Physical Explanations of Non-Darcy Effects for Fluid Flow in Porous Media

Abstract

Summary The nature of non-Darcy flow in porous media is analyzed by means of the volumetric averaging approach for a medium modelled by diverging converging capillaries. To clarify the mechanism responsible for the nonlinearity, a physical explanation is deduced for the dispersion term in the averaged momentum equation. With the present periodical model, the numerically obtained microscopic flow fields, in association with the macroscopic coefficients calculated by the average momentum balance, indicated quantitatively that the microscopic inertial phenomena, which lead to distorted pore velocity and pressure fields, is the fundamental reason for the onset of non-Darcy effects as filtration velocity increases. Introduction On the macroscopic scale, fluid flow in a porous medium at low velocity (specific discharge) is generally described by Darcy's law which presents a linear relationship between the driving force, and the filtration velocity, U. However, as the filtration velocity is raised beyond a certain value, numerous experimental observations have confirmed that Darcy's law should be replaced by another time-honored empirical formula, the Forchheimer equation, (1) to account for the nonlinear effects for the medium considered. In this equation, k denotes the Darcy's law permeability, is an experimentally derived parameter called the inertial coefficient, and and represent the density and the dynamic viscosity of the fluid, respectively. Both k and are thought to be material constants in the range of validity of Eq. (1). Proceeded from a consideration that the permeability may alternatively be treated as being velocity dependent, a new form of Forchheimer equation has been proposed: (2) where k = is the velocity dependent permeability and Fo = is called the Forchheimer number which replaces the Reynolds number as a dimensionless criterion to indicate when microscopic effects lead to significant macroscopic inertial effects. It has long been a common interest for many researchers to clarify the physical reason for the onset of the nonlinearity. Early descriptions attributed the nonlinearity to the occurrence of turbulence. However, experiments have indicated that when the macroscopic velocity gradually increases, the nonlinear phenomena appear much before the onset of real turbulence in porous media flow. Thus, it can be concluded firmly that the deviations from Darcy's law are not initiated by changes of flow regimes. A diversity of opinions as to why nonlinearity at high flow rates occurs still exists. In their paper, Hassanizadeh and Gray performed an order of magnitude analysis for the averaged momontum equation and concluded that the microscopic viscous force is the source for the onset of nonlinearity.

Experimental Investigation on Hydraulic Properties of Granular Sandstone and Mudstone Mixtures

“…was also called velocity coefficient [26], inertial resistance coefficient [27,28], turbulence factor [29,30], inertial coefficient [31], and so on. Although can be determined by (5a) and (5b), it is important to hold that is a function of the porous medium such as and [32].…”

Theoretical and Experimental Investigation of Water in Oil Transverse Dispersion in Porous Media

“…The mechanism of transverse dispersion is important for evaluating the size of mixing zone. Ma and Ruth (1997) and Greenkorn et al (1964) presented how momentum theory explains dispersion using flow streamline diverting and converging in theoretical models and physical experiments in microscopic scale. A simplified bifurcated flow model associated the momentum balance theory is developed to demonstrate the stream distortion between water and oil flow in this paper.…”

“…Non-Darcy flow coefficient β (Forchheimer, 1901) is commonly regarded as a parameter of intrinsic properties of porous media to describe the intensity of the change. Al-Rumhy et al (1996), Huang and Ayoub (2006), Haro (2007) and Ma and Ruth (1997) show that tortuosity and core heterogeneity are key factors inducing non-Darcy by In a water oil segregated flow (Figure 1), if the flow driven by inertia force diverts from the main flow across the W/O contact, transverse dispersion may occur. Considering capillary and gravity effects, criteria of transverse dispersion zone near wellbore was developed in this paper.…”

Theoretical and Experimental Investigation of Water-in-Oil Transverse Dispersion in Porous Media

“…Al-Rumhy and Kalam (1996), Huang and Ayoub (2008), Ma and Ruth (1997), and Haro (2007) indicate that tortuosity and core heterogeneity are the key factors inducing mixing flow. The mixing process was identified previously for uneven concurrent-laminar flows in porous media (Duan and Wojtanowicz 2006).…”