Specific case of general multi-phase pressure diffusion assuming the equivalent single-phase diffusion with constant dynamic parameters, thus resulting in linear partial differential equation:
(1) | \phi c_t \partial_t p - \nabla \big( M \cdot ( \nabla p - \rho \cdot \mathbf{g} ) \big) = q_t(\mathbf{r}) \delta(\mathbf{r}) |
where
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| relative oil mobility at reference pressure p_{\rm ref} and temperature T_{\rm ref} | ||
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| standard gravity | ||
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All the above dynamic properties are calculated at reference pressure p_{\rm ref} and temperature T_{\rm ref} thus making (1) a linear partial differential equation.
The reference temperature T_{\rm ref} is more or less in practical cases.
The choice of the reference pressure p_{\rm ref} depends on the task.
For accurate modelling of early time pressure response (ETR) it is recommended to use initial bottom hole pressure at the moment of the test:
(23) | p_{\rm ref} = p_{wf}(t=0) |
For accurate modelling of late time pressure response (LTR) it is recommended to use current formation pressure:
(24) | p_{\rm ref} = p_e |
The above equations (1) – (22) is generalization of original model ( [1], [2]) to the case of Volatile Oil.
Для случае черной нефти ( R_v = R_{vn} = R_{vp} = 0) некторые из вышеприведенных формул упрощаются:
| суммарный отбор воды, нефти и газа в пластовых условиях | ||
| эффективная сжимаемость пласта с мультифазным насыщением как функция насыщенности при опорном давлении P_{\bf ref} | ||
| эффективная проводимость пород как функция насыщенности при опорном давлении P_{\bf ref} | ||
| гравитационная компонента потока при опорном давлении P_{\bf ref} |
Для режима двухфазной фильтрации недонасыщенной нефти (а именно в этом случае модель Перрина дает наиболее удовлетворительное количественное приближение) вышеприведенные формулы упрощаются еще больше:
| суммарный отбор воды, нефти и газа в пластовых условиях | ||
| эффективная сжимаемость пласта с мультифазным насыщением как функция насыщенности при опорном давлении P_{\bf ref} | ||
| эффективная проводимость пород как функция насыщенности при опорном давлении P_{\bf ref} | ||
| гравитационная компонента потока при опорном давлении P_{\bf ref} | ||
| гидропроводность пласта при опорном давлении P_{\bf ref} |
Despite the fact that in many practical cases the Linear Perrine multi-phase diffusion model leads to inaccurate pressure predictions it still:
- provides fair understanding on how pressure responds to multi-phase intakes to or offtakes from formation
- introduce a concept of multiphase transmissibility, diffusivity, compressibility, mobility and total sandface flowrate which are very helpful in multiphase dynamic analysis
- provides fair understanding on how the above properties depend on reservoir porosity, permeability, saturation, compressibility and PVT/SCAL model
This makes Linear Perrine multi-phase diffusion model a helpful analytical and methodological tool for multiphase dynamic analysis.
One should remember that high content of light oil and gas in reservoir and high drawdowns deteriorate the accuracy of Linear Perrine multi-phase diffusion model.
More accurate pressure estimations are provided by the following pressure diffusion models:
- Pseudo-linear multi-phase diffusion model
- Non-linear multi-phase diffusion model
- Volatile Oil and Black Oil dynamic flow models
See also
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing
[ Pressure diffusion models ] [ Linear Perrine multi-phase diffusion @model derivation ]
Reference
- Perrine, R.L. 1956. Analysis of Pressure Buildup Curves. Drill. and Prod. Prac., 482. Dallas, Texas: API.
- SPE-1235-G, Martin, J.C., Simplified Equations of Flow in Gas Drive Reservoirs and the Theoretical Foundation of Multiphase Pressure Buildup Analyses, 1959