The general form of non-linear single-phase pressure diffusion @model with the finite number of sources/sinks is given by:
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where
p(t, {\bf r}) | reservoir pressure | t | time |
\rho({\bf r},p) | fluid density | {\bf r } | position vector |
\phi({\bf r}, p) | effective porosity | {\bf r }_k | position vector of the k-th source |
c_t({\bf r},p) | total compressibility | \delta ( \bf r ) | Dirac delta function |
q_k(t) | volumetric flowrate of the k-th source | \nabla | gradient operator |
M = k / \mu | phase mobility | { \bf g } | gravity vector |
k | formation permeability to a given fluid | ||
\mu | dynamic viscosity of a given fluid |
The alternative form is:
(3) | \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
\displaystyle \Psi(p) =2 \, \int_0^p \frac{p \, dp}{\mu(p) \, Z(p)} | Pseudo-Pressure |
\mu(p) | dynamic fluid viscosity |
Z(p) | fluid compressibility factor |
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.
Wellbore storage model | Well model | Reservoir model | Boundary model |
---|---|---|---|
Constant | Skin-factor | Homogeneous | Infinite |
Fair | Vertical well | Dual-porosity | Circle No Flow |
Rate-dependant | Dual-permeability | Circle Constant Pi | |
Limited entry well | Anisotropic reservoir | Single fault | |
Horizontal well | Multi-layer reservoir | Parallel faults | |
Slanted well | Linear-composite | Intersecting Faults | |
Radial-composite |
See also
Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Dynamics / Pressure Diffusion / Pressure Diffusion @model