Average Permeability of Flow Units in Parallel and in Series (Linear Flow)

Series Linear Flow-Harmonic Average

Consider linear flow for flow units in series as shown in the diagram below:

Flow going through flow units in series can be treated as resistors in series, where flow rate ( $q$ ) represents current ( $I$ ), pressure ( $P$ ) represents voltage ( $V$ ), and flow resistance ( $\frac{L}{kA}$ ) represents resistance ( $R_e$ ). For resistors in series, the following rules apply:

The current is the same across each resistor.
The sum of the potential differences across individual resistors is equal to the total voltage drop across all resistors.

This analogy can be extended to fluid flow, i.e.:

flow rate is the same across each flow unit for flow units in series.
the sum of the pressure drops across individual flow units is equal to the total pressure drop across all flow units

Knowing these rules, we will proceed to the derivation of average permeability for this system. Using Rule 2 the following expression can be derived:

(1) $\begin{equation*}\ \triangle P_t =\sum_{i=1}^{3}\triangle P_i= \triangle P_1 + \triangle P_2 + \triangle P_3 \end{equation*}$

Using Darcy’s law, the total pressure drop ( $\triangle P_t = P_i-P_F$ ) across the entire system is given by the following expression:

(2) $\begin{equation*} \triangle P_t =\frac{qu}{A}\frac{L}{k_{avg}} \end{equation*}$

Using Darcy’s law, the total pressure drop across a flow unit ( $\triangle P_i$ ) is given by the following expression:

(3) $\begin{equation*} \triangle P_i =\frac{qu}{A}\frac{L_i}{k_i} \end{equation*}$

Substituting Equations (2) and (3) into Equation (1) we can develop the following expression:

(4) $\begin{equation*} \frac{qu}{A}\frac{L}{k_{avg}} =\frac{qu}{A}\frac{L_1}{k_1} + \frac{qu}{A}\frac{L_2}{k_2} + \frac{qu}{A}\frac{L_3}{k_3} \end{equation*}$

Using Rule 1 and realizing the cross sectional area of each flow unit is identical, Equation (4) can be reduced to the following in terms of the average permeability:

(5) $\begin{equation*} {k_{avg} =\frac{L}{\frac{L_1}{k_1} + \frac{L_2}{k_2} + \frac{L_3}{k_3}} \end{equation*}$

A more general equation that describes the average permeability of $n$ flow units arranged in series is the following:

$\boxed{k_{avg} = \frac{\sum_{i=1}^{n}L_i}{\sum_{i=1}^{n}\frac{L_i}{k_i}}}$

Parallel Linear Flow-Weighted Average

Consider linear flow for flow units in parallel as shown in the diagram below:

Again we apply the same analogy. For resistors in parallel, the following rules apply:

The voltage drop is the same across each resistor.
The sum of the current across individual resistors is equal to the total current across all resistors.

This analogy can be extended to fluid flow, i.e.:

pressure drop is the same across each flow unit for flow units in parallel.
the sum of the flow rates across individual flow units is equal to the total flow rate across all flow units when they are in parallel.

Knowing these rules, we will proceed to the derivation of average permeability for this system. Using Rule 1 the following expression can be derived:

(6) $\begin{equation*}\ q_t =\sum_{i=1}^{3}q_i= q_1 + q_2 + q_3 \end{equation*}$

Using Darcy’s law, the total flow rate (q_t) across the entire system is given by the following expression:

(7) $\begin{equation*} q_t =\frac{k_{avg}wh_T}{u}\frac{\triangle P}{L} \end{equation*}$

Using Darcy’s law, the total flow rate across a flow unit ( $q_i$ ) is given by the following expression:

(8) $\begin{equation*} q_i =\frac{k_{i}wh_i}{u}\frac{\triangle P}{L} \end{equation*}$

Substituting Equations (7) and (8) into Equation (6) we can develop the following expression:

(9) $\begin{equation*} \frac{k_{avg}wh_T}{u}\frac{\triangle P}{L} =\frac{k_{1}wh_1}{u}\frac{\triangle P}{L} + \frac{k_{2}wh_2}{u}\frac{\triangle P}{L} + \frac{k_{3}wh_3}{u}\frac{\triangle P}{L} \end{equation*}$

Using Rule 1 and assuming the width of each flow unit is identical, Equation (9) can be reduced to the following in terms of the average permeability:

(10) $\begin{equation*} k_{avg} =\frac{k_{1}h_{1} + k_{2}h_{2} + k_{3}h_{3}}{h_T} \end{equation*}$

A more general equation that describes the average permeability of $n$ flow units arranged in parallel is the following:

$\boxed{k_{avg} = \frac{\sum_{i=1}^{n}k_{i}h_{i}}{h_T}}$