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We start with reservoir pressure diffusion outside wellbore:

\frac{\partial (\rho \phi)}{\partial t} + \nabla \, ( \rho \, {\bf u}) = \sum_k m_k(t) \cdot \delta({\bf r}-{\bf r}_k)

\int_{\Sigma_k} \, \rho(p) \, {\bf u} \,  d {\bf A} = m_k(t)

where

	well-reservoir contact of the -th well
	normal vector of differential area on the well-reservoir contact, pointing inside wellbore
	mass rate at -th well
	sandface flowrate at -th well
	fluid density as function of reservoir fluid pressure

Then use the following equality:

d(\rho \, \phi) = \rho \, d \phi + \phi \, d\rho = \rho \, \phi \, \left( \frac{d \phi }{\phi} +  \frac{d \rho }{\rho}  \right) 
= \rho \, \phi \, \left( \frac{1}{\phi} \frac{d \phi}{dp} \, dp +  \frac{1}{\rho} \frac{d \rho}{dp} \, dp  \right) 
= \rho \, \phi \, (c_{\phi} \, dp + c \, dp) = \rho \, \phi \, c_t \, dp

to arrive at:

\rho \, \phi \, c_t  \cdot \frac{\partial p}{\partial t} + \nabla \, ( \rho \, {\bf u}) = 0

\int_{\Sigma_k} \, {\bf u} \,  d {\bf A} = q_k(t)

where

total compressibility

We start with :

\phi \cdot c_t \cdot \partial_t p + \nabla  {\bf u}  
+ c \cdot ( {\bf u} \, \nabla p)
= \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)

{\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g})

and neglect the non-linear term for low compressibility fluid or equivalently to a constant fluid density: .

Together with constant pore compressibility this will lead to constant total compressibility .

Assuming that permeability and fluid viscosity do not depend on pressure and one arrives to the differential equation with constant coefficients:

\phi \, c_t \cdot \partial_t p + \nabla  {\bf u}  
= \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)

{\bf u} = - \frac{k}{\mu} \cdot ( \nabla p - \rho \, {\bf g})