Slow seepage of polar fluids through porous media

A study of slow seepage of polar fluids through porous media is made using smoothed continuity equation and Darcy’s equation in a porous medium. A transformation and approximate solution of the governing equations was carried out and its analysis showed that increase in both porosity and permeability result in an increase in the pressure of the fluid. Comparison with other studies also showed reasonable agreement.


Introduction
Fluids are sometimes found in the void spaces of earth materials. The amount of void space available for fluid flow is the effective porosity. The grain size, sorting, grain shape and clay and organic content of the earth materials determine the void spaces available for fluids to flow. The nature of the void spaces also gives an insight to the possible availability of natural resources. Porosity greatly affects fluid movement and exchange which are important for organisms that live in the soil. The transport of nutrients and contaminants will also be affected by porosity and because flow of fluids through porous media is common in nature and has many applications in engineering and earth sciences, its study cannot be overemphasized. The study of fluid flow through porous medium has been in conjunction with other parameters [1], [2], [3], [4], [5], [6], [7] [8], [9] and [10]. In the listed studies, a combination of continuity, Navier-Stokes and energy equations were used which in our view diminished the effect or otherwise of porosity and permeability in the various studies. Our aim in this study is to determine the effect of porosity and permeability on fluid flow. This in our view will broaden the study of fluid flow through porous media and also add to existing literatures.

Formalism
For flow of fluid through porous medium, the smoothed continuity equation and the Darcy's equation respectively are are respectively the porosity, permeability, density of fluid, fluid viscosity, superficial velocity, pressure of fluid, time and acceleration due to gravity. (1) and (2) results in

Combination of Equations
We write the equation of state for this study following the argument of [2] as Where 0  is the fluid density at unit pressure, m and  are integers.

Method of Solution
We approximate 2  by ignoring powers of  greater than unity using Taylor series expansion about 0 and reduce equation (5) To solve equation (6) If we put equation (7) into equation (6) , we get The solution of (9) and imposition of the boundary conditions of equation (8) as well as substitute back into equation (7)

Results and discussion
In order to get physical insight and numerical validation of the problem, an approximate value of acceleration due to gravity    Figure 1 shows the relationship between the fluid pressure and the space coordinate as porosity increases. As the effective porosity increases so also the pressure of the fluid decreases as a result of increase in the surface area of the void spaces. This result is also in agreement with an earlier study of [2]. Figure 2 shows that increase in permeability also result to a decrease in fluid pressure and this result laid credence to an earlier study of [5]

Conclusions
Generally, pressure of fluid reduces as porosity and permeability of materials are increased. In most of the studies listed in the literature, particularly studies that combined porosity and other parameters, permeability is not considered which confirmed our position that for effective description of porosity and permeability, the application of Darcy's model is of necessity.