We start with reservoir pressure diffusion outside wellborethe reservoir flow continuity equation:
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| anchor | rho_dif |
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| alignment | left |
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| \frac{\partial (\rho \phi)}{\partial t} + \nabla \, ( \rho \, {\bf u}) = \sum_k \dot m_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
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percolation model:
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| {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g}) |
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and the reservoir boundary flow condition:
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| anchor | qGamma |
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| alignment | left |
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| {\rm F}_{\Gamma}(p, {\bf u}) = 0 |
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where
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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 |
| mass flowrate at -th well | LaTeX Math Inline |
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| body | \dot m_k(t) = \rho(p) \cdot q_k(t) |
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