# A Preliminary Study on a Pumping Well Capturing Groundwater in an Unconfined Aquifer with Mountain-Front Recharge from Segmental Inflow

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## Abstract

**:**

## 1. Introduction

## 2. Conceptual and Mathematical Models of Groundwater Flow

#### 2.1. Conceptual Model with Simplifications

_{R}) across each segment is assumed to be equal and steady. In the natural state, groundwater divides are developed between different pluvial fans. In considering of the symmetry, a block of double pluvial fans is selected as an interested study area (Figure 2b) where the groundwater flow along Divide-2 could be affected by the pumping well.

_{w}.

#### 2.2. Mathematical Model and Lumped Parameters

^{−1}], h is the relative groundwater level [L], L is the distance between the upper and lower boundaries [L], B and D refer to the half-widths of a fan block and an inflow segment [L], respectively, along the x-direction, h

_{c}is the groundwater level on the discharge boundary [L], q

_{0}refers to the flow rate across per-unit width of the inflow segment [L

^{2}T

^{−1}], Q

_{R}is the total inflow from one segment [L

^{3}T

^{−1}], Q

_{w}is the pumping rate of the well [L

^{3}T

^{−1}], δ is the Dirac delta function [L

^{−1}], x

_{w}and y

_{w}are the coordinates of the well [L].

_{c}value is approximately determined as the effective thickness of the gravel to coarse sands limited by the clayey sediments (Figure 1b). The Q

_{R}value is specified from streamflow data of rivers in the eastern Qaidam Basin [17]. The Q

_{w}in fact is the total pumping rate of several wells that allocated in a relatively small area around the position of (x

_{w}, y

_{w}). Pumping wells were penetrated into the aquifer at a maximum depth around 100 m but can be plausibly considered as fully penetrating wells in this study. In the modeling investigation, we will check the effect of the well position by replace (x

_{w}, y

_{w}) in Table 1 with other values that are limited by B and L (Figure 2b).

_{c}= 100 m and changing the values of D, L, Q

_{R}, and Q

_{w}, as well as the x

_{w}and y

_{w}values. Then the results are expressed with dimensionless variables in Equation (6) and lumped parameters in Equation (7) to represent more general behaviors. Consequently, the result of the spatial distribution of groundwater level, h(x, y), is expressed as h

_{D}(x

_{D}, y

_{D}) to indicate a solution of Equation (8). Dimensionless results are useful for analyzing other sites with a different size and/or different physical parameters.

## 3. Numerical Methods

#### 3.1. Numerical Solution of Groundwater Flow

_{0}Δx is settled in each cell along this boundary, which is an equivalent implement of Equation (4). The pumping rate Q

_{w}is specified to a well-block including the place of (x

_{w}, y

_{w}).

^{3}/d. The grid resolution and the criteria have been checked to obtain accurate modeling results without excessive computational cost. The dimensionless result, h

_{D}(x

_{D}, y

_{D}), are obtained from the MODFLOW output, h(x, y), by using Equation (6).

#### 3.2. Particle Tracking Method for Streamlines and Travel Time

_{D}refers to the dimensionless travel length (=l/B). The travel time is an integral of that:

## 4. Results of the Catchment Zone

#### 4.1. General Shape and Classification

_{w}), which will reduce the area of the I zone and enlarge the area of the IV zone. When the λ value is small, like that shown in Figure 8a, the II and IV zones will be limited in small areas in the vicinity of the discharge boundary. While if the λ value is close or equal to 1.0, the I zone will significantly shrink to a narrow small area that encloses a short part of the Y-axis, as shown in Figure 8d. The IV zone extends to the X-axis in this situation, indicating that the constant head boundary could be connected with the catchment zone for a well pumped heavy even it is very close to the X-axis.

#### 4.2. Dependency of Shape Factors on Controls

_{1}whereas the source head width along the right segment is d

_{2}. The total length of the source head is defined as d

_{1}+ d

_{2}, which satisfies the following equation:

_{1}+ d

_{2}) and r/(d

_{1}+ d

_{2}).

_{wD}. It can be seen that w/d is positively correlated with the y

_{wD}value, almost in a linear manner. The r/d value is generally less than 1.0 and also increases with the decreasing α value as shown in Figure 10e, indicating that the distance from the well to the stagnation point is generally less than the width of source head. Similar to the relationship between w/d and y

_{wD}, r/d increases with the increasing y

_{wD}value, as shown in Figure 10f, however, the relationship becomes nonlinear when the λ value is large.

_{1}+ d

_{2}) value could be significantly larger than 1.0, indicating that the width of the catchment zone at the well center could be significantly larger than the effective width of the source head. Both Figure 11a,b show that w/(d

_{1}+ d

_{2}) increases with the decreasing α value in a nonlinear manner. The w/(d

_{1}+ d

_{2}) value also increases with the increasing λ value as shown in Figure 11a, however, it is not sensitive to the change in the β value as shown in Figure 11b. The relationship between w/(d

_{1}+ d

_{2}) and y

_{wD}is negative and nonlinear, as is clearly shown in Figure 11c,d. In particular, Figure 11c indicates that a smaller α value leads to a larger range of w/(d

_{1}+ d

_{2}) with respect to the same range of y

_{wD}. This effect seems can be also leaded by the change in the λ value, as shown in Figure 11d, whereas the impact is not significant. Figure 11e exhibits the negative nonlinear relationship between r/(d

_{1}+ d

_{2}) and α, where r/(d

_{1}+ d

_{2}) is less than 1.0 in most of the situations. The relationship between r/(d

_{1}+ d

_{2}) and y

_{wD}is a bit complex as shown in Figure 11f where r/(d

_{1}+ d

_{2}) does not simply increase with the decreasing y

_{wD}value but they could have a positive relationship when y

_{wD}is high, especially for situations of a large λ value. In particular, r/(d

_{1}+ d

_{2}) is not sensitive to the change in the λ value when y

_{wD}is sufficiently small (less than 0.4).

## 5. Travel Time Analyses for Capture Zones

#### 5.1. General Travel Time Distribution

_{D}= 1, 2, 3, etc.) in the map. In the type-I and type-II catchment zones shown in Figure 12a,b, the capture zone could be significantly stretched to the upstream area along the middle line between the sides of the catchment zone. In the type-III (Figure 12c) and type-IV (Figure 12d) catchment zones, this stretch effect also exists for relatively small travel times, whereas the capture zone will be stretched to the upstream area along the sides of the catchment zone for relatively large travel times, because the double source heads are not in the middle. The capture zone will be preferentially stretched toward a closer recharge boundary when the pumping well is not rightly located at Divide-2.

#### 5.2. The Relationship between the Travel Time and the Size Factor

_{D}, and the symmetry axis have an intersection point. R denotes the distance between this intersection point and the well. It certainly increases with the increasing t

_{D}as a function whereas the function is controlled by the well location and parameters.

_{D}can be expressed as

_{y}is the Darcy velocity [LT

^{−1}] in the Y-direction along the Y-axis. In the vicinity of the pumping well, the flow is almost in a radial form where the radial velocity (oriented toward the well), V

_{r}, depends on the radial distance from the well, r, as follows

_{w}− y| in the model. Accordingly, in the vicinity of the pumping well along the Y-axis, V

_{y}can be approximately estimated by

_{c}for small drawdown condition. Substituting Equations (16) and (21) into Equation (19), we have

_{D}can be checked by a formula as follows

_{1}+ d

_{2}) is used in the F() function to replace R/d.

_{D}/(αλ) versus R/d for capture zones in the type-I catchment. As indicated, γt

_{D}/(αλ) is almost linearly dependent on R/d. An increase in the α value will cause a decrease in the slope of the curve, as indicated in Figure 13b. The slope of curves also decreases with the increasing λ value, as shown in Figure 13c, but the response is not significant as that influenced by the α value. Figure 13d shows that the curves are not sensitive to y

_{wD}even the increase in y

_{wD}could increase the slope. It is clearly indicated that the relationship between γt

_{D}/(αλ) and R/d, i.e., the function F(R/d), is almost independent on λ and y

_{wD}when R/d is less than 1.0 but still significantly depends on α. Different curves of γt

_{D}/(αλ) versus R/(d

_{1}+ d

_{2}) are shown in Figure 14b–d for capture zones in the type-III catchment. Unlike those shown in Figure 13, these curves are significant nonlinear. When R/d is less than 1.0, the curves are close to each other and are not sensitive to the changes in α, λ and y

_{wD}. Otherwise the curves are significantly influenced by these parameters. Note that the intersection of the X-axis and Divide-2 is a stagnation point for the Type-III catchment, which limits the value of R/(d

_{1}+ d

_{2}) but leads to an infinite t

_{D}value because of the tiny V

_{y}value in the vicinity of the stagnation point. As a result, the curves are approximately vertical to the horizontal axis when R/(d

_{1}+ d

_{2}) is close to the maximum value. According to Equation (18), both α and λ increase d

_{1}+ d

_{2}so that the bound of R/(d

_{1}+ d

_{2}) decrease in Figure 14b,c, respectively, with the increasing α and λ values. However, an increase in the y

_{wD}value will enlarge the range of R and consequently increase the bound of R/(d

_{1}+ d

_{2}), as shown in Figure 14d.

## 6. Discussions on the Application

_{R}(λ < 1.0), the catchment will be a type-I catchment if the position is far enough away the discharge boundary and close enough to the Y-axis. When the position of the well is fixed, it would be a type-I catchment if the pumping rate is not too high. In the example of Figure 1a, when Q

_{w}is larger than 20 × 10

^{4}m

^{3}/d, the possibility of forming a type-III catchment is high, indicating that the pumping site should be moved westward to maintain a type-I catchment zone.

_{1}+ d

_{2}value of a type-II catchment are fixed according to Equations (17) and (18), respectively. Thus, the changes in the shape factors, w/d and r/d, for the type-I catchment, or w/(d

_{1}+ d

_{2}) and r/(d

_{1}+ d

_{2}) for the type-III catchment, can be considered to check the catchment size. The relationships between these shape factors and the parameters are shown in Figure 9 and Figure 10. Note that in the type-I catchment the w/d value increases with y

_{wD}, whereas in the type-III catchment the w/(d

_{1}+ d

_{2}) value decreases with y

_{wD}. The r/d and r/(d

_{1}+ d

_{2}) values could be nonlinearly dependent on y

_{wD}.

_{1}+ d

_{2}) value is recommended for the same security level to reduce the protection area. As indicated in Figure 13, R/d increases with the dimensionless travel time, γt

_{D}/(αλ), almost following a linear manner in the type-I catchment. However, R/(d

_{1}+ d

_{2}) nonlinearly increases with γt

_{D}/(αλ) in the type-III catchment with a limitation that positively depends on y

_{wD}, as shown in Figure 14. In application, one should notices the conversion between the dimensionless travel time t

_{D}and the real travel time, t, according to Equation (16).

_{c}would exist on a natural discharge boundary because of the unsteady groundwater flow. When the seasonal fluctuation of groundwater level in the natural state (generally less than 2 m at the site in Figure 1a) is significantly less than h

_{c}(>100 m at the site in Figure 1), the mean annual state could be adopted in analyses to represent a steady state flow. Vertical recharge to or discharge from water table should be also checked. At the site in Figure 1, the climate is extremely dry (mean annual precipitation is less than 50 mm, whereas mean potential evaporation is higher than 1500 mm) so that infiltration recharge could be ignored. Depth of water table in the pluvial fan area is generally larger than 10 m, which does not yield a significant loss of groundwater from evapotranspiration. The model in this study is false if there is a river flowing across the whole study area and providing persistent leakage recharge to groundwater. More comprehensive models are required to analyze capture zones for a complex pumping site, however, with more uncertain parameters.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- De Smedt, F. Analytical solution for capture and catchment zones of a well located on a groundwater divide. Water Resour. Res.
**2014**, 50, 736–740. [Google Scholar] [CrossRef] - Muskat, M.; Meres, M.W. The flow of homogeneous fluids through porous media. Physics
**1936**, 7, 346–363. [Google Scholar] [CrossRef] - Bear, J. Hydraulics of Groundwater; McGraw-Hill: New York, NY, USA, 1979; 567p. [Google Scholar]
- Javandel, I.; Tsang, C.F. Capture-zone type curves: A tool for aquifer cleanup. Groundwater
**1986**, 24, 616–625. [Google Scholar] [CrossRef] - Shan, C. An analytical solution for the capture zone of two arbitrarily located wells. J. Hydrol.
**1999**, 222, 123–128. [Google Scholar] [CrossRef] - Grubb, S. Analytical model for estimation of steady-state capture zones of pumping wells in confined and unconfined aquifers. Groundwater
**1993**, 31, 27–32. [Google Scholar] [CrossRef] - Newsom, J.M.; Wilson, J.L. Flow of ground water to well near a stream-effect of ambient ground-water flow direction. Groundwater
**1989**, 26, 703–711. [Google Scholar] [CrossRef] - Asadi-Aghbolaghi, M.; Rakhshandehroo, G.R.; Kompani-Zare, M. Analytical solutions for the capture zone of a pumping well near a stream. Hydrogeol. J.
**2011**, 19, 1161–1168. [Google Scholar] [CrossRef] - Intaraprasong, T.; Zhan, H. Capture zone between two streams. J. Hydrol.
**2007**, 338, 297–307. [Google Scholar] [CrossRef] - Samani, N.; Zarei-Doudeji, S. Capture zone of a multi-well system in confined and unconfined wedge-shaped aquifers. Adv. Water Resour.
**2012**, 39, 71–84. [Google Scholar] [CrossRef] - De Smedt, F. Analytical solution for the catchment zone of a well located near a groundwater divide in a recharged semi-confined aquifer. J. Hydrol.
**2014**, 519, 1271–1277. [Google Scholar] [CrossRef] - Simpson, M.J.; Clement, T.P.; Yeomans, F.E. Analytical model for computing residence times near a pumping well. Groundwater
**2003**, 41, 351–354. [Google Scholar] [CrossRef] - Chapuis, R.P.; Chesnaux, R. Travel time to a well pumping an unconfined aquifer without recharge. Groundwater
**2006**, 44, 600–603. [Google Scholar] [CrossRef] - Zhou, Y.X.; Haitjema, H. Approximate solutions for radial travel time and capture zone in unconfined aquifers. Groundwater
**2012**, 50, 799–803. [Google Scholar] [CrossRef] [PubMed] - Shafer, J.M. Reverse pathline calculation of time-related capture zones in nonuniform flow. Groundwater
**1987**, 25, 283–298. [Google Scholar] [CrossRef] - Pollock, D.W. Semianalytical computation of path lines for finite-difference models. Groundwater
**1988**, 26, 743–750. [Google Scholar] [CrossRef] - Wang, Y.; Guo, H.; Li, J.; Huang, Y.; Liu, Z.; Liu, C. Survey and Assessment on Groundwater Resources and Relevant Environmental Problems in the Qaidam Basin; Geological Publishing House: Beijing, China, 2008. (In Chinese) [Google Scholar]
- Wilson, J.L.; Guan, H. Mountain-block hydrology and mountain-front recharge. In Groundwater Recharge in a Desert Environment: The Southwestern United States; Phillips, F.M., Hogan, J., Scanlon, B., Eds.; The American Geophysical Union: Washington, DC, USA, 2004. [Google Scholar]
- McDonald, M.G.; Harbaugh, A.W. Techniques of Water-Resources Investigations of the United States Geological Survey Chapter A1: A Modular Three-Dimensional Finite Difference Ground-Water Flow Model; United States Government Printing Office: Washington, DC, USA, 1988.
- Harbaugh, A.W.; McDonald, M.G. User’s Documentation for MODFLOW-96, an Update to the U.S. Geological Survey Modular Finite-Difference Ground-Water Flow Model; Office of Ground Water U.S. Geological Survey: Reston, VA, USA, 1996; p. 56.
- Hill, M.C. Preconditioned Conjugate-Gradient 2 (PCG2): A Computer Programfor Solving Ground-Water Flow Equations; U.S. Geological Survey: Denver, CO, USA, 1990; p. 43.
- Pollock, D.W. Documentation of Computer Programs to Compute and Display Pathlines Using Results from the U.S. Geological Survey Modular Three-Dimensional Finite Difference Ground-Water Flow Model; U.S. Geological Survey: Reston, VA, USA, 1989.
- Pollock, D.W. User Guide for MODPATH Version 6—A Particle-Tracking Model for MODFLOW; U.S. Geological Survey: Reston, VA, USA, 2012.

**Figure 1.**Characteristics of pluvial fans in the Qaidam Basin: (

**a**) A satellite photo of a pluvial fan, with delineations of the groundwater inflow segment, discharge zone and potential catchment zone of a pumping well; (

**b**) A typical profile map of the general hydrogeological conditions (modified from [17]).

**Figure 2.**Conceptual model of the unconfined aquifer in a pluvial fan: (

**a**) uniform distribution of pluvial fans in the plan view; (

**b**) a representative area with a pumping well in the gray zone; (

**c**) the profile between P1 and P2.

**Figure 7.**General well-location zones with respect to different types (from I to IV) of the catchment zone.

**Figure 8.**Change in well-location zones with different values of λ when the other parameters are specified as α = 0.4, β = 1 and γ = 1: (

**a**) λ = 0.1; (

**b**); λ = 0.3; (

**c**): λ = 0.5; (

**d**): λ = 1.0.

**Figure 10.**Dependency of dimensionless shape factors on control parameters for x

_{wD}= 0 in the type-I catchment zone: (

**a**) Curves of w/d versus α with different λ values; (

**b**) Curves of w/d versus α with different β values; (

**c**) Curves of w/d versus y

_{wD}with different α values; (

**d**) Curves of w/d versus y

_{wD}with different λ values; (

**e**) Curves of r/d versus α with different λ values; (

**f**) Curves of r/d versus y

_{wD}with different λ values.

**Figure 11.**Dependency of dimensionless shape factors on control parameters for x

_{wD}= 1 in the type-III catchment zone:(

**a**) Curves of w/d versus α with different λ values; (

**b**) Curves of w/(d

_{1}+ d

_{2}) versus α with different β values; (

**c**) Curves of w/(d

_{1}+ d

_{2}) versus y

_{wD}with different α values; (

**d**) Curves of w/(d

_{1}+ d

_{2}) versus y

_{wD}with different λ values; (

**e**) Curves of r/(d

_{1}+ d

_{2}) versus α with different λ values; (

**f**) Curves of r/(d

_{1}+ d

_{2}) versus y

_{wD}with different λ values.

**Figure 12.**Travel time (t

_{D}, dimensionless) distribution in the catchment zone: (

**a**) type-I; (

**b**) type-II; (

**c**) type-III; (

**d**) type-IV. Dashed lines are the contours of t

_{D}. Parameter values are specified in the model as α = 0.4, β = 1, γ = 1, λ = 0.2.

**Figure 13.**The size factor of a capture zone (

**a**) with limited travel time in a type-I catchment for a well on the Y-axis and the curves of γt

_{D}/(αλ) versus R/d with varying α (

**b**), λ (

**c**) and y

_{wD}(

**d**).

**Figure 14.**The size factor of a capture zone (

**a**) with limited travel time in a Type-III catchment for a well on Divide-2 and the curves of γt

_{D}/(αλ) versus R/d with varying α (

**b**), λ (

**c**) and y

_{wD}(

**d**).

**Table 1.**Characteristics of the site in Figure 1a.

B (km) | D (km) | L (km) | h_{c} (m) |
---|---|---|---|

6–11 | 2–4 | 8–11 | 100–160 |

Q_{R} (×10^{4} m^{3}/d) | q_{0} (m^{2}/d) | K (m/d) | Porosity |

28–36 | 35–90 | 30–90 | 0.25–0.31 |

Q_{w} (×10^{4} m^{3}/d) | x_{w} | y_{w} | |

6.0–10.0 | 1.8 | 4.7 |

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## Share and Cite

**MDPI and ACS Style**

Li, H.; Wang, X.-s.
A Preliminary Study on a Pumping Well Capturing Groundwater in an Unconfined Aquifer with Mountain-Front Recharge from Segmental Inflow. *Water* **2019**, *11*, 1243.
https://doi.org/10.3390/w11061243

**AMA Style**

Li H, Wang X-s.
A Preliminary Study on a Pumping Well Capturing Groundwater in an Unconfined Aquifer with Mountain-Front Recharge from Segmental Inflow. *Water*. 2019; 11(6):1243.
https://doi.org/10.3390/w11061243

**Chicago/Turabian Style**

Li, Haixiang, and Xu-sheng Wang.
2019. "A Preliminary Study on a Pumping Well Capturing Groundwater in an Unconfined Aquifer with Mountain-Front Recharge from Segmental Inflow" *Water* 11, no. 6: 1243.
https://doi.org/10.3390/w11061243