# Accurate Empirical Calculation System for Predicting the Seepage Discharge and Free Surface Location of Earth Dam over Horizontal Impervious Foundation

## Abstract

**:**

## 1. Introduction

## 2. Governing Equation of Numerical Calculation

## 3. Concept of Equivalent KZ Flow

## 4. Calculation Accuracy of Conventional Empirical Methods

## 5. Interpolation-Equivalent KZ Flow

#### 5.1. Creating Basic Tables

#### 5.2. Application Examples of Basic Tables to General Dams

#### 5.3. Special Cases When the Basic Table Cannot Be Used

#### 5.4. Theoretical Consideration on the Seepage of Rectangular Dams

#### 5.5. Summary of This Section

## 6. Conclusions and Discussion

- (i)
- For a trapezoidal dam with a vertical entrance face, the equivalent KZ flow is $\mathrm{KZI}\left({q}_{e},{B}_{e}\right)$; the applicable ranges are ${R}_{u}\le {R}_{b}$, $0.20\le {R}_{u}\le 0.50,$ and $0.20\le {R}_{b}\le 1.00$.
- (ii)
- For the symmetrical dam center core, the equivalent KZ flow is $\mathrm{KZI}\left({q}_{e},{B}_{e}\right)$; the applicable ranges are${R}_{u}\le {R}_{d}$, $0.20\le {R}_{u}\le 0.50,\mathrm{and}0.20\le {R}_{d}\le 1.40$.

## Funding

## Conflicts of Interest

## Abbreviations

APIM | Algebraic Polynomial Interpolation Method |

BPP | boundary potential parabola |

BPIM | Boundary Polynomial Interpolation Method |

VFDM | Finite Difference Method |

FS | Free Surface |

FSP | Free Surface Parabola |

IFDM | Interpolation Finite Difference Method |

PDE | Partial differential Equation |

FDE | Finite Differential Equation |

SOR | Successive Over Relaxation |

FTCS scheme | Forward Time, lefted Space scheme |

TMSD scheme | Time Marching Successive Displacement scheme |

Nomenclature | |

${A}_{de}$ | deficit area |

${A}_{e}$ | equivalent area ($={A}_{K}-{A}_{de})$ |

${A}_{K}$ | KZ flow area |

d | distance defined in Figure 1 |

H | front water level, entrance face head |

$h$ | total head ($positionhead,y+pressurehead,p$) |

KZ0(A, O) | basic expression of KZ flow |

KZ1(C,${A}_{e}$) | area-equivalent KZ flow |

$\mathrm{KZ}2\left(C,{B}_{n}\right)$ | discharge-point-equivalent KZ flow |

$\mathrm{KZ}3\left(C,{q}_{t}\right)$ | discharge-equivalent KZ flow |

$\mathrm{KZ}b\left(C,O\right)$ | basic KZ flow |

$\mathrm{KZc}\left(A,O\right)$ | criteria KZ flow |

KZ flow | Kozeny flow |

$\mathrm{KZI}\left(C1,C2\right)$ | interpolation-equivalent KZ flow determined by conditions C1 and C2 |

$\mathrm{KZ}n\left({q}_{n},{B}_{n}\right)$ | numeric-equivalent KZ flow |

${R}_{b}$ | basic aspect ratio = H/d |

${R}_{d}$ | down-side aspect ratio |

${R}_{E}$ | equivalent aspect ratio |

${R}_{u}$ | up-side aspect ratio |

${L}_{c}$ | point C determination criteria length |

${L}_{h}^{*}$ | dimensionless half-decrease distance |

${l}_{wi,j}$ | wall-distance factor |

${m}_{v}$ | point C determination factor |

p_{k} | Boundary point used in numerical calculation |

${P}_{k}^{*}$ | dam configuration point |

point $A$ | entrance point |

point ${A}_{c}$ | point C determination criteria point |

point B | discharge point |

$\mathrm{point}{B}_{n}$ | discharge point defined by numerical calculation |

Point ${B}_{b}$ | discharge point of basic KZ flow, $\mathrm{KZ}b\left(C,O\right)$ |

point ${B}_{e}$ | discharge point estimated by interpolation |

point C | starting point of the basic parabola |

point ${C}_{a}$ | starting point of the basic parabola after A. Casagrande |

point C_{e} | starting point estimated by interpolation |

point ${C}_{n}$ | starting point defined by numeric-equivalent KZ flow |

point O | origin of KZ coordinate system |

$q$ | discharge |

${q}_{b}$ | discharge of basic KZ flow, $\mathrm{KZ}b\left(C,O\right)$ |

${q}_{Ca}$ | discharge calculated from A. Casagrande method |

${q}_{E}$ | discharge of equivalent trapezoidal dam with vertical entrance face |

${q}_{e}$ | discharge estimated by interpolation |

${q}_{e}^{*}$ | discharge calculated from KZI$\left(C,{B}_{e}\right)$ |

${q}_{LC}$ | discharge calculated from L. Casagrande method |

${q}_{n}$ | numerically calculated discharge |

${q}_{SV}$ | discharge calculated from Schaffernak–Van Iterson method |

${q}_{t}$ | theoretical discharge |

$s$ | stream function |

$yB$ | y-coordinate value of point B |

$y{B}_{n}$ | y-coordinate value of point ${B}_{n}$ |

$y{B}_{e}$ | y-coordinate value of point ${B}_{e}$ |

${R}_{b}$ | basic aspect ratio=H/d |

${R}_{E}$ | equivalent aspect ratio |

$\alpha $ | discharge angle |

${\alpha}_{b}$ | acceleration factor of the TMSD scheme |

${\alpha}_{bmax}$ | theoretical maximum acceleration factor of the TMSD scheme (=2.00) |

${\alpha}_{bopt}$ | optimum acceleration factor |

$\Delta t$ | time difference width |

$\Delta {t}_{c}$ | criteria time difference |

$\Delta x$ | $x$-direction difference width |

$\Delta y$ | $y$-direction difference width |

$\theta $ | entrance angle |

${\theta}_{c}$ | criteria entrance angle |

${\phi}_{i,j}$ | time-interval adjustment factor |

## Appendix A. The Schaffernak–Van Iterson Method

## Appendix B. L. Casagrande’s Method

## Appendix C. Two-Dimensional Interpolation Methods

**Figure A2.**Illustration for the two-dimensional interpolation methods: bilinear and bicubic interpolation methods, (

**a**) schematic for bicubic interpolation, (

**b**) schematic for bilinear interpolation.

## References

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**Figure 1.**Example of flow net of a Kozeny (KZ) flow [5].

**Figure 3.**Seepage flow of trapezoidal dam, $\theta ={\theta}_{c}=78.14\xb0,{R}_{b}=1.43,\mathsf{\alpha}=180\xb0$, in input data, PT: potential calculation condition, SF: stream function calculation condition (

**a**) numerical calculation, (

**b**) criteria KZ flow, KZc$(A,{O}_{C})$, ${B}_{n}\left(27.18,0.00\right)$, ${B}_{Ca}\left(27.22,0.00\right)$, ${B}_{c}\left(27.26,0.00\right)\left(={F}_{c}\right)x=a{y}^{2}+c$, in order to be a consistent calculation method. This in itself is a hypothesis; nevertheless, it is considered to be a reasonable one based on several numerical calculation results.

**Figure 4.**Seepage flow of trapezoidal dam, $\theta =33.69\xb0,{R}_{b}=1.75,\mathsf{\alpha}=180\xb0$, (Fukuchi, 2018), (

**a**) numerical calculation, (

**b**) basic KZ flow: KZb$\left(C,{O}_{b}\right)$, ${B}_{n}\left(26.84,0.00\right)$, ${B}_{Ca}\left(26.87,0.00\right)$, ${B}_{b}\left(26.93,0.00\right)\left(={F}_{b}\right)$.

**Figure 5.**Seepage flow of trapezoidal dam, $\theta =33.69\xb0,{R}_{b}=1.75,\mathsf{\alpha}=50\xb0$, (

**a**) numerical calculation, (

**b**) area-equivalent KZ flow: KZ1($C,{A}_{e}$), (

**c**) numeric-equivalent KZ flow: KZn(${q}_{n},{B}_{n}$), cf. discharge and discharge point calculated from the equivalent cross-section, ${C}_{E}{C}_{E}^{\u2019}OG$: ${q}_{E}=1.99\left(-0.7\%\right)$, $y{B}_{E}=2.79\left(-0.6\%\right)$.

**Figure 11.**(

**a**) Discharge ${q}_{n}$ depending on ${R}_{b}$ and $\mathsf{\alpha}$. (

**b**) Discharge point $y{B}_{n}$ depending on ${R}_{b}$ and $\mathsf{\alpha}$. (

**c**) Starting point of FSP depending on ${R}_{b}$ and $\mathsf{\alpha}$, red line: ${R}_{b}-\left|\overline{AC}\right|$.

**Figure 12.**Calculation examples of interpolation-equivalent KZ flow, (

**a**) $\mathsf{\alpha}=75\xb0$, (

**b**) $\mathsf{\alpha}=135\xb0$.

**Figure 13.**Calculation examples of interpolation-equivalent KZ flow, (

**a**) $\mathsf{\alpha}=75\xb0$, (

**b**) $\mathsf{\alpha}=135\xb0$.

**Figure 14.**Trapezoidal dam, difference width, $\Delta x=\Delta y=0.50$, (

**a**) numeric-equivalent KZ flow, (

**b**) interpolation-equivalent KZ flow, ${q}_{E}=1.72\left(0.5\%\right)$ in Figure 14b.

**Figure 15.**Trapezoidal dam with drain, difference width, $\Delta x=\Delta y=0.50$, interpolation-equivalent KZ flow, corresponding to Figure 5, ${q}_{E}=2.00\left(-0.2\%\right)$.

**Figure 16.**Calculation example of interpolation-equivalent KZ flow, trapezoidal dam with vertical entrance face and small value of ${R}_{u}$.

**Figure 18.**$yB$ value of rectangular dam, depending on ${R}_{b}$, discharge point: (

**a**) calculated by numerical calculation, (

**b**) numeric value of theoretical solution.

**Figure 19.**Calculation example of the equivalent KZ flow, $K{Z}_{f}\left({q}_{t},{B}_{f}\right)$, rectangular dam.

**Figure 20.**Relationship between discharge $q$ and discharge point $yB$, $q$: theoretical solution, discharge point$yB$: (

**a**) numerical calculation, (

**b**) numeric value of theoretical solution.

**Figure 21.**Relationship between discharge $q$ and discharge point $yB$ depending on discharge angle $\alpha $.

(A) | (B) Discharge | (C) R.E. (%) | ||||||
---|---|---|---|---|---|---|---|---|

$\alpha (\xb0)$ | ${R}_{u}$ | ${q}_{n}$ | ${q}_{Ca}$ | ${q}_{SV}$ | ${q}_{LC}$ | ${q}_{Ca}$ | ${q}_{SV}$ | ${q}_{LC}$ |

26.6 | 0.000 | 0.500 | 0.500 | 0.447 | *** | 0.0 | −10.6 | |

30 | 0.268 | 0.350 | 0.236 | 0.333 | 0.308 | −32.4 | −4.5 | −11.9 |

60 | 1.423 | 0.259 | 0.236 | 0.255 | 0.234 | −8.8 | −1.3 | −9.7 |

90 | 2.000 | 0.250 | 0.236 | 0.250 | 0.222 | −5.7 | 0.0 | −11.5 |

(A) | (B) yB-Value | (C) R.E. (%) | ||||||
---|---|---|---|---|---|---|---|---|

$\alpha (\xb0)$ | ${R}_{u}$ | $y{B}_{n}$ | $y{B}_{Ca}$ | $y{B}_{SV}$ | $y{B}_{LC}$ | $y{B}_{Ca}$ | $y{B}_{SV}$ | $y{B}_{LC}$ |

26.6 | 0.000 | 1.000 | *** | 1.000 | 1.000 | *** | 0.0 | 0.0 |

30 | 0.268 | 0.781 | 0.562 | 0.578 | 0.615 | −21.8 | −20.3 | −16.6 |

60 | 1.423 | 0.300 | 0.281 | 0.148 | 0.270 | −1.9 | −15.2 | −3.0 |

90 | 2.000 | 0.173 | 0.176 | 0.000 | 0.222 | 0.2 | −17.4 | 4.8 |

${\mathit{R}}_{\mathit{b}}$ | ${\mathit{\alpha}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{q}}_{\mathit{m}\mathit{a}\mathit{x}}$ | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 120 | 150 | 180 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.50 | 63.4 | 2.000 | 1.367 | 1.115 | 1.000 | 0.872 | 0.822 | 0.797 | |||||

0.75 | 53.1 | 1.333 | 0.920 | 0.768 | 0.703 | 0.668 | 0.620 | 0.599 | 0.580 | ||||

1.00 | 45.0 | 1.000 | 0.722 | 0.588 | 0.540 | 0.515 | 0.500 | 0.480 | 0.468 | 0.455 | |||

1.25 | 38.7 | 0.800 | 0.487 | 0.441 | 0.420 | 0.408 | 0.401 | 0.389 | 0.383 | 0.375 | |||

1.50 | 33.7 | 0.667 | 0.441 | 0.378 | 0.356 | 0.344 | 0.338 | 0.334 | 0.327 | 0.323 | 0.320 | ||

2.00 | 26.6 | 0.500 | 0.350 | 0.284 | 0.267 | 0.259 | 0.255 | 0.252 | 0.250 | 0.248 | 0.246 | 0.244 | |

2.50 | 21.8 | 0.400 | 0.237 | 0.216 | 0.208 | 0.206 | 0.202 | 0.201 | 0.200 | 0.198 | 0.197 | 0.197 | |

3.00 | 18.4 | 0.333 | 0.245 | 0.185 | 0.175 | 0.171 | 0.169 | 0.168 | 0.167 | 0.167 | 0.166 | 0.165 | 0.165 |

3.50 | 15.9 | 0.286 | 0.179 | 0.154 | 0.148 | 0.146 | 0.145 | 0.144 | 0.143 | 0.143 | 0.142 | 0.142 | 0.142 |

4.00 | 14.0 | 0.250 | 0.146 | 0.132 | 0.128 | 0.127 | 0.126 | 0.126 | 0.125 | 0.125 | (0.125) | (0.125) | (0.125) |

4.50 | 12.5 | 0.222 | 0.125 | 0.116 | 0.114 | 0.113 | 0.112 | 0.112 | 0.112 | 0.111 | (0.111) | (0.111) | (0.111) |

${\mathit{R}}_{\mathit{b}}$ | ${\mathit{\alpha}}_{\mathit{m}\mathit{i}\mathit{n}}$ | $\mathit{y}{\mathit{B}}_{\mathit{m}\mathit{a}\mathit{x}}$ | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 120 | 150 | 180 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.50 | 63.4 | 1.000 | 0.885 | 0.752 | 0.624 | 0.402 | 0.204 | 0.000 | |||||

0.75 | 53.1 | 1.000 | 0.855 | 0.683 | 0.563 | 0.472 | 0.291 | 0.148 | 0.000 | ||||

1.00 | 45.0 | 1.000 | 0.866 | 0.654 | 0.523 | 0.441 | 0.351 | 0.237 | 0.113 | 0.000 | |||

1.25 | 38.7 | 1.000 | 0.665 | 0.508 | 0.414 | 0.337 | 0.278 | 0.188 | 0.091 | 0.000 | |||

1.50 | 33.7 | 1.000 | 0.745 | 0.531 | 0.415 | 0.337 | 0.277 | 0.228 | 0.153 | 0.079 | 0.000 | ||

2.00 | 26.6 | 1.000 | 0.781 | 0.495 | 0.378 | 0.300 | 0.253 | 0.212 | 0.174 | 0.118 | 0.057 | 0.000 | |

2.50 | 21.8 | 1.000 | 0.540 | 0.379 | 0.293 | 0.238 | 0.197 | 0.166 | 0.132 | 0.091 | 0.047 | 0.000 | |

3.00 | 18.4 | 1.000 | 0.800 | 0.429 | 0.306 | 0.240 | 0.198 | 0.169 | 0.135 | 0.114 | 0.079 | 0.041 | 0.000 |

3.50 | 15.9 | 1.000 | 0.614 | 0.352 | 0.257 | 0.204 | 0.170 | 0.139 | 0.114 | 0.088 | 0.068 | 0.037 | 0.000 |

4.00 | 14.0 | 1.000 | 0.479 | 0.307 | 0.226 | 0.184 | 0.153 | 0.128 | 0.099 | 0.082 | (0.055) | (0.027) | (0.000) |

4.50 | 12.5 | 1.000 | 0.414 | 0.277 | 0.201 | 0.162 | 0.142 | 0.115 | 0.089 | 0.066 | (0.044) | (0.022) | (0.000) |

**Table 5.**Basic table, starting point of FSP(n) ${C}_{n}$ depending on ${R}_{b}$ and $\mathsf{\alpha}$, shown in the value ${R}_{b}-\left|\overline{A{C}_{n}}\right|$.

${\mathit{R}}_{\mathit{b}}$ | ${\mathit{\alpha}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{R}}_{\mathit{b}}-\left|\overline{\mathit{A}\mathit{C}}\right|$ | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 120 | 150 | 180 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.50 | 63.4 | 0.298 | 0.402 | 0.327 | 0.306 | 0.253 | 0.229 | 0.209 | |||||

0.75 | 53.1 | 0.587 | 0.640 | 0.596 | 0.585 | 0.582 | 0.569 | 0.560 | 0.576 | ||||

1.00 | 45.0 | 0.865 | 0.900 | 0.864 | 0.863 | 0.860 | 0.869 | 0.846 | 0.861 | 0.880 | |||

1.25 | 38.7 | 1.135 | 1.130 | 1.135 | 1.137 | 1.146 | 1.151 | 1.130 | 1.139 | 1.148 | |||

1.50 | 33.7 | 1.401 | 1.393 | 1.396 | 1.400 | 1.409 | 1.414 | 1.419 | 1.406 | 1.403 | 1.399 | ||

2.00 | 26.6 | 1.923 | 1.910 | 1.920 | 1.925 | 1.931 | 1.929 | 1.931 | 1.936 | 1.923 | 1.928 | 1.919 | |

2.50 | 21.8 | 2.437 | 2.431 | 2.437 | 2.442 | 2.444 | 2.446 | 2.444 | 2.453 | 2.451 | 2.451 | 2.453 | |

3.00 | 18.4 | 2.947 | 2.932 | 2.942 | 2.950 | 2.952 | 2.950 | 2.947 | 2.959 | 2.954 | 2.932 | 2.959 | 2.952 |

3.50 | 15.9 | 3.454 | 3.430 | 3.454 | 3.458 | 3.458 | 3.457 | 3.456 | 3.460 | 3.466 | 3.451 | 3.447 | 3.443 |

4.00 | 14.0 | 3.960 | 3.930 | 3.959 | 3.961 | 3.960 | 3.959 | 3.960 | 3.969 | 3.967 | (3.967) | (3.967) | (3.967) |

4.50 | 12.5 | 4.464 | 4.435 | 4.458 | 4.464 | 4.461 | 4.454 | 4.454 | 4.454 | 4.472 | (4.472) | (4.472) | (4.472) |

**Table 6.**Trapezoidal dam with $\mathsf{\theta}=90$°, and small value of ${R}_{u}$, discharge ${q}_{n}$ depending on ${R}_{u}$ and ${R}_{b}$.

${\mathit{R}}_{\mathit{u}}\downarrow ,{\mathit{R}}_{\mathit{b}}\to $ | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 | 0.90 | 1.00 |
---|---|---|---|---|---|---|---|---|---|

0.20 | 2.500 | 1.871 | 1.501 | 1.255 | 1.078 | 0.945 | 0.842 | 0.759 | 0.691 |

0.30 | (2.201) | 1.667 | 1.350 | 1.136 | 0.981 | 0.862 | 0.770 | 0.696 | 0.634 |

0.40 | *** | (1.539) | 1.250 | 1.055 | 0.914 | 0.806 | 0.721 | 0.652 | 0.595 |

0.50 | *** | *** | (1.183) | 1.000 | 0.867 | 0.766 | 0.686 | 0.621 | 0.567 |

**Table 7.**Trapezoidal dam with $\mathsf{\theta}=90$°, and small value of ${R}_{u}$, discharge point $y{B}_{n}$ depending on ${R}_{u}$ and ${R}_{b}$.

${\mathit{R}}_{\mathit{u}}\downarrow ,{\mathit{R}}_{\mathit{b}}\to $ | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 | 0.90 | 1.00 |
---|---|---|---|---|---|---|---|---|---|

0.20 | 0.850 | 0.849 | 0.841 | 0.838 | 0.833 | 0.832 | 0.832 | 0.831 | 0.831 |

0.30 | (0.798) | 0.774 | 0.764 | 0.759 | 0.754 | 0.752 | 0.748 | 0.747 | 0.743 |

0.40 | *** | (0.723) | 0.694 | 0.682 | 0.679 | 0.676 | 0.674 | 0.672 | 0.670 |

0.50 | *** | *** | (0.646) | 0.624 | 0.622 | 0.611 | 0.612 | 0.608 | 0.602 |

${\mathit{R}}_{\mathit{u}}\downarrow ,{\mathit{R}}_{\mathit{d}}\to $ | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 | 0.90 | 1.00 | 1.20 | 1.40 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.20 | 2.500 | 1.886 | 1.526 | 1.287 | 1.115 | 0.986 | 0.885 | 0.803 | 0.737 | 0.632 | 0.557 |

0.30 | (2.175) | 1.667 | 1.367 | 1.163 | 1.013 | 0.901 | 0.812 | 0.740 | 0.681 | 0.589 | 0.522 |

0.40 | *** | (1.515) | 1.250 | 1.071 | 0.938 | 0.837 | 0.757 | 0.692 | 0.639 | 0.555 | 0.493 |

0.50 | *** | *** | (1.166) | 1.000 | 0.882 | 0.788 | 0.715 | 0.654 | 0.605 | 0.527 | 0.469 |

${\mathit{R}}_{\mathit{u}}\downarrow ,{\mathit{R}}_{\mathit{d}}\to $ | 0.20 | 0.30 | 0.40 | 0.50 | 0.60 | 0.70 | 0.80 | 0.90 | 1.00 | 1.20 | 1.40 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.20 | 0.850 | 0.840 | 0.835 | 0.815 | 0.802 | 0.792 | 0.787 | 0.760 | 0.751 | 0.729 | 0.692 |

0.30 | (0.777) | 0.774 | 0.766 | 0.757 | 0.724 | 0.716 | 0.707 | 0.676 | 0.669 | 0.634 | 0.601 |

0.40 | *** | (0.706) | 0.694 | 0.686 | 0.653 | 0.645 | 0.641 | 0.610 | 0.603 | 0.568 | 0.545 |

0.50 | *** | *** | (0.666) | 0.624 | 0.616 | 0.580 | 0.574 | 0.550 | 0.543 | 0.519 | 0.496 |

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## Share and Cite

**MDPI and ACS Style**

Fukuchi, T.
Accurate Empirical Calculation System for Predicting the Seepage Discharge and Free Surface Location of Earth Dam over Horizontal Impervious Foundation. *Eng* **2020**, *1*, 60-95.
https://doi.org/10.3390/eng1020005

**AMA Style**

Fukuchi T.
Accurate Empirical Calculation System for Predicting the Seepage Discharge and Free Surface Location of Earth Dam over Horizontal Impervious Foundation. *Eng*. 2020; 1(2):60-95.
https://doi.org/10.3390/eng1020005

**Chicago/Turabian Style**

Fukuchi, Tsugio.
2020. "Accurate Empirical Calculation System for Predicting the Seepage Discharge and Free Surface Location of Earth Dam over Horizontal Impervious Foundation" *Eng* 1, no. 2: 60-95.
https://doi.org/10.3390/eng1020005