Properties of a Natural Fracture and Its Skins From Reservoir Well Tests

SPE Journal ◽  
2013 ◽  
Vol 19 (03) ◽  
pp. 390-397 ◽  
Author(s):  
M.. Prats ◽  
R.. Raghavan
Keyword(s):  
History Matching ◽  
Flow Resistance ◽  
Short Interval ◽  
Natural Fracture ◽  
Disperse Flow ◽  
Well Test ◽  
Transient Pressure ◽  
Matrix Permeability ◽  
Wide Range ◽  
Well Tests

Summary Two well tests are described that are aimed at the in-situ determination of the flow capacity (permeability-thickness product) of a natural fracture and the flow resistance of its skins at the boundaries with the reservoir matrix. Fracture skins tend to disperse flow, thus affecting the distribution of tracers in reservoir tests and contaminants and trace elements in aquifers. We are unaware of any other analytical procedure aimed at obtaining the properties of a natural fracture and its skins from subsurface measurements. Neither well test has been implemented. The well tests are modeled after previously reported analytical expressions for the transient pressure distributions in a three-region composite reservoir in a uniform-thickness reservoir in which (1) the natural fracture is represented by a thin middle region of relatively high permeability, (2) the pressure disturbance is caused by producing from a short interval in one of the outer regions, and (3) the response is measured relatively near the fracture. The source and sensor may be on the same side or on opposite sides of the fracture, distinguishing the two tests. Visualizing special completions in a horizontal well intersecting a natural fracture normally, pressure responses are given for both tests for a wide range of fracture/matrix permeability ratios and skin flow resistances for a source 190 ft from the fracture and 10 ft from the sensor and on either side of the fracture, both at the midplane of the reservoir. A simple graphical procedure, not intended to replace history matching or regression where field data are available, illustrates how the two unknowns—permeability-thickness product of a natural fracture and the flow resistance of its skins—may be estimated from two representative values of an assumed measured pressure response.

Effect of Scaleup and Aggregation on the Use of Well Tests To Identify Geological Properties

2005 ◽  
Vol 8 (03) ◽  
pp. 248-254 ◽  
Author(s):  
Olubusola O. Thomas ◽  
Rajagopal S. Raghavan ◽  
Thomas N. Dixon
Keyword(s):  
History Matching ◽  
Small Scale ◽  
Well Test ◽  
Transient Pressure ◽  
Fine Grid ◽  
Pressure Transient ◽  
Number Of Layers ◽  
Pressure Test ◽  
Well Tests ◽  
The Impact

Summary This paper discusses specific issues encountered when pressure tests are analyzed in reservoirs with complex geological properties. These issues relate to questions concerning the methodology of scaleup, the degree of aggregation, and the reliability of conventional methods of analysis. The paper shows that if we desire to use pressure-transient analysis to determine more complex geological features such as connectivity and widths of channels, we need a model that incorporates reservoir heterogeneity. This complexity can lead to significantly more computational effort in the analysis of the pressure transient. The paper demonstrates that scaleup criteria, based on steady-state procedures, are inadequate to capture transient pressure responses. Furthermore, the number of layers needed to match the transient response may be significantly greater than the number of layers needed for a reservoir-simulation study. The use of models without a sufficient number of layers may lead to interpretations that are in significant error. The paper compares various vertical aggregation methods to coarsen the fine-grid model. The pressure-derivative curve is used as a measure of evaluating the adequacy of the scaleup procedure. Neither the use of permeability at a wellbore nor the average layer permeability as criteria for the aggregation was adequate to reduce the number of layers significantly. Introduction The objectives of this paper are to demonstrate the impact of the detailed and small-scale heterogeneities of a formation on the flow characteristics that are obtained from a pressure test and how those heterogeneities affect the analysis of the pressure test. The literature recognizes that special scaleup procedures are required in the vicinity of wells located in heterogeneous fields. Our work demonstrates that these procedures apply only to rather small changes in pressure over time and are usually inadequate to meet objectives for history-matching well tests. Using a fine-scale geological model derived by geological and geophysical techniques, this work systematically examines the interpretations obtained by various aggregation and scaleup techniques. We will demonstrate that unless care is taken, the consequences of too much aggregation may lead to significant errors on decisions concerning the value of a reservoir. Current scaleup techniques presume that spatial (location of boundaries, location of faults, etc.) variables are maintained. In analyzing a well test, however, one of our principal objectives is to determine the relationship between the well response and geometrical variables. We show that a limited amount of aggregation will preserve the spatial and petrophysical relationships we wish to determine. At this time, there appears to be no method available to determine the degree of scaleup a priori. Because the objective of well testing is to estimate reservoir properties, the scaleup process needs to be made a part of the history-matching procedure. By assuming a truth case, we show that too much vertical aggregation may lead to significant errors. Comparisons with traditional analyses based on analytical techniques are made. Whenever an analytical model is used in the analysis, unless otherwise stated, we use a single-layer-reservoir solution.

Extensive Well-Testing Operations Provide Insight Into Khuff Reservoirs

2021 ◽  
Vol 73 (02) ◽  
pp. 52-53
Author(s):  
Chris Carpenter
Keyword(s):  
Production Performance ◽  
Abu Dhabi ◽  
Well Testing ◽  
Well Test ◽  
Natural Fractures ◽  
Apparent Lack ◽  
Petrophysical Analysis ◽  
Wide Range ◽  
Well Tests ◽  
Insight Into

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 203441, “Lessons Learned From Extensive Well-Testing Operations in Khuff Formations Offshore Abu Dhabi,” by Florian Hollaender, SPE, Schlumberger, and Mahmoud Basioni and Ahmed Yahya Al Blooshi, ADNOC, et al., prepared for the 2020 Abu Dhabi International Petroleum Exhibition and Conference, Abu Dhabi, held virtually from 9-12 November. The paper has not been peer reviewed. An extensive appraisal campaign was performed in the Khuff reservoirs offshore Abu Dhabi, with multiple appraisal wells drilled in different fields. Those wells were evaluated using detailed logging campaigns and then subjected to well tests, usually through drillstem testing for targeted intervals. The interpretation of well tests, combined with advanced petrophysical analysis, formation-test data, and production logs, provided insight into the nature of the Khuff reservoirs. A wide range of responses was observed, from tight to highly productive, but not necessarily with clear previous indications of deliverability or inflow intervals. Overview of the Khuff Formations The key characteristics of the Khuff formations offshore Abu Dhabi have been well-documented in previous work and can be summarized by the following: Low porosity and permeability carbonate reservoirs, where natural fractures are critical contributors to flow Properties vary widely laterally, with significant uncertainty regarding connectivity Variations in stress and petrophysical properties can be significant and affected by diagenetic and tectonic history These reservoirs present significant challenges for development planning. Previous studies have shown that it can be difficult to relate production performance to standard petrophysical analysis directly and that the presence of fractures - in particular, critically stressed fractures - in the vicinity of the wellbore is an essential factor for production performance. Productivity also was found to vary by several orders of magnitude within the same reservoir depending on the field and lateral location of a given well. The presence of natural fractures has been recognized as a major contributor to flow in tight gas reservoirs; however, this raises several questions related to assessing formation potential. First, the nature of the fractures must be evaluated. Some will contribute to production, while others will remain sealed. Equally importantly, identifying zones with promising porosity developments is not a solid indicator of production expectations. Well-Test Observations With more than 20 drillstem tests performed in the Khuff reservoirs during a 4-year period, the first observation is the wide range of reservoir responses encountered, with an apparent lack of consistency within a given reservoir or field.

Use of Exclusion Zones in Mapping and Modeling Fracture Corridors

2010 ◽  
Vol 13 (04) ◽  
pp. 679-687 ◽  
Author(s):  
Sait I. Ozkaya
Keyword(s):  
Distance Matrix ◽  
Well Testing ◽  
Well Test ◽  
Exclusion Zone ◽  
Fracture Permeability ◽  
Reservoir Evaluation ◽  
Lost Circulation ◽  
Matrix Permeability ◽  
Water Breakthrough ◽  
Well Tests

Summary Fracture corridors are fault-related, subvertical, tabular fracture clusters that traverse the entire reservoir vertically and extend for several tens or hundreds of meters horizontally. Conductive fracture corridors may have significant permeability and may profoundly affect reservoir-flow dynamics. Therefore, it is important to map conductive fracture corridors deterministically for reservoir evaluation and well planning. Deterministic mapping of fracture corridors requires locating fracture corridors and assigning to them length, orientation, fluid conductivity, and connectivity. Estimation of orientation, length, and—especially—connectivity is a major challenge in fracture-corridor mapping. An exclusion zone is a region that cannot have a conductive fault or fracture corridor passing through. Borehole images, open-hole logs, flow profiles, and lost-circulation data can be used to identify horizontal wells with no fracture-corridor intersection. Well tests, production/injection history, Kh ratio (permeability times thickness) well-test/core ratio, first water arrival, and oil-column-thickness maps can be used to identify vertical “matrix” wells that do not intersect fracture corridors. Adjacent matrix wells may be surrounded by inferred exclusion zones. The confidence level of inferred exclusion zones depends on factors such as interwell distance, matrix permeability, width, orientation, and spacing of fracture corridors. Overlapping of exclusion zones from independent data sources such as well testing and oil-column thickness have higher confidence than non-overlapping zones. Only borehole images provide orientation and only well tests provide length of fracture corridors. In the absence of well testing and borehole imaging, exclusion zones provide constraints and aid both in locating fracture corridors and assigning them orientation and length. Perhaps the most significant contribution of exclusion zones to fracture-corridor mapping is in identifying interconnected and isolated fracture corridors. An interconnected network of fracture corridors may extend laterally for several kilometers as major fracture permeability pathways, which not only improve pressure support, bottom upsweep of oil, but also cause rapid water breakthrough.

Numerical Simulation of Droplet Generation in Series Co-Flow Capillaries

2009 ◽  
Author(s):  
Yanxi Song ◽  
Jinliang Xu
Keyword(s):  
Reynolds Number ◽  
Disperse Phase ◽  
Weber Number ◽  
Three Dimensional ◽  
Continuous Phase ◽  
Disperse Flow ◽  
Double Emulsions ◽  
Wide Range ◽  
In Series ◽  
Dispersed Phases

We study the production and motion of monodisperse double emulsions in microfluidics comprising series co-flow capillaries. Both two and three dimensional simulations are performed. Flow was determined by dimensionless parameters, i.e., Reynolds number and Weber number of continuous and dispersed phases. The co-flow generated droplets are sensitive to the Reynolds number and Weber number of the continuous phase, but insensitive to those of the disperse phase. Because the inner and outer drops are generate by separate co-flow processes, sizes of both inner and outer drops can be controlled by adjusting Re and We for the continuous phase. Meanwhile, the disperse phase has little effect on drop size, thus a desirable generation frequency of inner drop can be reached by merely adjusting flow rate of the inner fluid, leading to desirable number of inner drops encapsulated by the outer drop. Thus highly monodisperse double emulsions are obtained. It was found that only in dripping mode can droplet be of high mono-dispersity. Flow begins to transit from dripping regime to jetting regime when the Re number is decreased or Weber number is increased. To ensure that all the droplets are produced over a wide range of running parameters, tiny tapered tip outlet for the disperse flow should be applied. Smaller the tapered tip, wider range for Re and we can apply.

A Novel Well Test Data Analyzer and Process Optimizer Using Artificial Intelligence and Machine Learning Techniques

2021 ◽  
Author(s):  
Nagaraju Reddicharla ◽  
Subba Ramarao Rachapudi ◽  
Indra Utama ◽  
Furqan Ahmed Khan ◽  
Prabhker Reddy Vanam ◽  
...  
Keyword(s):  
Artificial Intelligence ◽  
Machine Learning ◽  
Real Time ◽  
Test Data ◽  
Oil Field ◽  
Well Testing ◽  
Well Test ◽  
Time Data ◽  
Real Time Data ◽  
Well Tests

Abstract Well testing is one of the vital process as part of reservoir performance monitoring. As field matures with increase in number of well stock, testing becomes tedious job in terms of resources (MPFM and test separators) and this affect the production quota delivery. In addition, the test data validation and approval follow a business process that needs up to 10 days before to accept or reject the well tests. The volume of well tests conducted were almost 10,000 and out of them around 10 To 15 % of tests were rejected statistically per year. The objective of the paper is to develop a methodology to reduce well test rejections and timely raising the flag for operator intervention to recommence the well test. This case study was applied in a mature field, which is producing for 40 years that has good volume of historical well test data is available. This paper discusses the development of a data driven Well test data analyzer and Optimizer supported by artificial intelligence (AI) for wells being tested using MPFM in two staged approach. The motivating idea is to ingest historical, real-time data, well model performance curve and prescribe the quality of the well test data to provide flag to operator on real time. The ML prediction results helps testing operations and can reduce the test acceptance turnaround timing drastically from 10 days to hours. In Second layer, an unsupervised model with historical data is helping to identify the parameters that affecting for rejection of the well test example duration of testing, choke size, GOR etc. The outcome from the modeling will be incorporated in updating the well test procedure and testing Philosophy. This approach is being under evaluation stage in one of the asset in ADNOC Onshore. The results are expected to be reducing the well test rejection by at least 5 % that further optimize the resources required and improve the back allocation process. Furthermore, real time flagging of the test Quality will help in reduction of validation cycle from 10 days hours to improve the well testing cycle process. This methodology improves integrated reservoir management compliance of well testing requirements in asset where resources are limited. This methodology is envisioned to be integrated with full field digital oil field Implementation. This is a novel approach to apply machine learning and artificial intelligence application to well testing. It maximizes the utilization of real-time data for creating advisory system that improve test data quality monitoring and timely decision-making to reduce the well test rejection.

scholarly journals An approximation of behind-casing hydraulic conductivity between layers from transient pressure analysis

DYNA ◽  
2019 ◽  
Vol 86 (210) ◽  
pp. 108-114
Author(s):  
Freddy Humberto Escobar ◽  
Angela María Palomino ◽  
Alfredo Ghisays Ruiz
Keyword(s):  
Hydraulic Conductivity ◽  
Transient Analysis ◽  
Test Interpretation ◽  
Well Test ◽  
Transient Pressure ◽  
Pressure Transient Analysis ◽  
Pressure Transient ◽  
Transient Pressure Analysis ◽  
Pressure Analysis

Flow behind the casing has normally been identified and quantified using production logging tools. Very few applications of pressure transient analysis, which is much cheaper, have been devoted to determining compromised cemented zones. In this work, a methodology for a well test interpretation for determining conductivity behind the casing is developed. It provided good results with synthetic examples.

scholarly journals Improved Method for Measuring the Permeability of Nanoporous Material and Its Application to Shale Matrix with Ultra-Low Permeability

Materials ◽  
2019 ◽  
Vol 12 (9) ◽  
pp. 1567 ◽  
Author(s):  
Taojie Lu ◽  
Ruina Xu ◽  
Bo Zhou ◽  
Yichuan Wang ◽  
Fuzhen Zhang ◽  
...  
Keyword(s):  
High Pressure ◽  
Fluid Transport ◽  
Effective Means ◽  
Clean Energy ◽  
Nanoporous Materials ◽  
Environmental Research ◽  
Low Permeability ◽  
Matrix Permeability ◽  
Wide Range ◽  
The Matrix

Nanoporous materials have a wide range of applications in clean energy and environmental research. The permeability of nanoporous materials is low, which affects the fluid transport behavior inside the nanopores and thus also affects the performance of technologies based on such materials. For example, during the development of shale gas resources, the permeability of the shale matrix is normally lower than 10−3 mD and has an important influence on rock parameters. It is challenging to measure small pressure changes accurately under high pressure. Although the pressure decay method provides an effective means for the measurement of low permeability, most apparatuses and experiments have difficulty measuring permeability in high pressure conditions over 1.38 MPa. Here, we propose an improved experimental method for the measurement of low permeability. To overcome the challenge of measuring small changes in pressure at high pressure, a pressure difference sensor is used. By improving the constant temperature accuracy and reducing the helium leakage rate, we measure shale matrix permeabilities ranging from 0.05 to 2 nD at pore pressures of up to 8 MPa, with good repeatability and sample mass irrelevance. The results show that porosity, pore pressure, and moisture conditions influence the matrix permeability. The permeability of moist shale is lower than that of dry shale, since water blocks some of the nanopores.

A History Matching Framework to Characterize Fracture Network and Reservoir Properties in Tight Oil

2019 ◽  
Vol 142 (4) ◽  
Author(s):  
Xu Shiqian ◽  
Li Yuyao ◽  
Zhao Yu ◽  
Wang Sen ◽  
Feng Qihong
Keyword(s):  
History Matching ◽  
Fracture Network ◽  
Future Application ◽  
Tight Oil ◽  
Oil Reservoir ◽  
Fracture Permeability ◽  
Reservoir Properties ◽  
Matrix Permeability ◽  
Tight Oil Reservoir ◽  
Better Than

Abstract Accurately characterizing hydraulic fracture network and tight oil reservoir properties can lay the foundation for the production forecast and development design. In this work, we proposed a history matching framework for tight oil. We first use the Hough transform method to characterize complex fracture network from microseismic data. Then, we put the fracture network into an embedded discrete fracture model (EDFM) to build a tight oil reservoir simulation model. After that, we further couple whale optimization algorithm (WOA) and EDFM to match the field production data. In this way, we can accurately estimate reservoir properties, including matrix permeability and porosity, as well as fracture permeability. We apply the framework to two-field applications in China. One is fractured vertical well in the Songliao Basin of Daqing oilfield. The other one is multi-stage fractured horizontal well in the Jimsar Sag of the Xinjiang oilfield. Results show that if we do not consider tight oil characteristics, the estimated fracture permeability, matrix permeability, and matrix porosity will underestimate 73%, 20%, and 47%, respectively. Because we apply WOA to history matching for the first time, we compare the performance of WOA with ensemble–smoother with multiple data–assimilation (ES-MDA). When we fit six parameters, ES-MDA performs better than WOA. However, when we fit three parameters, WOA performs better than ES-MDA. In addition, for engineering problem, WOA performs well on both convergence speed and stability. Therefore, WOA is recommended in the future application of history matching.

An Investigation of Wellbore Storage and Skin Effect in Unsteady Liquid Flow: I. Analytical Treatment

1970 ◽  
Vol 10 (03) ◽  
pp. 279-290 ◽  
Author(s):  
Ram G. Agarwal ◽  
Rafi Al-Hussainy ◽  
H.J. Ramey
Keyword(s):  
Skin Effect ◽  
Initial Period ◽  
Outer Boundary ◽  
Well Test ◽  
Time Data ◽  
Wellbore Storage ◽  
Straight Line ◽  
Pan American ◽  
Well Tests ◽  
Short Time

Agarwal, Ram G., Pan American Petroleum Corp. Tulsa, Okla., Pan American Petroleum Corp. Tulsa, Okla., Al-Hussainy, Rafi, Junior Members AIME, Mobil Research and Development Corp., Dallas, Tex., Ramey Jr., H.J., Member AIME, Stanford U. Stanford, Calif. Abstract Due to the cost of extended pressure-drawdownor buildup well tests and the possibility of acquisitionof additional information from well tests, the moderntrend has been toward development of well-testanalysis methods pertinent for short-time data."Short-time" data may be defined as pressureinformation obtained prior to the usual straight-lineportion of a well test. For some time there has been portion of a well test. For some time there has been a general belief that the factors affecting short-timedata are too complex for meaningful interpretations. Among these factors are wellbore storage, variousskin effects such as perforations, partial penetration, fractures of various types, the effect of a finiteformation thickness, and non-Darcy flow. A numberof recent publications have dealt with short-timewell-test analysis. The purpose of this paper isto present a fundamental study of the importance ofwellbore storage with a skin effect to short-timetransient flow. Results indicate that properinterpretations of short-time well-test data can bemade under favorable circumstances. Upon starting a test, well pressures appearcontrolled by wellbore storage entirely, and datacannot be interpreted to yield formation flowcapacity or skin effect. Data can be interpreted toyield the wellbore storage constant, however. Afteran initial period, a transition from wellbore storagecontrol to the usual straight line takes place. Dataobtained during this period can be interpreted toobtain formation flow capacity and skin effect incertain cases. One important result is that thesteady-state skin effect concept is invalid at veryshort times. Another important result is that thetime required to reach the usual straight line isnormally not affected significantly by a finite skineffect. Introduction Many practical factors favor short-duration welltesting. These include loss of revenue during shut-in, costs involved in measuring drawdown or buildupdata for extended periods, and limited availabilityof bottomhole-pressure bombs where it is necessaryto survey large numbers of wells. on the other hand, reservoir engineers are well aware of the desirabilityof running long-duration tests. The result is usuallya compromise, and not necessarily a satisfactoryone. This situation is a common dilemma for thefield engineers who must specify the details of specialwell tests and annual surveys, and interpret theresults. For this reason, much effort has been givento the analysis of short-time tests. The term"short-time" is used herein to indicate eitherdrawdown or buildup tests run for a period of timeinsufficient to reach the usual straight-line portions. Drawdown data taken before the traditional straight-lineportion are ever used in analysis of oil or gas portion are ever used in analysis of oil or gas well performance. Well files often contain well-testdata that were abandoned when it was realized thatthe straight line had not been reached. This situationis particularly odd when it is realized that earlydata are used commonly in other technologies whichemploy similar, or analogous, transient test. It is the objective of this study to investigatetechniques which may be used to interpret informationobtained form well tests at times prior to the normalstraight-line period. THEORY The problem to be considered is the classic oneof flow of a slightly compressible (small pressuregradients) fluid in an ideal radial flow system. Thatis, flow is perfectly radial to a well of radius rwin an isotropic medium, and gravitational forces areneglected. We will consider that the medium isinfinite in extent, since interest is focused on timesshort enough for outer boundary effects not to befelt at the well. SPEJ p. 279

An Investigation by Numerical Methods of the Effect of Pressure-Dependent Rock and Fluid Properties on Well Flow Tests

1972 ◽  
Vol 12 (03) ◽  
pp. 267-275 ◽  
Author(s):  
R. Raghavan ◽  
J.D.T. Scorer ◽  
F.G. Miller
Keyword(s):  
Flow In Porous Media ◽  
Nonlinear Flow ◽  
Dimensionless Time ◽  
Well Test ◽  
Effect Of Pressure ◽  
Wide Range ◽  
Fluid Properties ◽  
Basic Equations ◽  
Diffusivity Equation ◽  
Time Substitution

Abstract Well test analyses of unsteady-state liquid flow have been based primarily on the linearized diffusivity equation for idealized reservoirs. Studies of pressure behavior of heterogeneous reservoirs have been highly restricted, and no general correlations have been developed for systems in which reservoir porosity, permeability and compressibility, together with fluid density and viscosity, are treated as functions of pressure. A second-order, nonlinear, partial-differential equation results when variations of the above parameters are considered. in the present study, this equation was reduced by a change of variables to a form similar to the diffusivity equation, but with a pressure- (or potential-) dependent diffusivity. pressure- (or potential-) dependent diffusivity. By making this transformation, the solutions to the linear diffusivity equation may be used to obtain solutions to nonlinear flow equations in which reservoir and fluid properties are pressure dependent. This paper provides correlations in terms of dimensionless potential and dimensionless time for a closed radial-flow system producing at a constant rate. The solutions obtained have been correlated with the conventional van Everdingen and Hurst solutions. It also has been shown that the solutions can be correlated with the transient drainage concept introduced by Aronofsky and Jenkins, even though no theoretical basis exists whereby their validity can be proved. In fact, the latter correlation provides a better approximation to the nonlinear provides a better approximation to the nonlinear equation than the van Everdingen and Hurst solutions for large values of dimensionless time. Substitution of the potential described has many important consequences in addition to those already mentioned. Usually, the second-degree pressure gradient term is neglected by assuming that pressure gradients in the reservoir are small. in the present study, these gradients are handled rigorously. Moreover, the selection of parameters such as "average reservoir compressibility" is avoided. Introduction The concept that the porous medium is absolutely rigid and nondeformable is a valid assumption for a wide range of problems of practical interest. It has been long realized that in many problems this assumption leads to certain discrepancies, however, and that the use of "average" properties of the medium would reduce these errors. Considerable research effort has been made to study the effect of pressure-dependent rock characteristics (compressibility, pressure-dependent rock characteristics (compressibility, porosity, permeability) and fluid properties porosity, permeability) and fluid properties using analytical and /or numerical techniques. As a result, numerous methods of solution have been outlined in principle, and a larger number of particular problems have been solved by means of particular problems have been solved by means of high-speed digital computers. Rowan and Clegg give a thorough review of the basic equations governing fluid flow in porous media, showing how the form of the equation changes depending on which of the parameters are taken as functions of pressure of space variables. They also discuss the implications of linearizing the basic equations. Bixel et al. have treated problems involving a single linear and a single problems involving a single linear and a single radial discontinuity. Mueller has considered the transient response of nonhomogeneous aquifers in which permeability and other properties vary as functions of space coordinates. Carter and Closmann and Ratliff have considered the problem of composite reservoirs and studied pressure response and oil production. SPEJ p. 267

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