Analytical Type-Curve Method for Hydraulic Parameter Estimation in Leaky Confined Aquifers with Fully Enclosed Rectangular Cutoff Walls

Abstract

In deep excavation dewatering engineering, fully enclosed cutoff walls are widely implemented to improve the efficiency of dewatering in the pit and prevent adverse environmental impacts such as land subsidence and damage to adjacent infrastructure. However, the presence of such impermeable barriers fundamentally alters flow dynamics, rendering conventional aquifer test interpretation methods inadequate. This study presents a novel closed-form analytical solution for transient drawdown in a leaky confined aquifer bounded by a rectangular, fully enclosed cutoff wall under constant-rate pumping. The solution is rigorously derived by applying the mirror image method within a superposition framework, explicitly accounting for the barrier effect of the curtain. A type-curve matching methodology is developed to inversely estimate key aquifer parameters—transmissivity, storativity, and vertical leakage coefficient—while incorporating the geometric and boundary effects of the curtain. The approach is validated against field data from a pumping test conducted at a deep excavation site in Wuhan, China. Excellent agreement is observed between predicted and measured drawdowns across multiple observation points, confirming the model’s fidelity. The proposed solution and parameter estimation technique provide a physically consistent, analytically tractable, and computationally efficient framework for interpreting pumping tests in constrained aquifer systems, thereby improving predictive reliability in dewatering design and supporting sustainable groundwater management in urban underground construction.

1. Introduction

To analyze flow in leaky confined aquifer systems, Hantush and Jacob [1] first presented an analytical drawdown solution involving pumping from a fully penetrated well in a semi-confined aquifer. Subsequently, under the assumption of the leaky aquifer’s horizontal infinite extent, numerous mathematical models for pumping tests have become available to accommodate various conditions [2,3,4,5,6,7]. For example, Hantush [8] considered a pumping well with partial penetration and obtained analytical solutions to investigate the drawdown response in a leaky confined aquifer. Perina [9] provided semi-analytical solutions for investigating groundwater flow in an infinite leaky confined aquifer with a recirculation well.

In groundwater-rich areas, deep excavation construction can induce significant deformation of the foundation pit and surrounding environments [10,11,12,13,14]. To mitigate such risks, cutoff walls are commonly employed in deep dewatering projects to control groundwater flow and prevent geotechnical hazards [15,16,17,18,19,20,21,22,23,24]. Due to the existence of the cutoff wall in aquifers, the above existing solutions cannot take account of the block effect of the curtain, resulting in some errors [25]. Unfortunately, only a few studies have focused on the influence of the finite impermeable or recharge boundary in a horizontal direction in a leaky confined aquifer. Zhou et al. [26] gave a semi-analytical solution for flow in a closed leaky confined aquifer with a fully penetrated well. Feng and Zhan [7] and Feng et al. [27] offered semi-analytical solutions for constant-head and constant-rate pumping tests in a bounded aquitard-confined aquifer system, with either an impermeable or constant-head outer boundary, respectively. The available studies have shown that the impact of radial finite boundary conditions must be considered.

Determining accurate hydraulic parameters is essential for the reliable design and optimization of dewatering systems in deep excavations [21,28,29]. Commonly employed approaches include laboratory testing, field-based aquifer tests, numerical modeling, and, more recently, data-driven techniques such as artificial intelligence and machine learning [2,30,31,32,33]. The field pumping test is widely regarded as the most effective and reliable way [34,35,36,37]. When estimating semi-confined aquifer parameters using field data, solutions for a leaky aquifer system are often applied [5,38]. For example, Hantush [39] applied the inflection point method and type-curve method for the identification of hydraulic parameters in leaky confined aquifers on the basis of the drawdown formula developed by Hantush and Jacob [1]. Hantush [3] determined the parameters of a leaky confined aquifer by applying the solution of drawdown with a partial penetration well. Singh [40] used the field data at the monitoring well and employed a straightforward computational approach to assess the hydraulic parameters of the leaky aquifer. Li and Qian [41] estimated the hydraulic parameter values in leaky confined aquifers with the help of the method of curve-matching based on data from transient discharge rate and recovery tests. Yeh and Huang [42] utilized the solution of Hantush and Jacob [1] in conjunction with the extended Kalman filter to ascertain the parameters for the semi-confined aquifer. De Smedt [43] gave a methodology for estimating aquitard storage using pumping tests in a leaky aquifer. However, these studies assume a leaky confined aquifer with an infinitely horizontal extent, disregarding the impact of the finite boundary. Consequently, there is a pressing need for a method to estimate parameters in leaky confined aquifers while considering the fully bounded outer boundary, particularly in the context of deep foundation pit dewatering engineering.

The objective of this study is to obtain a closed-form analytical drawdown solution for a leaky confined aquifer enclosed by a rectangular impermeable barrier, resulting from pumping at a partially penetrated well. Subsequently, a novel type-curve method is provided to ascertain the values of hydraulic parameters for the leaky confined aquifer. Finally, the method is validated through an in situ pumping test, demonstrating that the drawdown curve generated by adopting the estimated parameters closely matches the measured drawdown data.

2. Methodology

2.1. Solution of Drawdown in an Infinite Leaky Confined Aquifer

2.1.1. Mathematical Model

Figure 1 demonstrates an abstraction well with a constant discharge rate Q in an infinite leaky confined aquifer. The well is partially screened from depth d to d + l. The confined aquifer has a uniform thickness (M). The conceptual model adopted in this study follows that of Hantush [3], under the following key assumptions:

(1)

The pumped aquifer is homogeneous and isotropic, with constant hydraulic properties.

(2)

The hydraulic head in the overlying water table aquifer is kept constant during the pumping period.

(3)

The groundwater flow obeys Darcy’s law.

(4)

The horizontal flow and storage in the aquitard are negligible and the leakage across the aquitard is represented as a distributed source or sink term in the aquifer.

(5)

The well is treated as a line sink of negligible radius, with no hydraulic head loss considered across the screen.

(6)

The radial specific discharge across the well face is uniformly distributed.

Thus, the flow in the leaky confined aquifer system is governed by

$${\displaystyle \frac{{\partial }^{2}s\left(r,z,t\right)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial s\left(r,z,t\right)}{\partial r}}+{\displaystyle \frac{{\partial }^{2}s\left(r,z,t\right)}{{\partial }^{2}z}}-{\displaystyle \frac{s\left(r,z,t\right)}{B^2}}={\displaystyle \frac{S_s}{K}}{\displaystyle \frac{\partial s\left(r,z,t\right)}{\partial t}}$$

(1)

where s is drawdown in the pumped aquifer; r and z are the radial and vertical distance, respectively; t denotes pumping time; B = (TM′/K′)^1/2 represents leaky factor, T = KM refers to aquifer transmissivity; and S_s represents the aquifer specific storage. K and M are the hydraulic conductivity and thickness of the aquifer, respectively; K’ and M’ denote the hydraulic conductivity and thickness of the aquitard, respectively; and the boundary conditions can be written as follows:

$$s\left(r,z,0\right)=s\left(\infty ,z,t\right)=0$$

(2)

$${\displaystyle \frac{\partial s}{\partial z}}\left(r,0,t\right)={\displaystyle \frac{\partial s}{\partial z}}\left(r,M,t\right)=0$$

(3)

$$\underset{r\to 0}{\lim }r{\displaystyle \frac{\partial s}{\partial r}}=\left\{\begin{array}{c}-{\displaystyle \frac{Q}{2\pi Kl}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}d\le z\le l+d\\ 0,\hspace{1em}0<z<d or l+d<z<M\end{array}\right.$$

(4)

Figure 1. Schematic illustration of a pumping well that partially penetrates an infinitely leaky confined aquifer.

Figure 1. Schematic illustration of a pumping well that partially penetrates an infinitely leaky confined aquifer.

2.1.2. Drawdown Solution

Hantush [3] solved the boundary-value problem listed in Equations (1)–(4) and gave the close-formed drawdown solution as follows:

$$\begin{array}{l}s={\displaystyle \frac{Q}{4\pi T}}\left\{W\left(u,{\displaystyle \frac{r}{B}}\right)+{\displaystyle \frac{2M}{\pi l}}\right.{\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}\left\{\sin \left[\frac{\left(l+d\right)n\pi }{M}\right]-\sin \left(\frac{n\pi d}{M}\right)\right\}}\\ \left.\cdot \cos \left({\displaystyle \frac{n\pi z}{M}}\right)\cdot W\left(u,\sqrt{{\left({\displaystyle \frac{r}{B}}\right)}^2+{\left({\displaystyle \frac{n\pi r}{M}}\right)}^2}\right)\right\}\end{array}$$

(5)

where the variable u = r²S/4Tt, W(u, r/B) refers to the well function for leaky confined aquifer

$$W\left(u,{\displaystyle \frac{r}{B}}\right)={\displaystyle {\int }_u^{\infty }\frac{1}{y}}{e}^{-y-{\displaystyle \frac{r^2}{4B^{2}y}}}\mathrm{d}y$$

(6)

and

$$W\left(u,\sqrt{{\left({\displaystyle \frac{r}{B}}\right)}^2+{\left({\displaystyle \frac{n\pi r}{M}}\right)}^2}\right)={\displaystyle {\int }_u^{\infty }\frac{1}{y}}{e}^{\left(-y-{\displaystyle \frac{{\left(\frac{n\pi r}{M}\right)}^2+{\left(\frac{r}{B}\right)}^2}{4y}}\right)}\mathrm{d}y$$

(7)

It should be noted that the integrals in Equations (6) and (7) are evaluated numerically using the NIntegrate function implemented in Wolfram Mathematica.

The drawdown solution in an observation well screened between d′ and d′ + l′ as follows:

$$s={\displaystyle \frac{Q}{4\pi T}}W\left(u,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r}{M}},{\displaystyle \frac{r}{B}}\right)$$

(8)

where

$$W\left(u,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r}{M}},{\displaystyle \frac{r}{B}}\right)=W\left(u,{\displaystyle \frac{r}{B}}\right)+\epsilon \left({\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r}{M}},{\displaystyle \frac{r}{B}}\right)$$

(9)

with

$$\begin{array}{l}\epsilon \left({\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r}{M}},{\displaystyle \frac{r}{B}}\right)={\displaystyle \frac{2M^2}{{\pi }^{2}l{l}^{\prime }}}{\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}}\left\{\sin \left[{\displaystyle \frac{\left(l+d\right)n\pi }{M}}\right]-\sin \left({\displaystyle \frac{n\pi d}{M}}\right)\right\}\cdot \\ \left\{\sin \left[{\displaystyle \frac{\left(l^{\prime }+d^{\prime }\right)n\pi }{M}}\right]-\sin \left({\displaystyle \frac{n\pi d^{\prime }}{M}}\right)\right\}\cdot W\left(u,\sqrt{{\left({\displaystyle \frac{r}{B}}\right)}^2+{\left({\displaystyle \frac{n\pi r}{M}}\right)}^2}\right)\end{array}$$

(10)

2.2. Solution of Drawdown in a Rectangular Leaky Confined Aquifer Bounded by Cutoff Wall

Figure 2 shows the plan view of a well situated within a leaky confined aquifer enclosed by a rectangular impermeable barrier. One can derive the drawdown solution for the entirely enclosed aquifer by applying the image method and the principle of superposition. Given the configuration with four fully impermeable boundaries created by the cutoff wall, a series of n image wells are introduced according to the mirror image method. These image wells are conceptualized as abstraction wells, each having the same discharge rate as the actual pumping well, to simulate the effects of these boundaries on groundwater flow. Therefore, the final drawdown solution takes the form

$$\begin{array}{l}s={\displaystyle \frac{Q}{4\pi T}}W\left(u_0,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r_0}{M}},{\displaystyle \frac{r_0}{B}}\right)+{\displaystyle \frac{Q}{4\pi T}}W\left(u_1,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r_1}{M}},{\displaystyle \frac{r_1}{B}}\right)\\ +{\displaystyle \frac{Q}{4\pi T}}W\left(u_2,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r_2}{M}},{\displaystyle \frac{r_2}{B}}\right)+\dots +{\displaystyle \frac{Q}{4\pi T}}W\left(u_n,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r_n}{M}},{\displaystyle \frac{r_n}{B}}\right)\end{array}$$

(11)

One can rewrite Equation (11) as

$$\begin{array}{l}s={\displaystyle \frac{Q}{4\pi T}}\left[W\left(u_0,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r_0}{M}},{\displaystyle \frac{r_0}{B}}\right)+\right.W\left(u_1,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r_1}{M}},{\displaystyle \frac{r_1}{B}}\right)\\ \left.+W\left(u_2,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r_2}{M}},{\displaystyle \frac{r_2}{B}}\right)+\dots +W\left(u_n,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r_n}{M}},{\displaystyle \frac{r_n}{B}}\right)\right]\end{array}$$

(12)

in which r₀ denotes the radial distance from the pumping well to the observation well, and r_i (i = 1, 2, …, n) refers to the distance from the observation well to image well i. Notably, in theory, an infinite series of image wells exists; however, for practical applications, a finite number (typically three or four) is sufficient to achieve accurate results [2,31].

Further, the variable u_i in Equation (12) can be written as

$$u_0={\displaystyle \frac{r_0^{2}S}{4Tt}},u_1={\displaystyle \frac{r_1^{2}S}{4Tt}},u_2={\displaystyle \frac{r_2^{2}S}{4Tt}},\dots ,u_n={\displaystyle \frac{r_n^{2}S}{4Tt}}$$

(13)

Accordingly, the following expressions are achieved:

$${\displaystyle \frac{u_1}{u_0}}={\displaystyle \frac{\frac{r_1^{2}S}{4Tt}}{\frac{r_0^{2}S}{4Tt}}}={\displaystyle \frac{r_1^2}{r_0^2}},{\displaystyle \frac{u_2}{u_0}}={\displaystyle \frac{\frac{r_2^{2}S}{4Tt}}{\frac{r_0^{2}S}{4Tt}}}={\displaystyle \frac{r_2^2}{r_0^2}},\dots ,{\displaystyle \frac{u_n}{u_0}}={\displaystyle \frac{\frac{r_n^{2}S}{4Tt}}{\frac{r_0^{2}S}{4Tt}}}={\displaystyle \frac{r_n^2}{r_0^2}}$$

(14)

$${\displaystyle \frac{\frac{r_1}{B}}{\frac{r_0}{B}}}={\displaystyle \frac{r_1}{r_0}},{\displaystyle \frac{\frac{r_2}{B}}{\frac{r_0}{B}}}={\displaystyle \frac{r_2}{r_0}},\dots ,{\displaystyle \frac{\frac{r_n}{B}}{\frac{r_0}{B}}}={\displaystyle \frac{r_n}{r_0}}$$

(15)

and

$${\displaystyle \frac{\frac{r_1}{M}}{\frac{r_0}{M}}}={\displaystyle \frac{r_1}{r_0}},{\displaystyle \frac{\frac{r_2}{M}}{\frac{r_0}{M}}}={\displaystyle \frac{r_2}{r_0}},\dots ,{\displaystyle \frac{\frac{r_n}{M}}{\frac{r_0}{M}}}={\displaystyle \frac{r_n}{r_0}}$$

(16)

Assuming α_i = r_i/r₀ (i = 1, 2, …, n), Equations (14)–(16) become

$$\begin{array}{l}{u}_1={\alpha }_1^2{u}_0,u_2={\alpha }_2^2{u}_0,\dots ,u_n={\alpha }_n^2{u}_0\\ {\displaystyle \frac{r_1}{B}}={\alpha }_1{\displaystyle \frac{r_0}{B}},{\displaystyle \frac{r_2}{B}}={\alpha }_2{\displaystyle \frac{r_0}{B}},\dots ,{\displaystyle \frac{r_n}{B}}={\alpha }_n{\displaystyle \frac{r_0}{B}}\\ {\displaystyle \frac{r_1}{M}}={\alpha }_1{\displaystyle \frac{r_0}{M}},{\displaystyle \frac{r_2}{M}}={\alpha }_2{\displaystyle \frac{r_0}{M}},\dots ,{\displaystyle \frac{r_n}{M}}={\alpha }_n{\displaystyle \frac{r_0}{M}}\end{array}$$

(17)

Substituting Equation (17) into Equation (12) leads to

$$s={\displaystyle \frac{Q}{4\pi T}}\Phi \left(u_0,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r_0}{M}},{\displaystyle \frac{r_0}{B}},\alpha \right)$$

(18)

where the new well function Φ(u₀, l/M, l′/M, d/M, d′/M, r₀/M, r₀/B, α) is expressed as

$$\begin{array}{l}\Phi \left(u_0,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r_0}{M}},{\displaystyle \frac{r_0}{B}},\alpha \right)=W\left(u_0,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\displaystyle \frac{r_0}{M}},{\displaystyle \frac{r_0}{B}}\right)+\\ W\left({\alpha }_1^2{u}_0,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\alpha }_1{\displaystyle \frac{r_0}{M}},{\alpha }_1{\displaystyle \frac{r_0}{B}}\right)+W\left({\alpha }_2^2{u}_0,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\alpha }_2{\displaystyle \frac{r_0}{M}},{\alpha }_2{\displaystyle \frac{r_0}{B}}\right)\\ +\dots +W\left({\alpha }_n^2{u}_0,{\displaystyle \frac{l}{M}},{\displaystyle \frac{l^{\prime }}{M}},{\displaystyle \frac{d}{M}},{\displaystyle \frac{d^{\prime }}{M}},{\alpha }_n{\displaystyle \frac{r_0}{M}},{\alpha }_n{\displaystyle \frac{r_0}{B}}\right)\end{array}$$

(19)

2.3. Estimation for Hydrogeological Parameters Using the Type-Curve Method

The estimation of hydraulic parameters via type-curve matching requires the construction of a dimensionless type curve, plotting Φ(u₀, l/M, l′/M, d/M, d′/M, r₀/M, r₀/B, α) against 1/u₀ on log-log scale. This curve serves as a reference for matching field drawdown data to infer aquifer properties.

The detailed steps of the type-curve fitting method [38] are as follows:

Step 1: Plot a type curve family of Φ(u₀, l/M, l′/M, d/M, d′/M, r₀/M, r₀/B, α) versus 1/u₀ in log-log coordinates using Equation (19).

Step 2: Generate a log-log plot illustrating the observed drawdown s against pumping time t using a scale consistent with that of the type curve.

Step 3: Keeping the coordinate axes parallel at all times, superimpose the two plots until the best fit of data curve and one of the family of type curves is obtained.

Step 4: Choose a common point, known as the fitting point, either arbitrarily on the overlapping segment of the curve or at any location where the type curves and field data exhibit close alignment. At this fitting point, read the corresponding values of r₀/B, Φ(u₀, l/M, l′/M, d/M, d′/M, r₀/M, r₀/B, α), 1/u₀, s, and t.

Step 5: Substitute the values of Φ(u₀, l/M, l′/M, d/M, d′/M, r₀/M, r₀/B, α) and s, along with the available Q, into Equation (18) and solve for T.

Step 6: Substitute 1/u₀ and t into u₀ = $r_0^2$S/4Tt and calculate the value of S.

Step 7: Substitute T and B into B = (TM′/K′)^1/2 and determine the value of M′/K′. Then, utilizing the obtained value of M′/K′ and M′, calculate the value of K′.

3. Case Study for Pumping Test

3.1. Test Site

Field pumping tests were carried out within a deep rectangular excavation located in Wuhan, China. The excavation is fully enclosed by a reinforced concrete diaphragm wall, which functions as an impermeable cutoff barrier. The wall has a thickness of 1.2 m and plan-view dimensions of 25 m (width) and 180 m (length). The layout of the pumping and observation wells used the field tests is illustrated in Figure 3.

In the test region, the pumped aquifer has a total thickness of 31.4 m and consists of fine sand and silty sand. It is underlaid by impermeable bedrock and overlaid by a 7.4 m-thick layer of silty clay that functions as an aquitard. Figure 4 presents the stratigraphic profile, the penetration depth of the cutoff wall, and the configuration of the pumping and observation wells.

3.2. Pumping Test

Two single pumping tests, each with a duration of 420 min, were conducted in the test area. In this study, drawdown data from the pumping test in well W1, shown in Figure 3, were selected for the hydraulic parameter estimation of the leaky confined aquifer, owing to the completeness and quality of the recorded dataset. The pumping well was operated at a constant discharge rate of Q = 33.1 m³/h, with l = 10 m and d = 18.4 m. The structure parameters for observation well H1 are l′ = 10 m and d′ = 16.4 m; the total thickness of the aquifer is M = 31.4 m, as shown in Figure 4. The measured drawdowns during the pumping period are listed in

Table 1. Pumping test data at observation well H1.

t (min)	0	3	10	15	20	25	30	60	90
s (m)	0	0.66	1.66	2.2	2.68	2.86	3.00	3.38	3.68
t (min)	120	150	180	210	240	270	300	360	420
s (m)	3.87	4.00	4.00	4.00	4.00	4.00	4.00	4.00	4.00

.

3.3. Parameter Estimation

The results of the pumping test performed in well 1 were Q = 33.1 m³/h; M = 31.4 m; l = l′ = 10 m; d = 18.4 m; d′ = 16.4 m; r₀ = 11.58 m. To account for the no-flow boundaries imposed by the fully enclosed cutoff wall, the method of mirror images is employed. The extent of required mirror image systems is guided by the empirical influence radius R = 10 sK^1/2, with K denoting the hydraulic conductivity of the pumped aquifer and s the drawdown in the pumping well—commonly estimated using Siechardt’s formula [2,44]. In this study, three iterations of image well reflection are implemented to sufficiently approximate the boundary conditions [2,31]. Image wells located beyond the radius of influence are excluded, resulting in a total of 13 image wells, which are deemed sufficient for drawdown calculations. The values of the distances r_i from the image wells to the observation well can be easily obtained and are listed in

Table 2. The distance r_i (m) from observation well H1 to image well.

r1	r2	r3	r4	r5	r6	r7
36.578	61.578	86.578	86.894	62.021	37.319	13.741
r8	r9	r10	r11	r12	r13	-
15.327	39.128	63.852	63.422	38.422	13.422	-

. Additionally, the values of α_i are provided in

Table 3. Value of α_i.

α1	α2	α3	α4	α5	α6	α7
3.1593	5.3185	7.4778	7.7642	5.3568	3.2233	1.1868
α8	α9	α10	α11	α12	α13	-
1.3238	3.3795	5.5149	5.4778	3.3185	1.1593	-

.

In this study, point A in Figure 5 on the overlapping part of the curve is chosen as the calculation point. One can read the values of coordinates of A as r₀/B = 0.1, Φ(u₀, l/M, l′/M, d/M, d′/M, r₀/M, r₀/B, α) = 39.17, 1/u₀ = 1000, s = 4.0 m, and t = 270 min. Following the procedure outlined above for the type-curve matching method, the hydraulic parameters can be obtained as follows: aquifer transmissivity T = 619.36 m²/d, storage coefficient S = 0.0035, and K′ = 0.34 m/d.

To enhance the reliability of the type-curve matching method, the calculated drawdown values at various pumping times are compared with the corresponding observed field data, using the estimated hydraulic parameters. To quantitatively assess the goodness of fit between the calculated and observed values, the mean square error (MSE) is first calculated as follows [45,46]:

$$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(s_i-s_{ci}\right)}^2}$$

(20)

where n is the number of drawdown observations, s_i denotes the observed drawdown, and s_ci represents the calculated drawdown at the i-th observation point. Note that an MSE value less than or equal to 10⁻² is considered indicative of acceptable parameter estimation in the context of type-curve matching in this study.

Subsequently, three additional errors are employed to assess the discrepancy between observed and calculated drawdowns: the mean error (ME), the standard error of estimates (SEE) and the root-mean-square error (RMSE) [42,45,46].

The mean error (ME) is defined as

$$ME={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left(s_i-s_{ci}\right)}$$

(21)

The standard error of estimates (SEE) is given by

$$SEE=\sqrt{{\displaystyle \frac{1}{\upsilon }}{\displaystyle \sum _{i=1}^n{\left(s_i-s_{ci}\right)}^2}}$$

(22)

where υ is the degree of freedom, defined as the number of observed data points minus the number of estimated parameters.

The root-mean-square error (RMSE) is calculated as

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(s_i-s_{ci}\right)}^2}}$$

(23)

Figure 6 compares the calculated and observed drawdown values over time. The resulting error metrics for this fit are MSE = 8 × 10⁻³ m, ME = 5.1 × 10⁻² m, SEE = 9.8 × 10⁻² m and RMSE = 8.9 × 10⁻² m, indicating a close agreement between the model predictions and field measurements. Based on this comparison and error analysis, it can be concluded that the proposed method yields accurate and robust estimates of hydraulic parameters, demonstrating its validity and reliability for practical application.

4. Discussion

The analytical solution developed in this study can be applied to estimate hydraulic parameters of leaky confined aquifers using field drawdown data from a single pumping test conducted within a fully penetrated rectangular cutoff wall. Furthermore, the solution provides a robust framework for investigating transient flow dynamics in such systems, enabling systematic evaluation of the influences of key factors—including the leakage coefficient, well partial penetration, and geometric dimensions of the impermeable cutoff wall—on drawdown response.

Nevertheless, several limitations of the current study warrant acknowledgment and present opportunities for future work.

First, the solution is built upon the classical framework by Hantush [2] for flow toward a partially penetrating well under constant discharge in an isotropic leaky confined aquifer. Consequently, effects such as aquifer anisotropy, wellbore storage, aquitard storage, and near-well skin zones are not explicitly incorporated, despite their potentially significant impact on transient flow behavior [47,48,49]. The inclusion of these mechanisms would enhance the model’s realism and applicability to complex field conditions.

Second, the type-curve matching method employed for parameter estimation, while illustrative, relies on manual or semi-automated curve fitting. This process could be significantly improved through integration with advanced optimization algorithms or machine learning techniques to enable robust, objective, and efficient identification of the best-fit parameter set [33,50,51].

Third, this study is limited to type-curve analysis for hydraulic parameter estimation. Future research may explore advanced inverse methods—such as machine learning or numerical models integrated with evolutionary optimization algorithms—to achieve more accurate parameter inversion [32,33,52].

Fourth, for high-capacity pumping scenarios, non-Darcian flow may become significant in the vicinity of the wellbore, where hydraulic gradients are large. The present model assumes Darcian flow; thus, its validity under such conditions may be limited. A nonlinear head-loss relationship, such as the Forchheimer equation or Izbash law, should be considered in future extensions to account for inertial effects at high flow velocities [53,54].

Fifth, while fully penetrated rectangular cutoff walls are commonly implemented in geotechnical and dewatering practice, other geometries (e.g., circular, L-shaped, or partially penetrating barriers) may also be encountered. The extension of the solution to more general barrier configurations would broaden its practical utility. In addition, the partially penetrated cutoff walls should be taken into consideration [55,56,57].

Sixth, this study does not account for the influence of pre-existing underground structures. In practice, various infrastructures—such as metro stations, building foundation piles, or underground diaphragm walls—are often present near foundation pits and may significantly alter groundwater seepage patterns by introducing additional water-blocking effects [58,59,60].

Finally, the model does not address constant-head (i.e., controlled drawdown) boundary conditions, which are frequently imposed during deep excavation dewatering projects. Developing a solution for constant-head pumping tests within confined aquifers bounded by cutoff walls would complement the current work and improve its relevance to engineering practice [7].

5. Conclusions

This study presents a novel analytical solution for transient flow to a partially penetrating well of constant discharge in a leaky confined aquifer bounded by a fully penetrating rectangular cutoff wall. The solution is derived by combining the image method with the principle of superposition, enabling the treatment of the closed-boundary condition imposed by the impermeable barrier. Based on the derived drawdown solution, a type-curve matching methodology is proposed for the identification of key hydraulic parameters—such as transmissivity, storativity, and the leakage coefficient—using drawdown data recorded in a pumping test performed within the cutoff wall.

Application to a field case study demonstrates that the proposed analytical framework is both reliable and practical for estimating aquifer properties in real-world engineering settings, particularly in dewatering or groundwater management projects involving cutoff walls. The results confirm the feasibility of using the solution for interpreting pumping test data under complex boundary conditions.

While the current model assumes idealized conditions (e.g., isotropic aquifer properties and Darcian flow), it provides a physically sound and computationally efficient tool for preliminary aquifer characterization and supports an improved understanding of transient flow behavior in bounded, leaky aquifer systems.