, of 0.00012-m and an average time step, dt, of 4e-3-h had to be used to reduce numerical oscillations.

## 4 Numerical Simulation

Three sensitivity studies were performed. The first one involved soil response in terms of both soil suction and degree of saturation in consideration of SWCC variations alone and for irrigation flux; Table 2, runs 1, 2 and 3. The h-formulation of Richard’s equation incorporated the slope of SWCC, namely dθ/dψ, where difference in the slope of all considered curves was negligible for suctions greater than 100 kPa, see Fig. 2c. Therefore, a sensitivity study for the PE flux condition was considered unnecessary since identical results in terms of soil suction were expected with all SWCCs. The second sensitivity study was aimed at quantifying the effect of k(h) variability during irrigation (infiltration); Table 2, runs 1, 4 and 5. The final study was a sensitivity analysis for the PE surface flux condition and for variability in the k(h) function; Table 2, runs 6, 7 and 8.
Table 2

Summary of modeled scenarios

#

Flux (m/h)

SWCC

k(h)

p

1

Irrigation = 0.001

F1

F1

12

2

Irrigation = 0.001

F2

F1

12

3

Irrigation = 0.001

F3

F1

12

4

Irrigation = 0.001

F1

F1

8

5

Irrigation = 0.001

F2

F2

12

6

PE = 0.0002

F1

F1

8

7

PE = 0.0002

F1

F1

12

8

PE = 0.0002

F2

F2

12

### 4.1 Hysteresis in SWCC

Soil response due to a 50-h irrigation flux of 0.001 m/h, and for SWCCs described by F1, F2 and F3 and k(h) obtained with F1 and p = 12, are presented in Fig. 3. It is observed that pore water pressure variation with depth is almost identical in all considered scenarios, where the depth of influence is about 0.2-m (Fig. 3a). Similarly, the instantaneous (actual) flux profiles show that soil surface saturation occurred at almost the same time (7th hour of analysis), and is followed by almost identical absorption rate for the cases considered (Fig. 3c). It is no surprise then that the total absorbed flux is almost identical in all three scenarios: 0.0248, 0.0268 and 0.0248-m for F1, F2 and F3, respectively, as only one k(h) function was used. A difference between results is observed when the soil response is considered in terms of degree of soil saturation, Fig. 3b, which is consistent with the volumetric water content versus suction relationships presented in Fig. 2a.

It is concluded that soil response in terms of matric suction is independent of SWCC as long as the slopes of SWCC are similar. In the analyzed clayey soil’s, F1′, F2′ and F3′ curves, (the slopes of the SWCC curves are almost identical for suctions larger then about 30 kPa. The potential variation in degree of saturation is obtained from post processing of numerical modeling results in terms of suction.

### 4.2 Uncertainty in k(h)

#### 4.2.1 Infiltration

Soil response due to 50-h irrigation flux of 0.001 m/h, and varying unsaturated soil properties, as described in Table 2, runs 1, 4 and 5, is presented in Fig. 4. It is observed that different pore water pressure variation with depth is obtained for each scenario (Fig. 4a). In general, a steeper k(h) slope produces smaller depth of influence (wetting) and reduced effect of the diffusion component manifested in decreased spreading and sharper wetting front. The change in k(h) also has a consequence on stability and convergence of the solution, with the steeper function resulting in greater challenges in achieving stability and convergence. Cases characterized by advective dominated mass transfer exhibit more numerical instabilities, requiring the implementation of tighter mesh and smaller time step with consequences on run time and required computational resources.

The depth of influence of wetting is about 0.2, 0.42 and 0.03-m for scenarios 1, 4 and 5, respectively. The instantaneous absorbed flux profiles show that soil surface saturation occurred at different times as well, Fig. 4c, with consequences on the total absorbed flux; 0.0248, 0.0313 and 0.0119-m for 1, 4 and 5, respectively.

The presented analyses illustrate that numerical solution of moisture flow though soil is very sensitive to variations in k(h). It is suggested that a range of k(h) functions be modeled to bound the range of soil response.

#### 4.2.2 Evaporation

Soil response to the 500-h PE flux of 0.0002 m/h, and varying unsaturated soil properties, as described in Table 2, runs 6, 7 and 8, is presented in Fig. 5. As for the infiltration analyses, different pore water pressure variation with depth is obtained with each scenario, Fig. 5a. The profiles converge at the soil surface at about 200 000 kPa matric suction. The depth of influence was found to be 0.25-m, 0.92-m and 0.18-m for scenarios 6, 7 and 8, respectively. Figure 5b soil saturation with depth. The characteristics of degree of saturation (S) variation with depth due to k(h) uncertainty are the same as obtained for the irrigation (infiltration) boundary flux condition. As before, the data illustrates the importance of the k(h) estimate on the numerical solution. For engineering purposes, matric suction changes beyond 1,500 kPa bear little significance on volume change, since insignificant soil volume change is expected for suction changes in this very dry range. However, within the context of numerical modeling, appropriate estimation of soil response due to PE is as important as that resulting from infiltrating flux since it determines how much water can enter the profile during next irrigation event.

## 5 Conclusions

The quality of a numerical solution of moisture flow though unsaturated soil described with Richard’s equation, in part, depends on the use of properly described unsaturated soil properties, namely the Soil Water Characteristic Curve, SWCC, and the unsaturated hydraulic conductivity, k(h). The variability of the SWCC has been studied by many authors (see Sect. 2.1). Currently it is well understood that the SWCC can vary over two orders of magnitude of matric suction at a given degree of saturation. On the other hand, k(h) variability is more challenging to quantify, especially for clayey soils for which there is limited available data above 100 kPa suction. Certainly the variability of k(h) can span over one to two orders magnitude due to hysteresis. For the purpose of this paper, a range of SWCC and k(h) was considered for three sensitivity studies: (1) varied SWCC, one k(h), irrigation flux, (2) varied k(h) and varied SWCC for irrigation flux, and (3) varied k(h) and varied SWCC for potential evaporation flux.

The numerical solution in terms of matric suction is independent of SWCC variation alone (k(h) being the same in all cases). The SWCC serves as the relationship for translation of suction into terms of engineering significance such as degree of saturation, however. The conclusion of insensitivity of the matric suction solution to variation in the SWCC has a mathematical bases. In the h-form of Richard’s equation, (1), a derivative of the SWCC is used. For the family of considered SWCC curves, the SWCC derivatives had almost the same slopes for suctions larger then 30 kPa, and hence the numerical solutions were almost independent of the SWCC used in the analysis. Since the numerical solution for suction was not sensitive to variations in the SWCC (because the slopes of the selected SWCC curves were similar), the significance of SWCC variability on variation on degree of saturation or water content can be quantified with post-processing.

The numerical solution is very sensitive to k(h) variability, as expected. Additionally, there are consequences on numerical stability and convergence that are related to the assumed k(h) function. In general, a steeply sloped k(h) function produces smaller depth of influence and reduced effect of the diffusion component of flow, manifested in decreased spreading and sharper wetting front. For unsaturated flow conditions characterized by advection-dominated mass transfer, there are greater issues related to numerical instabilities which often require the implementation of tighter mesh and smaller time step with associated negative consequences on run time and required computational resources. In order to determine the range of possible soil responses, it is necessary to identify a range of potential k(h) variability followed by numerical analysis for the upper and lower band of the k(h) functions.

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## Copyright information

© Springer Science+Business Media B.V. 2009

## Authors and Affiliations

1. 1.Tao EngineeringMesaUSA
2. 2.School of Sustainability and the Built EnvironmentArizona State UniversityTempeUSA
3. 3.Deptartment of Mathematics and StatisticsArizona State UniversityTempeUSA