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We start with
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| anchor | PZ |
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| page | Single-phase pressure diffusion @model |
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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}) |
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}) |
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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