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 |
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- -+ 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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| {\rm F}_{\Gamma}(p, {\bf u}) = 0 |
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where
q_k(t) | volumetric flowrate of the LaTeX Math Inline |
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body | k | -th sourceM = / \muphase mobilityformation permeability to a givenk | \mu | dynamic viscosity of a given fluid
The alternative form is to write down equations
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and LaTeX Math Block Reference |
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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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( {\bf u} \, \nabla p)
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| {\bf u} = - M \cdot ( \nabla |
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| \int_{\Sigma_k} \, {\bf |
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| {\rm F}_{\Gamma}(p, {\bf |
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where
--uriencoded--\displaystyle \Psi(p) =2 \, \int_0%5ep \frac%7Bp \, dp%7D%7B\mu(p) \, Z(p)%7D | Pseudo-Pressure\mu(p) | dynamic fluid viscosityZ(p) | fluid compressibility factor | ...
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Physical models of pressure diffusion can be split into two categories: Newtonian and Rheological (non-Newtonian) based on the fluid stress model.
Mathematical models of pressure diffusion can be split into three categories: Linear, Pseudo-linearLinear and Non-linear.
These models are built using Numerical, Analytical or Hybrid pressure diffusion solvers.
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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 ]