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 |
m_k(t) | mass rate at k-th well m_k(t) = \rho(p) \cdot q_k(t) |
q_k(t) | sandface flowrate at k-th well |
\rho(p) | fluid density as function of reservoir fluid pressure p |
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 |
Then use the following equality:
(6) | \nabla \, ( \rho \, {\bf u}) = ( \rho \, \nabla \, {\bf u}) + (\nabla \, \rho, {\bf u}) = |
See also
Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / Single-phase pressure diffusion @model