The general form of non-linear single-phase pressure diffusion model is given by:
| (1) | \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 M on reservoir pressure p and spatial pressure gradient \nabla p:
| (2) | M = k_{air}({\bf r}) \, M_r(p, \nabla p) |
and non-linear dependence of compressivity \beta and compressibility c_t on reservoir pressure p :
| (3) | \beta = c_t({\bf r},p) \cdot \phi({\bf r},p) |
| (4) | c_t({\bf r},p) = c_r({\bf r},p) + \sum_\alpha s_\alpha({\bf r}) c_\alpha(p) |
where
M(p, \nabla p) | Fluid mobility as function of reservoir pressure p and spatial pressure gradient \nabla p |
|---|---|
M_r(p, \nabla p) | Relative mobility as function of reservoir pressure p and spatial pressure gradient \nabla p |
\beta(p) | Compressivity as function of reservoir pressure p |
c_t({\bf r},p) | Total compressibility as function of reservoir pressure p and location \bf r |
c_r({\bf r},p) | Rock compressibility as function of reservoir pressure p and location \bf r |
c_\alpha(p) | \alpha-phase compressibility as function of reservoir pressure p for \alpha = \{ w, \, o, \, g \} |
s_\alpha({\bf r}) | \alpha-phase reservoir saturation for \alpha = \{ w, \, o, \, g \} |
\phi_e({\bf r}, p) | Effective porosity as function of reservoir pressure p and location \bf r |
k_{air}({\bf r}) | Formation permeability at initial formation pressure p_0 as function of location \bf r |
\mu(p_0) | |
\xi (p, |\nabla p|) | Some function of reservoir pressure p and spatial pressure gradient \nabla p with the following asymptotic behaviour: \xi (p \rightarrow p_0, |\nabla p| \rightarrow 0) \rightarrow 1 |
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 (1) model.
Forcheimer
Pressure diffusion equation is going to be:
| \с_t \phi_e \frac{\partial p}{\partial t} = \nabla ( \frac{k(\nabla p)}{\mu} \nabla p) |
where
k(\nabla p) | Dynamic fluid viscosity as function of reservoir pressure p |
|---|---|
k(p) | Formation permeability as function of reservoir pressure p |
c_f(p) | Total compressibility as function of reservoir pressure p |
See also
Pressure diffusion / Pressure Diffusion @model / Single-phase pressure diffusion model / Non-linear single-phase pressure diffusion @model