We start with reservoir pressure diffusion outside wellbore:
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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 m_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
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| \int_{\Sigma_k} \, {\bf u} \, d {\bf A} = q_k(t) |
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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 rate at -th well | LaTeX Math Inline |
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| body | m_k(t) = \rho(p) \cdot q_k(t) |
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Then use the following equality:
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| anchor | rhophi |
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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 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
We start with
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| anchor | PZ |
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| page | Single-phase pressure diffusion @model |
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:
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| \phi \cdot c_t \cdot \partial_t p + \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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| {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g}) |
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and neglect the non-linear term | LaTeX Math Inline |
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| body | --uriencoded--c \cdot ( %7B\bf u%7D \, \nabla p) |
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for low compressibility fluid or equivalently to a constant fluid density: | LaTeX Math Inline |
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| body | \rho(p) = \rho = \rm const |
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Together with constant pore compressibility this will lead to constant total compressibility | LaTeX Math Inline |
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| body | c_t = c_r + c \approx \rm const |
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Assuming that permeability and fluid viscosity do not depend on pressure and | LaTeX Math Inline |
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| body | \mu(p) = \mu = \rm const |
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one arrives to the differential equation with constant coefficients:
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| \phi \, c_t \cdot \partial_t p + \nabla {\bf u}
= \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
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| {\bf u} = - \frac{k}{\mu} \cdot ( \nabla p - \rho \, {\bf g}) |
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See also
Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model / Single-phase pressure diffusion @model
