\phi \cdot \partial_\tau \Psifrac{\partial \Psi}{\partial \tau}

 -   
 \nabla \cdot \left( k \cdot   \vec \nabla \Psi   \right) 
  = 0

-\frac{k}{\mu} \, \int_{\Sigma} \, \nabla p \, d {\bf A\Sigma} = q(t)

where

...

In some practical cases the complex 

\tau(t) = \frac{t}{\mu \, c_t}

and one can write the diffusion equation as:

\phi \, c_t \, \mu \cdot \frac{\partial \Psi}{\partial \tau} -   
 \nabla \cdot \left( k \cdot   \vec \nabla \Psi   \right) 
  = 0

which is a treat it as a differential equation with linear coefficients.

But during the early transition times the pressure drop is usually high and the complex 

In this case one can use Pseudo-Time, calculated by means of the bottom-hole pressure

\tau(t) = \int_0^t \frac{dt}{\mu(p_{BHP}(t)) \, c_t(p_{BHP}(t))}

to correct early-time transient  behaviour.

In case of the ideal gas equation of state, the  Z-factor has a unit value:

\Psi(p) = \frac{p^2}{\mu}

\tau(t) = \frac{1}{\mu} \int_0^t p_{BHP}(t) dt

See also

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Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model

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