TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractThe storage capacity ratio (ω) measures the flow capacitance of the secondary porosity and the interporosity flow parameter (λ) is related to the heterogeneity scale of the system. Currently, both parameters λ and ω are obtained from well test data by using the conventional semilog analysis, type-curve matching or the TDS Technique. Warren and Root showed how the parameter ω can be obtained from semilog plots. However, no accurate equation is proposed in the literature for calculating fracture porosity. This paper presents an equation for the estimation of the λ parameter using semilog plots. A new equation for calculating the interporosity flow parameter, the storage capacity ratio and fracture porosity from the coordinates of the minimum point of the trough on the pressure derivative is presented. The influence of the wellbore storage on the trough was investigated and a new equation was derived to correct the coordinates of the minimum point. The equations are applicable to both pressure buildup and pressure drawdown tests. The interpretation of these pressure tests follows closely, the classification of naturally fractured reservoirs into four types, as suggested by Nelson.The paper also discusses new procedures for interpreting pressure transient tests for three common cases: (a) the pressure test is too short to observe the early-time radial flow straight line and only the first straight line is observed, (b) the pressure test is long enough to observe the late-time radial flow straight line, but the first straight line is not observed due to inner boundary effects, such as wellbore storage and formation damage, and (c) Neither straight line is observed for the same reasons, but the trough on the pressure derivative is well defined. Analytical equations are derived in all three cases for calculating permeability, skin, storage capacity ratio and interporosity flow coefficient, without using type curve matching.In naturally fractured reservoirs, the matrix pore volume, therefore the matrix porosity, is reduced as a result of large reservoir pressure drop due to oil production. This large pressure drop causes the fracture pore volume, therefore fracture porosity, to increase. This behavior is observed particularly in reservoir where matrix porosity is much greater than fracture porosity. Fractures in reservoirs are more vertically than horizontally oriented, and the stress axis on the formation is also essentially vertical. Using these principles, a new method is introduced for calculating fracture porosity from the storage capacity ratio, without assuming the total matrix compressibility is equal to the total fracture compressibility.Several numerical examples are presented for illustration purposes.
The application of reservoir simulation to mimic the behavior of reservoirs is widely accepted in the petroleum industry. The technique involves planning, constructing and operating a model whose behavior approximates the behavior of the actual (real) reservoir. Often, mathematical equations (i.e., a model) are developed and employed to understand the performance of the real (physical) reservoir. In this paper, a two-dimensional numerical reservoir simulator is developed for expansion-drive reservoirs. Cumulative production at bubble point is computed as a sum of expansion from each cell, based on average reservoir pressure and oil rate from the well. The average reservoir pressure is weighted based on stock tank oil that remains in the reservoir and each cell. Subsequently, plots were generated for average reservoir pressure versus time and cumulative production from each cell versus time. Also, wellbore pressure at mid-horizontal well was plotted against time on Cartesian and semi-log graphs. Furthermore, area plots were generated using color differentiation to illustrate how the reservoir pressure is depleting with time. Simulation studies performed using the developed simulator show a rapid pressure decline as would be expected for an expansion-drive reservoir due to the insignificant compressibilities that typify rocks and liquids. In addition, the reservoir volume estimated from the Cartesian and semi-log plots correctly matched the material balance estimate. Finally, the estimated recovery factor at bubble point (about 2% of the stock tank oil originally in place, STOIIP) agrees with published values in widely-circulated literatures - demonstrating the weakness of rock and liquid expansion as a primary drive mechanism. The reservoir simulator developed in this work can be used by reservoir engineers to understand depletion in under-saturated oil reservoirs whose primary production is driven by rock and liquid expansion.
Although the effect of partial penetration of an infinite conductivity hydraulic fracture has been considered in a homogeneous reservoir, there is no study in similar problem in naturally fractured reservoirs. This paper presents the analysis of the solution to such problem in naturally fractured reservoirs. The method of analysis with or without type curve that enables us to evaluate the permeability in the three principal axes directions is also presented. The solution to the mathematical model was obtained in Laplace domain with elliptical flow model. Several type curves were generated to study the pressure behavior. Both the early linear and pseudo-radial flow regimes are observed. The duration of the early linear flow regime is a function of the natural fractures storativity ratio, interporosity flow coefficient and the dimensionless hydraulic fracture's height. The effect of the dimensionless hydraulic fracture's height on the duration of the linear flow becomes negligible as its dimensionless height approaches unity. Therefore there is no single unique value of a dimensionless time for the end of the linear flow regime as in the case of homogeneous reservoir. Raghavan et al (1978) determined the end of the linear flow regime in fully penetrating hydraulic fractures in homogeneous reservoir to be 0.016. This value is based on the dimensionless pressure drop only. In this study, this value was found to be 0.01 and it was evaluated with pressure derivative curve which is more accurate. Two simulated examples were used to validate the method of the analysis developed. The results obtained are in agreement with the input data. Introduction Hydraulic fracturing in the oil industry has contributed sizable reserves to the overall hydrocarbon reserves in the world. All the tight hydrocarbon reservoirs have to be fractured before they can be producible. These reservoirs are often produced with fully penetrating hydraulic fracture. A fully penetrating hydraulic fracture in a reservoir with water and hydrocarbon in contact will lead to an early or immediate water production. The only method of preventing unwanted fluid at the wellbore in a hydraulic fracture is to carry out partially penetrating hydraulic fracturing. Anderson and Stahl (1967) have shown by actual measurement that hydraulic fracture may not penetrate the entire formation thickness even when it is intended to do so. According to Tinsley et al (1969), the entire height of the hydraulic fracture may not be producing in addition to partial penetration. Moreover, not all the fractured height is propped open by propants. The unpropped height may be healed and close completely. Therefore micro seismic and production logging tools are necessary to determine the effective height of the fracture. Raghavan et al (1978) first presented the solution to a partially penetrating hydraulic fracture in a homogeneous reservoir. Their solution is based on the Green's function product solution technique presented by Gringarten and Ramey (1973). Several type curves for evaluation were presented without any example. Rodriguez et al (1984) presented type curve method of analysis for finite and infinite conductivity based on a numerical method for homogeneous isotropic system. They did not investigate the effect of vertical position on the wellbore pressure. The effect of the transition flow regime on the duration of the linear flow regime makes it necessary to study the behavior of transient flow in naturally fractured reservoirs. Moreover, the method employed by Raghavan and Rodriguez cannot be applied directly to naturally fractured reservoirs because of the transfer function. The problem has to be solved in Laplace domain before inversion to the real time domain. In this study, the elliptical flow model was applied to compute the dimensionless pressure of a partially penetrating hydraulic fracture at the wellbore. The effect of the vertical position of the fracture on the computed wellbore pressure was fully investigated.
A new method for analyzing productivity index (PI) on vertical wells is presented. Well performance is often measured in terms of the well's productivity which is dependent on a number of factors such as the reservoir's configuration, the type of completion, petrophysical and fluid properties, formation damage, etc. The effect of partial completion is the main focus of this study since almost all vertical wells are partially completed due to the reasons of water coning or gas cap issue, etc. Productivity of a well is usually evaluated on the long time performance behavior, thus the pseudo-steady state (late time) approach has been employed for the calculation of the productivity index. Closed system (no-flow boundary) and constant pressure boundary (mixed boundaries) cases are investigated. Several key factors have been tested on productivity index such as pseudo skin, shape factors, penetration ratio, reservoir drainage area and etc. The effects of these factors have been analyzed on productivity index. Shape factors of various well positions in bounded reservoirs were computed and compared with the results obtained by Dietz.
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