The general form of non-linear single-phase pressure diffusion @model with the finite number of sources/sinks wells is given by:
| LaTeX Math Block |
|---|
| \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) |
| | LaTeX Math Block |
|---|
| {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g}) |
| | LaTeX Math Block |
|---|
| anchor | qGamma |
|---|
| alignment | left |
|---|
| {\rm F}_{\Gamma}(p, {\bf u}) = 0 |
|
where
| q_k(t) | sandface flowrates of the -th sourceM = / \muphase mobilityformation permeability to a givenk | \mu | | | reservoir boundary |
| LaTeX Math Inline |
|---|
| body | --uriencoded--%7B\rm F%7D_%7B\Gamma%7D(p, %7B\bf u%7D) |
|---|
|
| |
dynamic viscosity of a given fluid
The alternative form is to write down equations
| LaTeX Math Block Reference |
|---|
|
and | LaTeX Math Block Reference |
|---|
|
in reservoir volume outside wellbore and match the solution to the fluid flux through the well-reservoir contact:
...
...
...
...
...
( {\bf u} \, \nabla p)
= 0 |
| | LaTeX Math Block |
|---|
| {\bf u} = - M \cdot ( \nabla |
|
...
...
...
...
|
| LaTeX Math Block |
|---|
| \int_{\Sigma_k} \, {\bf |
|
...
...
...
...
...
| | LaTeX Math Block |
|---|
| anchor | qGamma |
|---|
| alignment | left |
|---|
| {\rm F}_{\Gamma}(p, {\bf |
|
...
...
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 | ...
...
| borderColor | wheat |
|---|
| bgColor | mintcream |
|---|
| borderWidth | 7 |
|---|
...
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.
...
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.
...
Fractured vertical well
...
...
Multifrac horizontal well
...
See also
...
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 ]
