Statistical & Numerical Density Derivatives Application in Oil and GasWell Test Interpretation

Abstract

Traditional transient well test analysis has been largely based on draw down solution, which works for the reservoir engineering problems of isothermal, uniform and single phase flow in porous media. After so many years of efforts on multi-phase flow approach, methods such as pseudo-pressure approach has been limited. Numerical well testing approach for multi-phase flow problems is the only method currently under further investigation. Presented in this study are three analytical approaches. (1) Statistical pressure derivative which utilises the 2nd differencing of pressure and time series since pressure change and subsurface flow rate are non stationary series, then integrates the residual of its 1st differences using simple statistical functions such as sum of square error SSE, standard deviation, moving average MA and covariance of these series to formulate the model. (2) Pressure-density equivalent algorithm for each fluid phase, which is derived from the fundamental pressure-density relationship and its derivatives used for diagnosing flow regimes and calculating permeability. (3) Density transient analytical DTA solution derived with the same assumptions as (2) above, but the density derivatives for each fluid phase are used along with the semi-log density versus time plot to derive permeability for each fluid phase. (2) and (3) are solutions for multi-phase flow problems when the fluid density is treated as a function of pressure with slight change in density. The first method demonstrated that for high water production well, a good radial stabilization can be identified for good permeability estimation without smoothing the data. Also it showed that in cases investigated, the drawdown fingerprint can be replicated in the build-up pressure response, hence a good match of the data and a better radial flow diagnosis. The second and third methods can, not only derived each individual phase permeability, the derivative response from each phase is visualised to give much clearer picture of the true reservoir response, which in return ensures that the derived permeability originates from the formation radial flow. These approaches were tested with synthetic and field data. The synthetic studies demonstrated that the calculated numerical density derivatives on the diagnostic plot yield much clearer reservoir radial flow regime and give more confident formation permeability estimation. The study also discovered that in the cases investigated, the heavier the fluid such as water, the better permeability estimation from the weighted average pressure-density equivalent derivatives. In order to support further field application of this approach, field data sets were identified and analysed using the developed methods. In this case, the conventional pressure derivative diagnostic method failed to identify the radial flow, hence unable to estimate the reservoir permeability. In contrast, the three methods: statistical pressure, fluid phase numerical density and pressure-density equivalent derivatives gave very clear radial flow stabilizations on the diagnostic plot, from which the reservoir permeability was derived, which matched the up scaled core permeability from the same formation. The presented approaches provide an estimation of the individual fluid phase and formation effective permeabilities, reflecting the contribution of each phase to flow at a given point.