ABSTRACT

The purpose of this study is to evaluate the applicability of some equations used in the calculation of seepage from an earth dam placed on an impervious base by using the Hele-Shaw Viscous Fluid Physical Model. For the laboratory experiments an Hele-Shaw Analog Model with 70 cm long, 50 cm high, 0.8 cm thick and 3 cm space between glass plate was prepared. Measurements were made by settling dam bodies prepared for 11 different slope conditions with 50 cm base long in the model. The measurements were made with 3 levels at the downstream side for 3 different levels at the side upstream. The values measured from model were corrected according to 22 °C by using kinematic viscosity (u = 2.72 cm^{2}/sec) obtained from temperature-viscosity relationship. Correlation and *t*-tests were conducted between the corrected model values and formula values. Obtained values from solutions of Dupuit, Schaffernak-Van Iterson, L. Casagrande and Pavlovsky were compared with values of model. Correlation coefficient were found as follows respectively : Between model and Dupuit solution 0.9927, model and Schaffernak-Van Iterson solution 0.9618, model and L.Casagrande solution 0.9618 and model and Pavlovsky solution 0.9426. Average values obtained from model and formula was compared with *t*-tests. As the result of the comparison from *t*-test model values and formula values in all slopes, the probability level were found to be P>0.05. Any statistical difference was not found between the model and formula values from correlation and *t*-tests. According to these results, the Dupuit Equation was the most appropriate for calculations.

Keywords: Seepage, Hele-Shaw model, Dupuit, Schaffernak-Van Iterson, L.Casagrande, Pavlovsky.

The authors have chosen to indicate the units of all physical quantities by the unit type indicators such as(L)for length(T)for time, etc. Although distracting at times, the clarity provided by the authors is retained—Ed.

INTRODUCTION

Movement of water through soils is often problematic in practice. Water flow in soils can be classified as seepage or groundwater flows. Some examples of problems are the flow of water towards the body of a dam, leakage of water from the sides of a stream, and seepage from channels. Earth dams are constructed to have reservoirs or detain stream water for special purposes. Filling materials in an earth dam should be stable against seepage.

Analytical solutions of groundwater flow problems are difficult for all but the simplest cases. Therefore, in addition to analytical solutions, physical or numerical models are used, especially when boundary conditions are complicated. A physical model is a transfer of a physical object to the laboratory with a known scale. Experimental results obtained from the physical models help defining the principles of the model preparation. Using the equations derived based on these principles is helpful for the solution of the real case problems. Prototype usually refers to the real object whereas model is a scaled form of the prototype based on a full scale. The relationship between prototype and model is the relation between the two similar systems whereas the relationship between prototype and analog is the relation between the two different physical systems.

The prototype of the Hele-Shaw Analog Models uses the same differential equations for the simulation of water flow in porous medium and flow of viscous liquid between closely-placed parallel plates (Harr, 1962). The Hele-Shaw Analog is commonly used to show two-dimensional laminar flow of water in porous soils (Todd, 1964). The Hele-Shaw models can be set up horizontally or vertically based on the type of the problem (Bear, 1972). Similarity principles are used to relate the prototype and the model. The reflection of all properties of the prototype in the model can be managed by the similarity. Although several types of similarity exist, geometric similarity, kinematic similarity, and dynamic similarity are the three most important. Scale or physical models and prototypes must not have the same physical dimensions. Physical model belongs to completely different physical categories based on the similarity between prototype and physical model (Gemalmaz, 1985). The main reason of the analogy between the two systems (prototype and physical model) is that characteristic equations (Heat and Mass Conservation, Darcy Law, and Ohm Law) of the two systems describe the same physical principles. Therefore, analogy is possible between the groundwater flow and the electric or heat transfer because the Laplace equation, which is a general equation representing flow in porous media, and equations for the electric and heat transfer have the same form as explained above (de Wiest, 1967).

Even though a variety of models exist for the simulation studies, Hele-Shaw Viscous Liquid Analog Models are one of the most used for the simulation of water movement in the porous media. Two-dimensional flow problems such as flow to drain pipes or channels, seepage from earth dams, and seepage below weirs are easily studied using Hele-Shaw Physical Models. Tsay and Hoopes (1998) developed a numerical model for calculation of the groundwater mounding, the rise of water table above its regional level in a local area of an aquifer, using SAE 50 oil as the fluid. The model predictions were verified with tests in a Hele-Shaw model for situations with and without a regional flow, with and without heterogeneity, and for two recharge rates. Whitaker (1994) used numerical methods to develop the interface between the two fluids with different viscosities in a Hele-Shaw cell.

Several studies explored the capacity of the Dupuit models by comparing exact three-dimensional solutions to the Dupuit solutions. These include infiltration at the top of an aquifer system (Strack, 1984 and Haitjema, 1987) and flow in multi-aquifer systems (Maas, 1987 and Bakker, 1999). Clement et al. (1996) compared the performance of three models; the Dupuit-Forchheimer, the fully saturated, and the variably saturated flows in unconfined, steady-state flow through porous media for different soil properties, problem dimensions, and flow geometries. The Dupuit-Forchheimer equation should be applied carefully to determine flow to wells because flow velocities into a well would be overestimated. The performance of the Dupuit-Forchheimer equations in modeling unconfined flow problems has also been investigated by Hantush (1964). The results indicated that the Dupuit-Forchheimer formula estimated the flow discharge within 1-2% of the experimental data.

Water movement through the bodies of earth dams can be studied using the Hele-Shaw models. Seepage in earth dams is a serious problem. The effect of seepage on an earth dam might appear in different ways. Water flowing through the body of an earth dam and emerging downstream causes downstream filling materials to liquify. As a result, the downstream face can be slowly deteriorated. Even though a number of models have been developed for the simulation of water flow in porous media including the seepage from earth dams, testing or validation of these models is lacking due to the shortage of the experimental data.

The objectives of this study were to: (1) calculate the amount of seepage from a physical model of an earth dam using the Hele-Shaw Analog Model, and (2) compare the results obtained from the solutions of Dupuit, Schaffernak and Van Iterson, L. Casagrande, and Pavlovsky equations with the results measured from the model.

MATERIALS AND METHODS

Equations for Describing Seepage in Earth Dams

A variety of equations have been developed for the calculation of the amount of seepage from earth dams placed on an impervious base. Each of these equations has been based on the Dupuit-Forchheimer assumptions. The four main equations are: (1) Dupuit, (2) Schaffernak and Van Iterson, (3) L. Casagrande, and (4) Pavlovsky (Harr, 1962).

Dupuit Equation

The following equation (Darcy Equation) is given to find the amount of seepage from any vertical profile of the dam (Figure 1).

(1) |

**Figure 1.** Figure used for the Dupuit Solution (Harr, 1962).

where *q* is the Darcy *flux* or *flow rate* (L^{3} T^{-1}), K is the hydraulic conductivity or *permeablilty* (L T^{-1}), *h*_{1} is the upstream head, *h*_{2} is the downstream head, and *L* is the length of the flow path. This equation shows that the free surface is parabolic in shape. The resulting free water surface is generally known as Dupuit parabola (Harr, 1962).

Schaffernak and Van Iterson Equation

Schaffernak and Van Iterson developed the first approximate method in 1916 for the computation of the seepage surface. The following equation was used for calculation of seepage flux from an earth dam with zero downstream head (Figure 2) (Harr, 1962).

(2) |

**Figure 2.** Figure used for Schaffernak and Van Iterson Solution

where *q* is the Darcy *flux* or *flow rate* (L^{3} T^{-1}), *K* is the hydraulic conductivity or *permeability* (L T^{-1}), *h* is the length of the seepage surface (L), and a is the angle of the downstream slope. The length of the seepage surface (a) is calculated as:

(3) |

where *d* is the seepage length (L) and *h* is the upstream head (L).

L. Casagrande Equation

In this method, *q* is expressed by assuming that the hydraulic slope dy/dx is equal to dy/ds, where *S* is the length measured along the free surface, (Figure 3) as:

**Figure 3.** Figure used for L. Casagrande Solution

where *q* is the Darcy *flux* or *flow rate* (L^{3} T^{-1}), *K* is the hydraulic conductivity or *permeability* (L T^{-1}), *S* is the length of the seepage surface (L), and a is the angle of the downstream slope. The length of the seepage surface (*a*) using upstream head, *h* (L), is calculated as:

(4) |

where *d* is the seepage length (Harr, 1962).

Pavlovsky Equation

Pavlovsky developed two equations, with and without water at the downstream face of the dam, for the measurement of the seepage from eart dams placed on impervious bases (Figure 4).

Without Water in Downstream

No flow other than the seepage through the dam is assumed and seepage flux in case of no water head in the downstream is calculated as:

**Figure 4.** Figure used for Pavlovsky Solution

where *q* is the Darcy *flux* or *flow rate* (L^{3} T^{-1}), *K* is the hydraulic conductivity or *permeability* (L T^{-1}), *b* is the top width of the dam (L), *h*_{d} is the height of earth dam (L), *h*_{w} is the upstream water head (L), m_{1} = m = cot(b), and *a* is the seepage surface. To calculate the values of *a*_{0} and *h* from these two equations, is calculated for a given value of *h* from each equation. The relation between *a*_{0} and *h* for each equation is plotted. The intersection of the curves on the graph gives the values of *a*_{0} and *h*.

Downstream Head > 0

Seepage flux in the case of non-zero downstream head is calculated as:

(5) |

where *q* is the Darcy *flux* or *flow rate* (L^{3} T^{-1}), *K* is the hydraulic conductivity or *permeability* (L T^{-1}), *h*_{w} is the upstream water head (L), *h*_{d} is the height of earth dam (L), *m* = cot(b)

(6) |

(7) |

where *h*_{0} is the downstream water head (L), *b* is the top width of the dam (L), *m*_{1} = *m* = cot(b) The value of *a*_{0} is obtained from graph drawn by trial and error as a result of putting the values of *a*_{0} in Equation 5 into the Equation 6 or plotting *f*(*a*_{0}) as a function of *a*_{0} (Harr, 1962).

Theoretical Principles of Hele-Shaw Models

That flow paths obtained by the Hele-Shaw model represented two-dimensional potential flow was proved using mathematical principles (Harr, 1962; Omay, 1967). When a liquid flows between two parallel plates (Figure 5) with the plates a distance 2a apart, velocity in the direction of z will be zero, that is, *v*_{z} = 0.

**Figure 5.** Flow between the two parallel plates.

When the flow is very slow or viscosity very high, Reynolds number is very small, which means turbulent flow is not significant. Therefore, hydrodynamic terms can be ignored because viscous forces are large when compared to the inertia forces. Therefore, only active mass force will be the gravity and the direction of this force will be Fx=0, Fy=-g and Fz=0.

Further mathematical details of the theory may not be of interest to the casual user, but essential for a full understanding for any further developments. Therefore these are presented in the Appendix (a 52k MS-Word DOC file).

LABORATORY EXPERIMENTS

A Hele-Shaw Viscous Liquid Model was prepared for this study in the laboratory of the Department of Agricultural Structures and Irrigation, the Faculty of Agriculture, Atatürk University, in Turkey. The model consists of the two parallel glass plates and with dam models made from glass placed between the parallel glass plates. The glass plates were vertical as shown in Figures 6a, 6b, and 6c. The lengths, heights, and thicknesses of the glass plates were 70, 50, and 0.8 cm, respectively. The glass plates were placed 3 cm apart from each other. The model was placed on a U profile made from iron. Two holes, 5 cm from the edges of the frame with the interior diameters of 1.3 cm, were open under the frame of U profile to control the entrance and exit of motor oil into/from the model. Oil seals were used to prevent leakage in the model. The seals with the same dimension as plate were placed between the frame of U profile and the glass plates with adhesive to prevent leakage. Discharge control and storage systems were placed at the entrance of the model, whereas discharge measurement and collection systems were placed at the exit. There was 2 mm opening between the bodies of the dam and the glass plates. Plastic pipes were used for the transportation of the oil between the tank and the discharge control system, between the discharge control system and the model, and between the model and the discharge measurement system. Kinematic viscosity of the oil was 3.311 cm^{2}/s in 20 °C. The oil coming from the storage tank was first transported to the cup which kept the level of the oil in the entrance of the model at a certain level. Then, it was transmitted from the cup into the model through the entrance hole located under the model. After the oil entering the model reached a certain level, it was taken into the measurement cup by observing the 2 mm opening and the conditions of the exit hole located the opposite site of the model.

**Figure 6a.** Schematic view of the model

**Figure 6b.** The cross section of the glass plates

**Figure 6c.** The cross section of the model

**Figure 7.** The relationship between viscosity and temperature

Although a variety of viscosimeters have been developed to measure viscosity, most of them operate with the same principles, that is, Poiseuille’s Law (Gemalmaz, 1985; Anapali, 1987). Poiseuille’s Law is based on the determination of the time for a flow of a known volume of liquid to pass through a capillary pipe. The amount of the oil passing through the capillary pipe was collected in the measurement cup and measured by keeping a constant head, 5.4 cm, of the oil in the cone. Viscosity as a function of temperature is given in Figure 7.

The following equation was used for the calculation of viscosity:

(8) |

where u is the kinematic viscosity of the oil (L^{2} T^{-1}), *i* is the hydraulic gradient (L L^{-1}), *g* is the gravitational acceleration (L T^{-2}), *r* is the interior radius of the capillary pipe (L), *A* is the cross-sectional area of the capillary pipe (L2), *Q* is the discharge of the oil from the capillary pipe for a given time (L^{3} T^{-1}).

A variety of bodies of the dam made from glass were used in the model. The bodies of the dam were prepared for 11 different slopes. Base length and top width of the dam for each slope were 50 and 5 cm, respectively, in the bodies of the dam. The dam heights for chosen slopes at 50 cm base length and 5 cm top width are given in Table 1. The thicknesses of the glasses for the bodies of the dam were 5 mm.

**Table 1.** The model dam heights for a given slope

Upstream slope | Downstream slope | Dam height (h_{d}), cm |

1:1 | 1:1 | 22.50 |

1:1.5 | 1:1.5 | 18.00 |

1:2 | 1:2 | 11.25 |

1:1 | 1:1.5 | 18.00 |

1:1.5 | 1:1 | 18.00 |

1:1 | 1:2 | 15.00 |

1:2 | 1:1 | 15.00 |

1:1.5 | 1:2 | 12.86 |

1:2 | 1:1.5 | 12.86 |

1:1.5 | 1:2.5 | 11.25 |

1:2.5 | 1:1.5 | 11.25 |

**Table 2.** The levels of the oil for a given slope

Slope | The level of the oil in the upstream (cm) |
The level of the oil in the downstream (cm) |

1:1 | 20 | 0 7 10 |

15 | 0 5 7 | |

5 | 0 2 3 | |

1:1.5 | 14 | 0 5 7 |

10 | 0 4 7 | |

5 | 0 2 3 | |

1:2 | 10 | 0 4 7 |

7 | 0 2 4 | |

5 | 0 2 3 | |

1:1/1:1.5 | 15 | 0 4 7 |

10 | 0 4 7 | |

5 | 0 2 3 | |

1:1.5/1:1 | 15 | 0 4 7 |

10 | 0 4 7 | |

5 | 0 2 3 | |

1:1/1:2 | 14 | 0 4 7 |

10 | 0 4 7 | |

5 | 0 2 3 | |

1:2/1:1 | 14 | 0 4 7 |

10 | 0 4 7 | |

5 | 0 2 3 | |

1:1.5/1:2 | 10 | 0 4 7 |

7 | 0 2 4 | |

5 | 0 2 3 | |

1:2/1:1.5 | 10 | 0 4 7 |

7 | 0 2 4 | |

5 | 0 2 3 | |

1:1.5/1:2.5 | 10 | 0 4 7 |

7 | 0 2 4 | |

5 | 0 2 3 | |

1:2.5/1:1.5 | 10 | 0 4 7 |

7 | 0 2 4 | |

5 | 0 2 3 | |

The oil with known viscosity was put into the storage tank. The oil was sent from the tank to the level-adjusting cup, then, to the model. There was 2 mm opening between the glass plates and the bodies of the dam. The amount of the oil leaked from the openings was measured by means of the measurement cup in the exit of the model. Since there is a direct relationship between viscosity and the size of the opening, temperature was measured at the entrance and exit of the model. Measurements were made for different temperatures. The values measured in the model were corrected to a standard temperature of 22°C by using the viscosity (u = 2.72 cm^{2}/sec) obtained from viscosity-temperature relationship (Figure 7). The level of oil sent from the tank to the level-adjusting cup and from the cup into the model was adjusted to enable measurements in the upstream section of the model. Measurements were made by settling dam bodies prepared for 11 different slope conditions with 50 cm base long in the model. The measurements were made with 3 levels at the downstream side for 3 different levels at the side upstream. The amount of the oil and temperature were measured together. The amount the oil flowed from the 2 mm openings of the bodies of the dam was measured with time and expressed as q (cm^{3}/sec)

RESULTS AND DISCUSSION

The measurements obtained from different combinations of the upstream head, the downstream head, and the slope and the results obtained from the solutions of Dupuit, Schaffernak-Van Iterson, L. Casagrande, and Pavlovsky equations were used in the statistical analyses. The original hydraulic conductivity or *permeability* of the model, *K*_{m}, was used in all equations with no change. As shown in Table 3, the amount of seepage (discharge) increases as the slope and the upstream head increase.

(9) |

Correlation and *t*-tests were conducted between the corrected model values and theoretical values. The Statistical Analysis System (SAS) was used for the computation of the correlation coefficients whereas Minitab statistical computer program was used for the computation of *t*-tests. The results of the correlation analyses showed a significant relationship between the model and the formula values for all slopes. The correlation coefficients between the all values of the model and the formulas are given in Table 3. The correlation coefficients shows significant relationship between the model and the equation values. The most significant relationship was determined between the model the Dupuit equation (0.9927).

**Table 3.** The correlation coefficients among all values of the model and formulas

Model | Dupuit | Schaffernak and Van Iterson | L.Casagrande | Pavlovsky |

Cor. Coef. | 0.9927 | 0.9618 | 0.9618 | 0.9426 |

**Table 4.** Probability values between the model and the formulas obtained from *t*-tests

Slope | Dupuit | Schaffernak and Van Iterson | L.Casagrande | Pavlovsky |

1:1 | 0.92 | 0.91 | 0.97 | 0.71 |

1:1.5 | 0.96 | 0.88 | 0.92 | 0.71 |

1:2 | 0.38 | 0.51 | 0.51 | 0.57 |

1:1/1:1.5 | 0.86 | 0.80 | 0.83 | 0.65 |

1:1.5/1:1 | 0.64 | 0.79 | 0.82 | 0.56 |

1:1/1:2 | 0.97 | 0.78 | 0.81 | 0.37 |

1:2/1:1 | 0.69 | 0.88 | 0.93 | 0.28 |

1:1.5/1:2 | 0.95 | 0.76 | 0.76 | 0.85 |

1:2/1:1.5 | 0.88 | 0.80 | 0.82 | 0.39 |

1:1.5/1:2.5 | 0.87 | 0.69 | 0.70 | 0.42 |

1:2.5/1:1.5 | 0.86 | 0.86 | 0.88 | 0.25 |

CONCLUSIONS

The applicability of several theoretical equations to predict seepage through an earth dam placed on an impervious base by using the Hele-Shaw Viscous Fluid Physical Model was evaluated in this study. The results showed that each of the four equations (Dupuit, Schaffernak and Van Iterson, L. Casagrande, and Pavlovsky) can be used in the calculation of the seepage from the body of an earth dam placed on an impervious base. The Pavlovsky equation was the least compatible with the model for seepage computations. The Schaffernak and Van Iterson and L. Casagrande equations did not produce acceptable solutions when there was head in the downstream. However, the results of these equations were very close to the model results when the head in the downstream was zero. The Dupuit equation was the most appropriate formula for the seepage calculations. The Dupuit equation can be safely used in the seepage calculations from an earth dam because of the ease of its use, giving results in different heads in the downstream, and its very high compatibility with the model.

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