Investigating changes in coefficients of the non-Darcy flow exponential relationship within rockfill materials under different flow conditions

Introduction
As we know, the calculation of hydraulic gradient is highly important in the analysis of steady flow inside rockfill materials. Binomial and exponential relationships are used to calculate the hydraulic gradient based on the non-Darcy flow velocity, and the binomial relationship is more accurate and efficient than the exponential relationship.
Since it is necessary to use an exponential relationship in the two-dimensional analysis of non-Darcy flow in coarse porous media, in the past, researchers have provided relationships to calculate the coefficients m and n of the exponential relationship based on the coefficients a and b of the binomial relationship. In some previous studies, Vmax= 1 has been considered, even though the maximum flow velocity depends on the physical characteristics of the pebbles and the characteristics of the flow and is not necessarily equal to one. For this reason, in this research, by designing and equipping the laboratory and recording the laboratory data, the maximum velocity based on the values of a, b and Re of the analytical model of Ahmed and Sunada(1969) is proposed.
As mentioned above, various researchers tried to calculate the coefficients of the exponential relationship using the values of a and b in the binomial relationship. One of the most important relationships is presented by George and Hansen (1992) as follows.
n=(5a+6bV_max)/(5a+3bV_max ) (1)
m=(5a+4bV_max )(4a+3bV_max )/(4(5a+3bV_max ) (V_max )^(n-1) ) (2)
Further, by stating that in the coarse-grained porous medium, the slope of the energy line (Sf) is equal to the hydraulic gradient (i), it can be stated that one of the most important parameters in the investigation of the flow in the gravel medium in free flow and under pressure is the calculation of it is a hydraulic gradient. In this research, using the coefficients of the binomial relationship, we presented a solution to calculate the values of m and n in the exponential relationship with better accuracy. Therefore, considering that the exponential relationship is used in the two-dimensional analysis of the non-Darcy flow in porous gravel media, this can play a significant role in reduction of the error of hydraulic gradient calculation.

Methodology
In the current research, the laboratory data recorded in the hydraulic laboratory of the Faculty of Civil Engineering of Zanjan University were used. For this purpose, an attempt was made to design and set up a test device and perform tests on different gravel materials. Experiments were carried out in a laboratory flume with the ability to tilt, with dimensions of 1m×1m and a length of 15m, and the length of 2.2m of the mentioned flume is filled with rockfill. The walls of the flume are made of plexiglass, and to measure the piezometric height along the porous media, 23 piezometers are used on the bottom of the channel, which are arranged at certain distances from each other and along them. The water flow in the channel is created by a pump with a maximum flow capacity of 90 liters per second. In order to create a porous media, three types of rockfill materials with small, medium and large diameters have been used in the experiments. During the tests, to ensure a stable flow, the pump was working for about 10 minutes with the desired flow and after the stability of the flow, the desired parameters were measured. These parameters include the piezometric height at the location of 23 piezometers as well as the water depth at the location of each piezometer. Piezometric values are read using a calibrated table. The water depth was also measured and recorded directly by a ruler.

Results and Discussion
Since the exponential relationship is only accurate for a certain range of Reynolds numbers and the user area recommended for this relationship by its providers is only non-quiet flow conditions, therefore, if the exponential relationship is used in the two-dimensional solution of the equations, there will be a large error will enter the calculations. To avoid this problem, various researchers have tried to convert the binomial relationship into an exponential relationship. If the minimum flow velocity Vmin and the maximum flow velocity Vmax in the conversion area of the binomial relationship is in the form of an exponential relationship. In order to convert the two mentioned relations, relations (1) and (2) can be used. According to the conducted tests, in most cases, Vmin is considered zero and Vmax value is assumed to be equal to one, while the maximum flow velocity depends on the physical characteristics of the pebbles and the characteristics of the flow and is not necessarily equal to one. Therefore, Vmax can be calculated from the following relationship according to Ahmed and Sunada's(1969) analytical model and the definition of the Reynolds number as Re=ρVd/μ.
V_max=Re_max a/b (3)
By using relations (1), (2) and (3), it is possible to take advantage of the accuracy of the binomial relation and the practical property of the exponential relation in the two-dimensional analysis in porous media.
If relations (1) and (2) are used in the calculation of the coefficients of the exponential relationship of steady flow in gravel materials, the average relative error between the calculated and recorded hydraulic gradients in the laboratory assuming Vmax=1 (according to previous research) in fine gravel materials, medium and coarse are calculated to be 21.95%, 22.98% and 21.97%, respectively. While if relation (3) is used (the solution presented in the current research), the average relative error values of the hydraulic gradient are equal to 11.39, 14.69 and 19.72%, respectively.

Conclusion
In general terms, by using relations (1) and (2) and using the relation proposed in the present study instead of Vmax=1, the average values of the relative error of the hydraulic gradient in fine, medium and coarse rockfill materials have decreased to 10.56, 8.29 and 2.25%, respectively, which indicates the high accuracy and efficiency of the proposed solution.

Keywords: Non-Darcy flow, exponential relation, binominal relation, rockfill, hydraulic gradient.