Analytical Methods for Single-Phase Oil Flow: Accounting for Changing Liquid and Rock Properties
Authors: Stalgorova Ekaterina , Mattar Louis 
Published: April 2016

Abstract
Many analytical methods are available and widely used in the industry to analyse single-phase flow in the reservoir. Analytical modeling, flowing material balance, type curve analysis and various straight-line diagnostic plots are some of the examples.
All these methods are based on the diffusivity equation. Analytical solutions for the diffusivity equation with various boundary conditions are readily available only for the cases when the coefficients in this equation are constant.
Coefficients of the diffusivity equation are functions of fluid and rock properties (such as density, total compressibility, viscosity and permeability). Traditionally the approach was: For single-phase liquid flow (oil or water): variation of liquid and rock properties with pressure is assumed to be negligible. Under this assumption the diffusivity equation has constant coefficients and can be solved analytically.For singe-phase gas flow: use pseudo-pressure and pseudo-time formulations to account for the variations of gas and rock properties with pressure and to convert the diffusivity equation to a form that has constant coefficients.
This approach has worked well for conventional oil plays. However these days we are producing oils with higher gas content; on top of that, there are reservoirs with higher pressure drawdowns and geomechanical effects (rock compressibility and permeability vary with pressure).
The objective of this study is to demonstrate that the assumption of liquid and rock properties being ‘almost constant’ is no longer valid for the cases we face these days and to discuss possible ways to account for change in liquid and rock properties.
For this study we used synthetic datasets (oil rates and flowing pressure), generated using a numerical multi-frac model based on realistic oil and rock properties for volatile oil in an unconventional reservoir. First, we used the traditional approach to analyse this dataset using linear straight-line analysis, flowing material balance analysis and analytical modeling. Analysis results were significantly different from the correct ones (correct results are known since we used synthetic cases).
Next, we modified all three analyses, to account for the fact that oil and rock properties vary with pressure. The modification is similar to what is traditionally done for gas cases (using pseudo-pressure and pseudo-time formulations). We also applied an additional modification to account for changes in fluid properties when pressure drops below the bubble point (two-phase oil pseudo-pressure).
We observed that the traditional approach may give significantly wrong results (in a particular example the OOIP estimated using flowing material balance was about 2 times larger than the actual OOIP, and the estimated linear flow parameter xf√k was 30% higher than the actual xf√k).
For the cases where pressure remains above the bubble point, we observed that the suggested modification (using oil pseudo-pressure and pseudo-time) brings answers obtained from analyses very close to the actual ones.
For the cases where the pressure drops below the bubble point, we observed that the suggested modification (using two-phase oil pseudo-pressure and pseudo-time) results in answers that are still not completely accurate. However, results obtained using the suggested modification are much closer to the actual answers, as compared to the traditional approach.
Using the suggested modifications generates more reliable results when applying analytical methods to the case of single-phase oil flow and significantly improves estimates when applying analytical methods to the case of two-phase flow (oil below the bubble point).