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We start with outside wellbore:

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

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

where

	well-reservoir contact of the -th well
	normal vector of differential area on the well-reservoir contact, pointing inside wellbore

and use to arrive at:

\rho \, 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

Let's assume Darcy flow and ignore gravity forces:

 {\bf u} = \frac{k}{\mu} \nabla \, p

so that diffusion equation becomes:

\rho \, c_t  \cdot \frac{\partial p}{\partial t} + \nabla \, ( k \cdot \frac{\rho}{\mu} \, \nabla  \, p) = 0

\frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf p} \,  d {\bf A} = q_k(t)

Let's express the density via Z-factor:

\rho = \frac{M}{RT} \, \frac{p}{Z(p)}

where

and assuming the fluid temperature does not change over time and space during the modelling period:

\rho \, c_t \, \mu  \cdot \frac{p}{\mu \, Z} \cdot \frac{\partial p}{\partial t} + \nabla \, ( k \cdot \frac{p}{\mu \, Z} \, \nabla  \, p) = 0

\frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf p} \,  d {\bf A} = q_k(t)