Non Darcy Flow

The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.

En los Yacimientos No Convencionales tipo Shale Gas se hace muy difícil una predicción del comportamiento de los fenómenos, que rigen el transporte del gas como difusión, la adsorción, y el flujo no Darcy, ya que estos no se rigen por los modelos conocidos en yacimientos convencionales. La adsorción es un mecanismo relacionado al almacenamiento del gas, pero la desorción contribuye con el flujo mediante la entrega, la difusión contribuye desde la migración primaria con la dispersión del gas y cuando se inicia el proceso de producción actúa también con la entrega del gas desde la matriz hasta la fractura, mientras que el flujo No Darcy opera en las fracturas. La presente investigación tiene como objetivo principal describir los fenómenos de transporte del gas en estos yacimientos. Se creó un modelo de yacimiento construido a través del simulador GEM, con datos representativos de un yacimiento tipo Shale Gas como lo es Eagle Ford, para estudiar el impacto que tienen estos fenómenos sobre la curva de producción acumulada de gas. Partiendo de un modelo base, el cual contempló todos los fenómenos juntos y luego se evaluó el efecto de cada fenómeno por separado. Obteniendo como resultado, que el gas acumulado disminuye en un 33,2% cuando no se considera el flujo no Darcy, mientras que sin la adsorción el efecto observado fue contrario con un aumento del 7%. Con la difusión no se obtuvieron cambios significativos. Por lo que se concluyó que todos los fenómenos tienen un impacto sobre el acumulado de gas, siendo el flujo no Darcy el de mayor importancia, seguido por la adsorción y finalmente la difusión, lo cuales deben ser simulados en conjunto.

Fluid flow in isotropic porous media with one porosity scale is a well understood process and a common scenario in numerous simulations published in the literature. However, there exists a class of porous materials that exhibit two porosity scales with strong permeability contrast between the two scales. Examples of such materials are aggregated soils and fractured sedimentary rocks such as shale. In sedimentary rocks, fluid could flow through the micro-fractures at the larger scale as well as through the nanometer-size pores of the rock matrix at the smaller scale. In this paper, we shall refer to the larger and smaller pores of sedimentary rocks as the micro-fractures and nanopores, respectively. Due to preferentially oriented micro-fractures in the rock, fluid could flow predominantly in the direction of the discontinuities, resulting in an anisotropic flow pattern at the larger scale. We idealize such material as a transversely isotropic medium with respect to fluid flow. In addition, the nanopores of sedimentary rocks such as shale are so small that Darcy's law may not hold at this scale. To better understand the impact of non-Darcy flow on the overall flow pattern, we present a hydromechanical model for materials with two porosity scales that accommodates both transverse isotropy at the larger scale and non-Darcy flow at the smaller scale. Even though this study is motivated by shale properties, the discussion revolves around a generic material with two porosity scales whose fluid flow characteristics are similar to those of shale. The overarching goal of this paper is to better understand the impacts of transverse isotropy and non-Darcy flow on the fluid flow pattern in this material.

A unified similarity transformation has been proposed which can deduce a class of possible similarity solutions during forced-, free- and mixed-convection flow of non-newtonian inelastic fluids through porous media under Darcy and non-Darcy conditions. The velocity scale using the slip velocity at the wall under mixed-convection flow has been utilized for generating a modified Peclet number which correctly transforms to the appropriate Peclet and Rayleigh numbers under limiting conditions. The unified approach is seen to reveal the existence of three limiting regimes lined by forced convection, Darcy free convection and Forchheimer free convection together with three intermediate regimes of Darcy mixed convection, Darcy-Forchheimer free convection and Forchheimer mixed convection. Exact solutions have been found for each of the regimes and the effects of non-newtonian character on the heat transfer characteristics have been deciphered.

Numerical unsteady predictions are carried out for two-dimensional natural convective heat transfer in a saturated porous square domain sandwiched between two finite wall thicknesses. The horizontal boundaries of the cavity are adiabatic and the vertical walls are maintained at fixed different temperatures Th and Tc. In the core cavity (porous region), the extension of the Darcy model/Forchheimer–Brinkman-extended Darcy model with the Boussinesq approximation is used to solve the momentum equations as well as the energy and continuity equations. The conduction equation is employed to solve for the temperature distribution in the finite thickness wall layers. The nondimensional equations are
solved by using the finite volume approach and the pressure velocity coupling is treated
via the SIMPLE algorithm applicable in the porous media. The results are presented for
different values of the nondimensional governing parameters, including the modified
Rayleigh number (100
Ra
1000), Darcy Number (107
Da
102), thermal conductivity
ratio (0.1
Kr
10), and the ratio of wall thickness to height (0.1
D
0.4).
A correlation to evaluate the average Nusselt numbers on the left wall interface of the
porous cavity is proposed as a function of modified Rayleigh number, Darcy number, as well
as a number of physical, geometrical and material property variables.