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We start with the general from of :

\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})

\int_{\Gamma} \, {\bf u} \,  d {\bf \Sigma} = q_\Gamma(t)

where

	reservoir pressure		time
	fluid density		position vector
	effective porosity		position vector of the -th source
	total compressibility		Dirac delta function
	reservoir fluid mobility		gradient operator
	formation permeability to a given fluid		gravity vector
	dynamic viscosity of a given fluid		fluid velocity under Darcy flow
	sandface flowrates of the -th source		reservoir boundary
	flow through the reservoir boundary , which is aquifer or gas cap

Let's neglect the non-linear term for low compressibility fluid which is equivalent to assumption of nearly 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 \cdot c_t \cdot \partial_t p + \nabla  {\bf u}  
= \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)

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

\int_{\Gamma} \, {\bf u} \,  d {\bf \Sigma} = q_\Gamma(t)

or

\phi \cdot c_t \cdot \partial_t p + \nabla  {\bf u}  = 0

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

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

\int_{\Gamma} \, {\bf u} \,  d {\bf \Sigma} = q_\Gamma(t)