Interference And Pulse Test Analysis By Nonlinear Regression

Abstract A mathematical model is developed that describes fluid flow and pressure behavior in a reservoir consisting of two permeable zones separated by a zone of low permeability, Or a "tight zone." This model can be used to design and to interpret buildup, vertical, interference, and pulse tests conducted in a single well or multiple wells across lithological strata. Dimensionless pressure functions and corresponding parametric type curves are derived to interpret vertical interference test data for tight-zone vertical penneability. Application of these type curves is illustrated using field data from two vertical interference tests. Test results obtained with the tight-zone model are shown to compare favorably with results obtained by usingcomputer simulations andBurns' method based on the uniform anisotropy assumption. Computer simulation using a numerical model also shows that high near-wellbore conductivity from a packer leak or poor cement job could not have adversely affected test results. The model presented and the type-curve interpretation method outlined are accurate for designing and interpreting single-well vertical interference tests across low-permeability zones. Introduction The knowledge of vertical flow properties across a low-permeability stratum is becoming increasingly important in reservoir development, especially when enhanced recovery projects are proposed for stratified reservoirs. Vertical well testing is a technique commonly used to determine values for the in-situ vertical permeability of a formation. Either the vertical interference or vertical pulse test may be used, depending on the amount of time required to obtain the necessary pressure response. The method of vertical interterence testing first was introduced by Burns,1 and later developed by Prats.2 Burns' model is based on the assumption of a homogeneous, infinite-acting reservoir with an average vertical permeability smaller than horizontal permeability. Four geometric parameters are used to computer-generate a type curve for analyzing the test data. One difficulty is that each type curve generated is specific to the four geometric parameters and, hence, to the well completion used. The analysis method proposed by Prats uses a plotting technique that does not require computer solutions. However, his technique is restricted by a point-source assumption; that is, the perforated production and observation intervals must be short compared with the distance between them. The most widely used vertical pulse test analysis technique was developed by Falade and Brigham.3–5 Briefly, the method uses sets of correlation curves relating a dimensionless pulse length and dimensionless pulse amplitude. Corrections can be made to account for the upper and lower formation boundaries. It should be noted that the times as given in the Falade and Brigham technique4,5 are too low by a factor of four.6 A second vertical pulse test analysis method, published by Hirasaki,7 is less general in that it considers only the situation with perforations at the upper and lower boundaries. Both methods use a point-source assumption. All previous vertical interference1,2 and vertical pulse3,4,7 test interpretation techniques were developed to determine vertical permeability in a homogeneous single-layer reservoir. These methods may be applied to stratified reservoirs where permeability contrasts are known to occur; however, they may yield misleading results in these cases where the homogeneous reservoir assumption is not justified. This paper presents an analytical model and interpretation technique to analyze vertical interference test data for tight-zone vertical permeability in a reservoir consisting of two permeable zones separated by a tight zone or a zone of low permeability. Pressure response data in the observation zone are plotted in a ?p vs. ?t format on log-log coordinates and matched against one of two type curves. The result of this match is a value for horizontal permeability in the upper and lower layers and a value for the effective vertical permeability across the tight zone. The type curves included are applicable for a wide range of thickness ratios between the permeable and low-permeability layers. Additionally, use of the model is not restricted by a point-source assumption.

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Using Particle Swarm Optimization (PSO) Algorithm in Nonlinear Regression Well Test Analysis and Its Comparison with Levenberg-Marquardt Algorithm

International Journal of Applied Metaheuristic Computing ◽

10.4018/ijamc.2016070101 ◽

2016 ◽

Vol 7 (3) ◽

pp. 1-23 ◽

Cited By ~ 8

Author(s):

Meisam Adibifard ◽

Gholamreza Bashiri ◽

Emad Roayaei ◽

Mohammad Ali Emad

Keyword(s):

Nonlinear Regression ◽

Heuristic Algorithms ◽

Initial Point ◽

Pso Algorithm ◽

Well Test ◽

Well Test Analysis ◽

Test Analysis ◽

Marquardt Algorithm ◽

Levenberg Marquardt ◽

The Cost

Since two of the most important disadvantages of the classical nonlinear regression methods, such as Levenberg-Marquardt (LM), are to calculate error derivative function and use an initial point to get the results, PSO algorithm, which lies in the category of population based meta-heuristic algorithms, is used in this study to implement nonlinear regression in well test analysis. Root Mean Square Error (RMSE) over pressure and pressure derivative data are used in the cost function formulation and the multi-objective problem is reduced to single objective one by including the weight for each of the cost functions related to pressure and pressured derivative data. The superiority of the procedure developed in this study is verified through a simulated drawdown test and one field case. Error comparison over estimated reservoir parameters and analysis of 95% confidence interval reveal that implemented PSO algorithm can be used accurately to estimate reservoir properties.

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