Analysis of Water Surface Profiles in Coarse-Grained Porous Media with Radial Flow Using the Gradually Varied Flow Theory

Document Type : Research Article


1 Assistant Professor, Department of Civil Engineering, Bu-Ali Sina University, Hamedan, Iran

2 Associate Professor, Department of Civil Engineering, University of Zanjan, Zanjan, Iran

3 . PhD Student of Hydraulic Structures, Department of Civil Engineering, University of Zanjan


Non-darcy flows into two categories: parallel flows (such as gravel dams, gabions, etc.) and radial flows (such as flows near wells drilled in coarse-grained alluvial beds, etc.) are divided. In the first category, streamlines are almost parallel so that there is no curvature or contraction of streamlines in the plan view. This type of flow is found in both pressurized and free-surface modes. Radial non-darcy flow analysis has many applications in the fields of civil engineering, geology, oil, and gas. The equations governing the radial non-darcy flow are solved using numerical methods of finite differences, finite elements and finite volumes. Solving these equations requires boundary conditions and a lot of data and is almost bulky, time consuming and costly. While, gradually varied flow theory, requires much less data and is easier and less expensive. For this reason, in the present study, for the first time, using experimental data recorded in a large-scale (almost real) device, the application of the gradually varied flow theory in radial non-darcy flows with free surface has been investigated. In other words, since the calculation of water surface profiles in a radial rockfill is of great importance. In the present study, using large-scale (almost real) experimental data and the gradually varied flow theory, the water surface profile in radial non-darcy flow with free surface and in steady state has been investigated.
In the present study, due to the compatibility of cylindrical coordinates and its adaptation to the physics of problems related to radial flows, a device has been constructed in the laboratory of Bu Ali Sina University in the form of a semi-cylinder with a diameter of 6 meters and a height of 3 meters. The dimensions of this device are made on a large scale and the effects limitations have practically no effect on the testing process. To measure piezometric pressure, piezometric grids have been used. The device has a volume of 14,000 liters and a capacity of materials weighing approximately 40 tons. Four pumps are installed in parallel at the top of the device to generate the required flow. Coarse-grained river materials with a diameter between 2 to 10 cm, a porosity of 40%, a Cu of 2.13, and a Cc of 1.016 have been used. To perform the tests, the model is first filled to a certain height (53, 60, 70, 85, 95, 110, 120, 140, 150, and 160 cm) by pumping operations. The flow rate created in these experiments is in the range of 49.94 to 53.16 L/s.
Results and Discussion
One-dimensional analysis of steady-non-darcy flow using gradually varied flow theory and two-dimensional analysis using Parkin equation solution. Most research has been done in parallel flow rockfills. Also, solving the Parkin equation in both parallel and radial flows requires a lot of data such as boundary conditions upstream and downstream, as well as the boundary condition of the water surface profile, and the calculation process is complex and time-consuming. The gradually varied flow theory requires much less data than solving the Parkin equation, and the water surface profile obtained from it is also used as the main boundary condition in solving the Parkin equation. In other words, calculating the water surface profile in a radial rockfill is very important to studying the movement of water. Also, the water surface profile is the main boundary condition in the two-dimensional analysis of steady flow (solving the Parkin equation), and with it, upstream and downstream boundary conditions will be practically available. For this reason, in the present study, using large-scale (almost real) experimental data and the gradually varied flow theory, the water surface profile in the case of radial non-darcy flow has been calculated. To calculate the flow depth at different points (water surface profile) using the gradually varied flow theory, the amount of flow depth at one point and the coefficients m and n must be available. Since the flow depth measurement in the well (downstream of the desired interval) can be measured, in the present study, the calculations started from the downstream (depth of flow in the well).
If the gradually varied flow theory is used to calculate the water surface profile in the case of radial non-darcy flow with a free surface, the mean relative error in the case of pumped heights is 53, 60, 70, 85, 95, 110, 120, 140, 150 and 160 cm are equal to 1.56, 0.96, 0.61, 0.45, 0.28, 0.19, 0.13, 0.16, 0.11 and 0.05 are calculated, respectively. In other words, the average mean relative error (MRE) of calculating the water surface profile for different heights of pumped water is equal to 0.45%. Also, according to the obtained results, the greater the depth of water pumped upstream, the higher accuracy of the gradually varied flow theory.
Radial Non-Darcy Flow, Steady Flow, One-Dimensional Analysis, Gradually Varied Flow Theory.


Ahmed, N. and Sunada, D.K. (1969). Nonlinear flow in porous media. Journal of the Hydraulics Division, 95(6), 1847-1858.‏
Arbhabhirama, A. and Dinoy, A.A. (1973). Friction factor and Reynolds number in porous media flow. Journal of the Hydraulics Division. ASCE, 99(6), 901-915.
Bari, R. and Hansen, D. (2002). Application of gradually-varied flow algorithms to simulate buried streams. Journal of Hydraulic Research, 40(6), 673-683.
Bazargan, J. and Shoaei, S.M. (2006). Application of gradually varied flow algorithms to simulate buried streams.‏ IAHR J. of Hydraulic Research, 44(1), 138-141.
Bazargan, J. and Shoaei, S.M. (2010). Analysis of non-darcy flow in rock fill materials using gradually varied flow method. Journal of Civil and Surveying Engineering, 44(2), 131-139. (In Persian)
Forchheimer, P. (1901). Wasserbewagung Drunch Boden, Z.Ver, Deutsh. Ing., 45, 1782-1788.
Gudarzi, M., Bazargan, J. and Shoaei, S. (2020). Longitude Profile Analysis of Water Table in Rockfill Materials Using Gradually Varied Flow Theory with Consideration of Drag Force. Iranian Journal of Soil and Water Research, 51(2), 403-415. (In Persian)
Hansen, D., Garga, V.K. and Townsend, D.R. (1995). Selection and application of a one-dimensional non-Darcy flow equation for two-dimensional flow through rockfill embankments. Canadian Geotechnical Journal, 32(2), 223-232.‏
Jamie, M., Ahmadianfar, I. and Raeisi Isa Abadi, A. (2019). A Numerical IMPES Discontinuous Galerkin method for Immiscible Groundwater Contaminations Flow Using Lax-Wendroff scheme. Journal of Water and Soil Conservation, 26(2), 1-27. (In Persian)
Leps, T.M. (1973). Flow through rockfill, Embankment-dam Engineering: Casagrande volume edited by Hirschfeld, R.C. and Poulos, S.J., John Wiley and Sones, New York, pp. 87-107.
McWhorter, D.B., Sunada, D.K. and Sunada, D.K. (1977). Ground-water Hydrology and Hydraulics. Water Resources Publication.‏ LLC. U.S Library.
Norouzi, H., Bazargan, J., Azhang, F. and Nasiri, R. (2022). Experimental study of drag coefficient in non-darcy steady and unsteady flow conditions in rockfill. Stochastic Environmental Research and Risk Assessment, 36(2), 543-562.‏
Pasupuleti, S., Kumar, P. and Jayachandra, K. (2014). Quantification of effect of convergence in porous media flow.‏ 5th International Conference on Porous Media and Their Applications in Science, Engineering and Industry, pp. 1-7.
Sadeghian, J. (2013). Analysis of radial flows in coarse alluvial beds. Ph.D. Thesis, College of Agriculture and Natural Resources, University of Tehran. (In Persian)
Sadeghian, J., Khayat Kholghi, M., Hoorfar, A. and Bazargan, J. (2013). Comparison of binomial and power equations in radial non-Darcy flows in coarse porous media. Journal of Water Sciences Research, 5(1), 65-75.‏
Sadeghian, J., Khayat Kholghi, M., Hoorfar, A., bazargan, J. (2014). Experimental study of radial non-Darcy flows in coarse alluvial beds. Iranian Water Researches Journal, 8(2), 11-21. (In Persian)
Saeedi, H., Akbarpour, A., Baghvand, A., Niksokhan, M.H. and Sadeghi Tabas, S. (2016). Simulation-Optimization Quantitative and qualitative model operation of aquifer in order to adjust pollutant concentrations using Cuckoo algorithm. Journal of Water and Soil Conservation, 23(5), 87-103. (In persian)
Scheidegger, A.E. (1958). The physics of flow through porous media. Soil Science, 86(6), 355.‏
Sedghi-Asl, M. and Ansari, I. (2016). Adoption of extended dupuit–Forchheimer assumptions to non-darcy flow problems. Transport in Porous Media, 113(3), 457-469.
Shayannejad, M. and Ebrahimi, A. (2020). Hydraulic investigation of non Darcy radial flow in unconfined aquifers in steady state. Iranian Journal of Irrigation & Drainage, 13(6), 1580-1588.  (In Persian)
Stephenson, D.J. (1979). Rockfill in hydraulic engineering. Elsevier scientific publishing company.‏ Distributors for the United States and Canada.
Venkataraman, P. and Rao, P.R.M. (2000). Validation of Forchheimer's law for flow through porous media with converging boundaries. Journal of Hydraulic Engineering, 126(1), 63-71.‏
Ward, J.C. (1964). Turbulent flow in porous media. Journal of the Hydraulics Division, 90(5), 1-12.‏