Published in Vadose Zone Journal 4:196-205 (2005)
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Validity of the Generalized Richards Equation for the Analysis of Pumping Test Data for a Coarse-Material Aquifer
Aly I. El-Kadi*
Department of Geology and Geophysics and Water Resources Research Center, 1680 East-West Road, University of Hawai‘i at Manoa, Honolulu, Hawai‘i 96822
* Corresponding author (elkadi{at}Hawaii.edu)
Contribution of the Department of Geology and Geophysics and Water Resources Research Center, University of Hawai‘i, Honolulu, HI. WRRC Contributed Paper CP-2005-01.
Received 31 December 2003.
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ABSTRACT
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This paper presents an examination of the validity of the generalized Richards equation (GRE), which includes unsaturation and compressibility effects, in the analysis of a well-documented, three-dimensional aquifer test. The potential effects of wellbore storage and monitoring-well delayed response were included in the analysis. The uniqueness of the solution was also examined by testing the potential success of fully saturated models in simulating the drawdown measurements. The solution of the GRE closely matched the field-measured drawdowns with some parameters that were close to their independently measured values. The aquifer-test analysis can thus provide accurate estimates for some average aquifer parameters, namely, horizontal and vertical hydraulic conductivities and specific storage. However, the model is not fully validated due to the need for calibrated soil hydraulic parameters. In general, it is possible to account for early time discrepancies by using an inflated fitting value for the specific storage. However, good accuracy was obtained using a physically based value for such a parameter when wellbore storage is considered. Sensitivity of results to values of saturated conductivity again confirmed the great importance of obtaining accurate estimates of such values. Finally, the study showed that saturated flow models did not provide results as accurate as those provided by the GRE model. Classifying the aquifer material as coarse can be misleading, considering that its effective soil properties is that of a finer texture. As such, unsaturated flow effects should not be overlooked.
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INTRODUCTION
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AQUIFER-TEST ANALYSIS is a popular technique for estimating aquifer parameters. Analytical and numerical solutions are used to estimate regional values for hydraulic parameters through an inverse procedure. Solutions for unconfined aquifers differ from those for confined aquifers because of the existence of a water table and water flow in the unsaturated zone. Examples of analytical solutions for aquifer-test analysis under idealized conditions include the early works by Boulton (1954)(1963), Boulton and Streltsova (1975), Dagan (1967), and Neuman (1972)(1974). Other solutions addressed deviations from ideal conditions by including partial well penetration and factors that cause delayed monitoring-well responses (Dagan, 1967; Neuman, 1974.) More recently, Moench (1995)(1997) and Moench et al. (2001) introduced new solutions that account for the deficiencies of the earlier solutions, especially those that assume instantaneous release of water from the unsaturated zone. Moench (1995)(1997) included an exponentially declining drainage term from the unsaturated zone. Further improvement was introduced by Moench et al. (2001), who added a series of exponential terms. The solutions also considered storage in the pumped well and monitoring wells and the effects of skin at the pumped-well screen. The model of Moench et al. (2001) was validated by using elaborate three-dimensional drawdown data collected in 1990 at a Cape Cod site in Massachusetts (see also Moench, 2004.)
Numerical solutions of the Richards (1931) equation can describe realistic field applications under various hydrogeological conditions where the assumptions needed for analytical solutions are not valid. An example application that is close to this current one is documented by Halford (1997). The study compared two hypothetical models of an unconfined aquifer, namely, a fully saturated one and a variably saturated one. The study indicated that the latter model provided a more accurate conceptual framework for the aquifer and yielded more accurate estimates of the actual aquifer parameters. The aquifer used in the study was radially symmetric and allowed the use of two-dimensional models. Their simulations served as a guideline for the analysis of aquifer tests conducted at Cecil Field Naval Air Station in Jacksonville, FL. The current study is also radially symmetric but characterized by parameters that were independently estimated. Moench (2003) estimated soil-water characteristics from aquifer-test data for the same site. The paper estimated such properties through a numerical solution of the GRE. The results showed that field-scale heterogeneity can greatly affect soil characteristics. Such an important conclusion can reduce the usefulness of core samples as means of estimating field-scale soil hydraulic properties. The numerical solution used by Moench (2003) did not, however, account for the effects of delayed monitoring-well responses and wellbore storage.
The objective of this paper is to build on the results of Moench (2003) to further assess the validity of the GRE in modeling the well-documented Cape Cod site. In addition, this paper examines the potential effects of delayed monitoring-well responses and wellbore storage on the accuracy of the solution. Finally, uniqueness of the GRE model is examined by comparing the field drawdown measurements to those obtained from a model that ignores unsaturated flow effects. Models that only consider saturated flow are easier to use, and their data requirements are less extensive. Hence, it is always of interest to examine their success in analyzing aquifer-test data. Such models usually use a specific yield value to account for the complicated dynamics of the unsaturated zone.
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The Site
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Figure 1
shows a map of the study site. The hydrogeology of the site is described in Moench et al. (2001), Hess et al. (1992), LeBlanc et al. (1991), Garabedian et al. (1991), and Rudolph et al. (1996). According to these studies, the aquifer is composed of unconsolidated glacial outwash sediments that were deposited during the recession about 15000 yr before present and that were part of the Wisconsin continental ice sheet that had previously covered New England. The unconsolidated sediments overlie crystalline bedrock at a depth of about 100 m. Clean, medium- to coarse-grained, high-permeability deposits cover fine-grained, relatively low-permeability material at about 50 m below the water table. The horizontal hydraulic conductivity in the western Cape Cod area ranges from 0.053 to 0.123 cm s–1 for the upper material and 0.004 to 0.025 cm s–1 for the lower material. The horizontal anisotropy for the upper and lower materials ranged from 3 to 10 and 30 to 100, respectively.
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Aquifer Test
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Details of the aquifer-test design and operation are described in Moench et al. (2001). Locations of the pumping well (Well F507-080) and monitoring wells are shown in Fig. 1. Table 1 lists pertinent information for the monitoring wells. The total depth of the pumping well was 24.4 m below the land surface. The depth to the top of the screen was about 4 m, and the length of the screen was about 14 m. The test continued for three days, 28 to 31 Aug. 1990. The pumping rate was about 0.02 m3 s–1. Drawdown data were collected manually using a steel tape in all wells and by the use of pressure transducers connected to data loggers in seven of the monitoring wells. Estimated hydraulic conductivity values from Moench et al. (2001) were in close agreement with those provided by Hess et al. (1992), who studied spatial variability of hydraulic conductivity for the same aquifer.
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THEORY
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The Governing Equation
For slightly compressible porous media and water, the generalized form of Richards (1931) equation can be written as follows:
 | [1] |
where h is hydraulic head, K is hydraulic conductivity tensor, 
is porosity, 
is water content, Ss is specific storage, t is time, Q is a source/sink, and C is moisture capacity defined as
 | [2] |
where 
denotes the pressure head. The specific storage is related to compressibility of the media and water.
The soil-water hydraulic characteristic function is defined by the van Genuchten (1978) equation as
 | [3] |
in which Se is defined as
 | [4] |
In Eq. [3] and [4], s and r are saturated and residual water content, respectively, and 
, n, and m are parameters, with n and m related by
 | [5] |
The hydraulic conductivity function is given by van Genuchten as
 | [6] |
Ks in Eq. [6] represents the saturated hydraulic conductivity.
Effects of Delayed Response of Monitoring Wells
Based on the works of Black and Kipp (1977), Moench (1997) proposed the following equation to account for the delayed response of monitoring wells:
 | [7] |
where h and hm are the estimated and measured values of head, respectively. The value of a is given by
 | [8] |
where
 | [9] |
and
 | [10] |
In Eq. [9] and [10], L is the screen length of the monitoring well; Kr and Kz are horizontal and vertical hydraulic conductivities, respectively; and rp is the radius of the monitoring well.
Equation [7] can be easily integrated to give:
 | [11] |
Equation [11] satisfies the initial condition h = hm = 0 at t = 0. Considering that hm reaches zero faster than h due to the delayed response of monitoring wells, it follows that hm/h 
0 as t 0.
Influence of Wellbore Storage
Based on the study of Moench et al. (2001), the effective pumping rate Qe can be estimated from
 | [12] |
where Q is the actual pumping rate, hw is the vertical average head in the well pore, and 
is the wellbore storage given by
 | [13] |
where rc is the effective diameter of the pumping well.
Independently Estimated Parameters
Aquifer-Test Data
As described earlier, Moench et al. (2001) used an analytical solution to estimate the hydraulic parameters for the aquifer, under the assumption of homogeneity and anisotropy. Moench et al. provided the following estimates for aquifer parameters:
- Specific yield = 0.26
- Saturated thickness = 52 m
- Horizontal hydraulic conductivity = 0.118 cm s–1
- Vertical conductivity = 0.072 cm s–1
- Specific storage = 4.27 x 10–7 cm–1
Saturated Hydraulic Conductivity Data
Hess et al. (1992) used a field method to assess variability of hydraulic conductivity. The values of horizontal hydraulic conductivity were estimated using flowmeter tests conducted in the upper 12 m of the saturated zone. Geostatistical techniques were used to estimate the vertical hydraulic conductivity and conductivity spatial structure. The following values of parameters were estimated:
- Average horizontal hydraulic conductivity = 0.123 cm s–1
- Average vertical hydraulic conductivity = 0.099 cm s–1
- Variance of logarithm of horizontal hydraulic conductivity = 0.24
- Horizontal correlation scale = 2.9–8.0 m
- Vertical correlation scale = 0.18–0.38 m
Unsaturated Soil Hydraulic Properties
Mace et al. (1998) used cores from the site to estimate parameters of the Brooks and Corey (1964) and van Genuchten (1978) equations. The authors assessed the effect of restricting the relationship between the parameters n and m of the van Genuchten equation based on Mualem and Burdine criteria as given by the equation. They also estimated the accuracy of predictions of the unsaturated hydraulic conductivity function based on both the estimated and fitted value of the residual water content. They concluded that the van Genuchten equation with m and n related by the Mualem criterion is most accurate. The use of an estimated value for the residual water content gave the best match with the unsaturated hydraulic conductivity data. Parameters recommended by Mace et al. (1998) are as follows:
- = 0.244 cm–1
- n = 2.494
- s = 0.35
- r = 0.03
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THE MODELS
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Two numerical models are used in the current analysis: VS2D and MODFLOW. VS2D (Lappala et al., 1987; Healy, 1990; Hsieh, 2000) is a finite-difference model and the problem was simulated as a two-dimensional axisymmetric domain. The governing equation is a two-dimensional form of Eq. [1] in the (r–z) domain, in which r is the radial distance and z is the vertical distance. MODFLOW is a three-dimensional, cell-centered, finite-difference, saturated-flow model developed by the United States Geological Survey (McDonald and Harbaugh, 1988). The governing equation for MODFLOW is a special case of Eq. [1] for a fully saturated aquifer where C = 0, = , and K = Ks. For unconfined layers, the specific yield, Sy, replaces the storage coefficient, which is defined as the aquifer thickness times Ss. The modeling package Groundwater Modeling System (GMS) was used as the working environment for MODFLOW.
For VS2D, the aquifer was modeled as a rectangle of size 300 by 60 m in the r and z directions, respectively. The 300-m distance greatly exceeds the 70 m where drawdown occurred, as reported by Moench et al. (2001), and the no-flow boundary assumed there should not affect the results. The initial water table was assumed to be 6.5 m below the ground level, as also reported by that study. The partially penetrated pumping well was simulated as a notch at the appropriate depth, and the pumping rate was divided equally over the screen length in a way that provided the total pumping rate reported in the study by Moench et al. A no-flow boundary was assigned at r = 300 m and at the bottom of the aquifer.
For MODFLOW, the region was modeled as a cylinder, with the well at the origin, and a radius of 300 m. The aquifer depth and the initial water table location were taken as those listed above.
A number of runs for VS2D were used to optimize the size of the finite-difference grid. A fine grid consisting of 117 columns by 103 rows was used as the benchmark, with grid spacing that ranged between 0.195 and 5 m in the r direction, and 0.16 and 0.76 m in the z direction. Based on these runs, the optimal grid that was used in all the calculations consisted of 80 rows by 103 columns, with cell sizes that varied between 0.52 and 5 m in the r direction, and 0.16 and 0.76 m in the z direction. The simulations gave identical results to those obtained by the finer grid of 117 columns by 103 rows (Fig. 2)
. The smaller spacing was placed near the pumping well and around the water table.
The MODFLOW grid consisted of 100 by 100 equal-sized cells of about 5.5 m each. The aquifer was divided into eight equal-sized layers of about 7.2 m each.
Parameter Estimation
Modeling efforts are always limited by the lack of data supporting the model of interest. An ideal situation for model validation would include availability of parameter values that are independently measured. However, even if such measurements exist, values are usually useful for a model that is developed under the same assumptions involved in setting up the measurement scheme. For example, pump tests provide regional values for parameters that are not, in general, suitable for detailed models describing heterogeneous systems. Column-scale experiments produce parameter values that are not suitable for larger-scale problems. Added difficulties include the fact that some parameters are not physically based and they are merely fitting parameters that are estimated through model calibration.
The Cape Cod site is ideal for testing the suitability of independently measured aquifer and soil parameters for validating the GRE. In this study, an attempt was made first to predict drawdowns without any parameter adjustments. Next, to improve on accuracy of prediction, an automatized parameter scheme was used to estimate the main parameters of the respective model. The parameter estimation program PEST was used in the process. For VS2D, the parameters estimated were horizontal and vertical hydraulic conductivities, residual water content, and van Genuchten parameters and n. Other parameters were known with less uncertainty or considered to have limited influence on results. For MODFLOW, attempts were made to estimate conductivities, storage coefficient, and specific yield. The best results were obtained by estimating only conductivity values. Attempts to optimize other parameters not only added a numerical burden to the calculations, but also provided unrealistic values for parameters without much improvement in results.
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RESULTS AND DISCUSSION
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Validity of the GRE
Table 2 lists various VS2D simulated cases completed for this study. The simulation results cover the following cases:- There are no compressibility effects (Ss = 0.0).
- Compressibility effects are included with a specific storage (Ss) value of 4.28 x 10–7 cm–1 estimated by Moench et al. (2001) and Moench (2003).
- Compressibility effects are included with a higher Ss value based on in an attempt to match the early drawdown data (Ss = 1.0 x 10–6 cm–1).
- Same as Case 2 above but with drawdowns adjusted to account for delayed monitoring-well responses.
- Same as Case 2 above but with drawdowns estimated based on an effective pumping rate, Qe, based on Eq. [12]. Figure 3 shows the effective pumping rate as a function of time based on the numerical integration of the equation, which used the actual drawdown data in the pumping well.
- Same as Case 2 above, but drawdowns are estimated based on the effective pumping rate and then adjusted to account for delayed monitoring-well responses. In addition, to assess uniqueness of the GRE solution, the two following cases were simulated for MODFLOW.
- A fully saturated, homogeneous, anisotropic aquifer, with the hydraulic conductivity and specific storage values that were used for Case 5. A value of 0.26 for Sy was used based on the estimate of Moench et al. (2001).
- An optimization technique was used to estimate values of horizontal and vertical hydraulic conductivities for a fully saturated, homogeneous, and anisotropic aquifer. The rest of the parameters were taken as those for Case 7.
The parameters of the van Genuchten equation that gave the best fit are 0.01 cm–1, 1.86, 0.35, and 0.03, for , n, s, and r, respectively. The saturated hydraulic conductivity values are 0.1 and 0.07 cm s–1 for Kr and Kz, respectively. In Fig. 4
, the soil-water hydraulic characteristic function is compared against that estimated by Moench (2003) who used the Brooks and Corey (1964) function rather than the van Genuchten (1978) function. The differences between the two forms do not seem to drastically affect the calculations, considering the relatively small drawdowns in the aquifer and the relatively shallow water table. However, there are differences in predictions at times larger than about 5 to 10 min (Fig. 5)
. Either function with the respective estimated parameters, as Moench (2003) also indicated, clearly represents a finer soil than the coarse-grained highly permeable material characterizes the site, with soil core parameters measured by Mace et al. (1998) (Fig. 4). Moench (2003) attributed such a discrepancy to the effects of restricted air entry or the existence of fine-grained material layers in the formation. However, additional research is needed to address this issue, namely, the scale-up of laboratory-measured hydraulic soil properties for use in field assessment. The values of saturated conductivity are very close to those estimated by Moench (2003) but smaller than those estimated by Hess et al. (1992), namely, 0.123 and 0.099 cm s–1 for the horizontal and vertical conductivities, respectively. The values are closer to those estimated by the analytical approach of Moench et al. (2001), which provided respective values of 0.118 and 0.072 cm s–1.
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Fig. 5. Comparison of solutions of the generalized Richards equation that utilized the van Genuchten and Brooks and Corey functions.
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The best overall results were obtained for Case 5, which included the use of an effective well-pumping rate. Results for all the cases simulated above roughly agreed after 1 to 10 min, when unsaturated flow dominated compressibility effects and the likely influence of wellbore storage and delayed monitoring-well responses diminished.
Figure 6
shows simulation results at monitoring well F505-032 where early drawdown monitoring data are available. In Fig. 6a, it is clear that compressibility effects cannot be neglected at early simulation times (Case 1). Accounting for wellbore storage by adjusting well pumping rate shows some influence on Case 1, but cannot by itself explain the error in results. Figure 6b compares between Cases 2, 3, and 5. Cases 2 and 3 fail to accurately predict drawdowns at early times. The use of an inflated specific storage improved on the accuracy. However, Case 5 showed a shape of the time–drawdown curve that agreed with that for the measured data. This indicates that such a case is more physically based and Ss is not merely a fitting parameter as Cases 2 and 3 assume. The value of Ss for Case 5 also agreed with that obtained by Moench et al. (2001) based on early time–drawdown data. The results confirm the fact that care must be taken in analyzing aquifer-test data at early times when the analysis can lead to an overestimated value for the specific storage (see, e.g., Halford, 1997).
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Fig. 6. Results for simulations and the measured drawdowns at early time for well F505-032: (a) Case 1 vs. Case 5, (b) Cases 2 and 3 vs. Case 5, (c) Cases 4 and 6 vs. Case 5.
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Figure 6c compares the results from Cases 4, 5, and 6. The accounting for delayed monitoring-well response, by itself (Case 4) or with wellbore storage (Case 6), did not improve on results. Including delayed monitoring-well responses with wellbore storage effects will reduce the quality of match for most wells, since results of Case 5 are very close to the measured data (Fig. 7, 8a, 8b, 9a, 9b, and 9c)
. From these figures, it is clear that accounting for delayed monitoring responses (i.e., Case 6) will reduce the quality of the match by underestimating the drawdown. Probably some improvement could occur for monitoring wells F504-032 and F504-080 (Fig. 8a and 8c) if a delay in response is considered. It should be also noted that the equations used to assess the delayed response is semi-empirical in nature and that probably some fitting can adjust the parameter a in Eq. [7] to get better representation of such effects. On the other hand, Eq. [12], and the resulting pumping rates in Fig. 3, are physically based, and they seem to represent the wellbore storage effects.
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Fig. 7. Results for simulations and the measured drawdowns for the duration of the aquifer test for well F377-037. The figure compares between the results of Cases 5, 7, and 8.
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Fig. 8. Results for simulations and the measured drawdowns for the duration of the aquifer test for monitoring cluster F504: (a) well F504-032, (b) well F504-060, and (c) well F504-080. The figure compares between the results of Cases 5 and 8.
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Fig. 9. Results for simulations and the measured drawdowns for the duration of the aquifer test for monitoring cluster F505: (a) well F505-032, (b) well F505-059, and (c) well F505-080. The figure compares between the results of Cases 5 and 8.
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Figures 7, 8, and 9 show results for Case 5 for other monitoring wells over the whole pumping period. Good overall matching with measurements can be seen.
Figure 10
compares drawdown results at three monitoring points that are farther away from the pumping well and where no early time–drawdown data existed. Excellent match can be seen for monitoring wells F434-060, F450-061, and F383-129. The respective radial distances from the center of the pumping well are about 12, 20, and 30 m. The GRE without compressibility effects, Case 1, successfully simulates the drawdown data. It is clear, however, that early time results are far off from those with compressibility. Therefore, the neglect of compressibility effects based on the lack of measured data can be misleading. Figures 8 and 10 show that the results for monitoring wells with long screens, namely, wells F504-032, F434-060, and F450-061, are also accurate. The values were estimated at the midpoint of the screen.
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Fig. 10. Example results for simulations and the measured drawdowns for the duration of the aquifer test for three monitoring points where no early test data exist: (a) well F434-060, (b) well F450-061, and (c) well F383-129. The figure compares between the results of Cases 1, 5, and 8.
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Figure 11
compares the estimated drawdowns at the pumping well against measurements. The model drastically underestimates drawdowns, which is most likely due to well losses. Moench et al. (2001) also indicated that pumping well–drawdown data are not appropriate for estimating aquifer parameters.
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Fig. 11. Results for simulations and the measured drawdowns for the duration of the aquifer test for the pumping well (Case 5).
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As stated earlier, the parameters for the soil hydraulic properties that gave the best fit were very different from those independently measured by Mace et al. (1998). Thus it seems not possible, at least for this case, to identify average parameters of the soil hydraulic properties through aquifer-test analysis. There a need to develop methods to independently identify soil hydraulic properties on the field scale, without the process of model calibration.
The results were very sensitive to the values of saturated hydraulic conductivity. Horizontal and vertical conductivity values that gave the best fit were 0.1 and 0.07 cm s–1, respectively, which are very close to those estimated by the numerical simulations of Moench (2003) of 0.11 and 0.064 cm s–1. Hess et al. (1992) estimated the average values of horizontal and vertical conductivities as 0.123 and 0.099 cm s–1, respectively. The study by Moench et al. (2001) estimated the respective conductivity values as 0.118 and 0.072 cm s–1. Close match between the estimates and those independently measured by Hess et al. is evident.
The close match of current estimates of conductivity values and those independently estimated by Moench et al. (2001), Hess et al. (1992), and Mace et al. (1998) supports the conclusions by Moench et al. that the aquifer test provides good estimates of average parameter values in the aquifer. The flowmeters used in the Hess et al. (1992) experiment were installed in wells in the cone of depression about 10 to 16 m northeast of the pumping well. The above estimates were geometric means of the measured values. Hess et al. (1992) also estimated horizontal and vertical conductivity correlation scales of 3.5 to 8.0 m and 0.19 to 0.38 m, respectively. The aquifer-test responses covered about 70 m in radial distance and 30 m in depth. This indicates that the pumping test covered sizes many times larger than the correlation scales of Hess et al. (1992). As also indicated by Moench et al. (2001), although the aquifer is heterogeneous in nature, it seems that the scale of variability and the size of the test justify the assumption of homogeneity and the results of the aquifer test provides reasonable estimates of average aquifer parameters. The discrepancy of results at some locations certainly indicates that variability of parameters can be a factor.
Uniqueness of the GRE
The obvious weakness of the GRE application in this study is the failure of the laboratory column methods in estimating the hydraulic properties of soil on the field scale and the large discrepancy between measured and calibrated soil parameters. However, the GRE is able to provide accurate results with hydraulic conductivity values that are close to those independently measured. It is of interest to also assess uniqueness of the GRE solution which combines both unsaturated flow and compressibility effects. The nature of the aquifer material, which is described as coarse, suggests the possibility of neglecting flow in the unsaturated zone without sacrificing accuracy. It is well known that applying fully saturated flow models is much easier than applying unsaturated flow models. The latter models require additional data relevant to the soil hydraulic properties. The nonlinear nature of the Richards equation causes additional computational difficulties, including the need for an iterative procedure in the numerical solution, for a relatively fine spatial mesh or grid and for smaller time steps. It is thus always tempting to ignore flow in the unsaturated zone by treating the aquifer as being fully saturated. The study by Brutsaert and El-Kadi (1984) proposed the use of an indicator in assessing the relative importance of the unsaturated zone. In the current notations, and using the van Genuchten equation, the indictor is given by
 | [14] |
This indicator represents the absolute value of the pressure head at an effective saturation, Se, of 0.5. For the current study, and based of the data of Mace et al. (1998), hp is about 5.6 cm, which describes relatively small capillary effects. Based on the fitting process in the current study, a value of hp of 168 cm is estimated. The study by El-Kadi (1985) provided average values for hp for a large number of soil samples that ranged between about 110 and 1177 cm. The smaller value for hp of 110 cm characterizes soils described by El-Kadi (1985) as sands. It is clear that the current modeling efforts necessitate the use of a soil with significant capillary retention characteristics compared with that described by Mace et al. (1998).
We simulated a fully saturated, homogeneous, anisotropic aquifer, with the hydraulic conductivity values that were used for the GRE solution, namely, 0.1 and 0.07 cm s–1 for the horizontal and vertical conductivities, respectively. The same value for Ss of 4.27 x 10–7 cm–1 was also used. A value for Sy of 0.26 was used based on the estimate of Moench et al. (2001). We labeled this simulation Case 7. Two runs were completed: with and without adjusting pumping rate to account for pumping wellbore storage. Figure 7 shows that adjusting the well pumping rate improves on the accuracy at early time, but the accuracy is poor compared to Case 5 at intermediate time, indicating the importance of the unsaturated flow.
Another simulation, Case 8, used an optimization technique to estimate values of horizontal and vertical hydraulic conductivities by treating the aquifer also as homogeneous and anisotropic. The rest of the parameters were taken as those for Case 7. As shown in Fig. 8, with adjusted pumping rate, there is some improvement over Case 7, yet the accuracy is not comparable to Case 5. The estimated values for hydraulic conductivity were 0.094 and 0.052 cm s–1 for horizontal and vertical conductivity, respectively. These values are slightly smaller than those used in Case 5, reflecting a reduction in conductivities due to the neglect of the unsaturated zone. Better match with measured values can be seen in Fig. 8 and 9 at monitoring clusters F504 and F505. There is no obvious reason for such higher accuracy at these locations, considering that accuracy cannot be correlated with the radial distance and the depth of the wells.
In conclusion, although the GRE provided more accurate results, we cannot conclude that it is fully validated for the site because calibration was necessary to estimate the soil hydraulic properties. The model is able to predict average aquifer parameters for conductivities and specific storage. Within the numerical scheme, the model is also able to account for wellbore storage and delay in monitoring well responses. More research should emphasize independently estimating the soil hydraulic properties on the field scale.
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SUMMARY AND CONCLUSIONS
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This paper examined the validity of the generalized Richards equation (GRE) in the analysis of a well-documented, three-dimensional aquifer test. The potential effects of wellbore storage and monitoring-well delayed responses were included in the analysis. In addition, a fully saturated model was applied to study the effects of ignoring flow in the unsaturated zone.
The paper demonstrated the success of the solution of the GRE, which also includes the effects of compressibility, in closely matching the field data. However, the model is not fully validated due to the need for calibrating the soil hydraulic properties. On the other hand, the model employed average values for hydraulic conductivities that were in good agreement with those independently measured. These include horizontal and vertical hydraulic conductivities and specific storage. Compressibility effects cannot be ignored at early times, in the range of 1 to 10 min. In general, it is possible to use an inflated fitting value for the specific storage Ss to account for early time discrepancy due potentially to wellbore storage and monitoring-well delayed responses. However, in the current application, good accuracy that included the use of a physically based value for Ss was possible by accounting for wellbore storage. An expression for estimating the effective well-pumping rate was used without any fitting. However, it seems for this specific application that the delayed response of monitoring would not improve on the accuracy of results.
The study again confirmed the great importance of obtaining accurate values for the saturated hydraulic conductivity. However, there is an urgent need to develop direct methods to estimate soil hydraulic properties on the field scale without model calibration. Finally, the study showed that a saturated flow model was not as accurate as the GRE model. Classifying the aquifer material as coarse can be misleading, considering that its effective soil properties are that of a finer texture. As such, unsaturated flow effects should not be overlooked.
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ACKNOWLEDGMENTS
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The author expresses his appreciation to Allen Moench for providing the aquifer-test data in an electronic format.
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REFERENCES
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- Brooks, R.H., and A.T. Corey. 1964. Hydraulic properties of porous media. Hydrology Paper No. 3. Colorado State Univ., Fort Collins, CO.
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R. W. Healy
Simulating Water, Solute, and Heat Transport in the Subsurface with the VS2DI Software Package
Vadose Zone J.,
May 27, 2008;
7(2):
632 - 639.
[Abstract]
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ABSTRACT
INTRODUCTION
