Introducing a Relationship to Estimate Hydraulic Gradient in Non-Darcy Turbulent Flow in Porous Media

Study of flow characteristics in rock porous media is one the most interesting issues for scientists and engineering dealing with river engineering works. So, there is no surprise that many models to describe the relationship between the flow velocity of clear water with hydraulic gradient, rock size, porosity, Reynolds number, and kinematic viscosity, have been introduced. Due to the large spaces between the coarse materials, flow velocity passing through the materials is high which in turn results in higher amounts of Reynolds number of flow. This type of flow classified as turbulent flow. Although Darcy law rules the flow in porous media, it is used for laminar flow in fine porous media and its application is not recommended for turbulent flows. Moreover, as the flow parameters in turbulent flows vary against time, the state of the flow is not steady. The equations describing the turbulent flows are obtained using equations defining basic concepts of hydrodynamics and turbulence effects. Due to complexity of the turbulent flow, these equations are described in the form of the partial differential equations. In order to introduce the specifications of this type of flow, various relationships have been provided by many researchers. However, their applications are confined to the limited conditions of porosity and size materials. In this study, we aim to provide a relationship which can be applied for a wide range of porosity and material size of porous media.

To describe the relation between effective hydraulic parameters in coarse porous media, we used dimensional analysis theorem of Buckingham. In this regard six dimensionless parameters have been provided from which a relationship including four constant parameters has been obtained. We used a part of (70 percent) several available sets of data, provided from Soil Conservation and Watershed Management Research Institute, Irrigation and Reclamation Engineering Department of the University of Tehran, and mostly from published results, to find the magnitude of the constant parameters. So, we introduced a new equation which expresses a relationship between hydraulic gradient, porosity, and Froud number. Finally, using the remained part of (30 percent) available data, we compared the results of the new equation with those obtained from available models.

To evaluate the new introduced equation and comparing the results obtained from the new equation and those obtained from available equations, we computed the magnitude of relative errors as well as the mean relative errors of the hydraulic gradient estimated from all equations versus the hydraulic gradients provided from field and laboratory observations. It is found that the new equation has the least mean of relative error (15.3 percent) among all equations. Moreover, for various magnitudes of rock size as well as porosity, we computed the mean relative error of estimated hydraulic gradients according to observed data. We found that the new equation has the second largest accuracy (with the mean error of 11.64%) among all evaluated models in this research. Finally, we developed two relationships between hydraulic gradient and Froud number using actual as well as apparent velocities. Again, it is found that the new relationship has the least mean of relative error (14.03 percent) among all equations.

Since all available equations introduced to express the flow characteristics in coarse porous media, can be used in a defined limits of porosity, rock size, etc., in this research we aimed to provide a new relationship which can be used for a wider range of porous media specifications. So, based on dimensional analysis and using several sets of available field and laboratory data, a new equation has been introduced in this research which can be used for a wide range of rock size, Reynolds number, and porosity; i.e. rock diameter of 0.5 to 20 cm, Reynolds number greater than 100, and porosity of 0.35 to 0.55. Moreover, we introduced two equations to demonstrate the relationship between hydraulic gradient and actual velocity as well as apparent velocity. When we evaluated the results obtained from the new relationship with those obtained from some available equations, we found that the relative error of the new equation is 14 percent, which illustrates that the error of the results produced by the new equation is less than those produced by the available equations.