The general form of non-linear single-phase pressure diffusion model is given by:
\beta({\bf r},p) \, \frac{\partial p}{\partial t} = \nabla \Big( M({\bf r},p, \nabla p) \cdot \nabla p \Big) |
with non-linear dependence of fluid mobility on reservoir pressure and spatial pressure gradient :
M = k_a({\bf r}) \, M_r(p, \nabla p) |
and non-linear dependence of compressivity and compressibility on reservoir pressure :
\beta = c_t({\bf r},p) \cdot \phi({\bf r},p) |
c_t({\bf r},p) = c_r({\bf r},p) + \sum_\alpha s_\alpha({\bf r}) c_\alpha(p) |
where
Fluid mobility as function of reservoir pressure and spatial pressure gradient | |
Relative mobility as function of reservoir pressure and spatial pressure gradient | |
Compressivity as function of reservoir pressure | |
Total compressibility as function of reservoir pressure and location | |
Rock compressibility as function of reservoir pressure and location | |
-phase compressibility as function of reservoir pressure for | |
Effective porosity as function of reservoir pressure and location | |
Absolute formation permeability at initial formation pressure as function of location | |
Some function of reservoir pressure and spatial pressure gradient with the following asymptotic behaviour: |
The same account for non-linearity can be applied for non-linear multi-phase pressure diffusion when Pressure Diffusion Model Validity Scope is met and multi-phase pressure dynamics can be modeled as effective single-phase pressure dynamics.
Below is the list of popular physical phenomena and their mathematical models which can be covered by model.
Pressure diffusion equation is going to be:
c_t(p) \, \phi({\bf r}) \, \frac{\partial p}{\partial t} = \nabla (M(p) \nabla p ) |
where
| Total compressibility as function of reservoir pressure and location | |
---|---|---|
Rock compressibility as function of location | ||
-phase compressibility as function of reservoir pressure for | ||
Fluid mobility as function of reservoir pressure | ||
Formation permeability as function of location | ||
Dynamic fluid viscosity as function of reservoir pressure | ||
Total compressibility as function of reservoir pressure | ||
Formation Volume Factor as function of reservoir pressure |
Pressure diffusion / Pressure Diffusion @model / Single-phase pressure diffusion model / Non-linear single-phase pressure diffusion @model