The general form of non-linear single-phase pressure diffusion @model is given by:
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anchor | PZ_singlephase |
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alignment | left |
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\phi \cdot c_t \cdot \partial_t p - \nabla \left( M \cdot ( \nabla p - \rho \cdot \mathbf{g} ) \right) - c \cdot M \cdot (\nabla p)^2 = \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k) |
where
| time |
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body | --uriencoded--%7B\rm r %7D |
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| position vector |
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body | --uriencoded--%7B\rm r %7D_k |
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| position vector of the -th source |
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body | --uriencoded--p(t, %7B\bf r%7D) |
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| reservoir pressure |
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body | --uriencoded--\phi(%7B\bf r%7D) |
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| reservoir porosity |
| total compressibility |
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body | --uriencoded--%7B \bf g %7D |
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| gravity vector |
| Dirac function |
The alternative form is:
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\phi \cdot c_t \cdot \mu \cdot \partial_t \Psi -
\nabla \cdot \left( k \cdot \Big( \vec \nabla \Psi - \frac{\rho^2}{\mu} \, {\bf g} \Big) \right)
= \sum_k q({\bf r}) \cdot \delta({\bf r}-{\bf r}_k) |
where
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-linear and Non-linear.
These models are built using Numerical, Analytical or Hybrid pressure diffusion solvers.
Many popular 1DR solutions can be approximated by Radial Flow Pressure Diffusion @model which has a big methodological value.
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.
See also
Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model