Controlling Seepage Flow Beneath Hydraulic Structures: Effects of Floor Openings and Sheet Pile Wall Cracks

Abstract

Using one opening (filter) within the floors of hydraulic structures is a known technique to relieve the seepage effects on their floors. In this study, a new method to control seepage flow by using two identical filters instead of one was tackled numerically. A comparative analysis of using one versus two filters was conducted for different thicknesses of the permeable stratum, apron size (b), filter length, and sheet pile wall depths. Results indicate that two filters are considerably more effective than using one where the overall uplift force, the maximum potential head, and the hydraulic exit gradient downstream of the floor are reduced to 42–56%, 42–51%, and 66–76%, respectively, compared to one filter, while slightly increasing seepage flow by 1–7%. Many reasons can lead to horizontal openings (cracks) appearing along the sheet pile walls beneath hydraulic structures. The current study tackled their effects on seepage flow for the first time and examined their impact on the floor. A crack in the upstream sheet pile wall can increase total uplift forces by up to 40%, while a crack in the downstream sheet pile wall can increase the hydraulic exit gradient by up to 230%

1. Introduction

The water levels upstream of any control or storage water structures are higher than the downstream water levels, which leads to seepage flow beneath the floors of those structures within the permeable strata. Seepage flow is one of the main factors affecting their floors’ design. Three main seepage effects should be secured for all hydraulic structures as follows: uplift forces, hydraulic exit gradients, and seepage flow. The uplift forces may cause floating or cracks in the floors. Hydraulic exit gradients downstream of the floor should be secured against piping phenomena.

Meanwhile, the quantity of the seepage flow should be considered under the storage dam to check its storage efficiency. However, this is not essential in weir/waterfall structures, water regulators, and water barrages. Figure 1 is a schematic that illustrates the effects of seepage beneath hydraulic structures in the case of a floor provided with one opening.

Bligh [1] developed the first empirical method for calculating seepage flow beneath hydraulic structures in 1910, assuming equal vertical and horizontal creep length effects. In 1935, Lane [2] developed another empirical method after he noticed that vertical creep is much more effective than horizontal creep length based on field results. Polubarinova-Kochina [3] studied many cases of hydraulic structures and calculated the uplift forces, the hydraulic exit gradient, and the seepage flow as the consequences of the groundwater flow beneath the hydraulic structures. Many research works tackled each one of the consequences, such as the uplift forces [4,5,6,7,8,9,10,11], the hydraulic exit gradient [6,7,8,9,10,11,12,13], and the seepage flow [7,14].

All hydraulic structures should be secured against uplift forces caused by creeping under the structures and the piping phenomena caused by the hydraulic exit gradient [15]. Securing against the quantity of seepage flow beneath hydraulic structures is only essential when the hydraulic structures are used for water storage.

The failure of the Narora Weir in India due to the uplift forces is a case in point [15]. The piping phenomena has caused the failure of many hydraulic structures, such as Ashely Dam in Massachusetts, Hauser Lake Dam in Montana, Elwha River Dam in Washington [16], and Menufia Regulator in Egypt, as stated in [15].

Many techniques, such as cutoffs, grout curtains, impermeable blankets, and drainage holes, could reduce the uplift forces and the hydraulic exit gradient. The first three techniques increase the water’s flow path length, while the last relies on relieving the uplift pressure.

Intermediate plane filters within the floor of hydraulic structures are also known as weep holes, drainage holes, and openings. Many studies have tackled the case of one opening, such as [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]. Only one research study has solved analytically the case of using two intermediate filters [40]. The feasibility of using two intermediate filters instead of one filter to control the seepage effects is one of the main objectives of this current study, as discussed later in this section.

Research [41,42] has studied hydraulic structures with two sheet pile walls in the middle and one intermediate filter in 3D. The porous layer’s thickness (T) was 0.375 times the floor length of the hydraulic structure (b). The differential head between the structure’s upstream and downstream sides was 1 m. The floor length (b) equals 16, 20, and 24 H (the differential head). Also, the top of the banks was 2 m above the canal bed. The results of the 2D and 3D models were comparable only when the canal width to differential head ratio was greater than 10.

Using intermediate filters considerably reduced the uplift by up to 72% compared to when no filter was in place [41]. Also, the maximum uplift reduction occurred when the filter was positioned 1.0 m downstream of the sheet pile for the studied variable range. In addition, the filter significantly reduced the exit gradient at the downstream end of the floor. Furthermore, the filter length did not introduce a further significant reduction to the uplift force.

Vertical intermediate filter cases under dams have also been tackled in some research, such as [5,43]. A study [43] determined the average uplift pressure numerically across the base of a gravity dam with a system of vertical filters with equal-spaced drains of uniform diameter. A study aimed to minimize the influence of water movement under hydraulic structures by using a filter [5]. The impact of the water head, the upstream sheet pile depth, the vertical filter depth, and the filter location were investigated on the uplift pressure, the exit gradient, and the seepage discharge. The results showed that the downstream position of the filter leads to increases in the seepage discharge and exit gradient. At the same time, the upstream location significantly reduces the uplift pressure.

Research [44] investigated the minimum required extension of the end boundaries (L) in the upstream and downstream, and its impacts on the uplift, water discharge, and hydraulic exit gradient in the case when the thickness of the permeable stratum is half the apron length (T/b = 0.5). The potential head at the inner side of the downstream end sheet pile was selected for comparison. This research numerically investigates the effectiveness of inclined double sheet pile walls under hydraulic structures, considering the influence of the upstream and downstream sheet pile walls’ depth, location, and inclination angle.

To the best of the researcher’s knowledge, the minimum extension of boundary limits on the upstream and downstream sides of the hydraulic structure apron has not been studied in detail earlier in the numerical modeling. Different values of L have been assumed in the research. L was considered to vary between 1.375 b and 0.75 b for T, which ranged between 0.375m and 0.25 m, respectively [42]. L is assumed constant and equals 2 b for different dimensions [45]. In research [46], L is considered equal to 0.42 b in the case of T = 0.32 b. L_min is investigated as 1 b for T/b = 0.5 with an expected error percentage of less than 1% for uplift, water discharge, and exit gradient [44].

Seepage flow through cutoffs and sheet piles is generally prohibited. However, seepage may occur through these features for some reasons, such as cracks, erosion, construction joints, and construction errors. A crack at the top of a sheet pile occurred due to excessive displacement, verified by visual inspection [47]. The effect of erosion of the sheet piles can cause some passage through them [48]. The loading mechanism, stress and deformation behavior, and damage distribution and cracking of the sheet pile wall were analyzed [49]. The effect of construction joints of the sheet piles on the seepage through them was discussed [50]. Passages with small dimensions may occasionally penetrate the sheet pile due to construction errors during the wall installation [51,52,53,54,55,56]. The consequences of such cracks have not yet been investigated. Therefore, the second main objective of the current research is to examine the implications of sheet pile cracks on the seepage flow beneath the hydraulic structure floors.

As mentioned above, two primary objectives of the current study were defined, which are as follows:

• Check the feasibility of the new proposed technique (using two horizontal filters instead of one) to control the seepage effects (the seepage potential heads, uplift forces, the hydraulic exit gradient, and the seepage flow).
• Determine the consequences of a crack on the upstream or the downstream end sheet pile walls beneath hydraulic structures.

However, to achieve those objectives, another goal should be tackled, including L_min, one of the primary inputs of the numerical simulation, which must be determined first since it was not comprehensively covered in earlier studies.

The goals of the current research can be encapsulated as follows:

• Investigate the value of L_min for various variables.
• Determine whether L_min is influenced by the study’s objectives (seepage potential heads, uplift, hydraulic exit gradient, and seepage flow) or not.
• Evaluate the effectiveness of using two filters compared to one on the total uplift force and maximum potential head beneath hydraulic structures.
• Assess the impact of using two filters versus one on the hydraulic exit gradient downstream of the floor.
• Investigate the effectiveness of using two filters in controlling seepage flow beneath the floor instead of one.
• Study the effect of a horizontal crack in the upstream sheet pile wall on the total uplift forces.
• Study the effect of a horizontal crack in the downstream sheet pile wall on the hydraulic exit gradient.

The research considered first the finite element model. Then, the minimum extension of the end boundary L_min upstream and downstream of the floor was discussed for the convergence check over a wide range of variables. After that, a sensitivity analysis for the size of the cells was conducted. Next, the model was verified by comparing the numerical model results with those undertaken by an analytical solution (conformal mapping). Then, the model was used for a comparative study using one and two intermediate filters with the same total length. The analysis included the entire uplift force, the maximum uplift head points and their locations, the seepage flow, and the hydraulic exit gradient downstream of the apron. Finally, a case study of a single horizontal crack in the sheet pile walls was studied. Many runs were conducted to investigate the effect of a crack in the upstream sheet pile wall on the uplift pressure forces beneath the floor and the impact of a crack in the downstream sheet pile on the hydraulic exit gradient.

2. The Finite Element Model

In granular soil, seepage flow follows Darcy’s law as illustrated in Equation (1) [16,57,58,59]:

q = KA ∂h/∂x

(1)

where:

• q is the seepage flow (m³/s);
• K is the hydraulic conductivity of the soil (m/s);
• A is the cross-section of water flow (m²);
• ∂h/∂x is the hydraulic gradient of the flow.

Darcy’s law is used along with mass conservation to derive the groundwater flow equation. Under steady-state flow conditions, these two equations express Laplace’s equation [60], a second-order partial differential Equation (2):

K_x (∂² h)/(∂x²) + K_y (∂² h)/(∂y²) = 0

(2)

where

• Kx is the hydraulic conductivity in the x direction;
• Ky is the hydraulic conductivity in the y direction.

Four-node rectangular cells are used in the numerical model design, which uses the subroutine library new_library [61]. The lengths located upstream and downstream of the hydraulic structure floor are considered equipotential lines equal to the water head upstream and downstream of the floor, respectively. Also, the intermediate filters are assumed to have an equipotential head equivalent to the water level downstream of the hydraulic structure gate. Other boundaries of the domain are believed to be streamlines (walls).

3. Convergence Check

This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, and the physical conclusions that can be drawn.

3.1. The Effect of Upstream and Downstream Lengths on the Accuracy of the Results

Since this finite element model could be used to determine the potential head and the seepage discharge, the current study has been devoted to obtaining the best possible results for both. Figure 2 shows a schematic section for the problem. The floor of a hydraulic structure, keeping an H operating head and having a length built over a permeable soil of T thickness, could have one or two filters and/or sheet piles, which can be any size. L_u and L_d, the limiting lengths of the soil on the upstream and downstream of the floor, respectively, are the main objectives in this step. Lu is assumed to be equal to L_d, and L expresses both. The minimum required length L_min, beyond which no significant change of either potentials or seepage discharges, has been determined over a wide range of parameters.

3.1.1. Parameters

Many runs were carried out for cases of no intermediate filters and cases of two filters. The parameters include thicknesses of the permeable stratum (T), cutoff depth (d), location of first filter b₁, and location of second filter b₂. The range of the parameters can be encapsulated as follows:

• L/b = 0.25, 0.5, 1.0, 2.0, 3.0, 4.0, 5.0;
• T/b = 0.5, 1.0, 1.5, and 2;
• f/b = 0 and 0.025;
• d₂ = d₁ = 0.25 b when T/b = 0.5;
• d₂ = d₁ = 0.25 b, 0.333 b for T/b = 1, 1.5, 2.0;
• b₁/b = 0.05 and 0.10;
• b₂/b = 0.50 and 0.60.

3.1.2. Assumptions and Limitations

Some assumptions and restrictions have been assumed in this study. They are listed below:

• The hydraulic structure width > 10 the operating head (2D model is recommended);
• d₁ = d₂ = d;
• L_u = L_d = L;
• One homogenous and isotropic stratum;
• 0.5 T ≥ d;
• 0.333 b ≥ d;
• b₁/b = 0.05;
• b₂/b = 0.50.

3.1.3. The Minimum Extension of the End Boundary for the Potential Head Calculations

The results revealed that the stratum and floor lengths markedly affect the length L, as shown in Figure 3. When L = 0.25 b, a considerable difference in the potential head is expected, and this difference decreases as L increases. L_min equals b when T/b = 0.5, as investigated in research [44], where the difference in the potential heads is reduced to less than 0.2%. L_min can be selected 1.25 b, 1.75 b, and 2 b for T equals 1 b, 1.5 b, and 2.0 b, respectively.

The results indicate that the locations of both upstream and downstream filters have an insignificant effect on L. Also, the limited sheet pile depths in the analyzed range have an inconsiderable effect on L. On the other hand, both intermediate filter location and length and sheet pile depths impact the potential head at point B. In the case of intermediate filters, the points downstream reach the best possible values before those on the upstream side.

3.1.4. The Minimum Extension of the End Boundary for the Seepage Flow Calculation

All the studied parameters calculated the total flow passing through the vertical plain BB^\, as shown in Figure 2. The seepage flow calculation is usually crucial in some cases, such as the seepage beneath dams, while it is not usually calculated for other structures, such as weirs or water regulators. In this research, L is estimated to be used in the design of the storage dam. As shown in Figure 4, the porous stratum’s thickness considerably affects the total flow. Lmin is 1.0 b, 1.6 b, 2.3 b, and 3.0 b in the case of T = 0.5 b, 1.0 b, 1.5 b, and 2.0 b, respectively, to obtain a difference in seepage flow less than 1.0%. The minimum extension of the end boundary for the case of T = 0.5 b is consistent with the [10] research results. The total flow beneath the apron is nearly doubled (increases by 94%) when T increases four times from 0.5 b to 2 b.

Like the potential head, the results indicate that the locations of both upstream and downstream filters have an insignificant effect on L. Also, the limited cutoff depths in the analyzed range have an inconsiderable effect on L. Conversely, the intermediate filters and their locations significantly affect the flow value. In comparison, the cutoff depths slightly impact the total seepage flow.

3.2. The Selection of Cell Sizes

Different sizes of rectangular cells have been selected. The potential heads along the apron have been calculated until their values are constant. Point B has been chosen as a reference, as shown in Figure 5, representing the case of T = b, d₁ = d₂ = 0.25 b, b₁/b = 0.05, b₂/b = 0.5, and f/b = 0.05. L_min was selected to be 2.0 b. Five different sizes of the four-node rectangular cells have been assumed, which are 7068, 9044, 12,108, 24,216, and 30,270. The potential head at point B achieves a constant value at 24,216 cells, as shown in Figure 5. Therefore, this mesh size has been used in the rest of the research.

4. Validation of the Finite Element Model

The finite element model has been applied over many variables to study seepage problems beneath hydraulic structures provided with two intermediate filters. The results are compared with those obtained from the analytical solution using the conformal mapping method for the same problem [40].

4.1. The Geometry of the Analytical Model

A hydraulic structure shown in Figure 2 is built on a porous soil of depth T. It keeps a head difference H, which causes seepage flow underneath the floor with b length provided with two end sheet pile walls, d₁ and d₂, and two equal intermediate filters, each equal f. The conformal mapping solution assumes the upstream and downstream lengths (L) are infinite.

4.2. The Studied Parameters

The finite element model has been applied to study the seepage problem for different parameter dimensions such as sheet pile depths, filter lengths, and permeable strata depths. These various cases have been studied over the following ranges:

• T/d₂ = 3, 4, and 6;
• b₁/b = 0.05 and 0.1;
• b₂/b = 0.50, 0.525, 0.55, 0.575, 0.60, and 0.625;
• f/b = 0.025, 0.05, 0.075, 0.10, and 0.125;
• d₂/d₁ = 1, 2, and 5;
• T/b = 0.5 and 1.

4.3. Discussion of the Results

The comparison study included all the variables covered by the analytical model, i.e., the potential heads at the key and the maximum uplift head, the hydraulic exit gradient just downstream of the floor, and the location of the top uplift head points.

4.3.1. The Potential Heads

The potential heads at key and the maximum uplift head points have been compared, as shown in Figure 6 and Figure 7 and

Table 1. (T/d₂ = 3, T/b = 0.75, d₂/d₁ = 2, b₁/b = 0.10, b₂/b = 0.50).

f/b=	Methods of Solution	hj/H2	Location of Lj
0.025	Analytical	0.246	0.271 b
	Numerical	0.239	0.264 b
0.05	Analytical	0.211	0.293 b
	Numerical	0.198	0.287 b
0.075	Analytical	0.178	0.310 b
	Numerical	0.173	0.305 b
0.100	Analytical	0.104	0.33 b
	Numerical	0.102	0.325 b
0.125	Analytical	0.093	0.348 b
	Numerical	0.092	0.350 b

. Figure 6 represents the relationship between the location of the upstream filter and the potential heads at points B and C when b/d₂ = 4, T/b = 1, d₂/d₁ = 5, f/b = 0.05, and b₂/b = 0.6. Figure 7 shows the relationship between the location of the downstream filter and the potential heads at points B and C when b/d₂ = 4, T/b = 1, d₂/d₁ = 1, f/b = 0.10, and b₁/b = 0.05. Table 1 represents the uplift head at the maximum uplift head point J. The maximum difference in results between the analytical and numerical solutions has been investigated over the studied range of variables. The maximum difference for points B and C, which have the highest potential head, is 1.7%. The maximum discrepancy for the top uplift head points is 3%. These relative ratios of 1.7% for point B and 3% for the other points represent less than 1% of the total operating head of the hydraulic structure.

4.3.2. Exit Gradients

The critical hydraulic exit gradient has been calculated at outlet point E. Figure 8 and Figure 9 represent the relation between the exit gradient at E and the upstream and downstream filter locations, respectively. Figure 8 illustrates the relationship between the exit gradient and the location of the upstream filter when b/d₂ = 4, T/b = 1, d₂/d₁ = 1, f/b = 0.05, and b₂/b = 0.5. Figure 9 shows the relationship between the exit gradient and the location of the downstream filter when b/d₂ = 4, T/b = 1, d₂/d₁ = 1, f/b = 0.1, and b₁/b = 0.1. The vertical axis is calculated as a function of the exit gradient at E, the downstream sheet pile depth, and the net water head acting on the hydraulic structure, as shown in Figure 8 and Figure 9. The hydraulic exit gradient is decreased slightly as the filter moves towards the downstream side, as shown for the upstream filter in Figure 8 and the downstream filter in Figure 9. Table 1 represents the discrepancy between the analytical and numerical solutions for filter lengths 0.025 b, 0.05 b, and 0.075.

Although the potential head is tiny at exit point E, the maximum discrepancy between the analytical and the numerical solutions is less than 3% over the studied range of variables.

4.3.3. Location of the Maximum Uplift Head Points

The maximum uplift head is located between the two filters. Table 1 shows the uplift head and the maximum uplift location for different filter lengths when T/d₂ = 3, T/b = 0.75, d₂/d₁ = 2, b₁/b = 0.10, and b₂/b = 0.50. The locations of the maximum uplift points are achieved before their potential heads. Then, their sites obtained from the numerical model are compared to the filter length for the analytical solution. A good agreement between the analytical and numerical solutions has been received. The maximum differences between the locations of the top uplift head points are less than 3%.

5. A Comparative Study between Using One and Two Intermediate Filters

A detailed study used the numerical model to compare floors with one and two intermediate filters having the same total length. The analysis included a wide range of variables: the depths of the porous layer, the upstream sheet pile, the downstream sheet pile, and the location of the upstream filter. In all cases, using two intermediate filters, the downstream filter was assumed to be in the middle of the apron. The filter length in the case of one is divided into two equal filters in the case of using two. That means the total sizes of the filters are constant, as shown in Figure 2 and Figure 10. Many aprons have been analyzed assuming no sheet pile, only one sheet pile at the upstream or the downstream end of the floor, and two end sheet pile walls. Also, aprons without intermediate filters and with one or two filters have been considered. This study answered whether using one filter or dividing it into two identical parts and using two intermediate filters is better. The variables were considered as follows:

• T/b = 1, 1.5, and 2;
• b₁/b = 0.00, 0.05, and 0.10, and b₂/b = 0.50;
• f/b = 0.05 for the case of one intermediate filter, and f/b = 0.025 for each filter for the case of two filters;
• d₁/d₂ = 0.0, 0.2, 0.5, 1.0, 2.0, and 5.0, and the case of d₂ = zero;
• b = (2, 3, 4, and 6) d1 if d1 > d2;
• b = (2, 3, 4, and 6) d2 if d₂ > d_1.

The comparative study includes the following three main items: the uplift force, the hydraulic exit gradient at the end of the floor, and the total seepage flow.

5.1. Uplift Pressures

The total uplift forces, the maximum uplift heads underneath floors, and their locations are the main factors that have been calculated to compare different cases. Figure 11 and Figure 12 illustrate the effect of the upstream and downstream sheet pile depths on the uplift beneath the floor for the possibilities of no filter, one filter, and two intermediate filters, respectively. Figure 11 illustrates the case when T/b = 1.5, d₂ = 0.25 b, b₁ = 0.05 b, b₂ = 0.5 b, and f/b = 0.025. Meanwhile, Figure 12 illustrates the case when T/b = 1.5, d₁ = 0.25 b, b₁ = 0.05 b, b₂ = 0.5 b, and f/b = 0.025. Figure 11 and Figure 12 clearly show that using two filters significantly reduces the uplift force compared to using one with the same total length. For the studied range of variables, the total uplift force and the maximum potential head underneath a floor provided with two filters are reduced to 26–39% and 31–41%, respectively, compared with those for the case of no filter. At the same time, the reductions are 42–56% and 42–51%, respectively, for the possibility of one filter. Figure 11 shows that the upstream sheet pile depths affect the total uplift forces and the maximum uplift head points. The effect on the maximum head is slightly higher in the case of two filters than in the case of one. The effect of the downstream sheet pile depth is insignificant when using two filters. At the same time, it is limited in the case of one filter, as shown in Figure 12.

The location of the maximum uplift points moves towards the downstream side as the upstream sheet pile depth increases, as shown in Figure 11. In contrast, their locations travel towards the upstream side as the downstream sheet pile depth increases for the case of one filter. It is nearly constant in the case of two filters, as shown in Figure 12. The location of the maximum uplift head points markedly moves along a vast distance compared to the possibility of two filters. In the case of two filters, the downstream filter controls the location of the maximum head located between the two filters. Therefore, the relative size of the maximum uplift (X/b) is within 20–30% in the case of two filters. On the other hand, in the case of one filter, the location of the maximum potential head freely moves downstream of the filter, where it is located between X/b = 30–80%.

5.2. The Total Seepage Flow

The total seepage flow passing through section BB^\, as shown in Figure 2 and Figure 10, generally increases with any decrease in the sheet pile depth and with an increase in the permeable stratum depth. The maximum seepage flow for all cases, with or without filter, occurs when no sheet pile is used, and the minimum when two end sheet pile walls are used. This study assumes the seepage flow for the possibility of no filters and sheet pile walls as a seepage flow q_to reference value. The seepage flow for any filter case is considered qt. Figure 13a represents the effect of the downstream sheet pile depth on the total seepage flow (d₁ = b/6), assuming that d2 changes from zero value (no downstream sheet pile) to d₂ = d₁ = b/6 for the case T/b = 2.0, b₁/b = 0.1, b₂/b = 0.5, and f/b = 0.025. Figure 13b represents the upstream sheet pile depth effect for the same variables.

Using filters considerably increases the seepage flow, even for small sizes. The seepage flow for the floor with intermediate filters increases with increasing the filter length. The total seepage flow slightly increases in the two filters case than in the one filter case, as shown in Figure 13. The downstream sheet pile depth has an inconsiderable effect on the total seepage flow in the case of two filters. In contrast, it has a very slight impact in the case of one filter, as shown in Figure 13a. The upstream sheet pile depth considerably affects the total seepage flow in the cases of one and two filters, where the entire seepage flow decreases as the depth of the upstream filter increases, as shown in Figure 13b. The seepage flows for the cases of one or two filters are increased to be 1.1–3 times the flow for the no-filter case. Using two intermediate filters increases the seepage flow to 101–107% compared to one with the same total length. The maximum difference occurs when d₁ = d₂ and the minimum when no sheet pile is used.

5.3. The Hydraulic Exit Gradients

Hydraulic exit gradients just downstream of hydraulic structures I_h, provided with or without filters, are decreased considerably by increasing d₂ and slightly by reducing d₁, as shown in Figure 14a,b, where the hydraulic exit gradients I_h are compared to that of I_ho for the case of no filter or sheet pile wall. Similarly, the exit gradient decreases when the depth of the permeable stratum decreases. Using intermediate filters markedly reduces I_h. This reduction increases as the lengths of the filters increase, and the filter moves downstream. Figure 14a represents the effect of the downstream sheet pile depth on the hydraulic exit gradient (d₁ = b/6), assuming that d2 changes from zero value (no downstream sheet pile) to d₂ = d₁ = b/6 for the case T/b = 1.0, b₁/b = 0.1, b₂/b = 0.5, and f/b = 0.025. Figure 14b represents the upstream sheet pile depth effect for the same variables.

Using filters considerably decreases the hydraulic exit gradient, even for small sizes. The exit gradient also decreases as the depth of the sheet pile increases for the cases of one and two filters. The downstream sheet pile depth considerably affects the exit gradient in both cases of one and two filters, as shown in Figure 14a, where the exit gradient decreases as the downstream sheet pile depth increases. In contrast, the upstream sheet pile depth slightly affects the exit gradient in the cases of one and two filters. The exit gradient decreases as the upstream sheet pile depth increases, as shown in Figure 14b. Using two intermediate filters reduces I_h to 48–62% compared to the case of no filter, while decreasing it to 67–77% in the case of one. Consequently, two intermediate filters reduce I_h to 66–76% compared with the possibility of one.

6. Case Study: The Effect of a Crack in Either Upstream or Downstream Sheet Pile Wall

A case study of a single horizontal crack in upstream or downstream end sheet pile walls was discussed below. First, the case study and the range of parameters studied were represented. Then, the effect of a crack in the upstream sheet pile wall hydraulic structures on the uplift pressure force and the impact of a crack in the downstream sheet pile wall on the hydraulic exit gradient were analyzed.

6.1. The Case Study

Hydraulic structures are commonly provided with sheet pile walls beneath their floors. The upstream end sheet pile walls are mainly used to reduce the uplift forces below the structure floor. Conversely, the downstream sheet pile walls are primarily used to reduce the hydraulic exit gradient; however, they increase the uplift forces under the hydraulic structures. In this study, the effect of a crack in the upstream sheet pile walls on the uplift forces and the impact of a crack on the downstream end sheet pile walls on the hydraulic exit gradients were investigated. The effect of a crack on either sheet pile wall on the total seepage flow is insignificant. The hydraulic structure shown in Figure 15 is built on a porous soil of depth T and keeps a head difference H, which causes seepage flow underneath the floor. It has a length equal to b and is provided with two end sheet pile walls, d1 and d2.

The finite element model has been applied to study the seepage problem for the case study. Different crack depths (y1 for the case of a crack in the upstream sheet pile wall and y2 for the case of the downstream one) were assumed. The study included the following parameter ranges:

• The cases of the upstream sheet pile wall crack

  i.

  d1/b = 0.5;

  ii.

  y1/d1 = 0.0, 0.25, 0.50, 0.75, and no crack;

  iii.

  d2/d1 = 0.0, 0.5, 0.75, and 1.00;

  iv.

  T/b = 0.75, 1.00, and1.25;

• The cases of the downstream sheet pile wall crack

  v.

  d2/b = 0.5;

  vi.

  y2/d2 = 0.0, 0.25, 0.50, 0.75, and no crack;

  vii.

  d1/d2 = 0.0, 0.5, 0.75, and 1.00;

  viii.

  T/b = 0.75, 1.00, and1.25.

The cracks’ width, dy, was assumed to be very small, where dy/d1 and dy/d2 = 0.167%. The crack is considered to be on the entire width of a wide sheet pile wall. Therefore, the case was studied as a two-dimensional problem. Many runs have been required to cover the above range of variables, where 6496–10,556 four-node planes were used.

6.2. Discussion of the Results

6.2.1. The Case of a Crack in the Upstream Sheet Pile Wall

Figure 16, Figure 17 and Figure 18 represent a comparison of the uplift forces in the standard design case (case of no crack) and the uplift forces for the instances of a crack in the upstream sheet pile wall in different locations to evaluate the increment of the uplift forces. As shown in Figure 16, the uplift heads considerably increased as the crack location was moved upward toward the apron.

Figure 17 represents the relationship between the floor length and the total uplift force ratio U^\/(Hb), showing the effect of a horizontal crack depth in the upstream sheet pile wall for different downstream sheet pile wall depths on the total uplift forces for the case of T/b = 1.25 and d1 = d2 = 0.5 b. The total uplift forces acting on the apron for the instances of cracked upstream sheet pile walls U^\ are represented by Equation (3) below:

$$U^{\backslash }={{\displaystyle \int }}_o^{b}f(h_p)dl$$

(3)

where:

• U^\ is the total uplift pressure forces per unit width of the floor;
• b is the length of the floor, as shown in Figure 15;
• h_p is the uplift pressure head;
• f(h_p) is the function between the location and its uplift pressure head under the floor;
• dl is an increment length of the floor;
• H is the head difference between upstream and downstream water levels of the hydraulic structure, as shown in Figure 15.

As expected, using downstream sheet pile walls increased the total uplift, and the value of the increment increased as the depth of the sheet pile wall increased. The maximum increment of the total uplift forces occurred when the crack was at the top of the sheet pile wall, where it varied between 21% and 23% for the case of d1 = d2 = 0.5 b. In comparison, it ranged between 35% and 40% for the cases of d2 = 0 and d1 = 0.5 b for the range of T/b = 0.75–1.25.

The relationship between the total uplift forces and the thickness of the porous layer is illustrated in Figure 18 for the case of d1 = d2 = 0.5 b, where the total uplift forces were decreased very slightly when the thickness of the impermeable layer increased for the different locations of a horizontal crack in the upstream sheet pile walls. The cases of the various positions of horizontal cracks in the upstream sheet pile walls have the same trend as the case of no crack.

6.2.2. The Case of a Crack in the Downstream Sheet Pile Wall

Figure 19 represents in its vertical axis the ratio between the hydraulic exit gradient (i_c) in the case of an appearing horizontal crack in the downstream sheet pile wall to the hydraulic exit gradient (i_o) in the case of no crack (standard design case) for the exact dimensions and the ratio between the depth of the downstream crack (y2) to the depth of the downstream sheet pile wall (d2) as a horizontal axis. The Figure illustrates the effect of the horizontal crack depth in the downstream sheet pile wall on the hydraulic exit gradient for different upstream sheet pile wall depths in the case of d2/b = 0.5 and T/b = 1. As shown in the Figure, the hydraulic exit gradient downstream of the floor increased when a horizontal crack appeared in the downstream sheet pile walls. The increase is maximum when the crack appeared on the top of the sheet pile wall, where the hydraulic exit gradient increased by 217–228% compared to its value for the case of no crack. The increment ratio was considerably reduced as the location of the horizontal crack in the downstream sheet pile wall traveled away from the apron. Then, it was slightly reduced, as shown in the Figure. The effect of upstream sheet pile wall depth on the hydraulic exit gradient is considerably limited, where the increase of the hydraulic exit gradient is slightly reduced as the depth of the upstream sheet pile increases.

7. Conclusions

A 2D finite element model has been used to simulate the seepage flow beneath hydraulic structures. It can be applied to hydraulic structures such as concrete dams, weirs, water regulator structures, spillways, and water barrages with and without sheet piles and intermediate filters. The model can determine the uplift forces, the hydraulic exit gradient, and the seepage flow rate under those structures.

The minimum boundary length (L_min) upstream and downstream of the floor is one of the main inputs in the numerical simulation and was only determined for a limited range of variables earlier by other studies. Therefore, L_min was deduced first for a broader range of variables in the current study. The current work revealed that L_min depends on the objective of any research. The study shows that the L_min required to determine the potential head and the hydraulic exit gradient is less than those required to determine the seepage flow, which may be necessary in some cases, such as storage dams. Also, the results revealed that the stratum and floor lengths markedly affect the length L. In calculating potential heads and the hydraulic exit gradient, L_min is b, 1.25 b, 1.75 b, and 2 b for T = 0.5 b, b, 1.5 b, and 2.0 b, respectively. In the case of the calculation of seepage flow, L_min is b, 1.6 b, 2.3 b, and 3.0 b for T = 0.5 b, b, 1.5 b, and 2.0 b, respectively.

A sensitivity analysis for the size of the cells was conducted. Then, the model verification was applied by comparing the numerical model results for the case of two intermediate filters with those undertaken by an analytical solution for a wide range of variables. The maximum difference in the potential heads between the analytical and numerical solutions was less than 1% of the total operating head. Meanwhile, the maximum discrepancy in the hydraulic exit gradient was less than 3%.

The effectiveness of the new method of using two filters to control the seepage effects was examined in this study. A comprehensive comparison between one and two intermediate filters having the same total length was tackled using the numerical model. The total uplift forces, the maximum uplift heads underneath floors, and their locations are the main factors that have been calculated to compare different cases. Both the entire uplift pressure force and the top uplift head underneath the hydraulic structures aprons with two filters were reduced to 42–56% and 42–51%, respectively, compared to the apron with one filter at 26–39% and 31–41%, compared with those for the case of no filter. However, the seepage flow in the cases with two filters increases to only 101–107% compared to one with the same total size. In addition, the location of the maximum regained potential head beneath the floor (the stagnation point) was considerably controlled when two filters were used. In contrast, it was not in the case of one filter, where it was uncontrolled and vastly moved along the floor length.

The results of this study demonstrate the superior effectiveness of utilizing two identical filters compared to a single filter of the same total length. Thus, it is recommended that further research be conducted to optimize the performance by selecting varying lengths for each filter rather than employing identical lengths. Moreover, future studies should investigate the optimum location of each filter along the floor of the hydraulic structures, as opposed to the limited locations considered in this study, to achieve optimal results.

Finally, the study analyzed the impact of an emerging horizontal crack in the upstream sheet pile walls on the total uplift forces and the downstream sheet pile walls on the hydraulic exit gradient for the first time. Results showed that as the crack moved upward toward the apron in the upstream wall, uplift pressures significantly increased, where the maximum uplift force increment, ranging from 21% to 40%, occurred when the crack was at the top of the upstream wall. Also, increasing the thickness of the impermeable layer slightly reduced total uplift forces across different crack locations. On the other hand, in the case of a crack in the downstream sheet pile wall, the hydraulic exit gradient increased significantly, especially at the top, with an increment of 220–230% compared to a no-crack scenario. However, this increase decreased as the crack location moved away from the apron.

Based on the results of this part of the study, which indicate a significant increase in the uplift force and the hydraulic exit gradient just downstream of the floor in the event of a crack in the sheet pile walls, it can be recommended to increase the safety factor for each or both parameters if the designer has concerns regarding the workers’ execution efficiency or the type of soil that may cause some problems in the sheet pile efficiency.