Jitendra Kikani

Perturbation Analysis of Stress-Sensitive Reservoirs (includes associated papers 25281 and 25292 )

The assumption of constant rock properties in pressure-transient analysis of stress-sensitive reservoirs can cause significant error in determination of reservoir transmissibility and storativity. One the other hand, inclusion of pressure-dependent rock properties makes the governing equation for the pressure in the reservoir nonlinear. These nonlinear can be treated only approximately by numerical means. If a permeability modulus is defined, the nonlinearities associated with the governing equation become weaker and an analytical solution in terms of a regular perturbation series can be obtained for a radial infinite-acting reservoir. Three terms in the perturbation series are derived to show the convergence and accuracy of the solution. The equation obtained for each order (zero, first, and second) in the perturbation series is solved exactly, and hence, the solution is exact to the third order. In this paper the effect of wellbore storage on the pressure behavior is investigated. First-order approximation for bounded systems is presented to show qualitative effects. A field example is analyzed to determine the permeability modulus and reservoir properties.

Pressure-Transient Analysis of Arbitrarily Shaped Reservoirs With the Boundary-Element Method

Two boundary-element-method (BEM) formulations are proposed for the solution of pressure-transient problems in homogeneous, anisotropic reservoirs. Pressure solutions in arbitrary reservoir shapes with multiple sources and/or sinks and a variety of constant and/or time-dependent boundary conditions can be generated. This technique is superior to numerical methods because it perserves the analytical nature of the solution and because numerical dispersion and grid-orientation effects are nonexistent. Procedures for the convolution and Laplace-domain solution procedures are compared, and problems illustrating the various aspects of the BEM are solved

Processing and Interpretation of Long-term Data from Permanent Downhole Pressure Gauges

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Abstract Long-term data from permanent gauges have the potential to provide more information about a reservoir than data from traditional pressure transient tests that last for a relatively small duration. Besides reducing ambiguity and uncertainties in the interpretation, long-term data also provide an insight on how reservoir properties may change as the reservoir is produced. This type of long-term surveillance provides the opportunity to look at the reservoir information in four dimensions rather than obtaining a glimpse or snapshot in time. However, the installation of permanent downhole gauges is only a recent phenomenon, and a methodology for the interpretation of the data has yet to be developed. The use of long-term data requires special handling and interpretation techniques due to the instability of in-situ permanent data acquisition systems, extremely large volume of data, incomplete flow rate history caused by unmeasured and uncertain rate changes, and dynamic changes in reservoir conditions and properties throughout the life of the reservoir. This study developed a multistep procedure for the processing and interpretation of long-term pressure data. The procedure was tested with several sets of simulated and actual field data. It was found to be an effective approach for the analysis of long-term pressure data from permanent gauges.

Modeling Pressure-Transient Behavior of Sectionally Homogeneous Reservoirs by the Boundary-Element Method

Pressure-transient solution in a piecewise homogeneous reservoir with arbitrary geometry of each region is proposed by the Laplace domain boundary-element method (BEM). Any number of such regions with different rock and fluid properties can be included in the solution procedure. This formulation can solve for impermeable barriers of any shape, size, and orientation, including large pressure-support sources (aquifers) and fluid-injection problems that show composite behavior. Any number of line-source wells can be included. Example solutions are verified against known analytical solutions. Injection-fluid-front velocity is calculated as part of the solution; thus, the fluid fronts can be tracked in a quasistationary sense. The analytical aspects of this procedure are discussed. BEM solutions are accurate because the method retains the free-space Greens function of the governing differential operator as a global weighting function in the integral equation.

Application of boundary element method to reservoir engineering problems

Abstract This paper presents the theory of the Boundary Integral Equation Method in general and discusses the application of this method in problems related to reservoir engineering. Integral equation methods have existed for quite some time in the mathematical literature but have been popularized only recently in engineering applications within such divers field as aeronautics, heat transfer, elasticity and groundwater hydrology. Reservoir engineering applications of this method are starting to be realized. The usefulness of the methodology lies in the fact that the solutions obtained are highly accurate and do not suffer from the usual drawbacks of the other domain type numerical schemes. Also complex reservoir geometries with multiple wells can be handled with ease due to the good boundary conformance obtained with the elements. Such desirable features are realized because the analytical nature of the solution is preserved due to the use of free space Greens function of the governing differential operator as the weighting function in the weighted residual approach. A collocation type method is used for the solution of the resulting integral equations. Also, since the method is a boundary procedure, the dimensions of the problem are reduced by one. This reduction in dimensionality is obtained in cases where there are no distributed sources/sinks in the problems domain and the initial conditions are homogeneous. The potential applications in reservoir engineering include rapid generation of streamlines and isochrones for steady-state single phase flow problems. Front tracking in the steady-state case can be done quickly and effectively for multi-well situations with arbitrarily shaped boundaries and a variety of boundary conditions. Application to pressure transient testing of arbitrary shaped, multi-well multi-rate reservoirs is also possible, under two different approaches. Both the above applications are demonstrated with a variety of examples, some of which are difficult to solve by analytical means.

Analysis of Pressure-Transient Tests for Composite Naturally Fractured Reservoirs

Naturally fractured reservoirs that undergo waterflooding or steam injection or that are acid-stimulated frequently can be modeled as a composite naturally fractured reservoir with matrix skin. A new analytical solution is presented in Laplace space for such a model that includes wellbore storage and skin. Design and analysis methods for pressure-transient tests based on the pressure derivative are also presented. Correlating parameters have been found that ease test design and analysis. Practical methods to determine the size of the inner zone are given. These methods are based on a scaled deviation time from the semilog straight line. In this paper a field example for an acidized well is presented.

New Approaches for Permanent Downhole Gauge (PDG) Data Processing

Abstract One of the major issues in processing permanent downhole gauge (PDG) data is that too many transients exist over a reasonable time period, say 6 months. A formula was proposed to predict the transients that may be detected or missed. Reasonable prediction was achieved via the formula. Noise usually exists in data recorded by PDGs. Denoising is thus one of the most important steps in PDG data processing. In order to denoise the data, the data noise level must be estimated beforehand. Unfortunately, the data noise level is typically case-dependent, and therefore, it is impossible to identify a universal value for the level that may be used for all the application scenarios. One appropriate approach to estimate the noise level is to first best fit the data, subtract the predicted pressure response from recorded values, and then calculate the noise level based on the difference. We propose to apply nonlinear regression via the Polytope method (Gill, Murray, and Wright, 1981) for best-fitting PDG data to determine the noise level. It is found that the new approach is superior to the least square error (LSE) linear regression as used by Khong (2001), because the bottom-hole wellbore pressure response in a well should be treated as a nonlinear function of time over the majority of the well production/injection/shut-in period. Unless a very small range of the data (say 2 h) is considered, a linear pressure response with time is not anticipated. Furthermore, with nonlinear regression through the Polytope approach, there is no strong restriction in data quantity and data density, hence, automatic detection of the data noise level can be implemented.

The Use of Boundary Element Method in Front Tracking for Composite Reservoirs

This paper demonstrates a new approach using the Boundary Element Method (BEM) to solve for pressure transient behavior in composite and sectionally homogeneous reservoirs. A boundary element solution is proposed in Laplace space to a piecewise homogeneous reservoir with arbitrary geometry of each region. Any number of such regions with different rock and fluid properties can be included in the solution procedure. This formulation can solve fluid injection problems which show composite behavior (as in steam injection and CO2 flooding). In addition, impermeable barriers of any shape and orientation as well as large pressure support sources (aquifers) can be included.

Integrated well performance evaluation for an undulating dual-lateral well

Abstract Integrated evaluation of bottomhole pressure (BHP) and water flow log (WFL) data from an undulating dual-lateral well provided significant insights into the contribution, flow performance, and injectivity of the well. This allowed for the setting of long-term performance characteristics for the well design in a multilayered reservoir. The injectivity test data (BHP pressure and injection profile) acquired in the first lateral and the drill stem testing (DST) data acquired in the second lateral were analyzed using a multilateral well modeling package designed for a multilayered reservoir. Useful and consistent interpretation not only increased confidence but also has assisted in making sound reservoir management decisions. Major observations out of this study include the following: - The average vertical to horizontal permeability ratio (k v /k h ), a difficult parameter to compute, in general, was determined with an excellent match spanning 10 injection and two fall-off periods in one lateral and four drawdowns and two buildups in the other. - Injection profiles at two different water injection rates were predicted and compared with results based on reservoir saturation tool (RST)/WFL log. Excellent agreement was achieved. It is demonstrated that the agreement could be further improved if a nonconstant correction factor was used in interpreting RST/WFL log data. - Well productivity and injectivity indices are dependent upon production/injection time and history. For both the well laterals, the late-time productivity/injectivity index equals around 11 STB/d/psi.