Statistical and numerical density derivatives: example of flow regime diagnosis and permeability k estimation

Abstract

Presented in this paper are three analytical approaches. (1) Statistical pressure derivative utilises the 2nd differencing of pressure and time series since pressure change and subsurface flow rate are nonstationary series and 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 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 is 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 multiphase flow problems when the fluid density is treated as a function of pressure with slight change in density. The first method demonstrated that for field and design data tested, a good radial stabilisation can be identified with good permeability estimation without smoothing the data. Also it showed that in cases investigated, near and far reservoir features can be diagnosed with clarity. However, 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 as seen in the synthetic data analysed which in return ensures that the derived permeability originates from the formation radial flow. Summarily, the three methods: statistical pressure, fluid-phase numerical density and pressure–density equivalent derivatives gave very clear radial flow stabilisations on the diagnostic plot, from which the reservoir permeability was derived.