Investigating the use of variable coefficients of Forschheimer's relation in the analysis of unsteady flow in porous media

Introduction
In Coarse-grained porous media, the size of particles and pores causes complications in the flow behavior, in a way that the flow has no layered state and the Darcy relation loses its validity. In these cases, the hydraulic gradient velocity relationship is nonlinear. The coefficients of the relationships have been examined by different researchers.
Surface water relations, also known as Saint-Venant relations, are among the best computational tools governing free surface water flows. Equations mentioned above were first used in 1871 by Adbemar Barre de Saint Venant in order to analyze unsteady flow with a free surface, and afterwards many researchers investigated and estimated the characteristics of free flows as well as flow in porous medium by using these equations
The main purpose of this research is to investigate characteristics of steady and un-steady flow in porous environment. By calculating the velocity values at each point and plotting the velocity graph against the hydraulic gradient, the coefficients of Forschheimer's binomial relation were obtained for each of the discharges. By examining the changes of these coefficients, linear relationships were obtained for the changes of the coefficients against the flow rate changes.
Methodology
In this study, the tilting laboratory channel of the Faculty of Civil Engineering of Zanjan University was used. In order to create a porous environment, 1.2 meters of the length of the channel has been selected and separated by two net separators. (Fig 1). The grading of pebbles used in this research is presented in Figure 2. Also, their physical characteristics are given in Table 1.
The experimental program of this research was carried out in two sections of steady and unsteady flow. In the steady part, water flowed with 10 different flow rates from 8.51 to 20.62 L/s. By arranging the coefficients and the flow rates and plotting them against each other, a function can be derived to calculate each of the coefficients a and b based on the flow rate. The graphs in Figure 4 illustrates these functions.
In the next part of the tests, the hydrograph in Figure 5 was passed through the porous media.
.Saint-Venant's equations were considered as governing equations and were solved using the method of characteristics. The equations were solved once by using fixed values and the other time by using the functions of Forschheimer coefficients.

Results and Discussion
The Saint-Venant equations for the problem were solved once by using the average values of the coefficients of Forschheimer's relation and again by using the functions of these coefficients. By solving the equations, the velocity and depth values and hence the flow rate at any moment and at any point of the porous medium were calculated. Table 3 shows the calculated flow rate error for two solution modes. In this table, the minimum and maximum values of the input hydrograph are compared with their corresponding values in the hydrograph at the point of 6 cm. Checking the error values shows that the calculated error of the maximum flow rate (which occurred at the peak of the hydrograph) in the case of constant Forschheimer coefficients is 13.36%, which is about 3 liters per second more than the actual hydrograph. In spite of this, there is only 2.16% error in the calculation hydrograph with variable coefficients of maximum discharge. This is also important in the minimum of hydrograph. So that the error value in the calculation hydrograph with fixed coefficients has 22% error (minimum of the hydrograph occurs at the beginning), while the corresponding value for the calculation mode with variable coefficients is only 1%.
As can be seen in Fig 9, the calculated profile is approximately always a little lower than the observed profile and the difference between these two profiles maximize at the time of the hydrograph peak, and then the difference decreases again as the discharge decreases.
Table No. 4 shows the percentage of the relative error for the observed and calculated flow profiles at different times. As mentioned, the maximum error among all times and all points is observed in 400 seconds and at the terminal point of the profile.
Conclusion
The results of the numerical solution in two cases of fixed and variable coefficients show that the percentage of relative error in the maximum and minimum discharge for the case of fixed coefficients is much more than the case of variable coefficients.
The results indicate that the maximum discharge go smaller as we move along the medium and also occurs at a later time.
The investigations determined that the average calculation error of the depth at the times of 100, 300 and 600 seconds is 4.88, 8.05 and 9.93 percent, respectively.
Keywords: porous media, Forschheimer's Relation, Unsteady Flow, Saint-Venant Equations, Method of Characteristics.