A cube of sandstone, 3 cm on each side, is made up of three layers that are each 1 cm thick and have permeability values of 1, 5 and 10 darcys.
For the linear flow of an incompressible fluid (μ = 1 cp) with a potential difference of 1 atm, determine the flow rates and apparent permeability for both the series and parallel flow directions.

The answers to the questions are;

Series flow

Flow rate =   cm³/s

Apparent permeability = 30/13 Darcy.

Parallel flow

Flow rates for each layer

q₁ =3 cm²

q₂ = 5 cm³/s

q₃ =  10 cm³/s

Total flow rate = q = 16  cm³/s  

Apparent permeability = 16/3 Darcy.

Explanation:

Here we have the flow rate given by

q =

Where:

q = Flow rate

K = Permeability

dP = Potential difference = 1 atm

μ = Viscosity

Series flow

 We note that the average permeability is given by

k' = =  = 30/13 Darcy

P₁ - P₂ =

Therefore, q = ,

here  A = 3 cm × 3 cm = 9 cm²

Hence q = = cm³/s.

Parallel flow  

The apparent permeability k' = ∑(kh)/∑h = (1 + 5 + 10)/(1 + 1 + 1) = 16/3 Darcy

q =

A = W × h  = 3 cm²

∴ we have q = = = 16 cm³/s.

Aliter

Therefore for each of the three layers, we have

q₁ =  A₁ =3 cm²

q₁ = = cm³/s     = 1 cm³/s

For q₂, we have

q₂ = =  5 cm³/s   and q₃ =   = 10 cm³/s

Therefore we have the total flow rate ∑q = q₁ + q₂ + q₃

=  1 cm³/s +5 cm³/s +10 cm³/s =  16  cm³/s.

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