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We start with
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| anchor | rho_dif |
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| page | Derivation of Single-phase Linear pressure diffusion @model |
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outside wellbore:
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and use
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| anchor | din_term |
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| page | Derivation of Single-phase 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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...
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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 p} \, d {\bf A} = q_k(t) |
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...
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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} \, 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} \, 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 |
In some practical cases the complex
can be considered as constant in time which makes
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a linear differential equation.
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