We start with reservoir pressure diffusion outside wellbore:
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where
\Sigma_k | well-reservoir contact of the k-th well |
d {\bf \Sigma} | normal vector of differential area on the well-reservoir contact, pointing inside wellbore |
Then use the following equality:
(3) | 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:
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where
c_t = с_\phi+ c |
We start with (Single-phase pressure diffusion @model:1):
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and neglect the non-linear term c \cdot ( {\bf u} \, \nabla p) for low compressibility fluid c \sim 0 or equivalently to a constant fluid density: \rho(p) = \rho = \rm const.
Together with constant pore compressibility c_r = \rm const this will lead to constant total compressibility c_t = c_r + c \approx \rm const.
Assuming that permeability and fluid viscosity do not depend on pressure k(p) = k = \rm const and \mu(p) = \mu = \rm const one arrives to the differential equation with constant coefficients:
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See also
Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / Single-phase pressure diffusion @model