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| body | --uriencoded--d %7B\bf \Sigma%7D |
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| normal vector of differential area on the well-reservoir contact, pointing inside wellbore |
and use Then use the following equality:
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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 |
to
| page | Derivation of Linear pressure diffusion @model |
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to arrive at:
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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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| \rho \, \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla \, ( k \cdot \frac{\rho}{\mu} \, \nabla \, p) = 0 |
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| \frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf \nabla p} \, p \cdot d {\bf A} = q_k(t) |
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Let's express the density via Z-factor:
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| \phi \, c_t \, \mu \cdot \frac{p}{\mu \, Z} \cdot \frac{\partial p}{\partial t} + \nabla \, ( k \cdot \frac{p}{\mu \, Z} \, \nabla \, p) = 0 |
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| \frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf p\nabla } \, p \cdot d {\bf A} = q_k(t) |
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or
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| \phi \, c_t \, \mu \cdot \frac{\partial \Psi}{\partial t} + \nabla \, ( k \cdot \nabla \, \Psi) = 0 |
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| \frac{k}{\mu} \cdot \int_{\Sigma_k} \, {\bf p\nabla } \, p \cdot d {\bf A} = q_k(t) |
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where
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| body | --uriencoded--\displaystyle \Psi(p) =2 \, \int_0%5ep \frac%7Bp \, dp%7D%7B\mu(p) \, Z(p)%7D |
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| Pseudo-Pressure |
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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:| LaTeX Math Block |
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\phi \cdot \frac{\partial \Psi}{\partial \tau} + \nabla \, ( k \cdot \nabla \, \Psi) = 0 |
See also
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Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / Pseudo-linear pressure diffusion @model
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