The general form of non-linear single-phase pressure diffusion @model with the finite number of sources/sinks wells is given by:
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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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| {\intrm F}_{\Gamma} \(p, {\bf u} \, d {\bf \Sigma} = q_\Gamma(t)) = 0 |
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where
source| q_\Gamma(t) | | --uriencoded--%7B\rm F%7D_%7B\Gamma%7D(p, %7B\bf u%7D) |
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flow through is
The alternative form is to write down equations
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and
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in reservoir volume outside wellbore and match the solution to the fluid flux through the
well-reservoir contact:
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| \phi \cdot c_t \cdot \partial_t p + \nabla {\bf u}
+ c \cdot ( {\bf u} \, \nabla p)
= 0 |
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| {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g}) |
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| \int_{\Sigma_k} \, {\bf u} \, d {\bf \Sigma} = q_k(t) |
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| anchor | qGamma |
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| alignment | left |
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| {\ |
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where
Physical models of pressure diffusion can be split into two categories: Newtonian and Rheological (non-Newtonian) based on the fluid stress model.
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The simplest analytical solutions for pressure diffusion are given by 1DL Linear-Drive Solution (LDS) and 1DR Line Source Solution (LSS)The table below shows a list of popular well and reservoir pressure diffusion models.
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Fractured vertical well
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Multifrac horizontal well
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See also
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Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model
[ Aquifer Drive Models ] [ Gas Cap Drive Models ]
[ Linear single-phase pressure diffusion @model ]
