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.
Compressible rocks
Pressure diffusion equation is going to be:
| (5) | c_t(p) \phi(p) \frac{\partial p}{\partial t} = \nabla ( \alpha(p) \nabla p) |
where
\phi(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