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Then use the following equality:
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| anchor | rhophi |
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| alignment | left |
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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 |
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| \rho \, \phi \, c_t \cdot \frac{\partial (p)}{\partial t} + \nabla \, ( \rho \, {\bf u}) = 0 |
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| \int_{\Sigma_k} \, {\bf u} \, d {\bf A} = q_k(t) |
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where
Let's assume Darcy flow with constant permeability
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| body | --uriencoded--\displaystyle \frac%7Bdk%7D%7Bdp%7D = 0 |
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and ignore gravity forces:
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\tau(t) = \int_0^t \frac{dt}{\mu (p_{BHP}(t )) \, c_t (p_{BHP}) } \, , \ \ p_{BHP} = p_{BHP}(t))}
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to correct early-time transient behaviour which turn equation
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into:
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