# Analytical Method for Groundwater Seepage through and Beneath a Fully Penetrating Cut-off Wall Considering Effects of Wall Permeability and Thickness

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Analytical Method

- Two-dimensional steady seepage flow is studied, and the analytical model is symmetrical taking the centerline of the cut-off wall as the axis of symmetry.
- The aquitard extends horizontally infinitely on both sides of the cut-off wall.
- The permeability ratio k’/k ≤ 1 is discussed here.
- The flow is approximated as one-dimensional and horizontal within the regions BCC’B’ and CDD’C’. The flow in a vertical leaky wall can be assumed to be normal to the wall, which is demonstrated by Strack et al. [33]. Here, BCC’B’ is a leaky wall region with permeability of k’, and CDD’C’ can be taken as a wall region with permeability of k.

_{1}) and quantity of seepage under the wall (flow rate q

_{2}) simultaneously. Based on the superposition principle, the drawdown S is the sum of one part caused by q

_{1}(noted as S

_{(q1)}) and another part caused by q

_{2}(noted as S

_{(q2)}). The relationship can be obtained as S = S

_{(q1)}+ S

_{(q2)}. The S

_{(q2)}and S

_{(q1)}can be obtained by two simpler models in Figure 2a and Figure 2b, respectively, which are called the model with only seepage under the wall and the one with only leakage through the wall body. The quantity of seepage under the wall in Figure 2a is identical to q

_{2}, and the quantity of leakage through the wall body in Figure 2b is identical to q

_{1}.

_{AB}and the head value h(x, y): S(x, y) = h

_{AB}− h(x, y). The drawdown S(x, y) of each point in the region A’B’C’D’A’ is the difference between the downstream head h

_{A’B’}and the head value h(x, y): S(x, y) = h

_{A’B’}− h(x, y).

_{1}, q

_{2}, and the total flow rate q (= q

_{1}+ q

_{2}) in Figure 1c.

#### 2.1. Mathematic Expression for Flow Rates

_{BC}= −$\overline{S}$

_{B’C’}and $\overline{S}$

_{CD}= −$\overline{S}$

_{C’D’}. According to the definition of drawdown, $\overline{S}$

_{BC}= h

_{AB}− $\overline{h}$

_{BC}, $\overline{S}$

_{CD}= h

_{AB}− $\overline{h}$

_{CD}, $\overline{S}$

_{B’C’}= h

_{A’B’}− $\overline{h}$

_{B’C’}, and $\overline{S}$

_{C’D’}= h

_{A’B’}− $\overline{h}$

_{C’D’}are obtained. Considering H = h

_{AB}− h

_{A’B’}, then

_{BC}and $\overline{S}$

_{CD}can be obtained by the superposition of average drawdowns on the segments BC and CD in Figure 2a and Figure 2b, respectively. They are expressed as Equation (3). For the convenience of subsequent discussion, the average drawdowns in Figure 2a,b are normalized by q

_{2}/k and q

_{1}/k, respectively.

_{1}) and (q

_{2}) denote the models in Figure 2a and Figure 2b, respectively. R denotes the normalized average drawdown.

_{1}and q

_{2}as variables, one can obtain the following:

_{BC(q2)}, R

_{CD(q2)}, R

_{BC(q1)}, and R

_{CD(q1)}) in Figure 2a,b should be determined prior to the flow rates.

#### 2.2. Determination of Normalized Average Drawdowns

#### 2.2.1. Analytical Solutions for the Approximate Models

_{0}is the head value at point C, which is expressed as

_{0}is determined by

_{2}is also identical. In Figure 5b,d, the head value along the segment AB is identical, and the flow rate q

_{1}is also identical.

#### 2.2.2. Approximate Expressions for R_{BC(q2)} and R_{CD(q2)}

_{CD(q2)}is the average head on the segment CD in Figure 3a.

_{CD(q2)}= −q

_{2}/k·R

_{1}. Then, considering that the upstream head in Figure 3a is zero, the normalized average drawdown on the segment CD is expressed as

_{2}value is determined by Equation (18).

#### 2.2.3. Approximate Expressions for R_{BC(q1)} and R_{CD(q1)}

_{BC(q1)}is the average head on the segment BC in Figure 3b.

_{BC(q1)}= −q

_{1}/k·R

_{2}. Then, considering that the upstream head in Figure 3b is zero, the normalized average drawdown on the segment BC is expressed as

_{1}. Substituting them into Equations (9) and (11), the head value on the segment CD is obtained:

_{1}= (−2/π)sinh

^{−1}[T/(s$\sqrt{{\xi}_{0}}$)]; when y = T, kh/q

_{1}= (−2/π)sinh

^{−1}[1/$\sqrt{{\xi}_{0}}$].

_{1}value is determined by Equations (24) and (25).

#### 2.2.4. Correction

_{BC(q2)}, R

_{CD(q2)}, R

_{BC(q1)}, and R

_{CD(q1)}, respectively.

_{BC(q2)}, the approximate values are very close to the exact values when w/T > 0.1 or d/T < 0.5. For R

_{CD(q2)}, the approximate values are close to the exact values on the whole; only when d/T ≥ 0.9 and w/T < 0.5 do the results show a large deviation between the approximate and the exact values.

_{BC(q2)}, R

_{CD(q2)}, R

_{CD(q1)}, and R

_{BC(q1)}are calculated by the approximate solutions, respectively. Then, the flow calculation results in Cases 2–5 are compared with those in Case 1, respectively, and the effects of the approximate values of the four normalized average drawdowns on the flow calculation results are studied.

_{BC(q2)}and R

_{CD(q1)}does not affect the calculation results. However, the approximate values of R

_{CD(q2)}and R

_{BC(q1)}exerts a notable effect on the calculation results.

_{BC(q2)}and R

_{CD(q1)}can be directly calculated by the approximate solutions, regardless of the value of w/d and w’/s. To avoid large errors in the flow calculation results, approximate solutions for R

_{CD(q2)}and R

_{BC(q1)}need to be corrected within the following scopes: s/T ≤ 0.1(or d/T ≥ 0.9) and w/T < 0.5 (for R

_{CD(q2)}); w’/T ≤ 0. 5 and s/T ≥ 2w’/T (for R

_{BC(q1)}). Outside the scopes, the approximate solutions can be directly used to calculate R

_{CD(q2)}and R

_{BC(q1)}.

_{CD(q2)}and R

_{BC(q1)}are the product of the approximate values and the correction coefficients. Data fitting is applied to obtain the correction coefficients suitable for the scopes of correction:

_{1}and β

_{2}are the correction coefficients for R

_{CD(q2)}and R

_{BC(q1)}, respectively.

_{CD(q2)}and R

_{BC(q1)}are

_{BC(q2)}and R

_{CD(q1)}, the integrals of Equations (19) and (26) need to be solved numerically. The numerical integration scheme seems unattractive to engineers. To facilitate the calculation, linear fitting is adopted to obtain the simplified formulas for R

_{BC(q2)}and R

_{CD(q1)}. Figure 8a,b present the fitting results for R

_{BC(q2)}and R

_{CD(q1)}, respectively. The fitting curves have large correlation coefficients and satisfactory accuracy. Finally, the approximate expressions for R

_{BC(q2)}and R

_{CD(q1)}are

#### 2.3. Implementation

_{BC(q2)}, R

_{CD(q2)}, R

_{BC(q1)}, and R

_{CD(q1)}can be determined by simple calculation. Then, substituting them into Equations (5) and (6), the flow rates q

_{1}, q

_{2}, and q are obtained.

_{1}and q

_{2}into the exact models in Figure 2a,c, and use the analytical solutions of the two models to obtain the drawdowns in the regions ABCDA and A’B’C’D’A’. Then, by the superposition of the drawdowns of the two models, the drawdowns at each point in Figure 1c can be obtained. Finally, the head value at each point is the difference between the upstream or downstream head and the obtained drawdowns. When w/d and w’/s are sufficiently large, the approximate models in Figure 3a,b can be used to replace the two exact models to determine drawdowns at each point.

## 3. Verification and Discussion

- The wall penetration depth in the aquitard is zero (s/T = 0).
- The cut-off wall fully penetrates the aquitard (s/T = 1).
- The wall permeability is very small compared to the aquitard permeability.
- The wall thickness is very small compared to the aquitard thickness.

#### 3.1. Comparison with the Numerical Method in General Cases

_{TPM}− q

_{FEM}|/q

_{FEM}× 100(%). The q

_{TPM}and q

_{FEM}represent the flow calculation results of TPM and FEM, respectively.

#### 3.2. Special Cases where the Ratio s/T Is Zero or One

#### 3.3. Effect of Wall Permeability

#### 3.4. Effect of Wall Thickness

^{−6}, 0.001, 0.1, 0.25, 0.5, 1, 2.5, and 5 are adopted here, which makes it easy to obtain the flow calculation results of Yakimov. Figure 13a–c show the results for w/T = 0.01, 0.1, 0.5, and 1, respectively.

## 4. Conclusions

- Based on the proposed method, the seepage problem, which simultaneously includes leakage through the wall body embedded in the aquitard and seepage under the wall, can be solved by a simple analytical method.
- The exact solutions for the exact models are applicable to all situations, in principle, but they involve Legendre’s elliptic integrals of the first and third kinds, which makes the solution complicated. The simplified solutions obtained from the approximate models can be applicable to most situations. For situations outside the scope of application, corrections are given through comparison with the exact solutions, and hence the simplified solutions can be applicable to more-general situations.
- When the wall penetration depth in the aquitard is very small or the wall permeability is very small compared to the aquitard permeability, leakage through the wall body can be neglected. If the cut-off wall fully penetrates the aquitard, seepage only occurs through the wall body. For these cases, the proposed method can be degenerated into models that do not need superposition of drawdowns and the degeneration formulas are applicable.
- The wall permeability and thickness have large effects on the flow calculation results. When the permeability ratio is large (k’/k > 0.01) and the thickness ratio is large (w/T > 0.01), the effects of the wall permeability and thickness should be considered at the same time.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Seepage problems with two types of cut-off walls: (

**a**) partially penetrating cut-off wall; (

**b**) fully penetrating cut-off wall; and (

**c**) seepage model in the aquitard.

**Figure 2.**Two exact models: (

**a**) model with only seepage under the wall; (

**b**) model with only leakage through the wall body; and (

**c**) equivalent model of (

**b**).

**Figure 5.**Consistency in the flow nets between the approximate and exact models when the ratios w/d and w’/s are sufficiently large: (

**a**,

**b**) show the calculated flow nets for the approximate models, respectively; (

**c**,

**d**) show the calculated flow nets for the exact models, respectively.

**Figure 6.**Calculation results of the four normalized average drawdowns: (

**a**) R

_{BC(q2)}, (

**b**) R

_{CD(q2)}, (

**c**) R

_{BC(q1)}, and (

**d**) R

_{CD(q1)}.

**Figure 7.**Calculation results of the normalized flow rate when (

**a**) s/T = 0.1, k’/k = 0.1 and (

**b**) s/T = 0.75, k’/k = 0.9.

**Figure 9.**Flow calculation results of representative examples for (

**a**) s/T = 0.1, (

**b**) s/T = 0.25, (

**c**) s/T = 0.5, and (

**d**) s/T = 0.75.

**Figure 10.**Comparison of flow calculation results between TPM and FEM when k’/k = 0.9: (

**a**) flow calculation results and (

**b**) relative error.

**Figure 12.**Comparisons of flow calculation results among Wang, TPM, and FEM for (

**a**) s/T = 0.25, (

**b**) s/T = 0. 5, and (

**c**) s/T = 0.75.

**Figure 13.**Comparisons of flow calculation results among Yakimov, TPM, and FEM for (

**a**) w/T = 0.01, (

**b**) w/T = 0.1, (

**c**) w/T = 0.5, and (

**d**) w/T = 1.

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**MDPI and ACS Style**

Mei, J.; Cao, H.; Luo, G.; Pan, H. Analytical Method for Groundwater Seepage through and Beneath a Fully Penetrating Cut-off Wall Considering Effects of Wall Permeability and Thickness. *Water* **2022**, *14*, 3982.
https://doi.org/10.3390/w14233982

**AMA Style**

Mei J, Cao H, Luo G, Pan H. Analytical Method for Groundwater Seepage through and Beneath a Fully Penetrating Cut-off Wall Considering Effects of Wall Permeability and Thickness. *Water*. 2022; 14(23):3982.
https://doi.org/10.3390/w14233982

**Chicago/Turabian Style**

Mei, Jinling, Hong Cao, Guanyong Luo, and Hong Pan. 2022. "Analytical Method for Groundwater Seepage through and Beneath a Fully Penetrating Cut-off Wall Considering Effects of Wall Permeability and Thickness" *Water* 14, no. 23: 3982.
https://doi.org/10.3390/w14233982