Skip to main content

Average Permeability for Linear Flow in Layered Beds

Average permeability for linear flow in layered beds is used in reservoir engineering, as it governs how fluids (oil, gas, or water) flow through porous media.  In layered reservoirs, where beds of different permeabilities are stacked, the average permeability depends on the flow geometry, specifically whether the flow is linear (parallel or perpendicular to the layering) and how the layers are arranged (in series or parallel).
 
Key Points about Radial Systems
 
Reservoirs  -  Flow may not be purely parallel or perpendicular, requiring more complex models (geometric averaging or numerical simulation).
Permeability  -  Usually measured in core samples or derived from well tests and can vary widely depending on the rock type (sandstone, carbonate, etc.).

  

Average Permeability for Linear Flow in Layered Beds Formula
\( k_{avg} \;=\;  \dfrac{  k_1 \cdot A_1 + k_2 \cdot A_2 + k_3 \cdot A_3    }{   A_1 + A_2 + A_3    }\)
Symbol English Metric
\( k_{avg} \) = Average Permeability for Linear Flow in Layered Beds \(mD\) -
\( k_1 \) = Permeability for Layer 1 \(mD\) -
\( A_1 \) = Area of Layer 1 \(ft^2\) -
\( k_2 \) = Permeability for Layer 2 \(mD\) -
\( A_2 \) = Area of Layer 2 \(ft^2\) -
\( k_3 \) = Permeability for Layer 3 \(mD\) -
\( A_3 \) = Area of Layer 3 \(ft^2\) -

       

Linear Flow Parallel to Layers (Arithmetric Average) Formula

  • When fluid flows parallel to the bedding planes (ialong the length of the layers), the layers act as if they are in parallel.  Each layer contributes to the total flow rate proportional to its permeability and thickness.
  • Arithmetric typically yields a higher average permeability because it emphasizes high-permeability zones.
  • The flow rate is additive across layers, and high-permeability layers dominate the total flow.
  • This is analogous to resistors in parallel in an electrical circuit, but permeability is a conductance like property, so the arithmetic average applies.
\( k_{avg} \;=\;  \dfrac{  \sum ( k_i \cdot h_i )    }{   \sum h_i    }\)
Symbol English Metric
\( k_{avg} \) = Average Permeability for Linear Flow in Layered Beds \(mD\) -
\( k_i \) = Permeability of the i-th Layer \(mD\) -
\( h_i \) = Thickness of the i-th Layer \(ft\) -
\( \sum h_i \) = Total Thickness of all Layers Combined \(ft\) -

   

Linear Flow Perpendicular to Layers (Harmonic Average) Formula

  • When fluid flows perpendicular to the bedding planes (through the layers stacked vertically), the layers act as if they are in series.  The total flow resistance is the sum of the resistances of each layer, and the average permeability is calculated as the thickness-weighted harmonic mean.
  • Harmonic yields a lower average permeability because it is limited by low-permeability zones.
  • The flow must pass through each layer sequentially, so low-permeability layers disproportionately restrict the overall flow.
  • This is analogous to resistors in series, where the inverse of permeability (resistance) is summed.
\( k_{avg} \;=\;    \dfrac{   \sum h_i   }{   \sum \left(  \dfrac{ k_i  }{  h_i }  \right)   }\)
Symbol English Metric
\( k_{avg} \) = Average Permeability for Linear Flow in Layered Beds \(mD\) -
\( \sum h_i \) = Total Thickness of all Layers Combined \(ft\) -
\( k_i \) = Permeability of the i-th Layer \(mD\) -
\( h_i \) = Thickness of the i-th Layer \(ft\) -

 

Piping Designer Logo 1