Analytical solution based on stream-aquifer interactions in partially penetrating streams

Yong HUANG*, Zhi-fang ZHOU, Zhong-bo YU

1. College of Earth Science and Engineering, Hohai University, Nanjing 210098, P. R. China

2. College of Hydrology and Water Resources, Hohai University, Nanjing 210098, P. R. China

1 Introduction

Analysis of interaction between groundwater and surface water is a very complicated problem in the evaluation and management of water resources. There has been some research on analytical solutions for stream-aquifer or stream-lake systems. For example, an analytical solution for groundwater table drawdown and stream depletion, which incorporates conductance and stream partial penetration, was described by Hunt (1999)and Hunt et al.(2001), but the Hunt solution is approximate and assumes that the stream width is close to zero.After the development of the Hunt model, an analytical stream-aquifer model was developed to predict the drawdown in an aquifer (Darama 2001; Fox et al. 2002). The impact of groundwater pumping on nearby streams was described by Butler et al. (2007)using a new semi-analytical solution. An analytical model of groundwater discharge from an unconfined aquifer to a lake was also developed (Li and Wang 2007). In addition, an analytical model of stream-aquifer interaction that considers the effects of the stream stage on the hydraulic head was proposed(Zlotnik and Huang 1999; Szilagyi et al. 2006). The effects of fluctuating stream stage on the adjacent alluvial valley aquifer were examined with a new analytical solution by Srivastava et al. (2006). A two-dimensional semi-analytical solution was presented by Kim et al. (2007)in order to analyze stream-aquifer interactions in a coastal aquifer where groundwater level responds to tide. Intaraprasong and Zhan (2009)improved the analytical solution of stream-aquifer interaction by simultaneously considering temporally and spatially variable stream stages, low-permeability streambeds, and pumping wells near the streams. Analytical solutions have also been developed based on the assumptions of negligible drawdown in the source bed of a leaky aquifer and horizontal flow in an aquifer of infinite extent (Swamee et al.2000; Zlotnik and Tartakovsky 2008). Furthermore, Christensen et al. (2009, 2010)developed a series of two- and three-dimensional solutions to identify the validity and range of applicability of these assumptions.

The objectives of this study were to derive an analytical solution of drawdown caused by pumping in an aquifer partially penetrated by two streams, to compare the analytical solution with a previously accepted solution, and to describe the effects of two streams on drawdown caused by pumping.

2 Analytical models

2.1 Hunt’s model

Based on the Theis solution, which calculates the drawdown caused by pumping groundwater from an aquifer in hydraulic connection with a stream that is typically assumed to be completely penetrating, and the Hantush solution, which differs from the Theis solution only by the inclusion of a vertical layer of semipervious material along the stream boundary,Hunt (1999)developed an analytical solution considering the effects of streambed clogging and partial stream penetration, shown in Fig. 1.

Fig. 1 Hunt’s model (1999)

Hunt made four assumptions for the calculation of this solution: (1)The aquifer is homogeneous and isotropic in all horizontal directions and of infinite extent. (2)Drawdown caused by pumping is small compared with saturated aquifer thickness, and the thickness of semipervious material of the streambed is small compared to that of the saturated aquifer. (3)Changes of stream stage created by pumping are small compared with changes of the groundwater level in the aquifer. (4)The well flow rate is constant during pumping.

According to the assumptions, the governing equation for the Hunt model (1999)can be expressed as

where T is the transmissivity of the aquifer, which is the product of hydraulic conductivity K and the saturated thickness of the aquifer h; s is the drawdown caused by pumping groundwater; x and y are the coordinates within the infinite domain; μ is the specific yield for an unconfined aquifer or the storage coefficient for a confined aquifer; t is the time since the start of pumping; Q is the constant pumping rate; δ is the Dirac Delta function; L is the distance from the pumping well to the stream; and C is the leakage coefficient of streambed semipervious material. The leakage coefficient is expressed as a function of the streambed hydraulic conductivity Ksb, the stream width W, and the thickness of the streambed layer M:

The initial groundwater table is constant before pumping, and there is no drawdown at an infinite distance from the pumping well. The drawdown is derived by Hunt using the Laplace transform in the domain. The result is

where θ is an integration variable; and E1is the well function of the Theis solution, which can be obtained using the pre-formulated tables of values.

2.2 Proposed analytical model

2.2.1 Problem description

Using the Hunt model, interaction between the stream and aquifer was examined by considering the effects of stream width and of two streams, one situated on either side of the pumping well. The conceptual hydrogeological model is shown in Fig. 2(a). According to the drawdown distribution caused by pumping and the recharge of the stream to the aquifer, the domain is subdivided into six separate domains, and drawdown is different for each domain.Because the pumping well located on one side of the streams, and domains I and VI are located on the other side of the streams and far from the pumping well, the drawdown in these domains is small compared to that in the pumping well. Domains II and V are close to the pumping well, but their drawdown is also not large on account of direct recharge from the stream. However, the drawdown of domains III and IV, which are the main study domains, is large compared with that of the other four domains.

Fig. 2 Conceptual hydrogeological model for two streams

2.2.2 Governing equation and calculation of drawdown

The origin of the coordinate system is located at the right boundary of the left stream and along a perpendicular line from the pumping well to the stream, as shown in Fig. 2(b). Note that the coordinate system is defined to mathematically express the governing differential equations within three separate domains:

Domain I: non-pumping well side of the left stream

Domain II: beneath the stream

Domain III: between the left stream and the pumping well

where h is the thickness of aquifer; s1, s2, and s3are the drawdowns of domain I, domain II and domain III, respectively; subscript L denotes the domain to the left of the pumping well;QLis the constant pumping rate from the pumping well without the right stream; and CLis the leakage coefficient of streambed semipervious material for the left stream, expressed as a function of the streambed hydraulic conductivity for the left stream KL, the width of the left stream WL, and the thickness of the streambed layer for the left stream ML:CL=KLWLML. Eq. (6)is integrated, and the drawdown function can be expressed as

where

where α is an integration variable, u is a dimensionless parameter, and r is the well radius. Incorporating the initial and boundary conditions into Eq. (6), the integral constants C1and C2can be determined as follows:

Thus, the drawdown function can be rewritten as

Based on the Fourier and Laplace transforms, the first, second, and third terms on the right side of Eq. (11)can be expressed as (Fox et al. 2002)

where β is an integration variable. Therefore, the drawdown of Eq. (6)is determined as follows:

The higher terms of Taylor series expansion are complex and ignored because the sum of these terms is very small compared with the first three terms in Eq. (15). Therefore, the analytical solution is simplified to three terms all relating to the well-known Theis well function. For domain IV, where DL≤x≤D, the same method is used as for domain II, and the origin is at the left boundary of the right stream, so the drawdown can be expressed as

where QRis the constant pumping rate from the pumping well without the left stream, CRis the leakage coefficient of streambed semipervious material for the right stream, WRis the width of the right stream,. When the recharge of two streams to an aquifer is taken into account, the total pumping rate Q is the sum of QLand QR: QL+QR=Q.

According to the theory of water yield balance, the pumping rate should be equal to the recharge rate when the groundwater flow is in a steady state. Therefore, QLand QRare calculated using Darcy’s Law. The result is

where HL0and HR0are the initial stream stages of the left and right streams, respectively,swellis the water table at the pumping well, and d is the width of calculated cross sections. If HL0=HR0and DL=D 2, the result is that QL=QR=Q 2. Also, if CL=CRand WL=WR, the result is that s3=s4=s 2.

For a given point, the value of the first term on the right side of Eq. (15)can be calculated easily according to the tables of values for the well function. The second term on the right side of Eq. (15)needs to be integrated for the variable α.

If

It is obvious that f(γ)is a continuous and integrated exponential function, and a smooth curve, but its value cannot be obtained easily through the integration. However, integration can be obtained using straightforward numerical techniques, such as the trapezoidal rule. Eq. (19)can be expressed as

where f (0)and f (1)are the function values of f(γ)at γ=0 and γ=1, respectively.Based on the L Hospitcl rule, we can obtain the result f (0 )= 0, and f (1)is rewritten as

Thus, the second term on the right side of Eq. (15)can be determined as follows:

A similar method is used to derive the analytical expression for the third term on the right side of Eq. (15):

Substituting Eqs. (22)and (23)into Eq. (15)and rearranging results in the following expression, we arrive at

The same method is applied to Eq. (16)to derive the analytical solution, and a similar result can be expressed:

Hence, Eq. (24)and Eq. (25)are the analytical solutions of drawdown in domains III and IV, respectively.

3 Results analysis

3.1 Comparison of analytical solutions

Table 1 Comparison between Hunt solution and proposed analytical solution

It can be seen in Table 1 that when CDLT= 0 (i.e., when there is no recharge flux from the stream), the two analytical solutions are equivalent, demonstrating that this proposed analytical solution can calculate the drawdown under the same conditions of the Hunt model.However, when CDLT increases to relatively large values (i.e., 0.1), deviations between the two analytical solutions become significant with the increase of tT (μ D). Note that the errors in Table 1 are the ratios of the difference between the proposed analytical solution and the Hunt solution to the Hunt solution, and the errors were obtained by multiplying the ratio by 100%. Deviations show that the stream width increases the recharge flux to the aquifer, so the drawdown becomes smaller than that of the Hunt model.

The proposed analytical solution also shows that there is a relationship between the drawdown of the aquifer and the stream width. Fig. 3 shows the dimensionless drawdown for the Hunt and proposed analytical solutions for CDLT= 0.1 and DLW ranging from 10 to 100. It can be seen from Fig. 3 that with the decrease of DLW (i.e., W increases and DLremains constant), sT Q decreases, which means that the drawdown decreases when T Q is constant. This may be caused by the greater recharge flux to the aquifer on account of the increase of the stream width. This result is in accordance with the practical cases. The calculated results also show that when the ratio of the distance between the streams and the pumping well to the stream width (i.e., DLW)is greater than 50, the difference between the Hunt solution and the proposed analytical solution is less than 1%. However, when DLW＜ 20, the Hunt solution deviates from the proposed analytical solution by approximately 5% at tT (μ D)= 10 and by approximately 10% at tT (μ D)= 100,showing that increased stream width results in greater deviations between the two analytical solutions. This demonstrates that the proposed analytical model is very close to the actual situation.

Fig. 3 Relationship of dimensionless drawdown functions sT Q to DLW for Hunt and proposed analytical solutions

As drawdown depends on the position on the plane, it is important to examine the results at other positions, not only at x DL= 0.2 and y DL= 0. The difference between the Hunt solution and the proposed analytical solution at different locations is discussed for tT (μ D)= 1.0 and CDLT= 1.0 and a contour map has been provided to indicate the effects of stream width on the Hunt solution (Fig. 4). It can be seen from Fig. 4 that the effects of stream width on drawdown are very significant near the stream, where the aquifer can receive lots of recharge from the stream, so the drawdown is very small. However, the effects of stream width on drawdown are not significant near the well far from the stream, where the drawdown from the proposed analytical solution is very close to that of the Hunt solution.

Fig. 4 Drawdown contours of sT Q for tT (μ D)= 1.0 and CDL T= 1.0 for Hunt and proposed analytical solutions

3.2 Effects of two streams on drawdown

Four cases were studied to examine the effects of two streams on drawdown. In each case,some parameters were involved, including initial stream stage, stream width, the distance between the stream and the pumping well, the stream recharge rate, and the leakage coefficient of streambed semipervious material (Table 2).

Table 2 Values of stream parameters and flux of pumping well in different cases

Case 1 shows that when the leakage coefficient of streambed semipervious material, the pumping rate, and the stream width of the two streams are the same, the drawdown caused by pumping is the same in both domains beside the pumping well, which means that s3= s4.When the pumping well is located at the midpoint of the two streams, the distribution of drawdown in the aquifer is symmetrical between the streams and pumping well. The results can also be calculated with Eqs. (24)and (25).

In Case 2 the two streams have the same leakage coefficient and stream width, but the distance from the pumping well to the left stream is half of the distance to the right stream.The drawdown in domains III and IV was calculated using the proposed analytical solution.The results are shown in Fig. 5(a). It can be seen that the drawdown to the left of the pumping well is large compared with that to the right of the pumping well (i.e., s3＞s4), which is due to the different pumping rates and the distance difference (DL=D 3). According to Eq. (17),when the pumping well is close to the left stream, QLand QRcan be obtained. If the total pumping rate (Q)is constant, the aquifer to the left of the pumping well will provide more water flux than the one to the right, but the recharge rates of the two streams are equal, so larger drawdown to the left of the pumping well is possible with the greater pumping rate.

Case 3 and Case 4 mainly relate to the sensitivity of the leakage coefficient of streambed semipervious material to drawdown in the aquifer during pumping. In these cases, the leakage coefficients of the left stream are larger than those of the right stream (i.e., CL= 10CRand CL= 100CR). The calculated drawdown is shown in Fig. 5(b)and 5(c). It can be seen that the drawdown to the left of the pumping well is small compared with that to the right of the pumping well, which may be caused by the large leakage coefficient of the left stream.According to Darcy’s Law, the recharge rate from the stream is positively proportional to the leakage coefficient. Also, when the difference in the leakage coefficients from the two streams is small enough, drawdown on both sides of the pumping well is also small. If the difference in the leakage coefficients is more than two orders of magnitude, considerable drawdown will occur to the left and right of the pumping well. Thus, when the pumping of groundwater is for the water supply near the stream, not only the transmissivity of the aquifer but also the leakage coefficient of streambed semipervious material will need to be considered.

Fig. 5 Relationship of dimensionless drawdown functions (sT Q versus tT (μ DL 2))in case of two streams

4 Conclusions

On the basis of the Hunt model, an analytical solution of drawdown has been developed for an aquifer partially penetrated by two streams. The proposed analytical solution modifies Hunt’s analytical solution and not only considers the effect of stream width on drawdown, but also takes the distribution of drawdown and the interaction of the two streams into account.The results of this study show that the proposed analytical solution agrees with the Hunt solution and errors between the two solutions are equal to zero without considering the effect of stream width. Also, deviations between the two analytical solutions increase with stream width. Comparison of analytical solutions shows that the Hunt solution deviates less than 5%from the proposed analytical solution for tT(μ DL2)＜ 100 as long as DLW≥ 20. However,when DLW ＜ 20, Hunt’s solution deviates from the proposed analytical solution by approximately 5% at tT (μ DL2)= 10 and approximately 10% at tT (μ DL2)= 100, which demonstrates that increased stream width results in greater deviations between the two analytical solutions.

Four cases were studied to examine the effect of two streams on drawdown. The results demonstrate that when the pumping well is located between the two streams, the distribution of drawdown in the aquifer is symmetrical between the streams and the pumping well. When the pumping well is close to a stream, large drawdown will occur in the region between the pumping well and the stream. Furthermore, the effect of the leakage coefficient of streambed semipervious material on drawdown is significant, which means that a large leakage coefficient will result in small drawdown.

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