Published in Soil Sci. Soc. Am. J. 67:1327-1333 (2003).
© 2003 Soil Science Society of America
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DIVISION S-1—SOIL PHYSICS
Estimating Temperature Effects on Water Flow in Variably Saturated Soils using Activation Energy
Fucang Zhanga,b,
Renduo Zhang*,a,b and
Shaozhong Kanga,c
a Key Lab. of Agricultural Soil and Water Engineering in Arid and Semiarid Areas, Northwest Science and Technology Univ. of Agriculture and Forestry, Yangling, Shaanxi, 712100, P.R. China
b Dep. of Renewable Resources, Univ. of Wyoming, Laramie, WY 82071-3354, USA, Also at Dep. of Water Resources, Wuhan Univ., Wuhan 430072, P.R. China
c College of Water Resources and Civil Engineering, China Agriculture Univ., Beijing, 100083, P.R. China
* Corresponding author (renduo{at}uwyo.edu).
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ABSTRACT
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Temperature effects on soil water flow are attributable to various factors in the soil water system, such as fluid viscosity, soil water content, and other soil physical and chemical properties. To account for the factors as a whole and quantify the temperature effects, the first objective was to apply the concept of activation energy and to estimate apparent activation energies of steady-state saturated flow and horizontal as well as vertical infiltration. The second objective was to predict the water flow processes under different temperatures. Soil column experiments were conducted to measure discharge of steady-state saturated flow and processes of cumulative infiltration using four soils at four temperatures. The parameters of the flow equations were related to temperature and apparent activation energies. Based on the relationships, apparent activation energies of the water flow processes were estimated using the experimental data. The sensitivities of temperature effects on the steady-state saturated flow and infiltration parameters were linearly related to the activation energy and inversely proportional to the absolute temperature. In general, temperature effects on the water flow processes were larger in the fine-textured soils and/or with higher soil water saturation. Using the parameters estimated from measured water flow processes at two temperatures, we predicted the processes at other temperatures and compared the predicted results with the measured data. The predicted results were highly correlated with the measured data with coefficients of determination (r2) larger than 0.990 and the relative errors of the predicted processes were within 12%.
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INTRODUCTION
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THE EFFECT OF temperature on water flow processes in soils, such as infiltration and evaporation, has long been recognized. Some researchers have investigated variations in measured infiltration rates attributable to temperature changes. Bouwer et al. (1974) noted increased infiltration rates in a basin during summer vs. winter. Taking into account temperature dependence of soil hydraulic properties, Hopmans and Dane (1984) investigated the influence of soil temperature on soil water distributions during and after infiltration. Hopmans and Dane (1986) also simulated soil-water movement with different temperature regimes and boundary conditions. Jaynes et al. (1988) and Jaynes (1990) measured infiltration and soil temperature in the field and observed that the infiltration rate varied during a 24-h period, with a maximum in the late afternoon and a minimum in the early morning. Constantz and Murphy (1991) studied the temperature dependence of ponded infiltration and found that infiltration rate increased three to four-fold between 5 and 60°C for their soils. Based on measured cyclic seepage rates from unlined irrigation canals, Mitchell et al. (1990) related the 24-h cycle variation of seepage rates to the diurnal temperature variation of water.
Various factors in the soil water system, such as fluid viscosity, soil water content, and soil physical and chemical properties, interact with temperature changes in the system, therefore, influence the temperature effects on soil water flow processes. Some researchers (Constantz, 1982; Hopmans and Dane, 1986) related temperature dependence of the flow transport to the temperature change of liquid viscosity, which in turn affect soil water hydraulic conductivity. However, it was observed that using the temperature change of liquid viscosity only was inadequate to describe the influence of the temperature dependence of the matric potential. Other factors, such as the soil water surface tension and the diffuse double-layer thickness, may have a role in affecting the flow processes under different temperatures (Constantz, 1982).
To account for the various factors as a whole, the concept of apparent activation energies, which are necessary to overcome energy barriers in flow process, is a useful means to quantify the temperature effects on water flow processes in soils. The activation energy concept was introduced into the field of chemical kinetics by Arrhenius (1889) to explain the temperature dependence of sucrose hydrolysis rate in various acids. Glasstone et al. (1941) extended the concept to fluid flow processes to interpret the change of fluid viscosities and diffusion coefficients with temperature. Biggar and Taylor (1960) related an infiltration parameter to temperature with the Arrhenius equation, from which they estimated activation energies and described the observed temperature dependence of infiltration data. However, the temperature-dependent parameter used had no physical interpretation. Anderson et al. (1963) applied the concept of apparent activation energy to study temperature fluctuations at the wetting front and water movement in the liquid and vapor phases. They obtained apparent activation energies of 17.93, 25.18, and 25.39 kJ mol-1 for Palo Verde sandy loam, Arizona bentonite, and a muck soil, respectively. The results of apparent activation energies of Anderson et al. (1963) were significantly different from those by Biggar and Taylor (1960), which ranged from 4 to 12 kJ mol-1 for infiltration in various fractions of Millville silt loam were considered too low by Anderson et al. (1963). Therefore, it is necessary to further explore the spectrum of apparent activation energies for various soil water processes under different conditions. Furthermore, there is little attempt to utilize the activation energy concept to predict the temperature-dependent water flow processes in soils. Thus, the first objective of this study was to estimate apparent activation energies of water flow processes in saturated and unsaturated soils, from which we evaluated temperature effects on the water flow processes. The second objective was to predict the water flow processes under different temperatures, applying the concept of apparent activation energy. The predicted water flow processes were compared with experimental data.
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MATERIALS AND METHODS
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Water Flow Equations and Apparent Activation Energies
The steady-state flow in soils can be described by the Darcy equation:
 | [1] |
where Q is the discharge (cm3 s-1), K is the hydraulic conductivity (cm s-1), A is the cross-section area (cm2), and 
H is the hydraulic gradient (cm cm-1). Eq. [1] can be expressed by
 | [2] |
in which K0 is the intrinsic hydraulic conductivity (cm2), reflecting the effect of soil matrix on water flow, 
is the fluid density (g cm-3), g is the gravitational acceleration (cm s-2), and 
is the water viscosity (g cm-1 s-1). The water viscosity changes with temperature, which may be quantified with the Arrhenius form of (Glasstone et al., 1941; Low, 1958)
 | [3] |
Here b is a constant, Ev is the apparent activation energy (J mol-1) for viscosity change of pure water, R is the gas constant (8.31451 J mol-1 K-1), and T is the absolute temperature in Kelvin. Assuming that the fluid density is independent of temperature, from Eq. [3] we have
 | [4] |
where B is a constant and Ed is the apparent activation energy (J mol-1) of steady-state flow, which is a lumped parameter to account for factors having temperature effects on water flow, such as fluid viscosity, soil water content, and soil properties. Using Eq. [4] and plotting ln Q vs. 1/T, we can evaluate Ed from the plot slope. In turn, Eq. [4] can be used to calculate steady state flow under different temperatures.
For horizontal infiltration, Philip (1957) gave the following approximation:
 | [5] |
where I(t) is the cumulative infiltration (cm) within time t (s), S is the sorptivity (cm s-1/2), relating to the soil matric potential. Vertical infiltration can be approximated by (Philip, 1957):
 | [6] |
where A0 is the steady-state infiltration rate (cm s-1), relating to the gravitational potential. The parameters in Eq. [5] and [6], S and A0, change with temperature. According to Philip (1957), the soil sorptivity can be related to the soil water viscosity as follows
 | [7] |
where 
is the surface tension (N cm s-2), 
the contact angle in radian, Si the intrinsic sorptivity (cm s-1/2). Invoking Eq. [3], we may express the sorptivity with the Arrhenius equation in the form of (Anderson et al., 1963)
 | [8] |
where F is a constant, Es is the apparent activation energy (J mol-1) of infiltration attributable to the soil matric potential. The activation energy can be estimated from a plot of ln S2 vs. 1/T. The parameter A0 is related to the gravitational potential and usually used to approximate the soil hydraulic conductivity. Therefore, similar to the derivation of Eq. [4], we can write the temperature change of A0 in the following form:
 | [9] |
where G is a constant, Eg is the apparent activation energy (J mol-1) of infiltration attributable to the gravitational potential.
Experiments
Four representative soils (designated with S1, S2, S3, and S4) collected in Shaanxi Province of China were used for the experiments. Some physical and chemical properties of the soils are listed in Table 1, such as particle-size distributions, soil classification, organic matter content, pH values, and cation-exchange capacity (CEC). The texture of the four soils ranged from loamy sand to sandy clay with clay content from 15 to 43%.
To prepare soil columns for the following experiments, we filled and packed the air-dried soil in a 1-cm interval into a soil column to keep a constant bulk density about 1.4 g cm-3. More specifically, we calculated the weight of 1-cm depth of the soils based on the cross-sectional area of the soil column (4 cm in diameter) and the bulk density. The weighted soil was filled into the soil column and packed carefully to a depth of 1 cm. Then we brushed the soil surface to have a good contact with added soil and repeated the procedure. For vertical soil columns, ceramic plates with large pores were used at the bottom to hold the soil.
Steady-state flow experiments were conducted in vertical soil columns of 15 cm in length. To achieve complete saturation in the soil, the soil column was saturated from the bottom using a positive pressure head. After the soil column was totally saturated, a 3.6-cm pressure head was kept at its top using a Mariotte bottle and the bottom was opened to the air. As the steady state was established with the constant pressure head (h) boundary conditions (h = 3.6 cm and 0 for the upper and lower boundaries, respectively), water collected from the bottom was used to calculate the discharge. The experiments were performed with the four soils and at four temperatures, 5, 20, 35, and 50°C. We set up one soil column for each of the soils and temperatures, totaling 16 soil columns. All experiments were conducted in a constant-temperature chamber with a temperature control precision ± 0.2°C. For each temperature, we repeated the experiment for five times using the same soil column.
Horizontal infiltration experiments were conducted in 20-cm long soil columns with a diameter of 5.8 cm. The inflow boundary was controlled at a -3.0-cm pressure head using a ceramic plate and a Mariotte bottle. Infiltrated water changing with time was measured from reading the graduated Mariotte bottle. The experiments were performed with the Soils S1 and S3 at four temperatures (5, 20, 35, 50°C). To verify gravity impact on soil water flow in the horizontal experiments, we plotted the cumulative infiltration vs. t1/2 and fitted the data with a linear equation with a zero interception for each case of the different soils and temperatures. For all cases, the linear equations fitted the data extremely well with coefficients of determination r2 > 0.998, which indicated that the gravity impact on soil water flow in the horizontal experiments was negligible.
Vertical infiltration experiments were conducted in 50-cm long soil columns with a diameter of 4.0 cm. The inflow boundary was controlled at a 2.0-cm pressure head using a graduated Mariotte bottle and infiltrated water changing with time was measured using the Mariotte bottle. The experiments were conducted with the Soils S1 and S3 at three temperatures (5, 20, 35°C). For the horizontal and vertical infiltration experiments, we set up one soil column for each of the soils and temperatures. Each experiment of the horizontal and vertical infiltration was repeated three times using the same column.
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RESULTS AND DISCUSSION
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Estimations of Apparent Activation Energies
Figure 1
presents the data of discharge of steady-state flow measured at different temperatures and in different soils. For all the measurements, the standard errors calculated based on the five measurements at each temperature were within 0.09 (cm3 min-1) and the relative errors were <5%. With the consistent repeated measurements, we mainly concentrated on the average results of the data in the following discussions. Using Eq. [4] and the measured discharge data at different temperatures, we calculated apparent activation energies of the steady-state flow. We plotted ln Q vs. 1/T and obtained a linear relationship as follows:
 | [10] |
where a is the intercept and Ed/R is the slope, from which we evaluated Ed. For example, for S1 we obtained Ed/R = 3182 °K, thus Ed = 26460 J mol-1 = 26.46 kJ mol-1 from the linear relationship with a coefficient of determination (r2) of 0.9928 (Fig. 1). Results for other soils are summarized in Table 2.
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Fig. 1. The linear relationship between log of the discharge (ln Q) of steady-state saturated flow vs. 1/T (T: the absolute temperature).
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Using a nonlinear optimization procedure (Marquardt, 1963) with the measured horizontal and vertical infiltration processes as well as Eq. [5] and [6], respectively, we obtained values of the infiltration parameters of S and A0 at different temperatures. Note that sorptivity (S) is a function of soil water content. The sorptivity values calculated were the average results corresponding to the initial water content and supply water content. As an example, the temperature dependence of S calculated from the vertical infiltration processes is shown in Fig. 2
, which includes the average S values based on the five measurements at each temperature and error bars. Using Eq. [8] and [9] and the estimated infiltration parameters, from the relationships of ln S2 vs. 1/T and ln A0 vs. 1/T, we calculated the apparent activation energies of horizontal and vertical infiltration. The results are also listed in Table 2.
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Fig. 2. The linear relationship between log of the sorptivity (ln S2) for the vertical infiltration vs. 1/T (T: the absolute temperature).
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The apparent activation energies of the steady-state saturated flow are larger than those of infiltration (Table 2). The high apparent activation energies of the steady-state flow may result from the saturated soil water content, which affects the specific heat capacity of the soil water system. Van Duin (1963) showed that changing from a dry condition to the saturated condition, the heat capacity of a sand soil increased 2.86 times and the heat capacity of a clay soil increased 3.18 times (Jury et al., 1991). The heat capacity per unit volume of a substance is defined as the quality of heat energy required to raise a unit volume of the substance one degree of temperature. Therefore, a higher heat capacity increases the energy barrier for temperature change. The first portion of apparent activation energies of the vertical infiltration and the apparent activation energies of the horizontal infiltration for Soils S1 and S3 were calculated using the same equation (Eq. [8]). However, the apparent activation energies (relating to the matric potential) of the vertical infiltration were higher than those of the horizontal infiltration because of the saturated boundary condition for the vertical infiltration verse the unsaturated boundary condition for the horizontal infiltration.
In general we obtained larger apparent activation energies for the fine-texture soils (S3 and S4), especially in the saturated flow. Because of larger specific surface area of fine texture soils, the thermal properties of a fine texture soil are quite different from those of a coarse-texture soil (Bowers and Hanks, 1962; Abu-Hamdeh and Reeder, 2000). More importantly, the saturated water content of a fine-texture soil is higher that that of a coarse texture soil, resulting in a larger heat capacity in the fine texture soil. The differences in the thermal properties would result in different temperature impacts on soil water flow. The highest apparent activation energy of saturated flow in S3 may result from other soil properties, such as its high organic matter content (Abu-Hamdeh and Reeder, 2000).
Prediction of Temperature Dependent Water Flow Processes
The discharge of steady-state saturated flow changing with the temperature is related to the apparent activation energy with Eq. [4] or the linear equation of Eq. [10]. Therefore, we can predict the discharge at any temperature based on measured data at two temperatures. The temperature dependent parameters of horizontal and vertical infiltration are related to the apparent activation energies with Eq. [8] and [9]. Similarly, we can predict horizontal and vertical infiltration processes at any temperature based on measured data at two temperatures.
Using the measured discharge data of steady-state saturated flow at temperatures 5 and 50°C (i.e. T = 278 and 323°K) and Eq. [4], we predicted discharge values of steady-state saturated flow at temperatures 20 and 35°C for the four soils. The predicted results were compared with the measured values in Table 3. The absolute relative errors of the prediction ranged from 0.36 to 14.33%, most of which were <7%. In the literature, temperature dependence of the flow transport was frequently related to the temperature change of the liquid viscosity (Constantz, 1982; Hopmans and Dane, 1986). However, besides the liquid viscosity, other soil and water parameters, such as soil water content, soil texture, soil organic matter, and others, may have impact on the temperature dependence of the flow processes in soils. If only the temperature change of water viscosity is considered, the apparent activation energy calculated from Eq. [3] is about 24 kJ mol-1. Using Eq. [4] with the apparent activation energy resulted in over estimated discharge of the steady-state saturated flow. A comparison between the estimated and measured discharge results shows that relative errors of the estimation ranged from 15 to 60% (Table 3).
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Table 3. Predicted and measured discharge values of steady state saturated flow at different temperatures for different soils.
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Using the measured cumulative values of horizontal infiltration at temperatures 5 and 50°C and Eq. [5] and [8], we predicted cumulative values of horizontal infiltration at temperatures 20 and 35°C for the soils of S1 and S3. The predicted and measured results are presented in Fig. 3 and 4
for S1 and S3, respectively. The predicted and measured results are highly correlated for all the cases with coefficients of determination (r2) larger than 0.990. The predicted cumulative values of horizontal infiltration for S3 at 20°C have an excellent agreement with the measured data. The predicted cumulative values of horizontal infiltration for S1 at temperatures 20 and 35°C and for S3 at 35°C were under estimated. Nevertheless, the prediction precision for all the cases is within 10% of absolute relative estimated errors. Similarly, we utilized the measured cumulative values of vertical infiltration at temperatures 5 and 35°C and Eq. [6], [8], and [9] to predict cumulative values of vertical infiltration at temperature 20°C for the soils of S1 and S3. As shown in Fig. 5
, the predicted results match the measured results very well. We also repeated the prediction procedure using the measured data with different combinations of two temperatures, such as 5 and 20, 5 and 35, 20 and 35, 20 and 50, and 35 and 50°C, the accuracy of the predictions were similar to that reported above.
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Fig. 3. Predicted and measured cumulative values of horizontal infiltration at temperatures of (A) 20 and (B) 35°C for Soil S1.
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Fig. 4. Predicted and measured cumulative values of horizontal infiltration at temperatures of (A) 20 and (B) 35°C for Soil S3.
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Fig. 5. Predicted and measured cumulative values of vertical infiltration for soils of (A) S1 and (B) S3 at temperature 20°C.
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Temperature effects on water flow in saturated and unsaturated soils can also be quantified in the following way. For the steady-state saturated flow, taking the derivative of Q (Eq. [4]) with respect to T, we have
 | [11] |
or
 | [12] |
Eq. [12] shows that the sensitivity of temperature effects on the steady-state saturated flow is linearly related to the activation energy and inversely proportional to the absolute temperature. In the same way, from Eq. [8] and [9], we have
 | [13] |
and
 | [14] |
Equations [13] and [14] show the sensitivity of temperature effects on the infiltration parameters, from which we can quantify the temperature effects on horizontal and vertical infiltration processes.
In this study, it was assumed that the apparent activation energies of the soil water flow are temperature-independent (Biggar and Taylor, 1960). An analysis of the water viscosity changing with temperatures from 5 to 40°C supported the assumption. Using the data of water viscosity changing with the temperatures and Eq. [3], we obtained an Ed value of 24 kJ mol-1 from the plot of ln 1/ vs. 1/T with a coefficient of determination (r2) of 0.9999. However, as mentioned above that the apparent activation energies account for various factors related to temperature effects on water flow, including fluid viscosity, soil water content, and soil physical and chemical properties. To further improve predictions of temperature dependent flow processes in soils, it may be essential to account for the temperature dependence of apparent activation energies, which should be a further research topic.
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SUMMARY AND CONCLUSIONS
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The concept of apparent activation energy was applied to quantify temperature effects on water flow processes, including steady-state saturated flow, and horizontal and vertical infiltration in variably saturated soils. The discharge of steady-state saturated flow and the parameters in the equations of horizontal and vertical infiltration were related to temperature and apparent activation energies. The relationships were derived mainly based on the temperature dependent relationship of water viscosity. Then other factors related to temperature effects on water flow, such as soil water content and soil properties, were accounted for implicitly with lumped apparent activation energies.
Apparent activation energies were estimated with measured data of the processes at temperatures 5 to 50°C for four soils. The ranges of apparent activation energies were 26 to 32 kJ mol-1 for the steady-state saturated flow and about 13 kJ mol-1 for sorptivity of horizontal infiltration. For vertical infiltration, the ranges of apparent activation energies were 15 to 25 kJ mol-1 for sorptivity and 5 to 10 kJ mol-1 for the steady-state infiltration rate. Sensitivity analyses showed that temperature effects on the steady-state saturated flow and the infiltration parameters were linearly related to the activation energy and inversely proportional to the absolute temperature. In general temperature effects on the water flow processes increase with soil clay content as well as soil water saturation, but soil water saturation plays more profound role.
Based on the temperature dependent relationships, the parameters of steady-state saturated flow, and horizontal and vertical infiltration can be calculated using measured processes at two temperatures. Then the processes at other temperatures can be predicted. Comparing with measured data of the processes for different soils at different temperatures, we examined the prediction procedure and obtained excellent predicted results. For all the cases, the predicted results were highly correlated with the measured data with coefficients of determination (r2) >0.990 and the relative errors of the predicted results were within 12%. Therefore, the procedure should be a reliable method to estimate various soil water processes changing with the temperature.
Received for publication July 18, 2002.
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REFERENCES
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ABSTRACT
INTRODUCTION
