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Direct estimation of hydraulic parameters relating to steady state groundwater flow
Environmental Modelling & Software 86 (2016) 50e55
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Direct estimation of hydraulic parameters relating to steady state groundwater flow Kerang Sun a, *, Mark N. Goltz b a b
CH2M HILL, Inc., 6 Hutton Center Drive, Santa Ana, CA 92707, United States Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, United States
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 January 2016 Received in revised form 26 July 2016 Accepted 13 September 2016
Groundwater as an important, life-sustaining resource for humankind is being threatened by massive over-extraction and wide-spread contamination. Wise development and protection of this crucial resource requires a thorough understanding of groundwater flow in the subsurface. This paper presents a novel direct method to estimate important hydraulic parameters characterizing steady state groundwater flows for confined and unconfined isotropic aquifers. The method is appropriate for application in aquifers where horizontal flow dominates. The governing equations for the direct estimation method are superposed extensions of the well-known Thiem equation governing steady-state radial flow toward a pumping well under confined and unconfined conditions. This new approach has the following advantages over conventional methods: (1) simultaneously provides estimates of both hydraulic conductivity and hydraulic gradient, (2) can be applied using historical data without the need to conduct a pumping test, and (3) is a simple analytical method that can be applied easily. Verification of the direct estimation method is achieved by applying it to hypothetical homogeneous and heterogeneous aquifers simulated by three-dimensional finite element models. The usefulness of the method is also demonstrated with data from an actual field site. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Two-dimensional groundwater flow Hydraulic conductivity Transmissivity Superposition Darcy's law Hydraulic gradient Confined aquifer Unconfined aquifer Pumping test Steady-state flow
1. Introduction Groundwater is the largest fresh water body on the Earth and supplies drinking water to approximately two billion people worldwide (Richey et al., 2015). It may be the only source of water in some arid regions (Clarke et al., 1996). To protect this crucial lifesustaining resource from the progressively escalating crisis of aquifer depletion (Clarke et al., 1996; Richey et al., 2015; Konikow, 2011; Wada et al., 2010; Werner and Gleeson, 2012), the increasingly frequent occurrence of aquifer contamination (Clarke et al., 1996), and the potential impacts of climate change (Taylor et al., 2013), we must have detailed knowledge of aquifer hydrogeologic properties. Such knowledge is obtained through aquifer characterization which typically requires substantial field efforts. Given the complex nature of subsurface flow along with financial and technological limitations, it is often difficult or impossible to
* Corresponding author. E-mail addresses: [email protected] (M.N. Goltz).
(K.
http://dx.doi.org/10.1016/j.envsoft.2016.09.014 1364-8152/© 2016 Elsevier Ltd. All rights reserved.
Sun),
mark.goltz@afit.edu
generate sufficient data from field testing to adequately characterize the subsurface in order to construct hydrogeological models that will support good management decision making. Hydraulic conductivity and hydraulic gradient are the key parameters needed to model groundwater flow. Aquifer pumping tests are conventionally used to obtain estimates of hydraulic conductivity, with piezometers used to determine gradient. A pumping test involves extracting groundwater at known rates and measuring the hydraulic responses at selected locations. The response data at monitoring wells are analyzed to estimate hydraulic conductivity values. Various deterministic and stochastic models have been developed to suit different pumping and aquifer conditions (Batu, 1998; Kruseman and de Ridder, 1994; Li et al., 2005). While other techniques such as geophysical surveys (Capriotti and Li, 2015; Hinnell et al., 2010; Singha et al., 2015), borehole flow meters (Li et al., 2008; Paillet, 1998; Paradis et al., 2011), and empirical estimation methods (Domenico and Schwartz, 1998) have been applied to indirectly determine aquifer hydraulic conductivity, pumping tests remain the prevailing characterization method. One advantage of pumping test is that it interrogates a relatively large subsurface volume; thus, the estimate
K. Sun, M.N. Goltz / Environmental Modelling & Software 86 (2016) 50e55
of hydraulic conductivity obtained from a pumping test is a largescale volume-average. However, conducting a pumping test is by no means a trivial task. Firstly, it costs both time and money to plan and execute a test. Secondly, there are situations where a meaningful pumping test cannot be conducted. For example, aquifers with low conductivity may not be able to sustain a large enough pumping rate to induce measureable drawdown at monitoring wells. Lastly, there are often regulatory and economic hurdles imposed by the need to dispose of the potentially contaminated water that was extracted during the test. An alternative method that can be applied to directly estimate aquifer hydraulic conductivity while avoiding pumping tests would be of great benefit. Here we present a novel approach to quantitatively estimate hydraulic conductivity and regional hydraulic gradient without the need to conduct a dedicated pumping test. So long as long term pumping data from at least one pumping well and monitoring data from at least three observation wells are available, the method can simultaneously obtain estimates of hydraulic conductivity and gradient of the background flow in the aquifer. While the method has the advantage of sampling a large subsurface volume, it avoids the disadvantages of pumping tests discussed above. 2. Method development The equations upon which the direct estimation method is based require a number of simplifying assumptions. The closer these assumptions come to describing the real physical system, the more appropriate it is to apply the direct estimation method to that system. The simplifying assumptions are: 1. Isotropic aquifer of infinite extent with a homogeneous hydraulic conductivity (K) field. 2. Steady-state flow conditions. 3. Two-dimensional flow. In the case of a confined aquifer, the aquifer thickness (b) is constant. In the case of an unconfined aquifer, the Dupuit assumptions hold. 4. Uniform, regional flow. In the case of a confined aquifer, constant Darcy velocity (q ¼ KVh ¼ Ki) where h is the hydraulic head and i ¼ Vh is the regional hydraulic gradient. In the case of an unconfined aquifer, constant total discharge per unit width of aquifer (Q ¼ K2 Vðh2 Þ ¼ KhVhÞ) For a confined aquifer, by applying Darcy's Law and superposition (Bear, 1979), the following expression may be obtained, which relates the difference in head at two monitoring well locations (Dh) to aquifer characteristics (hydraulic conductivity (K) and aquifer thickness (b)), pumping well locations (x0j,y0j), pumping rates (Qj), and the regional hydraulic gradient prior to pumping (i):
Dh ¼ i½ðx2 x1 Þcos a þ ðy2 y1 Þsin a
2 2 n x2 x0j þ y2 y0j 1 X þ Q ln 2 2 4pbK j¼1 j x1 x0j þ y1 y0j
disadvantages that were noted above. Equation (1) has also been derived using complex potential theory (Javendal et al., 1984). Defining transmissivity (T) as the product of hydraulic conductivity (K) and aquifer thickness (b), equation (1) can be rearranged to obtain:
T¼
where.Qj is the pumping rate of the jth well.(x0j,y0j) are the x and y coordinates of the jth pumping well.a is the angle in radians between the Darcy velocity vector and the positive x-axis.n is the number of pumping wells.(x1,y1) and (x2,y2) are the x and y coordinates of the two monitoring wells. Equation (1) has also been derived by Brooks et al. (2008), who used it as the foundation of a method, referred to as the Integral Pump Test (IPT) approach, to estimate contaminant flux. As the name implies, the Brooks et al. (2008) IPT approach involved conducting a number of pump tests that comes with all of the associated
iT½ðx2 x1 Þcos a þ ðy2 y1 Þsin a Dh 2 2 n x2 x0j þ y2 y0j 1 X Q ln þ 2 2 4pDh j¼1 j x1 x0j þ y1 y0j
(2)
If we're interested in the total discharge per unit width of aquifer prior to pumping (Q ) we see for a confined aquifer Q ¼ KbVh ¼ TVh ¼ Ti and we can rewrite equation (2) as:
T¼
Q ½ðx2 x1 Þcos a þ ðy2 y1 Þsin a Dh 2 2 n x2 x0j þ y2 y0j 1 X Q ln þ 2 2 4pDh j¼1 j x1 x0j þ y1 y0j
(3)
For an unconfined aquifer, so long as the Dupuit assumptions hold, Darcy's Law and superposition can again be applied to obtain (Bear, 1979):
Dh2 ¼ Vh2 ½ðx2 x1 Þcos a þ ðy2 y1 Þsin a
2 2 n x2 x0j þ y2 y0j 1 X þ Q ln 2 2 2pK j¼1 j x1 x0j þ y1 y0j
(4)
where Dh2 ¼ h22 h21 is the difference in the squares of the hydraulic heads measured relative to the aquifer bottom at two monitoring wells and Vh2 is the gradient of the square of the regional hydraulic head. Rearranging terms results in:
K¼
Vh2 K½ðx2 x1 Þcos a þ ðy2 y1 Þsin a
Dh2
2 2 x2 x0j þ y2 y0j þ Qj ln 2 2 2pDh2 j¼1 x1 x0j þ y1 y0j 1
n X
(5)
If we're interested in the total discharge per unit width of aquifer prior to pumping (Q ) we see for an unconfined aquifer Q ¼ K2 Vh2 ¼ KhVh and we can rewrite equation (5) as:
K¼ (1)
51
2Q ½ðx2 x1 Þcos a þ ðy2 y1 Þsin a
Dh2
2 2 x2 x0j þ y2 y0j þ Qj ln 2 2 2pDh2 j¼1 x1 x0j þ y1 y0j 1
n X
(6)
Note that equations (3) and (6) each have two unknowns. The unknowns in equation (3) are the transmissivity (T) and the total discharge per unit width of aquifer prior to pumping (Q ). The unknowns in equation (6) are the hydraulic conductivity (K) and the total discharge per unit width of aquifer prior to pumping (Q ). Once these unknowns are determined, for a confined aquifer, the hydraulic conductivity and the regional hydraulic gradient prior to pumping (i) can be determined by the following formulas, respectively:
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K. Sun, M.N. Goltz / Environmental Modelling & Software 86 (2016) 50e55
T b
(7a)
Q T
(7b)
K¼
i¼
For an unconfined aquifer, if an average aquifer depth is assumed, the regional hydraulic gradient prior to pumping (i) can be determined by equation (8):
i¼
Q Kb
(8)
To solve equations (3) and (6) for the unknowns, it is necessary to either have head data from at least three monitoring wells with one pumping scenario or head data from two monitoring wells with at least two pumping scenarios. Although the method requires data under pumping conditions, it is quite conceivable that useful historical data are available in many instances (for example, head data obtained during normal operation of agricultural or municipal supply wells), precluding the need to perform a dedicated pump test. It is noted that equations (1) and (4) are superposed extensions of the well-known Thiem equation governing steady-state radial flow toward a pumping well, where equation (1) is for confined flow and equation (4) for unconfined flow (Bear, 1979; Domenico and Schwartz, 1998).
The direct approach presented above is novel for several reasons. Firstly, although equations (1) and (4) are relatively straight forward extensions of well-known steady state flow models which were developed over 100 years ago (i.e., the Thiem equation superposed with Darcy's Law for the confined case and the Thiem equation superposed with the Dupuit-Forchheimer discharge formula for the unconfined case), the extension formulated in equation (4) for the unconfined case is, to our knowledge, new. Both extensions greatly expand the practical applicability of the well-known models. Secondly, the methodology of applying these formulas to simultaneously estimate both hydraulic conductivity and regional hydraulic gradient is new. Thirdly, the methodology is based upon application of historical data. The standard approach of most practitioners faced with constructing a site model is to conduct pump testing. If practitioners can be convinced of the usefulness of the direct estimation method, the considerable economic, regulatory, and practical consequences of pumping tests can be avoided. To summarize, the study is significant, as it presents a novel approach for practitioners to apply simple analytical formulas in order to use readily available historical data to help construct a site model that will facilitate decision making. In the next three sections, direct estimation method results are compared to results obtained from numerical modeling, and results from an actual site, to demonstrate the utility of the method in constructing a site model that can be used to make informed site management decisions.
Fig. 1. Model configuration with contours (m) of the simulated water table. P1 (3500 m, 3500 m) and P2 (3500 m, 4000 m) are pumping well locations, with each well pumping at 4000 m3/day; M1 (3550 m, 3500 m), M2 (4000 m, 3500 m) and M3 (6000 m, 3500 m) are monitoring well locations.
K. Sun, M.N. Goltz / Environmental Modelling & Software 86 (2016) 50e55
3. Method verification A numerical model was constructed to assist in verifying the direct estimation method. The model simulates an aquifer system measuring 7000 m 7000 m, and consisting of an upper unconfined aquifer with a saturated thickness of 40 me60 m averaging at 50 m, a lower confined aquifer of 50 m thick, and an aquitard of 5 m thick separating the two aquifers. Boundary conditions for the two aquifers are identical including constant head boundaries at the upstream and downstream ends with respective head values of 110 m and 90 m, and two no-flow side boundaries (Fig. 1). A homogeneous and isotropic hydraulic conductivity of 50 m/day is assumed for both aquifers, and for the aquitard, an extremely small value of 1 1012 m/day is assumed so that the model essentially simulates no hydraulic communication between the two aquifers. With these assignments, both aquifers have a uniform gradient of 0.002857 m/m and a total discharge per unit width of aquifer of 7.14 m2/day. Both aquifers have two fully-screened pumping wells, P1 and P2 operating at 4000 m3/day each, and three monitoring wells M1, M2 and M3. The model was implemented using the finite element modeling software FEFLOW (Diersch, 2014). The model
53
domain is laterally discretized into 198,864 triangular elements with 99,931 nodes. With each aquifer represented by one model layer and the aquitard represented by two layers, the four-layered model consists of 499,655 nodes and 795,456 elements. For each aquifer, the three monitoring wells were paired and the simulated heads, distances to the pumping wells, and pumping rates were processed to form a system of two equations according for confined aquifers as presented in equation (3), and for unconfined aquifers as presented in equation (6), and subsequently solved. For the confined aquifer, the estimation results are T ¼ 2587.42 m2/day, K ¼ 51.75 m/day, and ǭ ¼ 7.31 m2/day, slightly larger than their corresponding true values of 2500 m2/day, 50.0 m/ day and 7.14 m2/day. The gradient, i, calculated as the ratio between ǭ and T, is 0.002824, slightly smaller than its true value of 0.002857. Similar results were obtained for the unconfined aquifer, with K ¼ 51.79 m/day, and ǭ ¼ 7.31 m2/day; i ¼ 0.002823 was calculated assuming a saturated thickness of 50 m. Recognizing that the aquifer system with a finite domain simulated by the model violates the assumption of a laterally infinite flow system of the state functions, corrections were made by including image wells to account for boundary effects (Bear,
Fig. 2. The Gardena site.
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K. Sun, M.N. Goltz / Environmental Modelling & Software 86 (2016) 50e55
Table 1 Transmissivity and hydraulic conductivity values estimated from pumping tests at the Gardena site. Well ID
Test date
Test type
Test duration (min)
Thickness (m)
T (m2/day)
K (m/day)
E-9A E-9A E-10A E-12 M-9 M-17
Oct-06 Apr-11 Apr-11 Apr-11 May-88 May-88
pumping Specific capacity Specific capacity Specific capacity Pumping pumping
150 5 20 11 720 720
7.0 7.5 7.8 7.7 7.0 7.0
1.0e24.5 20.2 29.7 21.9 8.6e10.1 13.3e15.2
0.14e3.5 2.7 3.8 2.8 1.2e1.4 1.9e2.2
T: transmissivity; K: hydraulic conductivity.
Table 2 Aquifer parameters for the Gardena site estimated using the direct method. Well set
ǭ (m2/day)
T (m2/day)
K (m/day)
i
MW-15A, MW-1, MW-2 MW-20A, 18A, W-1 MW-4, MW-2, MW-5 W-1, W-3, W-4 MW-21, W-3, W-4 M-7, MW-15A, MW-5 M-7, MW-18A, W-4 M-7, MW-1, MW-2 MW-18A, MW-19A, MW-20A MW-19A, MW-20A, MW-2
0.057 0.122 0.129 0.118 0.147 0.084 0.136 0.057 0.099 0.125
11.3 17.2 25.0 14.7 20.1 14.6 23.5 11.2 12.1 13.7
1.6 2.5 3.6 2.1 2.9 2.1 3.4 1.6 1.7 2.0
0.0050 0.0071 0.0052 0.0080 0.0073 0.0058 0.0058 0.0051 0.0082 0.0092
Max Mean Min
0.147 0.107 0.057
25.0 16.3 11.2
3.6 2.3 1.6
0.0092 0.0067 0.0050
Table 3 Heterogeneous hydraulic conductivity field statistics (m/day). Maximum hydraulic conductivity Minimum hydraulic conductivity Mean hydraulic conductivity Hydraulic conductivity standard deviation
382 6.99 55.7 27.1
1972). After the correction, the accuracy of the estimation is improved, with the following estimated parameter values: T ¼ 2504.84 m2/day, K ¼ 50.10 m/day, ǭ ¼ 7.20 m2/day, and i ¼ 0.002876 for the confined aquifer; K ¼ 50.14 m/day, ǭ ¼ 7.21 m2/ day, and i ¼ 0.002875 for the unconfined aquifer. The remaining small errors are attributable to numerical errors. The direct estimation method is thus verified. 4. Application of the direct estimation method at an environmental site The direct estimation method was applied at an environmental site located in the City of Gardena, California, the United States. The Gardena site was a former manufacturing facility for gas furnace control valves operated from 1953 to 1991 (Fig. 2). Site investigation
commencing in 1984 detected volatile organic compound (VOC) contamination in groundwater. The current groundwater treatment system was constructed in 1990 and has been operational since 1991. The aquifer impacted by VOC contamination at the site is the semiperched aquifer described as a semiconfined aquifer extending to approximately 9.1e13.7 m (30e45 feet) deep where the Bellflower aquiclude, a competent vertical flow barrier, is encountered. Groundwater in the aquifer occurs at depths of 5.2e7.6 m (17e25 feet) and flows to the south-southeast. The aquifer is composed of highly uniform silty sand with an estimated saturated thickness of approximately 6.1e7.6 m (20e25 ft). Pumping tests previously conducted on selected wells yielded a transmissivity ranging from 1.0 to 29.7 m2/day and a hydraulic conductivity ranging from 0.14 to 3.8 m/day (Table 1). The Gardena Site is monitored on a quarterly basis. The site condition in the second quarter of 2009 was chosen to demonstrate the use of the direct estimation method. Assuming a groundwater flow direction of E70 S, the extraction rates of the containment wells and the pizometric heads at the monitoring wells in May 2009 (Fig. 2) were used to solve equation (3). Ten well sets each consisting of three monitoring wells were formed, resulting in ten different solutions (Table 2). ǭ: total discharge per unit width of aquifer; T: transmissivity; K: hydraulic conductivity; i: gradient; T and ǭ are estimated by the direct estimation method; K is calculated from the estimated T assuming a saturated thickness of 7.0 m; i is calculated as the ratio between the estimated ǭ and T. The estimated values for T and K (means of 16.3 m2/day and 2.3 m/day, respectively) compare well with the corresponding values derived from the aquifer tests (1.0e29.7 m2/day and 0.14e3.8 m/day, respectively). The small ranges of the T and K values estimated by the direct method are also consistent with the aquifer sediments described as highly uniform silty sand. The mean value of the estimated K of 2.3 m/day can be regarded as an estimate of the effective K for the Gardena site. i, calculated as the ratio between ǭ and T, represents the gradient of groundwater flow in the aquifer before pumping. The Darcy groundwater flux is determined as the product of K and i. If groundwater contamination is of
Table 4 Parameter estimates for heterogeneous aquifers using the direct method. Parameters
K (m/day)*
i
Ǭ (m2/day)
Values used in numerical model Direct method confined aquifer estimates QP1 ¼ QP2 ¼ 4000 m3/day QP1 ¼ QP2 ¼ 6000 m3/day QP1 ¼ QP2 ¼ 8000 m3/day Direct method unconfined aquifer estimates QP1 ¼ QP2 ¼ 4000 m3/day QP1 ¼ QP2 ¼ 6000 m3/day QP1 ¼ QP2 ¼ 8000 m3/day
55.7
0.00286
7.96
56.9 56 55.6
0.00285 0.00286 0.00287
8.11 8 7.96
57 56.1 55.7
0.00285 0.00286 0.00286
8.11 8.01 7.97
*Mean hydraulic conductivity.
K. Sun, M.N. Goltz / Environmental Modelling & Software 86 (2016) 50e55
concern, the product of Darcy groundwater flux and contaminant concentration is the mass flux of contaminant, a measure that has recently gained significant attention (Brooks et al., 2008; ITRC, 2010; Goltz et al., 2009; Sun, 2015). Recall that the value of i calculated by the direct method is the background regional hydraulic gradient prior to pumping. This calculated value can be compared to the gradient determined by measuring piezometric heads to reveal the change in the aquifer flow regime in response to the pumping stresses. For example, using the well set consisting of M-7, MW-18A, and W-4, the direct method estimates an i of 0.0058. In comparison, the gradient calculated using the piezometric heads measured at M-7 and W-4 is 0.0052. The pumping activity leads to a slightly reduced gradient.
5. Heterogeneous conditions To see how violation of the assumptions used to develop the direct method affects results, the numerical model described in Section 3 (Fig. 1) was revised by replacing the homogeneous hydraulic conductivity field in the unconfined and confined aquifers with heterogeneous hydraulic conductivity fields. The heterogeneous fields were generated using PMWIN software (Chiang, 2005). The horizontal hydraulic conductivity used in the simulation is lognormally distributed with a mean log-transformed value of 1.7, a variance of 0.2 and a correlation length of 7 m. The statistics of the K field are in Table 3. Assuming the mean hydraulic conductivity is the best estimate of conductivity (Zhang et al., 2007), the parameters that are being used by the numerical model, and which will be estimated by the direct method, are listed in the first row of Table 4. Three simulations were run in each of the aquifers. In each simulation, wells P1 and P2 were pumped at the same rate. Estimates of the aquifer parameters determined by the direct method for each of the three simulated pumping rates for both aquifers are shown in Table 4. As may be seen, the estimates are quite close to the “actual” parameter value used in the numerical model, with the quality of the estimate improving with increased pumping rates.
6. Conclusions The three demonstration examples show that the direct estimation method is capable of accurately estimating hydraulic conductivity and background regional hydraulic gradient in aquifers with steady state flow. These parameters can be used to construct a site model that will support decision making (e.g., decide on the placement and pumping rate of wells to control migration of a contaminant plume). Further study beyond the examples presented above is needed to determine the limits of applicability of the method. Areas for future investigation might include sensitivity analyses to evaluate the effect on the estimate of extraction well pumping rate(s) in relation to aquifer flow rate (as could be quantified by the capture zone width of one or multiple extraction wells) and the effect on the estimate of extraction/monitoring well locations. Given its simplicity and its ability to estimate important hydraulic parameters from historical data that are often available at a site, without the need to conduct a pumping test, which is the “goto” site characterization approach for practitioners, the direct estimation method proposed here adds a valuable alternative method to quantitatively assess groundwater flow.
55
Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.envsoft.2016.09.014.
References Interstate Technology & Regulatory Council (ITRC), 2010. Use and Measurement of Mass Flux and Mass Discharge. MASSFLUX-1. Interstate Technology & Regulatory Council, Integrated DNAPL Site Strategy Team, Washington, D.C. www. itrcweb.org. Batu, V., 1998. Aquifer Hydraulics: a Comprehensive Guide to Hydrogeologic Data Analysis. Willey-Interscience. Bear, J., 1972. Dynamics of Fluids in Porous Media. American Elsevier Publishing Company Inc., New York. Bear, J., 1979. Hydraulics of Groundwater. McGraw-Hill, New York. Brooks, M.C., et al., 2008. Changes in contaminant mass discharge from DNAPL source mass depletion: evaluation at two field sites. J. Contam. Hydrol. 102, 140e153. Capriotti, J., Li, Y., 2015. Inversion for permeability distribution from time-lapse gravity data. Geophysics 80, WA69eWA83. http://dx.doi.org/10.1190/geo20140203.1. Chiang, W.H., 2005. 3D-Groundwater Modeling with PMWIN, second ed. SpringerVerlag, Berlin. Clarke, R., Lawrence, A., Foster, S., 1996. Groundwater: a Threatened Resource. (United Nations Environment Programme Environment Library No. 15, Nairobi, Kenya). Diersch, Hans-Joerg, 2014. FEFLOW e Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media. Springer, Berlin Heidelberg. http://dx.doi.org/10.1007/978-3-642-38739-5. Domenico, P.A., Schwartz, F.W., 1998. Physical and Chemical Hydrogeology, second ed. John Wiley and Sons, New York. Goltz, M.N., et al., 2009. Validation of two innovative methods to measure contaminant mass flux in groundwater. J. Contam. Hydrol. 106, 51e61 (2009). Hinnell, A.C., et al., 2010. Improved extraction of hydrologic information from geophysical data through coupled hydrogeophysical inversion. Water Resour. Res. 46, W00D40. http://dx.doi.org/10.1029/2008WR007060. Javendal, J., Doughty, C., Tsang, C.F., 1984. Groundwater Transport: Handbook of Mathematical Models, Water Resources Monograph 10. American Geophysical Union, Washington, DC. Konikow, L.F., 2011. Contribution of global groundwater depletion since 1900 to sea level rise. Geophys. Res. Lett. 38, L17401. http://dx.doi.org/10.1029/ 2011GL048604. Kruseman, G., de Ridder, N., 1994. Analysis and Evaluation of Pumping Test Data. 2nd Edition. Volume 47. International Institute for Land Reclamation and Improvement. The Netherland, Wageningen. Li, W., Nowak, W., Cirpka, O.A., 2005. Geostatistical inverse modeling of transient pumping tests using temporal moments of drawdown. Water Resour. Res. 41, W08403. http://dx.doi.org/10.1029/2004WR003874. Li, W., Englert, A., Olaf, A.C., Vereecken, H., 2008. Three-dimensional geostatistical inversion of flowmeter and pumping test data. Ground Water 46, 193e201. http://dx.doi.org/10.1111/j.1745-6584.2007.00419.x. Paillet, F.L., 1998. Flow modeling and permeability estimation using borehole flow logs in heterogeneous fractured formations. Water Resour. Res. 34, 997e1010. http://dx.doi.org/10.1029/98WR00268. Paradis, D., Lefebvre, R., Morin, R.H., Gloaguen, E., 2011. Permeability profiles in granular aquifers using flowmeters in direct-push wells. Ground Water 49, 534e557. http://dx.doi.org/10.1111/j.1745-6584.2010.00761.x. Richey, A.S., et al., 2015. Quantifying renewable groundwater stress with GRACE. Water Resour. Res. 51, 5217e5238. http://dx.doi.org/10.1002/2015WR017349. Singha, K., Day-Lewis, F.D., Johnson, T., Slater, L.D., 2015. Advances in interpretation of subsurface processes with time-lapse electrical imaging. Hydrol. Process. 29 (6), 1549e1576. Sun, K., 2015. Numerical study identifying the factors causing the significant underestimation of the specific discharge rate estimated using the modified integral pumping test method in a laboratory experiment. J. Contam. Hydrol. 180, 1e11. http://dx.doi.org/10.1016/j.jconhyd.2015.07.001. Taylor, R.G., et al., 2013. Ground water and climate change. Nat. Clim. Change 3, 322e329. http://dx.doi.org/10.1038/nclimate1744. Wada, Y., et al., 2010. Global depletion of groundwater resources. Geophys. Res. Lett. 37, L20402. http://dx.doi.org/10.1029/2010GL044571. Werner, A., Gleeson, T., 2012. Regional strategies for the accelerating global problem of groundwater depletion. Nat. Geosci. 5, 853e861. http://dx.doi.org/10.1038/ ngeo1617. Zhang, Y., Person, M., Gable, C.W., 2007. Representative hydraulic conductivity of hydrogeologic units: insights from an experimental stratigraphy. J. Hydrology 339, 65e78.
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Direct estimation of hydraulic parameters relating to steady state groundwater flow Kerang Sun a, *, Mark N. Goltz b a b
CH2M HILL, Inc., 6 Hutton Center Drive, Santa Ana, CA 92707, United States Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, United States
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 January 2016 Received in revised form 26 July 2016 Accepted 13 September 2016
Groundwater as an important, life-sustaining resource for humankind is being threatened by massive over-extraction and wide-spread contamination. Wise development and protection of this crucial resource requires a thorough understanding of groundwater flow in the subsurface. This paper presents a novel direct method to estimate important hydraulic parameters characterizing steady state groundwater flows for confined and unconfined isotropic aquifers. The method is appropriate for application in aquifers where horizontal flow dominates. The governing equations for the direct estimation method are superposed extensions of the well-known Thiem equation governing steady-state radial flow toward a pumping well under confined and unconfined conditions. This new approach has the following advantages over conventional methods: (1) simultaneously provides estimates of both hydraulic conductivity and hydraulic gradient, (2) can be applied using historical data without the need to conduct a pumping test, and (3) is a simple analytical method that can be applied easily. Verification of the direct estimation method is achieved by applying it to hypothetical homogeneous and heterogeneous aquifers simulated by three-dimensional finite element models. The usefulness of the method is also demonstrated with data from an actual field site. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Two-dimensional groundwater flow Hydraulic conductivity Transmissivity Superposition Darcy's law Hydraulic gradient Confined aquifer Unconfined aquifer Pumping test Steady-state flow
1. Introduction Groundwater is the largest fresh water body on the Earth and supplies drinking water to approximately two billion people worldwide (Richey et al., 2015). It may be the only source of water in some arid regions (Clarke et al., 1996). To protect this crucial lifesustaining resource from the progressively escalating crisis of aquifer depletion (Clarke et al., 1996; Richey et al., 2015; Konikow, 2011; Wada et al., 2010; Werner and Gleeson, 2012), the increasingly frequent occurrence of aquifer contamination (Clarke et al., 1996), and the potential impacts of climate change (Taylor et al., 2013), we must have detailed knowledge of aquifer hydrogeologic properties. Such knowledge is obtained through aquifer characterization which typically requires substantial field efforts. Given the complex nature of subsurface flow along with financial and technological limitations, it is often difficult or impossible to
* Corresponding author. E-mail addresses: [email protected] (M.N. Goltz).
(K.
http://dx.doi.org/10.1016/j.envsoft.2016.09.014 1364-8152/© 2016 Elsevier Ltd. All rights reserved.
Sun),
mark.goltz@afit.edu
generate sufficient data from field testing to adequately characterize the subsurface in order to construct hydrogeological models that will support good management decision making. Hydraulic conductivity and hydraulic gradient are the key parameters needed to model groundwater flow. Aquifer pumping tests are conventionally used to obtain estimates of hydraulic conductivity, with piezometers used to determine gradient. A pumping test involves extracting groundwater at known rates and measuring the hydraulic responses at selected locations. The response data at monitoring wells are analyzed to estimate hydraulic conductivity values. Various deterministic and stochastic models have been developed to suit different pumping and aquifer conditions (Batu, 1998; Kruseman and de Ridder, 1994; Li et al., 2005). While other techniques such as geophysical surveys (Capriotti and Li, 2015; Hinnell et al., 2010; Singha et al., 2015), borehole flow meters (Li et al., 2008; Paillet, 1998; Paradis et al., 2011), and empirical estimation methods (Domenico and Schwartz, 1998) have been applied to indirectly determine aquifer hydraulic conductivity, pumping tests remain the prevailing characterization method. One advantage of pumping test is that it interrogates a relatively large subsurface volume; thus, the estimate
K. Sun, M.N. Goltz / Environmental Modelling & Software 86 (2016) 50e55
of hydraulic conductivity obtained from a pumping test is a largescale volume-average. However, conducting a pumping test is by no means a trivial task. Firstly, it costs both time and money to plan and execute a test. Secondly, there are situations where a meaningful pumping test cannot be conducted. For example, aquifers with low conductivity may not be able to sustain a large enough pumping rate to induce measureable drawdown at monitoring wells. Lastly, there are often regulatory and economic hurdles imposed by the need to dispose of the potentially contaminated water that was extracted during the test. An alternative method that can be applied to directly estimate aquifer hydraulic conductivity while avoiding pumping tests would be of great benefit. Here we present a novel approach to quantitatively estimate hydraulic conductivity and regional hydraulic gradient without the need to conduct a dedicated pumping test. So long as long term pumping data from at least one pumping well and monitoring data from at least three observation wells are available, the method can simultaneously obtain estimates of hydraulic conductivity and gradient of the background flow in the aquifer. While the method has the advantage of sampling a large subsurface volume, it avoids the disadvantages of pumping tests discussed above. 2. Method development The equations upon which the direct estimation method is based require a number of simplifying assumptions. The closer these assumptions come to describing the real physical system, the more appropriate it is to apply the direct estimation method to that system. The simplifying assumptions are: 1. Isotropic aquifer of infinite extent with a homogeneous hydraulic conductivity (K) field. 2. Steady-state flow conditions. 3. Two-dimensional flow. In the case of a confined aquifer, the aquifer thickness (b) is constant. In the case of an unconfined aquifer, the Dupuit assumptions hold. 4. Uniform, regional flow. In the case of a confined aquifer, constant Darcy velocity (q ¼ KVh ¼ Ki) where h is the hydraulic head and i ¼ Vh is the regional hydraulic gradient. In the case of an unconfined aquifer, constant total discharge per unit width of aquifer (Q ¼ K2 Vðh2 Þ ¼ KhVhÞ) For a confined aquifer, by applying Darcy's Law and superposition (Bear, 1979), the following expression may be obtained, which relates the difference in head at two monitoring well locations (Dh) to aquifer characteristics (hydraulic conductivity (K) and aquifer thickness (b)), pumping well locations (x0j,y0j), pumping rates (Qj), and the regional hydraulic gradient prior to pumping (i):
Dh ¼ i½ðx2 x1 Þcos a þ ðy2 y1 Þsin a
2 2 n x2 x0j þ y2 y0j 1 X þ Q ln 2 2 4pbK j¼1 j x1 x0j þ y1 y0j
disadvantages that were noted above. Equation (1) has also been derived using complex potential theory (Javendal et al., 1984). Defining transmissivity (T) as the product of hydraulic conductivity (K) and aquifer thickness (b), equation (1) can be rearranged to obtain:
T¼
where.Qj is the pumping rate of the jth well.(x0j,y0j) are the x and y coordinates of the jth pumping well.a is the angle in radians between the Darcy velocity vector and the positive x-axis.n is the number of pumping wells.(x1,y1) and (x2,y2) are the x and y coordinates of the two monitoring wells. Equation (1) has also been derived by Brooks et al. (2008), who used it as the foundation of a method, referred to as the Integral Pump Test (IPT) approach, to estimate contaminant flux. As the name implies, the Brooks et al. (2008) IPT approach involved conducting a number of pump tests that comes with all of the associated
iT½ðx2 x1 Þcos a þ ðy2 y1 Þsin a Dh 2 2 n x2 x0j þ y2 y0j 1 X Q ln þ 2 2 4pDh j¼1 j x1 x0j þ y1 y0j
(2)
If we're interested in the total discharge per unit width of aquifer prior to pumping (Q ) we see for a confined aquifer Q ¼ KbVh ¼ TVh ¼ Ti and we can rewrite equation (2) as:
T¼
Q ½ðx2 x1 Þcos a þ ðy2 y1 Þsin a Dh 2 2 n x2 x0j þ y2 y0j 1 X Q ln þ 2 2 4pDh j¼1 j x1 x0j þ y1 y0j
(3)
For an unconfined aquifer, so long as the Dupuit assumptions hold, Darcy's Law and superposition can again be applied to obtain (Bear, 1979):
Dh2 ¼ Vh2 ½ðx2 x1 Þcos a þ ðy2 y1 Þsin a
2 2 n x2 x0j þ y2 y0j 1 X þ Q ln 2 2 2pK j¼1 j x1 x0j þ y1 y0j
(4)
where Dh2 ¼ h22 h21 is the difference in the squares of the hydraulic heads measured relative to the aquifer bottom at two monitoring wells and Vh2 is the gradient of the square of the regional hydraulic head. Rearranging terms results in:
K¼
Vh2 K½ðx2 x1 Þcos a þ ðy2 y1 Þsin a
Dh2
2 2 x2 x0j þ y2 y0j þ Qj ln 2 2 2pDh2 j¼1 x1 x0j þ y1 y0j 1
n X
(5)
If we're interested in the total discharge per unit width of aquifer prior to pumping (Q ) we see for an unconfined aquifer Q ¼ K2 Vh2 ¼ KhVh and we can rewrite equation (5) as:
K¼ (1)
51
2Q ½ðx2 x1 Þcos a þ ðy2 y1 Þsin a
Dh2
2 2 x2 x0j þ y2 y0j þ Qj ln 2 2 2pDh2 j¼1 x1 x0j þ y1 y0j 1
n X
(6)
Note that equations (3) and (6) each have two unknowns. The unknowns in equation (3) are the transmissivity (T) and the total discharge per unit width of aquifer prior to pumping (Q ). The unknowns in equation (6) are the hydraulic conductivity (K) and the total discharge per unit width of aquifer prior to pumping (Q ). Once these unknowns are determined, for a confined aquifer, the hydraulic conductivity and the regional hydraulic gradient prior to pumping (i) can be determined by the following formulas, respectively:
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T b
(7a)
Q T
(7b)
K¼
i¼
For an unconfined aquifer, if an average aquifer depth is assumed, the regional hydraulic gradient prior to pumping (i) can be determined by equation (8):
i¼
Q Kb
(8)
To solve equations (3) and (6) for the unknowns, it is necessary to either have head data from at least three monitoring wells with one pumping scenario or head data from two monitoring wells with at least two pumping scenarios. Although the method requires data under pumping conditions, it is quite conceivable that useful historical data are available in many instances (for example, head data obtained during normal operation of agricultural or municipal supply wells), precluding the need to perform a dedicated pump test. It is noted that equations (1) and (4) are superposed extensions of the well-known Thiem equation governing steady-state radial flow toward a pumping well, where equation (1) is for confined flow and equation (4) for unconfined flow (Bear, 1979; Domenico and Schwartz, 1998).
The direct approach presented above is novel for several reasons. Firstly, although equations (1) and (4) are relatively straight forward extensions of well-known steady state flow models which were developed over 100 years ago (i.e., the Thiem equation superposed with Darcy's Law for the confined case and the Thiem equation superposed with the Dupuit-Forchheimer discharge formula for the unconfined case), the extension formulated in equation (4) for the unconfined case is, to our knowledge, new. Both extensions greatly expand the practical applicability of the well-known models. Secondly, the methodology of applying these formulas to simultaneously estimate both hydraulic conductivity and regional hydraulic gradient is new. Thirdly, the methodology is based upon application of historical data. The standard approach of most practitioners faced with constructing a site model is to conduct pump testing. If practitioners can be convinced of the usefulness of the direct estimation method, the considerable economic, regulatory, and practical consequences of pumping tests can be avoided. To summarize, the study is significant, as it presents a novel approach for practitioners to apply simple analytical formulas in order to use readily available historical data to help construct a site model that will facilitate decision making. In the next three sections, direct estimation method results are compared to results obtained from numerical modeling, and results from an actual site, to demonstrate the utility of the method in constructing a site model that can be used to make informed site management decisions.
Fig. 1. Model configuration with contours (m) of the simulated water table. P1 (3500 m, 3500 m) and P2 (3500 m, 4000 m) are pumping well locations, with each well pumping at 4000 m3/day; M1 (3550 m, 3500 m), M2 (4000 m, 3500 m) and M3 (6000 m, 3500 m) are monitoring well locations.
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3. Method verification A numerical model was constructed to assist in verifying the direct estimation method. The model simulates an aquifer system measuring 7000 m 7000 m, and consisting of an upper unconfined aquifer with a saturated thickness of 40 me60 m averaging at 50 m, a lower confined aquifer of 50 m thick, and an aquitard of 5 m thick separating the two aquifers. Boundary conditions for the two aquifers are identical including constant head boundaries at the upstream and downstream ends with respective head values of 110 m and 90 m, and two no-flow side boundaries (Fig. 1). A homogeneous and isotropic hydraulic conductivity of 50 m/day is assumed for both aquifers, and for the aquitard, an extremely small value of 1 1012 m/day is assumed so that the model essentially simulates no hydraulic communication between the two aquifers. With these assignments, both aquifers have a uniform gradient of 0.002857 m/m and a total discharge per unit width of aquifer of 7.14 m2/day. Both aquifers have two fully-screened pumping wells, P1 and P2 operating at 4000 m3/day each, and three monitoring wells M1, M2 and M3. The model was implemented using the finite element modeling software FEFLOW (Diersch, 2014). The model
53
domain is laterally discretized into 198,864 triangular elements with 99,931 nodes. With each aquifer represented by one model layer and the aquitard represented by two layers, the four-layered model consists of 499,655 nodes and 795,456 elements. For each aquifer, the three monitoring wells were paired and the simulated heads, distances to the pumping wells, and pumping rates were processed to form a system of two equations according for confined aquifers as presented in equation (3), and for unconfined aquifers as presented in equation (6), and subsequently solved. For the confined aquifer, the estimation results are T ¼ 2587.42 m2/day, K ¼ 51.75 m/day, and ǭ ¼ 7.31 m2/day, slightly larger than their corresponding true values of 2500 m2/day, 50.0 m/ day and 7.14 m2/day. The gradient, i, calculated as the ratio between ǭ and T, is 0.002824, slightly smaller than its true value of 0.002857. Similar results were obtained for the unconfined aquifer, with K ¼ 51.79 m/day, and ǭ ¼ 7.31 m2/day; i ¼ 0.002823 was calculated assuming a saturated thickness of 50 m. Recognizing that the aquifer system with a finite domain simulated by the model violates the assumption of a laterally infinite flow system of the state functions, corrections were made by including image wells to account for boundary effects (Bear,
Fig. 2. The Gardena site.
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Table 1 Transmissivity and hydraulic conductivity values estimated from pumping tests at the Gardena site. Well ID
Test date
Test type
Test duration (min)
Thickness (m)
T (m2/day)
K (m/day)
E-9A E-9A E-10A E-12 M-9 M-17
Oct-06 Apr-11 Apr-11 Apr-11 May-88 May-88
pumping Specific capacity Specific capacity Specific capacity Pumping pumping
150 5 20 11 720 720
7.0 7.5 7.8 7.7 7.0 7.0
1.0e24.5 20.2 29.7 21.9 8.6e10.1 13.3e15.2
0.14e3.5 2.7 3.8 2.8 1.2e1.4 1.9e2.2
T: transmissivity; K: hydraulic conductivity.
Table 2 Aquifer parameters for the Gardena site estimated using the direct method. Well set
ǭ (m2/day)
T (m2/day)
K (m/day)
i
MW-15A, MW-1, MW-2 MW-20A, 18A, W-1 MW-4, MW-2, MW-5 W-1, W-3, W-4 MW-21, W-3, W-4 M-7, MW-15A, MW-5 M-7, MW-18A, W-4 M-7, MW-1, MW-2 MW-18A, MW-19A, MW-20A MW-19A, MW-20A, MW-2
0.057 0.122 0.129 0.118 0.147 0.084 0.136 0.057 0.099 0.125
11.3 17.2 25.0 14.7 20.1 14.6 23.5 11.2 12.1 13.7
1.6 2.5 3.6 2.1 2.9 2.1 3.4 1.6 1.7 2.0
0.0050 0.0071 0.0052 0.0080 0.0073 0.0058 0.0058 0.0051 0.0082 0.0092
Max Mean Min
0.147 0.107 0.057
25.0 16.3 11.2
3.6 2.3 1.6
0.0092 0.0067 0.0050
Table 3 Heterogeneous hydraulic conductivity field statistics (m/day). Maximum hydraulic conductivity Minimum hydraulic conductivity Mean hydraulic conductivity Hydraulic conductivity standard deviation
382 6.99 55.7 27.1
1972). After the correction, the accuracy of the estimation is improved, with the following estimated parameter values: T ¼ 2504.84 m2/day, K ¼ 50.10 m/day, ǭ ¼ 7.20 m2/day, and i ¼ 0.002876 for the confined aquifer; K ¼ 50.14 m/day, ǭ ¼ 7.21 m2/ day, and i ¼ 0.002875 for the unconfined aquifer. The remaining small errors are attributable to numerical errors. The direct estimation method is thus verified. 4. Application of the direct estimation method at an environmental site The direct estimation method was applied at an environmental site located in the City of Gardena, California, the United States. The Gardena site was a former manufacturing facility for gas furnace control valves operated from 1953 to 1991 (Fig. 2). Site investigation
commencing in 1984 detected volatile organic compound (VOC) contamination in groundwater. The current groundwater treatment system was constructed in 1990 and has been operational since 1991. The aquifer impacted by VOC contamination at the site is the semiperched aquifer described as a semiconfined aquifer extending to approximately 9.1e13.7 m (30e45 feet) deep where the Bellflower aquiclude, a competent vertical flow barrier, is encountered. Groundwater in the aquifer occurs at depths of 5.2e7.6 m (17e25 feet) and flows to the south-southeast. The aquifer is composed of highly uniform silty sand with an estimated saturated thickness of approximately 6.1e7.6 m (20e25 ft). Pumping tests previously conducted on selected wells yielded a transmissivity ranging from 1.0 to 29.7 m2/day and a hydraulic conductivity ranging from 0.14 to 3.8 m/day (Table 1). The Gardena Site is monitored on a quarterly basis. The site condition in the second quarter of 2009 was chosen to demonstrate the use of the direct estimation method. Assuming a groundwater flow direction of E70 S, the extraction rates of the containment wells and the pizometric heads at the monitoring wells in May 2009 (Fig. 2) were used to solve equation (3). Ten well sets each consisting of three monitoring wells were formed, resulting in ten different solutions (Table 2). ǭ: total discharge per unit width of aquifer; T: transmissivity; K: hydraulic conductivity; i: gradient; T and ǭ are estimated by the direct estimation method; K is calculated from the estimated T assuming a saturated thickness of 7.0 m; i is calculated as the ratio between the estimated ǭ and T. The estimated values for T and K (means of 16.3 m2/day and 2.3 m/day, respectively) compare well with the corresponding values derived from the aquifer tests (1.0e29.7 m2/day and 0.14e3.8 m/day, respectively). The small ranges of the T and K values estimated by the direct method are also consistent with the aquifer sediments described as highly uniform silty sand. The mean value of the estimated K of 2.3 m/day can be regarded as an estimate of the effective K for the Gardena site. i, calculated as the ratio between ǭ and T, represents the gradient of groundwater flow in the aquifer before pumping. The Darcy groundwater flux is determined as the product of K and i. If groundwater contamination is of
Table 4 Parameter estimates for heterogeneous aquifers using the direct method. Parameters
K (m/day)*
i
Ǭ (m2/day)
Values used in numerical model Direct method confined aquifer estimates QP1 ¼ QP2 ¼ 4000 m3/day QP1 ¼ QP2 ¼ 6000 m3/day QP1 ¼ QP2 ¼ 8000 m3/day Direct method unconfined aquifer estimates QP1 ¼ QP2 ¼ 4000 m3/day QP1 ¼ QP2 ¼ 6000 m3/day QP1 ¼ QP2 ¼ 8000 m3/day
55.7
0.00286
7.96
56.9 56 55.6
0.00285 0.00286 0.00287
8.11 8 7.96
57 56.1 55.7
0.00285 0.00286 0.00286
8.11 8.01 7.97
*Mean hydraulic conductivity.
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concern, the product of Darcy groundwater flux and contaminant concentration is the mass flux of contaminant, a measure that has recently gained significant attention (Brooks et al., 2008; ITRC, 2010; Goltz et al., 2009; Sun, 2015). Recall that the value of i calculated by the direct method is the background regional hydraulic gradient prior to pumping. This calculated value can be compared to the gradient determined by measuring piezometric heads to reveal the change in the aquifer flow regime in response to the pumping stresses. For example, using the well set consisting of M-7, MW-18A, and W-4, the direct method estimates an i of 0.0058. In comparison, the gradient calculated using the piezometric heads measured at M-7 and W-4 is 0.0052. The pumping activity leads to a slightly reduced gradient.
5. Heterogeneous conditions To see how violation of the assumptions used to develop the direct method affects results, the numerical model described in Section 3 (Fig. 1) was revised by replacing the homogeneous hydraulic conductivity field in the unconfined and confined aquifers with heterogeneous hydraulic conductivity fields. The heterogeneous fields were generated using PMWIN software (Chiang, 2005). The horizontal hydraulic conductivity used in the simulation is lognormally distributed with a mean log-transformed value of 1.7, a variance of 0.2 and a correlation length of 7 m. The statistics of the K field are in Table 3. Assuming the mean hydraulic conductivity is the best estimate of conductivity (Zhang et al., 2007), the parameters that are being used by the numerical model, and which will be estimated by the direct method, are listed in the first row of Table 4. Three simulations were run in each of the aquifers. In each simulation, wells P1 and P2 were pumped at the same rate. Estimates of the aquifer parameters determined by the direct method for each of the three simulated pumping rates for both aquifers are shown in Table 4. As may be seen, the estimates are quite close to the “actual” parameter value used in the numerical model, with the quality of the estimate improving with increased pumping rates.
6. Conclusions The three demonstration examples show that the direct estimation method is capable of accurately estimating hydraulic conductivity and background regional hydraulic gradient in aquifers with steady state flow. These parameters can be used to construct a site model that will support decision making (e.g., decide on the placement and pumping rate of wells to control migration of a contaminant plume). Further study beyond the examples presented above is needed to determine the limits of applicability of the method. Areas for future investigation might include sensitivity analyses to evaluate the effect on the estimate of extraction well pumping rate(s) in relation to aquifer flow rate (as could be quantified by the capture zone width of one or multiple extraction wells) and the effect on the estimate of extraction/monitoring well locations. Given its simplicity and its ability to estimate important hydraulic parameters from historical data that are often available at a site, without the need to conduct a pumping test, which is the “goto” site characterization approach for practitioners, the direct estimation method proposed here adds a valuable alternative method to quantitatively assess groundwater flow.
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Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.envsoft.2016.09.014.
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