# Mathematical Treatment of Saturated Macroscopic Flow in Heterogeneous Porous Medium: Evaluating Darcy’s Law

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Darcy’ Law for Flow in a Homogeneous Saturated Porous Medium

^{3}/TL

^{2}), $\varphi $ is the scalar energy potential per unit volume of fluid and is defined here as always positive in space and time (FL/L

^{3}), p is the fluid pressure, (F/L

^{2}), $\rho $ is the mass density of the homogeneous fluid (FT

^{2}/L

^{4}), g is the gravitational scalar (L/T

^{2}), and z is the vertical Cartesian coordinate axis directed outward from the center of the earth (L), and $\nabla $ is the vector mathematical gradient operator (1/L).

#### 1.2. Heterogeneous Porous Material Defined

^{4}/FT), F

_{1}is a function of the variables in parenthesis, which is at least once differentiable (L

^{4}/FT), and x, y, and z are the Cartesian spatial coordinates with z oriented vertically away from the earth’s center (L). For later convenience, we included the second equality here, where k is the scalar intrinsic permeability of the saturated heterogeneous media (L

^{2}), $\mu $ is the dynamic viscosity of the homogeneous fluid (FT/L

^{2}), and f

_{1}is a function of the variables in parenthesis, which, again, is at least once differentiable (L

^{2}). We can also state that $\overline{d}$, is continuous and differentiable everywhere in the porous media. This statement is both the necessary and sufficient condition on $\overline{d}$.

^{3}/TL

^{2}), and P

_{e}is the macroscopic scalar effective porosity (L

^{3}/L

^{3}). With ${P}_{e}={F}_{2}\left(x,y,z\right)$, where F

_{2}is a function of space that is at least once differentiable (L

^{3}/L

^{3}).

#### 1.3. A Commonly Used but Questionable Law for Flow in a Heterogeneous Saturated Porous Medium

## 2. A Proposed Law for Flow in a Heterogeneous Saturated Porous Medium

## 3. An Extended Fluid Pore Velocity Expression in Heterogeneous Porous Materials

## 4. Conclusions and Recommendations

## Author Contributions

## Funding

## Conflicts of Interest

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Nelson, R.W.; Williams, G.P.
Mathematical Treatment of Saturated Macroscopic Flow in Heterogeneous Porous Medium: Evaluating Darcy’s Law. *Hydrology* **2020**, *7*, 4.
https://doi.org/10.3390/hydrology7010004

**AMA Style**

Nelson RW, Williams GP.
Mathematical Treatment of Saturated Macroscopic Flow in Heterogeneous Porous Medium: Evaluating Darcy’s Law. *Hydrology*. 2020; 7(1):4.
https://doi.org/10.3390/hydrology7010004

**Chicago/Turabian Style**

Nelson, R. William, and Gustavious P. Williams.
2020. "Mathematical Treatment of Saturated Macroscopic Flow in Heterogeneous Porous Medium: Evaluating Darcy’s Law" *Hydrology* 7, no. 1: 4.
https://doi.org/10.3390/hydrology7010004