The general form of the Linear single-phase pressure diffusion @model with the finite number of sources/sinks is given by:
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| \phi \cdot 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} = - M \cdot ( \nabla p - \rho \, {\bf g}) |
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alignment | left |
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| \int_{\Gamma} \, {\bf u} \, d {\bf \Sigma} = q_\Gamma(t) |
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or |
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| \phi \cdot c_t \cdot \partial_t p + \nabla {\bf u} = 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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| \int_{\Gamma} \, {\bf u} \, d {\bf \Sigma} = q_\Gamma(t) |
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or |
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| \phi \cdot c_t \cdot \partial_t p =
\nabla \big( M \cdot ( \nabla p - \rho \, {\bf g}) \big) |
| LaTeX Math Block |
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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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| \int_{\Gamma} \, {\bf u} \, d {\bf \Sigma} = q_\Gamma(t) |
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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.
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 / Single-phase pressure diffusion @model