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The general form of non-linear single-phase
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Pseudo-linear single-phase diffusion model
Non-linear single-phase diffusion models
Linear Rheological single-phase diffusion models
Multi-layer diffusion model
Anisotropic diffusion model
Dual permeability diffusion model
Dual porosity diffusion model
Multifrac horizontal well diffusion model
Horizontal well diffusion model
Hydraulic frac vertical well diffusion model
Slanted well model
Limited entry well model
Wellbore storage models
Diffusion boundary models
General form of 1DR pressure diffusion of low-compressibility fluid
Skin-factor
Line Source Solution (LSS) (model)
1DL low-compressibility diffusion in infinite homogeneous reservoir (model)
Low-compressibility newtonian single-phase diffusion (model)
pressure diffusion @model with the finite number of 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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anchor | qGamma |
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alignment | left |
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| {\rm F}_{\Gamma}(p, {\bf u}) = 0 |
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where
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: LaTeX Math Block |
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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) |
| LaTeX Math Block |
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anchor | qGamma |
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alignment | left |
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| {\rm F}_{\Gamma}(p, {\bf u}) = 0 |
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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).
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 ]
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