The general form of non-linear single-phase pressure diffusion @model with the finite number of 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
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: LaTeX Math Block |
---|
| \phi \cdot c_t \cdot \partial_t p + \nabla {\bf u}
+ c \cdot ( {\bf u} \, \nabla p)
= 0 |
| LaTeX Math Block |
---|
| {\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g}) |
|
LaTeX Math Block |
---|
| \int_{\Sigma_k} \, {\bf u} \, d {\bf \Sigma} = q_k(t) |
| LaTeX Math Block |
---|
anchor | qGamma |
---|
alignment | left |
---|
| {\rm F}_{\Gamma}(p, {\bf u}) = 0 |
|
where
Physical models of Physical models pf pressure diffusion can be split into two categories: Newtonian and Rheological (non-Newtonian) based on the fluid stress model.
Mathematical mdoels models of pressure diffusion can be split into three categories: Linear, Pseudo-linearLinear and Non-linear.
Solvers for the above problems can be These models are built using Numerical, Analytical or Hybrid pressure diffusion solvers.
Many popular 1DR solutions can be approximated by universal equation of 1DR pressure diffusionRadial Flow Pressure Diffusion @model which has a big methodological value.
The simplest analytical solutions for pressure diffusion is captured 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
...
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 ]
...