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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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| anchor | pre_filnal |
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| alignment | left |
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| \rho \, \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla \, ( \rho \, {\bf u}) = \sum_k \rho \, \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
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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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The left-hand side of equation
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can be transformed in the following wayThen use the following equality:
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| anchor | nabla_rho |
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| alignment | left |
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\nabla \, ( \rho \, {\bf u}) = \rho \, \nabla \, {\bf u} + (\nabla \rho, \, {\bf u}) = \rho \, \nabla \, {\bf u} + \frac{d\rho}{dp} \cdot (\nabla p, \, {\bf u}) = \rho \, \nabla \, {\bf u} + \rho \, c \cdot (\nabla p, \, {\bf u}) |
where
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| body | --uriencoded--\displaystyle c(p) = \frac%7B1%7D%7B\rho%7D \frac%7Bd\rho%7D%7Bdp%7D |
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is
fluid compressibility.
The right-hand side of equation
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can be transformed in the following way:| LaTeX Math Block |
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| anchor | right_hand_side |
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| alignment | left |
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\sum_k \rho(p(t, {\bf r})) \, \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)
= \sum_k \rho(p(t, {\bf r}_k)) \, \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)
= \sum_k \rho(p(t, {\bf r})) \, \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)
= \rho(p) \, \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
Substituting
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and | LaTeX Math Block Reference |
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into
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and reducing the density
one arrives to:
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