SUMMARY This paper presents a method of determining kg/ko, oil relative permeability, kro, and gas relative permeability krg, using oil and gas rate-time performance data from individual wells and from a total field. Advanced decline curve analysis is used to obtain original oil in place, N, and thus saturation; the Δp2 form of an oil inflow performance equation is used to determine kro below the bubble point pressure. The procedure was used on production data from several wells in a North Sea naturally fractured limestone volatile oil field. Results indicate the calculated oil and gas relative permeabi1ity curves differ from laboratory and correlation calculated curves. By analyzing the oil and gas relative permeability curves of each of the seven wells in the field, it was found that the degree of natural fracturing of a specific well influences the position of the oil and gas relative permeability curves. The results expressed as kg/ko curves appear to be consistent with the field case history findings of Arps for limestone reservoirs – i.e., as the degree of fracturing increases, the kg/ko curves become more unfavorable with respect to oil recovery. Initial pressure surveys on-each well determine its degree of fracturing while a later field-wide pressure survey confirms the oil-in-place calculated for each well using rate-time decline curve analysis. Pressure-time data to make these calculations is seldom available for all wells in a field or, when available, is much less frequent than rate-time data. In contrast, the principal calculation methods shown in this paper use rate-time data, thus taking advantage of the most frequently collected and the most widely available information.

Pressure drop characteristics in a system composed of two adjacent concentric regions of different permeability were studied. The differential equations for continuity of mass flow in the two regions were solved using the Laplace transformation and the necessary boundary conditions to give the pressure distribution in the composite reservoir. The resulting equation for pressure drop at the inner boundary was evaluated for a variety of composite reservoirs and compared with the results for a uniform reservoir. From this study it was found that under certain conditions the permeability in both zones, as well as the size of the inner zone, can be determined from the pressure drop curve. Introduction The theory for the pressure distribution and pressure build-up behavior of a well producing a single, slightly compressible fluid from infinite and finite homogeneous reservoirs was presented by Horner and Miller, Dyes and Hutchinson. Extensions on this original work to provide improved and extended interpretations and better agreement between theory and observed results have been made by Matthews, et al, van Everdingen, Gladfelter, et al, Stegemeier and Matthews, Hurst and Guerrero, and Perrine. More recently Lefkovits, et al, studied pressure build-up behavior in bounded reservoirs composed of stratified layers. Houpeurt has suggested various approaches to the general problem of variable permeability and porosity but presented no analytic solutions for particular permeability variations. Albert, Jaisson and Marion studied the finite composite reservoir and presented numerical solutions to the unsteady-state case and an analytical solution valid only for large times. They also studied the so-called pseudosteady state for several examples of radial permeability variations. Very similar examples have been treated in the unsteady state with application to pressure build-up by Loucks in an unpublished manuscript. More recently Hurst has presented the complete point-sink solution (valid for all times) for the infinite composite reservoir. He applied these solutions to interference between oil fields along with an even more elegant application of his explicit solution to the material-balance equation including water influx. Mortada approaches the same application by avoiding the point-source limitation but gives the solution for the aquifer region which is valid only for large times. Hopkinson, Natanson and Temple have treated both the finite and infinite composite reservoir obtaining the pressure distribution for the inner zone valid for large times. This paper presents a theoretical study of the pressure distribution in an infinite composite reservoir composed of two adjacent concentric regions of different permeability. The object was to determine the manner in which pressure drop at the inner boundary of a composite reservoir depends upon time, the permeability of each zone and the size of the inner zone. Expressions for the pressure distribution in both zones are developed which take into account the radius of the sink and are valid for small times as well as large times. It was felt that an understanding of the pressure drop behavior in various composite reservoirs would be of assistance in the interpretation of some pressure build-up curves which do not behave according to the theory derived for uniform systems. Often the region surrounding the wellbore is either more permeable or less permeable than the reservoir because of the various drilling and completion practices. The effects of reduced permeability due to drilling- fluid invasion and of increased permeability due to fracturing or acidizing need to be more carefully defined. Therefore, an equation for the pressure drop in a composite reservoir was developed, and the effects of both the permeability in each zone and the size of the inner zone were studied.

he appropriate characterization of the two-phase flow has been recently considered as a topic of interest at industrial level. The Computational Fluid Dynamics (CFD) is one of the techniques used for this analysis. Commonly, the Volume Of Fluid (VOF) model and the Eulerian model are used to model the two-phase flow. The mathematical formulations of these models cause differences in their convergence, computational time and accuracy. This article describes the differences between these two models for applications in the two-phase upward-flow. In order to accomplish this objective, the CFD models were validated with experimental results. This study modeled six experiments with an orthogonal (butterfly) grid. As a result, the Eulerian model shows mean square errors (13.86%) lower than the VOF model (19.04%) for low void fraction flows (< 0.25). Furthermore, it was demonstrated that Eulerian model performance is independent from grid, spending less computational time than the VOF model. Finally, it was determined that only the VOF model predicts the pattern flow.

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