Evaluating the effect of different diaphragm wall cracking scenarios on seepage and slope stability in Earth dams using experimental and numerical modeling approaches

Background Seepage control measures are essential in earth dams to enhance their safety against seepage and slope failures. A diaphragm is a thin impervious wall that is used in the core area of the dam to prevent the seep‑ age of water. Any crack in the diaphragm can reduce its efficiency and threaten the safety and stability of the dam. Extensive evaluation of the influence of diaphragm fracturing on dam safety is required to establish a safe design and offer the appropriate measures. This study investigates the effect of diaphragm cracking on the safety of the dam for seepage and slope stability. A permeability tank is first used to validate the numerical model; then, different scenarios for cracked diaphragm walls are studied numerically. The crack’s height, width, direction, and number are all investigated. The effect of having a downstream horizontal blanket drain with cracked internal diaphragms is also evaluated. Results The results showed that the internal diaphragm enhances the safety of earth dams as it reduced the seep‑ age discharge by 98% and increased the downstream slope safety by 34%. Considering the development of a crack in the internal diaphragm, the most critical location was at the base of the dam, wider cracks were more destructive, and the most critical direction was the horizontal crack. The fracturing of the diaphragm at multiple locations proved to be detrimental to the safety of the dam. Having three cracks caused the seepage discharge to reach 74% of its value for no diaphragm, the pore water pressure to raise drastically, and the downstream critical factor of safety to be higher than the case of no diaphragm by 8% only. Conclusions According to the findings, it is advised to continuously monitor the components of diaphragm‑type earth dams and perform the necessary maintenance works whenever any sign of damage or water seepage appears. Furthermore, using a well‑graded horizontal drain of sufficient length and high permeability with the internal dia‑ phragm is highly recommended as a mitigation measure if diaphragm fracturing occurs. This helps to quickly release the seeping water and keep the phreatic line away from the downstream slope.


Graphical abstract 1 Background
Earth dams are one of the most ancient and prevalent types of dams [1].They retain water in their upstream reservoir for multiple purposes like protecting from floods, providing water requirements in shortage times, generating hydro power, providing a sufficient navigation depth for ships, and diverting water to high-level intakes of upstream canals [2].Earth dams are the best choice when the foundation's strength is inadequate to withstand a gravity dam [3].In addition, they are built using natural materials with simple equipment and basic procedures.Earth dams have three main types: homogeneous (constructed using one kind of soil), zoned (comprises soils of various hydraulic conductivities, with the central core having the lowest hydraulic conductivity), and diaphragm (with a thin impervious internal wall) [1,4].Homogenous earth dams should only be utilized for low heights, as safety hazards and vulnerability to failure increase for high dams of large storage [5].Earth dams may collapse due to seepage failure, hydraulic failure, or structural failure [6].Seepage is one of the key elements that affect the stability of hydraulic structures [7].The majority of dam failure incidents are attributed to the seepage of water through the earth dam's body or its foundations [8].Hence, it is crucial to keep seepage through earth dams under control.This can occur by using drains, cores, and diaphragm walls.Such control measures enable the usage of steeper slopes and prevent the phreatic line from reaching the downstream face [1].Selecting the most suitable seepage control measure depends on the intended purpose/function of the earth dam.Drains and filters are employed to get rid of the seeping water quickly and avoid its build-up inside the dam's body [9].On the other side, cores and diaphragm walls are typically used in storage dams to block the seepage and prevent water loss from the dam's reservoir [10].
Drains and filters have a significant role in the safety and stability of earth dams [11,12].Their main functions are preventing the internal erosion of the dam's material and avoiding excessive pore water pressure from building-up inside the dam's body [13,14].They are constructed from high permeability granular materials with specific gradations that ensure that they release the water but retain the dam material's particles [15].Filters are usually placed between the shell material and the drain material as a transition zone, yet a single filter layer can sometimes be used to retain the dam material and perform as the drain at the same time [16].Geotextiles, which are a type of permeable fabrics, can be used between the dam material and the drain to act as the filter layer [17].Drains have many shapes and types including horizontal blanket drains, triangular rock toe drains, vertical and inclined chimney drains, and pipe drains [18].The most important charachteristic for a horizontal drain is its length [13,14,18], the same effective parameter was observed for the triangular toe drain [19], while the vertical drain is mainly affected by its height [20], and the length of parallel pipe drains is not recommended to exceed one-third of the dam's base [21].
A core is a low permeability zone inside the earth dam that significantly reduces seepage [22].The thickness of the core should be sufficient to prevent the seepage of water through the body of the dam [23][24][25].The core may be constructed using masonry, puddle clay, or concrete.Similarly, a diaphragm is a thin impervious wall in the central part of the dam that is used to prevent the seepage of water.The thin impervious diaphragm wall may be made of concrete or any other impervious material.The main difference between the zoned and diaphragm earth dam types is the thickness of the impervious part.The thickness of the diaphragm should be less than the height of the dam [1].On the other hand, a cutoff is a barrier that can be constructed on the upstream or downstream sides, or extended into the dam's foundation under the core to reduce seepage [26,27].Cutoffs are usually constructed by excavating trenches filled with an impervious material [28].Foundation cutoffs can be made by extending the core, or diaphragm wall deeper [5,22].Sometimes the words "diaphragm" and "cutoff " are used interchangeably with the same meaning of a thin impervious barrier inside or under the dam.
Concrete is commonly used to construct the internal diaphragm wall in earth dams [29].This can be attributed to its distinct properties, which include durability, low permeability, and structural stability [30].By manipulating the proportions of the concrete mix constituents and incorporating admixtures, it can be tailored to have enhanced microstructure, physical, and mechanical properties that fit for diverse applications, making concrete a key material in modern construction projects [31][32][33][34][35][36].Highly impermeable concrete is the most preferable for constructing internal diaphragm walls [37].This can be often achieved through a low water-to-cement ratio and proper curing [38].
Numerous earlier studies looked into the impact of employing an internal core on the safety of earth dams [39,40] and aimed to determine its preferable characteristics [23] and optimum geometry [41].The low permeability core proved to reduce the seepage discharge and lower the phreatic level [41].It also improves the dam's safety against seepage and slope failures [40].A transition zone can be added between the core and shell materials.The optimum ratio of inner, transition, and outer zones thicknesses, which reduces seepage and construction cost, was found to be 2:1.5:1.5 [42].It is also preferable to use materials of lower permeability in the outer shells and upstream transition zone [43].
Other studies assessed the effect of using an internal cutoff, grouted barrier, or sheet pile on the seepage parameters and slope stability of an earth dam.El Molla [44] numerically investigated the effect of using a vertical sheet pile inside the earth dam's cross section on the total seepage discharge and velocities.The sheet pile reduced the discharge, but the velocity increased at its upper end.Sazzad and Alam [45] applied finite element modeling to analyze the effect of using a bentonite grout curtain on seepage through earth dams constructed on an impervious base.The findings pointed out that seepage can be sufficiently controlled using a grout curtain near the upstream face combined with the impervious base.Attia et al. [46] studied the influence of adding a cutoff to the earth dam's cross section.They implemented a Hele-Shaw experimental apparatus to evaluate the effect of the cutoff penetration depth (measured from the crest of the dam) on the total seepage discharge, the drop in the total head, and the required length of the horizontal filter.The internal cutoff reduced the discharge and the needed length of the horizontal filter by 10%.Haghdoost et al. [47] studied numerically the effect of using an internal cutoff inserted from the crest of the dam on the hydraulic gradient, seepage discharge, and drop in the total head.The impact of the cutoff penetration depth was also evaluated.The internal cutoff caused a remarkable reduction in seepage discharge.Increasing its penetration depth further reduced the discharge and increased the drop in the total head and hydraulic gradient.
On the other side, any crack in the core, cutoff, or diaphragm reduces its efficiency and can be detrimental to the safety and stability of the dam.Park et al. [48] evaluated the filter's performance when both the filter and the core materials are cracked.The results showed that the current filter design criteria provide reliably performing filters even with cracks.Zomorodian and Moghadam [49] investigated experimentally the case of filter and core cracking in embankment dams.Their findings highlighted that increasing the time of compaction reduces the amount of eroded materials.Nayebzadeh and Mohammadi [50] evaluated the hydraulic fracturing problem and core arching phenomenon due to the excessive usage of clay in the core zone.The numerical model SIGMA/W was applied to study the case of Ghavoshan dam in Iran.They concluded that the inclined core shape is preferred from the point of view of stress, deformation, and resistance against hydraulic fracturing for the same dam width.
Also, Talukdar and Dey [51] studied the function of drainage blankets in earth dams with a cracked core.They concluded that drainage blankets do not reduce core cracking or stop its occurrence.However, such blankets control the movement of water inside the dam if the core cracked.In addition, Chen et al. [52] evaluated the effect of the number, position, and variability of defects in a polymer cutoff on the seepage characteristics using a 3D numerical model.Thickness and nonhomogeneity defects were considered.The results clarified that such defects negatively impact the effectiveness of the polymer cutoff and its seepage control ability.Aghajani et al. [53] investigated the impact of having a defective high permeability zone inside a clay core on the seepage characteristics.Demirdogen and Gunaratne [54] studied the effect of using different shapes of geomembrane barriers inside the section of an earth fill dam.The internal geomembranes reduced the seepage and pore water pressure, yet their defects hindered the stability of the downstream slope.Steep upstream slopes were also susceptible to instability when sudden drawdown of the reservoir was simulated.Fawzy et al. [55] carried out a study on the effect of using a grouted diaphragm to rehabilitate existing homogeneous earth dams.The diaphragm was considered to be grouted from the crest.Different grouting locations, thicknesses, and penetration depths were evaluated.The possible defects that can occur during the grouting process and construction were also considered.The studied defects included uneven diaphragm thickness, high permeability concrete mixtures, and the occurrence of a single crack.The results highlighted that diaphragm grouting is a very efficient method for earth dam rehabilitation, especially if it has deep penetration.On the other hand, construction defects may diminish its efficiency and threaten the dam's safety.It was also concluded that adding a triangular toe drain with the grouted diaphragm during earth dam rehabilitation resists the negative consequences of the different possible construction defects.Taleb et al. [56] used numerical modeling to assess the effect of fissured soil on the stability of earth dams.The study included various configurations and angles of fissured planes in the dam material.The performance of the earth dam with fissured soil was compared to the base case of un-defected soil.The results showed that the slope stability safety factor in all the fissured material cases was lower than that of the undefected soil.The findings indicated the importance of considering the effect of fissured soil during the design of dams.
Surveying the previous literature and research work revealed the necessity to perform a study that extensively evaluates the effect of diaphragm cracks on the dam's safety against seepage and slope failures, as this problem has not been sufficiently tackled before.Hence, this study aims to simulate seepage through earth dams with an internal diaphragm wall and estimate the effect of the diaphragm's cracking on the seepage and slope stability of the dam.The study focuses on existing diaphragmtype earth dams, which have a thin internal concrete wall that controls seepage (dams initially constructed with an internal diaphragm wall that almost fully blocks the seepage through the dam).This internal diaphragm wall can develop cracks during the dam's service time due to stresses, temperature changes, earthquakes, aging, or deterioration.The diaphragm cracks are assumed to occur after the dam is fully constructed and operated.The main research questions addressed in this paper are: What is the effect of diaphragm cracks on the dam's safety?Will the crack's characteristics (height, width, direction, and number) affect its negative consequences?Does using a horizontal blanket drain in diaphragm-type earth dams help to reduce the negative impacts in the case of diaphragm wall cracking?Accordingly, this study uses experimental and numerical modeling approaches to answer the three aforementioned questions.A permeability tank experimental model is first used to validate the numerical model; then, the study is completed numerically.Different scenarios for internal diaphragms with cracks are studied using SEEP/W and SLOPE/W numerical models.The height, width, inclination direction, and number of cracks are all investigated.In addition, the effect of having a downstream horizontal drain with cracked internal diaphragms is evaluated.The study offers useful insights for the designers of earth dams that help them to select a sustainable seepage control system which can withstand deterioration with time.

Dimensional analysis
Figure 1 displays the variables considered in this study, which are presented functionally as follows: where H is the maximum upstream water head on the dam, B is the width of the crest, Y is the height of the phreatic line over the dam's base, y is the height of the crack measured from the dam's base, t is the width of the crack in the diaphragm, L is the base width of the dam, X is the piezometer's distance from the beginning of the dam, hp is the piezometric height, D 50 is the median size of the dam material, N is the number of cracks, θ is the inclination angle of the crack measured from the horizontal plane, g is the gravitational acceleration, q is the total seepage discharge per unit width of the dam, v is the maximum velocity at the section just downstream the diaphragm, ρ w is the density of water, K is the hydraulic conductivity of the earth dam's material, K D is the hydraulic conductivity of the diaphragm's material, and FS is the critical factor of safety for downstream slope.
The dimensional analysis was performed using Buckingham's π theorem, with the repeating variables of H , ρ w and K , leading to the following π-terms: The piezometer's distance ( X ) is related to the length of dam's base ( L ).Also, the same shell and diaphragm mate- rials are used throughout the study; consequently, D 50 , ( 1) K , and K D are considered constant throughout the study.
In addition, B, H , ρ w , and g are also constant.Thus, the final dimensionless π-terms are:

The experimental work
In this study, a permeability tank (TecQuipment H312) was utilized to simulate water seepage through an earth dam with a cracked internal diaphragm wall.The tank is 1500 mm length, 600 mm height, and 180 mm width.It is connected to fourteen piezometers installed at equal spacing along the tank's base to measure the pore water pressure.Figure 2 shows a view of the experimental setup and the dimensions of the permeability tank.The smallscale earth dam model is constructed inside the tank, and the water levels are adjusted at both sides of the model using end chambers with adjustable overflow pipes.The tank gives trustworthy observations and allows drawing the flow lines and equipotential lines on its transparent glass side.A dye injection system is used to help in tracing the flow lines.Two different materials are used to construct the dam, the shell material and the diaphragm material.The used shell material is poorly graded sand with a very low percentage of fines, which is optimum to reduce the capillary effects and ensure reliable and accurate results.The hydraulic conductivity (K) of the shell material is (1.3 × 10 -2 cm/s), which according to Das [57] falls within An earth dam with a cracked internal diaphragm wall was constructed.The diaphragm is located at the middle of the dam's crest, while the crack has a height ratio (y/H) of 0.568 and a crack width ratio (t/H) of 0.01113.The value of the upstream water head was kept constant to simulate the case of a full reservoir, which is the most critical.The effect of having a crack in the diaphragm on the phreatic line and pore water pressure distribution was observed.The physical modeling experiment was designed to test and validate SEEP/W numerical model before utilizing it to investigate further cases and scenarios for the height of crack (y), width of crack (t), inclination direction of crack (θ), and number of cracks (N).
The dimensions and scale of the studied earth dam have been selected after Refaiy et al. [14] who used the same permeability tank to study the optimum geometry of downstream drains.The dam dimensions are based on Terzaghi and Strange's recommendations to ensure a safe section of the earth dam and stable slopes [1].For the experimental model, two main scaling techniques have been implemented as recommended by Heller [58] and Refaiy et al. [14], avoidance and compensation.This means that the dam's scale is taken as 1:25 to ensure a realistic experimental model and avoid scale effects.In addition, the same grain size is used for the soil of both the experimental model and real-life dam to improve the similarity between them.Hence, the experimental dam model dimensions are 30 cm height, 8 cm freeboard, 22 cm upstream water head, 8 cm top width, 2.5:1 upstream slope, 1.75:1 downstream slope, D 50 = 0.62 mm, and a diaphragm thickness of 2 cm.This corresponds to a real dam of 7.5 m height, 2 m freeboard, 5.5 m upstream water head, 2 m top width, 2.5:1 upstream slope, 1.75:1 downstream slope, D 50 = 0.62 mm, and a diaphragm thickness of 0.5 m, which is studied numerically.The downstream water level was considered to be dry in all the experiments and numerical simulations.A downstream slope protection was used in the experimental model to safeguard the stability of the dam and avoid its failure by piping during the experiment.The dimensions of the real dam and the scaled experimental model are shown in Fig. 3.
To perform the earth dam seepage experiment, the tank was first cleaned to guarantee that no tiny particles interfered with its precision.After that, the dimensions of the dam were drawn on the front glass to facilitate its construction and set the required upstream water level accurately.Next, the template of the diaphragm was formed by using plywood with the required dimensions and the diaphragm was cast using (Sikacrete R -114).A binding wire was used as a light reinforcement for the diaphragm.The crack was made in the diaphragm by placing sand with the required crack width at its location during the casting process.The cracked diaphragm was installed inside the permeability tank, resting on the tank's base.Silicone was applied where the diaphragm encountered the bottom, front, and back glass of the tank (Fig. 4a).This step is necessary to ensure the sealing of all the connections between the tank and diaphragm, avoid the flow of water from any opened space between Fig. 2 The experimental setup and permeability tank dimensions them, force the water to flow only through the crack, and maintain accurate results.After fixing the diaphragm, the filling material was added simultaneously to each side in successive layers with a height of 10 cm each (Fig. 4b) and each layer was compacted by five successive hits with a steel weight to mimic the same compaction method used while performing the hydraulic conductivity test.
A geotextile membrane was placed as a filter layer between the infill material (sand) and the slope protection material (aggregate) to prevent the shell material from washing away (Fig. 5a).After the dam construction, water is allowed into the tank through the inlet pipe with low discharge values to prevent any waves that affect the stability of the upstream slope (Fig. 5b).The upstream overflow pipe is set to the required upstream head to act as a shaft spillway that drains the excess water and keeps the upstream head constant (Fig. 5c).The dye injection technique was used to trace the path of the phreatic line.A very low injection rate is used and the dye is traced using a whiteboard marker to detect the phreatic line.This process is repeated at least five times to ensure accurate results.Figure 6 shows the process of dye injection and tracing.
The pore water pressures were monitored by piezometers.Before the usage of piezometers, they were checked to verify that no air bubbles were present inside their tubes.All the measurements and calculations were documented to be analyzed.The water was left to flow for a while to wash the dye out of the filling soil then the water supply was cut.

The numerical modeling
Numerical analysis is rapidly becoming the most significant practical tool for effectively assessing earth dam problems.In this research, the numerical models SEEP/W and SLOPE/W are used to study seepage The geotechnical analysis software package Geo-studio, produced by GEO-Slope international, includes the computer program SEEP/W.SEEP/W is a finite element model that can assess seepage beneath and through earth dams.It is used to simulate seepage flow and the distribution of pore water pressure inside porous materials.SEEP/W is based on Darcy's law, which asserts that the flow of water through both saturated and unsaturated soil follows [59]: where Q is the seepage discharge, K is the hydraulic conductivity, I is the hydraulic gradient, and A is the crosssectional area of flow.
To perform a seepage analysis run, a new SEEP/W project was created from the start page of Geo-studio with the type of analysis selected as a steady-state condition.The first step was to set the scale and units such that the minimum and maximum extents in SEEP/W correspond to those required for the analysis.In this study the vertical and horizontal scales are 1:200, the units are in SI system (length in meters, time in seconds, force in kilonewton, mass in kilograms), and the unit weight of water is 9.807 kN/m 3 .Next, the individual regions were drawn.After that, the materials were created and then assigned to the regions.In this study, the saturated/unsaturated model was selected to solve the problem.Two materials are considered in this study, the shell and diaphragm materials.The shell material (sand) has a hydraulic SEEP/W uses finite element numerical modeling that divides the entire domain of the model into smaller simpler parts known as a mesh.A mesh size of 0.25 m was used in the present study as shown in Fig. 8b.To compute the amount of flow through the earth dam, flux sections were used (Fig. 8a).Flux sections identify the locations where the software should compute and report the amount of flow passing through them during the solving process.After the model is set, the solver for SEEP/W was launched and the results were displayed.The results show the velocity vectors, the flow direction and rate, the location of the phreatic line, the total head contours, and the pore water pressure.Figure 9a shows a sample of the SEEP/W numerical model results.
SLOPE/W model is another module in the Geo-studio software that was used to perform the slope stability analysis.One of the most common applications for developing a seepage model in SEEP/W is to utilize it as a parent analysis for the slope stability model SLOPE/W.The major benefit of this process is the accurate representation of the seepage conditions and hydraulic conductivity which play an essential role in slope stability problems.The first step in the slope stability analysis was adding a new model depending on the results of the seepage model with the limit equilibrium method selection.After that, the pore water pressure conditions were selected to be from the parent analysis.The second step was to define the materials used in the study by adding the information of the unit weight, cohesion, and the angle of internal friction (phi) for each material (sand and concrete) then assigning each material to its region.Finally, the solution was started and the results were displayed.The lowest factor of safety is the one that should be considered.Typically, a factor of safety of 1.5 is an acceptable value for the dam's stability during operation [60].

Experimental observations
The experimental model was used to visualize how the cracked internal diaphragm interferes with the seepage flow field.Considering that the internal concrete diaphragm wall should totally block the seepage through the dam, cracking of the diaphragm was observed to be detrimental to its efficiency.The flow was more concentrated at the crack's location, and the injected dye was directed downward toward the crack.In addition, the phreatic level and the pore water pressure were high at the downstream side, especially near the crack.This can be due to the water seeking the least resistance, which in this case is the crack's location.Figure 10 shows the behavior of the injected dye and seepage flow at the cracked internal diaphragm wall.

Validation of the numerical model
Numerical modeling was conducted for the same experiment described before in the experimental work section.In order to test the ability of the numerical model in simulating earth dams with a cracked internal diaphragm, the phreatic line resulting from it was compared to that measured experimentally.The case of the internal diaphragm wall with a crack at the middle having a width ratio of t/H = 0.011 was considered.The maximum upstream head on the dam was used to study the most critical condition, which is the full reservoir (H = 5.5 m). Figure 11a shows the comparison between the phreatic lines obtained from the numerical model SEEP/W and that measured experimentally using the dye injection and visual tracing technique.Figure 11b shows a graph matching the phreatic line of the numerical and experimental models.The results show very good matching at all points except at (X/L) = 0.61, where the phreatic line was slightly higher in the experimental work.This slight discrepancy can be due to the dispersion that occurs in the injected dye as it meets the cracked internal diaphragm, which makes it challenging to take an accurate reading for the phreatic line at this location.These matching results give rise to the applicability and accuracy of SEEP/W numerical model in drawing the phreatic line for the case of earth dam with a cracked diaphragm.
Moreover, the results of the piezometer readings ratios (hp/H) were plotted against their locations ratio (X/L) for both the experimental and numerical models.Figure 11c shows the comparison between the numerical and experimental piezometers readings ratios (hp/H).The comparative analysis between the numerical and experimental results of the pore water pressure ratio inside the dam (hp/H) shows perfect agreement.Hence, the numerical model SEEP/W is considered accurate and reliable in simulating the pore water pressure.
Finally, after testing the results of SEEP/W numerical model, it proved to have high accuracy and reliability and was found to perform very well in simulating earth dams with a cracked internal diaphragm wall.Hence, SEEP/W numerical model was used to complete this study and model further cases and scenarios to cover all the aspects of the studied problem.

Results of the numerical model
The results of all the studied numerical runs are presented in the following subsections in the form of charts coupled with an explanation of the observations.

The effect of using the internal diaphragm wall
The results show that adding the diaphragm wall to the dam's cross section led to a decrease in the seepage discharge by more than 98% compared to the case of the dam without the diaphragm (Fig. 12a).The internal diaphragm wall also reduced the pore water pressure in the downstream side of the dam (Fig. 12b).The results of

The effect of the crack's height in a cracked internal diaphragm wall
Considering the cracking of the diaphragm wall, the results of the seepage discharge ratio (q/KH) obtained for each crack's height (y/H) show that the closer the crack is to the base of the dam (y/H = 0), the greater the value of the discharge ratio (q/KH).Figure 13a shows the results of (y/H) and (q/KH) values obtained from SEEP/W numerical model.It was also observed that increasing the crack's height ratio (y/H), while maintaining a constant crack's width ratio (t/H) tends to cause a very slight increase in the piezometers readings ratios (hp/H) at some piezometers locations ratios (X/L = 0.38 to 0.826), while no significant Fig. 10 The behavior of the injected dye in the case of cracked internal diaphragm influence is observed for the other (X/L) piezometer's locations refer to Fig. 13b.This shows that the influence of the crack's height ratio (y/H) on the pore water pressure (piezometric height ratio) (hp/H) occurs only around the diaphragm wall's zone.
The maximum velocity ratio downstream the diaphragm (v/K) occurred when the crack is located at the connection between the diaphragm and the base of the dam (y/H = 0).The difference between the maximum (at y/H = 0) and minimum (at y/H = 0.91) velocity ratio is approximately 60% as shown in Fig. 13c.
The results also show that the presence of a crack reduced the critical factor of safety of the downstream slope (FS) by an average of 11% compared to the case of using a diaphragm without cracks.When the crack occurred at the connection between the diaphragm and the base of the dam (y/H = 0) or at the surface of the water of the reservoir (y/H = 0.91) the critical factor of safety was slightly greater than that of the other crack heights as shown in Fig. 13d.

The effect of the crack's width in a cracked internal diaphragm wall.
Increasing the width of the crack (t/H) increased the discharge as shown in Fig. 14a.Furthermore, increasing t/H led to an increase in the pore water pressure in the downstream portion of the dam, represented by the piezometric height ratio (hp/H) as shown Fig. 14b.Also, increasing t/H first increased the velocity, but further enlargement in the crack's width leads to decreasing the velocity ratio (v/K) (refer to Fig. 14c).The maximum velocity occurred in the case of the crack's width (t/H = 0.075).This can be explained using the continuity equation, as the velocity is inversely proportional to the cross-sectional area (v = q/A).In addition, increasing t/H reduced the value of the critical factor safety of the downstream slope (FS) as shown in Fig. 14d.

The effect of the crack's direction in a cracked internal diaphragm wall
Two different crack directions were investigated, the crack inclined by θ = − 45°, and the crack inclined by θ = + 45°.The results of the two cases were compared to the results of the horizontal crack (θ = 0°) that has the same height and width.The crack's direction had a significant effect on the results.It was noticed that the value of the discharge ratio (q/KH) in case of θ = − 45° inclination was less by 35% than that of the horizontal crack.Also, the discharge ratio in the case of θ = + 45° inclined crack was 96% less than its value for the θ = − 45° inclined crack and 98% less than the horizontal crack as shown in Fig. 15a.Moreover, when the crack is inclined by θ = − 45°, the pore water pressure (piezometric height ratio) (hp/H) was reduced by about 25% compared to the case of horizontal crack.Also, for the crack inclined with θ = + 45°, the pore water pressure (piezometric height ratio) (hp/H) was reduced by more than 90% compared to the case of a horizontal crack at the same height as shown in Fig. 15b.In addition, for the case of a crack inclined by θ = − 45°, the velocity ratio (v/K) decreased by 47% compared to the velocity resulting from the horizontal crack.Also, the velocity ratio decreased by 98.5% for the crack inclined by θ = + 45° compared to the horizontal crack and as shown in Fig. 15c.
The critical factor of safety of the downstream slope decreased by 5% in case of θ = − 45° and decreased by 12.5% in the case of horizontal crack (θ = 0°), while in the case of crack inclined by θ = + 45° was found to be equal to that of the case of diaphragm without cracks as shown in Fig. 15d.

The effect of having multiple cracks in the internal diaphragm wall
The case of multiple cracks was also evaluated considering a number (N) of one, two, and three cracks having the same width.The discharge ratio (q/KH) increased as the number of cracks increased reducing the efficiency of diaphragm to control seepage as shown in Fig. 16a.q/KH for the case of three cracks (N = 3) reached 74% of its value in the case of no diaphragm, which shows the remarkable negative effect of cracks on the seepage control efficiency of the diaphragm.Also, increasing the number of cracks led to an increase in the pore water pressure represented by the piezometric height ratio (hp/H) as shown in Fig. 16b.The velocity ratio (v/K) had its greatest value in the case of one crack (N = 1), and it was reduced when two (N = 2) and three (N = 3) cracks were considered, but both had equal values (Fig. 16c).
The results also showed that having more cracks reduces the critical factor of safety for downstream slope stability (FS) (refer to Fig. 16d).Although the value of (FS) for the case of un-cracked diaphragm was 34% higher than having no diaphragm, the cases of one, two, and three cracks were higher than the case of no diaphragm by only 17%, 10%, and 8%, respectively.

The effect of using a downstream horizontal blanket drain with the cracked internal diaphragm wall
Finally, the effect of using a downstream horizontal blanket drain on the safety of the dam in the case of diaphragm cracking was investigated.The best performing downstream drain geometry and dimensions recommended by Refaiy et al. [14] were used, which is the horizontal drain with a length of L d = 2H and height of h d = 0.18H.The base case for diaphragm wall with a single horizontal crack (at a height of y/H = 0.545 and with a width of t/H = 0.0227) was modeled without and with the aforementioned drain.The analysis showed that in the case of having a downstream horizontal blanket drain, the discharge increased by 9.8% (refer to Fig. 17a) compared to the case of cracked diaphragm without the drain.In addition, the existence of the horizontal drain with the cracked diaphragm significantly decreased the pore water pressure (hp/H) at the downstream side of the dam as presented in Fig. 17b.The velocity ratio (v/K) also increased by approximately 3% in the case of the existence of horizontal drain compared to the value of the velocity measured at the same location when no drain existed (refer to Fig. 17c).Having the horizontal blanket drain at the downstream with the cracked diaphragm also increased the critical factor of safety of the downstream slope (FS) to reach 99.6% of its value in the case of diaphragm without cracks as illustrated in Fig. 17d.

Discussion
The results of this study emphasize the notable role of an internal diaphragm wall in controlling seepage through earth dams and enhancing their safety and stability.Nevertheless, any concrete element can be subject to deterioration and fracturing with time.Thus, this study extensively examined different possibilities of diaphragm wall cracking.A single crack at any height in the internal diaphragm wall increased the seepage discharge, pore water pressure, and velocity, while reducing the slope stability factor of the dam.This matches with the results of Taleb [56] who reported that fissured materials in earth dams reduce their stability and recommended accounting for it in the design.The results also agree with Demirdogen and Gunaratne [54] who reported a notable deficiency with any defect in seepage control geo-membrane barriers.
The diaphragm crack is thought to resemble an orifice, with the flow rate through it depending upon the head of water above the crack.The height of the crack showed to be a crucial factor, as cracking near the dam's base can have detrimental consequences of excessive seepage and downstream slope instability.This can be attributed to the bigger water head acting on the crack at this location, which leads to higher discharges and velocities compared to cracks of equal widths at higher positions.
The results clarified that a wider crack leads to greater discharge, higher pore water pressure, and less downstream slope stability.This can be due to the water flowing out of the crack at a higher rate as the crack's area enlarges which leads to raising the water level in the downstream portion of the dam, threatening its slope's stability [61].The horizontal crack proved to be is the most critical direction in terms of discharge, pore water pressure, velocity, and downstream slope stability.This can be due to the higher pore water pressures at the downstream part of the dam in this case.Also, the crack's inclination breaks the velocity of flow into horizontal and vertical components that depend on the crack's inclination angle, instead of mainly concentrating it toward the horizontal direction.
The findings also highlight the significant negative influence of diaphragm fracturing at multiple locations, as it reduces the diaphragm's efficiency, approaching the case of its nonexistence as the cracking increases (q/KH for the case of three cracks (N = 3) reached 74% Fig. 15 Effect of the crack's direction on a the discharge (q/KH), b the pore water pressure (hp/H), c the velocity (v/K), d the critical factor of safety for downstream slope stability (FS) of its value in the case of no diaphragm).This aligns with the conclusions reported by El-Molla and Eltarabily [62] regarding the cracking of concrete canal lining, as the lining layer lost its seepage control properties by a discharge approaching 88% of the unlined canal when cracked at multiple locations that covered 6% of the canal bed width.This can be considered a warning alert for designers, as they should make sure that cracking is controlled and implement a backup seepage control plan when using concrete structural elements to stop seepage.
Having a horizontal downstream drain with the cracked diaphragm wall enhances the dam's performance with respect to seepage discharge, pore water pressure, flow velocity, and downstream slope stability.This means that the drain can eliminate the negative effect of diaphragm cracking on the seepage and downstream slope stability of the dam.These results draw attention to the importance of using a properly designed drain as a mitigation measure to ensure the safety of earth dams with a diaphragm if fracturing occurred.This agrees with the results of Talukdar and Dey [51] who studied the effect of using a drainage blanket with a cracked core and concluded that such blankets are very helpful in lowering the phreatic level and reducing the negative effect of incidental core cracking.It also aligns with the recommendation by Fawzy et al. [55] to add a triangular toe drain when rehabilitating existing homogeneous earth dams by diaphragm grouting to enhance the dam's safety in the case of construction defects.Although this study considers the diaphragm material to be concrete, the results and conclusions can be applicable to any other low permeability material with similar characteristics to those of concrete.

Conclusions
This study implemented the experimental and numerical modeling approaches to simulate water seepage through an earth dam with an internal diaphragm wall without and with cracks.Comparing the results of the permeability tank and SEEP/W model showed an excellent agreement for the phreatic line and pore water pressure inside the dam.Hence, the study was completed numerically using the validated model.
The results of the study showed that using an internal diaphragm wall in earth dams is very beneficial for their safety, as it decreased the seepage discharge by more than 98%, lowered the pore water pressure at the downstream portion of the dam, significantly minimized the velocity downstream it to be v/K = 0.001, and increased the critical factor of safety of the downstream slope by 34%.
By assuming that there were cracks in the internal diaphragm, it was found that the most critical location of the crack (y/H) was near the base of the dam, while the least critical was near the water surface.When a crack occurs in the diaphragm near the base of the dam, it increases the seepage discharge and velocities and can lead to the washing out of the dam material.The width of crack (t/H) is also very effective, as the enlargement of the crack increased the seepage discharge and pore water pressure, threatening the safety of the dam against piping and slope stability.
In the case of cracks that are formed with an inclination, it was found that the inclination in the direction Fig. 17 Effect of using a horizontal downstream drain with the cracked diaphragm on a the discharge (q/KH), b the pore water pressure (hp/H), c the velocity (v/K), d the critical factor of safety for downstream slope stability (FS) of θ = + 45° is the least critical, followed by the crack inclined with θ = − 45°; then, the horizontal crack (θ = 0°) proved to be the most critical in terms of discharge, pore water pressure, velocity, and downstream slope stability.Also, investigating the influence of the number of cracks (N) highlighted that the fracturing of the diaphragm at multiple locations can be detrimental to the safety of the dam.Having only three cracks caused the diaphragm's efficiency to approach the case of its nonexistence (the seepage discharge to be 74% of its value for no diaphragm, the pore water pressure raised drastically, and the critical factor of safety of downstream slope was only 8% higher than its value in the case of no diaphragm).Based on the results of this study, it is strongly advised to continuously monitor the components of earth dams with a diaphragm and perform the necessary maintenance works whenever any sign of damage or water seepage appears.
Finally, installing a downstream horizontal drain in the section of earth dams with internal diaphragms showed to be crucial as a mitigation measure in the case of diaphragm deterioration and fracturing.The downstream horizontal drain reduced the pore water pressure in the downstream portion of the dam, while increasing the magnitude of seepage discharge by 9.8% only.It also eliminated the negative effect of diaphragm cracking on the downstream slope stability of the dam.Hence, using a properly graded horizontal blanket drain of sufficient length in earth dams with an internal diaphragm is highly recommended.

Fig. 1
Fig. 1 The variables considered in the present study

Fig. 3 Fig. 4 a
Fig. 3 Dimensions of the dam, a the real dam (in m), b the scaled experimental model (in cm)

Fig. 5 a
Fig. 5 a Adding the geotextile membrane to the surface of the exit face, b adding the downstream slope protection layer and letting the water enter from the inlet pipe, c the inlet overflow pipe draining the excess water to keep the upstream water head constant

( 4 )
Q = K I A conductivity of K = 0.0013 m/s.The diaphragm material has a hydraulic conductivity of K D = 10 −9 m/s.Setting up the boundary condition in the model is a very essential component as the solution greatly depends upon them.The boundary conditions used in this study are (Fig.8a):1.The potential seepage face boundary condition which allows locating the possible position of the exit face.2. The total head boundary condition which is represented by 5.5 m of water depth in the reservoir behind the dam.3. The zero-pressure boundary condition which is used to model drains and areas where pore water pressure dissipates instantly.

Fig. 6 a
Fig. 6 a The dye injection process, b tracing of the phreatic line Figure 9b shows a sample of the results of SLOPE/W model; the white arc depicts the critical slip surface.

Fig. 7
Fig. 7 Scheme of the numerical modeling scenarios

Fig. 8 a
Fig. 8 a The boundary conditions assigned to the studied dam and the flux sections, b the finite element mesh size used in the study

Fig. 9 a
Fig. 9 a Sample of SEEP/W results, b sample of SLOPE/W results (critical factor of safety for downstream slope)

Fig. 11 a
Fig. 11 a Comparison between the phreatic lines obtained from the numerical and experimental models for the case of cracked internal diaphragm, b the phreatic line of the numerical and experimental models (Y/H), c the pore water pressure ratio (hp/H) obtained from the numerical and experimental models

Fig. 12
Fig.12 Effect of using the diaphragm on a the discharge (q/KH), b the pore water pressure (hp/H), c the velocity (v/K), d the critical factor of safety for downstream slope stability (FS)

Fig. 13
Fig.13 Effect of the crack's height (y/H) on a the discharge (q/KH), b the pore water pressure (hp/H), c the velocity (v/K), d the critical factor of safety for downstream slope stability (FS)

Fig. 14
Fig.14 Effect of the crack's width (t/H) on a the discharge (q/KH), b the pore water pressure (hp/H), c the velocity (v/K), d the critical factor of safety for downstream slope stability (FS)

Fig. 16
Fig.16 Effect of the number of cracks (N) on a the discharge (q/KH), b the pore water pressure (hp/H), c the velocity (v/K), d the critical factor of safety for downstream slope stability (FS)