We start with reservoir pressure diffusion outside wellborethe reservoir flow continuity equation:
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| anchor | rho_dif |
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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:
| qk | \int_ | { | Gamma}} \,bf u} = - M \cdot ( \nabla p - \rho |
| (p) | u} \, d |
and the reservoir boundary flow condition:
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| anchor | qGamma |
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| {\rm F}_{\Gamma}(p, {\bf |
| A | m_{\Gamma}(t) |
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 rate 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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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 |
where
to arrive at:
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| anchor | pre_filnal |
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| \rho \, \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla \, ( \rho \, {\bf u}) = \sum_k \rho \, |
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\sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
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qkint{} \ \, d {\bf A} = q_{\Gamma}(t)where
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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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\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.
By using the Dirac delta function property:
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| body | f(x) \cdot \delta(x-x_0) = f(x_0) \cdot \delta(x-x_0) |
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the right-hand side of equation | LaTeX Math Block Reference |
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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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\sum_k \rho(p(t, {\bf r})) \cdot q_k(t) \cdot \delta({\bf r}-{\bf r}_k)
= \sum_k \rho(p(t, {\bf r}_k)) \cdot q_k(t) \cdot \delta({\bf r}-{\bf r}_k)
= \sum_k \rho(p(t, {\bf r})) \cdot q_k(t) \cdot \delta({\bf r}-{\bf r}_k)
= \rho(p) \cdot \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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| anchor | pre_filnal |
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| \phi \, c_t \cdot \frac{\partial p}{\partial t} + \nabla {\bf u}
+ c \cdot ( {\bf u} \, \nabla p) = \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
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qkint{} \ \, d {\bf A} = q_{\Gamma}(t)
See also
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Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / Single-phase pressure diffusion @model
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