Well-Test Analysis for a Well in a Constant-Pressure Square

1974 ◽  
Vol 14 (02) ◽  
pp. 107-116 ◽  
Author(s):  
Anil Kumar ◽  
Henry J. Ramey
Keyword(s):  
Constant Pressure ◽  
Outer Boundary ◽  
Fluid Injection ◽  
Well Test ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Mean Pressure ◽  
Long Time ◽  
Single Well ◽  
Well Tests

Abstract Very little information exists for analyzing well tests wherein a part of the drainage boundary is under pressure support from water influx or fluid injection. An idealization is the behavior of a well in the center of a square whose outer boundary remains at constant pressure. A study of this system indicated important differences from the behavior of a well in a square with a closed outer boundary, the conventional system. At infinite shut-in, the well with a constant-pressure boundary will reach the initial pressure of the system, rather than a mean pressure resulting from depletion. It is possible to compute the mean pressure in the constant-pressure case at any time during shut-in. Interpretative graphs for analyzing drawdown and buildup pressures are presented and discussed. This case is also of interest in analyzing well tests obtained from developed five-spot fluid-injection patterns. Introduction Moore at. first demonstrated the application of transient flow theory to individual well behavior in 1931. Classic studies by Muskat, Elkins, and Arps in the 1930's and 1940's set the stage for two important papers in 1950 that clearly elucidated the basics of modern well-test analysis. One paper by Horner 5 summarized methods for analyzing transient pressure data from wells in infinite reservoirs (new wells in large reservoirs), and a well in a closed, circular reservoir under depletion (fully developed fields). The second paper by Miller, Dyes, and Hutchinsons considered two cases for wells assumed to have produced a long time before shut-in for pressure buildup. One case assumed a closed circular drainage boundary, and the other case assumed a circular drainage boundary at constant pressure. The former would represent annual well tests for fully developed fields, and the latter would represent wells under full water drive in single-well reservoirs. Since 1950, several hundred papers and a monograph have developed the behavior of a constant-rate well in a closed drainage shape of almost any geometry. Key in this development was a classic study by Matthews, Brons, and Hazebroek. The constant-pressure outer-boundary drainage region problem introduced by Miller-Dyes-Hutchinson was reviewed by Perrine in 1955, discussed by Hazekoek el al. in 1958 in connection with five-spot injection patterns, and mentioned briefly by Dietz in 1965. The only other studies dealing with water-drive conditions (constant-pressure outer boundaries) appear in Ref. 7 (Page 44) and in papers by Earlougher et al., published in 1968. It is clear that this case was eitherconsidered totally unimportant, orstudiously avoided. Almost all effort was expended on studying closed outer boundary (depletion) systems.Another problem concerned the conventional assumptions involved in developing well-test analysis method. Even for the common closed (depletion) systems, field applications raised the question of the importance of assumptions. Homer method of graphing assumed the well had been produced a short time, whereas the Miller-Dyes-Hutchinson method assumed that production was long enough to reach pseudosteady state -a long time in many cases. Engineers involved in applications were further confused by differences in methods, as well as by the importance of the assumptions required for analytical solutions that established welltest methods. Recently, Ramey and Cobb showed that an empirical approach could be used to avoid assumptions (which were sufficient but unnecessary) inherent in many previous analytical studies. It was decided to apply this method to the limiting case of a well in a full-water-drive, single-well reservoir - a well in a constant-pressure square. This case is a rarity not often seen in practice. It is closely approached by either an injector or a producer in a developed fluid-injection pattern, by a single injector in an aquifer gas-injection storage test, or by some single-well reservoirs in extensive aquifers.The main point is that a well in a constant-pressure square sets a limiting condition similar to a full water drive. The more common case of a well in a partial-water-drive reservoir should lie between this behavior and that of a closed square. SPEJ P. 107^

Well Test Analysis for a Well in a Constant Pressure Square

1972 ◽  
Author(s):  
Anil Kumar ◽  
H. J. Ramey
Keyword(s):  
Constant Pressure ◽  
Outer Boundary ◽  
Initial Pressure ◽  
Fluid Injection ◽  
Well Test ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Water Influx ◽  
Mean Pressure ◽  
Well Tests

Abstract Very little information exists for analyzing well tests wherein a part of the drainage boundary is under pressure support from water influx or fluid injection. An idealization is the behavior of a well in the center of a square whose outer boundary remains at constant pressure. A study of this system indicated important differences from the behavior of a well in a closed outer boundary square, the conventional system. At infinite shut in, the constant- pressure boundary case well will reach the initial pressure of the system, rather than a mean pressure resulting from depletion. But it is possible to compute the mean pressure in the constant-pressure case at any time during shut in. Interpretative graphs for analyzing drawdown and buildup pressures are presented and discussed. This case is also of interest in analysis of well tests obtained from developed five-spot fluid injection patterns. Introduction Well-test analysis has become a widely used tool for reservoir engineers in the last twenty years. The initial theory was reported by Horner1 for unsteady flow of single phase fluids of small but constant compressibility to a well producing at a constant rate in -infinite and closed boundary reservoirs. Extension of the theory to the finite reservoir case involves specification of the outer boundary condition. The two most commonly observed conditions are: (1) no flow at the outer boundary corresponding to a closed or depletion reservoir, and (2) constant pressure at the outer boundary corresponding to complete water-drive.

Similar Constructive Method of Solution for the Nonlinear Percolation Model in Composite Reservoir

2014 ◽  
Vol 670-671 ◽  
pp. 678-682
Author(s):  
Feng Jiu Zhang ◽  
Xi Tao Bao ◽  
Shun Chu Li ◽  
Dong Dong Gui ◽  
Xiao Xu Dong
Keyword(s):  
Constant Pressure ◽  
Outer Boundary ◽  
Percolation Model ◽  
Constructive Method ◽  
Well Test ◽  
Analysis Software ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Method Of Solution ◽  
Quadratic Gradient

This paper presents a percolation model for the composite reservoir, in which quadratic-gradient effect, well-bore storage, effective radius and three types of outer boundary conditions: constant pressure boundary, closed boundary and infinity boundary are considered. With Laplace transformation, the percolation model was linearized by the substitution of variables and obtained a boundary value problem of the composite modified zero-order Bessel equation. Using the Similar Constructive Method this method, we can gain the distributions of dimensionless reservoir pressure for the composite reservoirs in Laplace space. The similar structures of the solutions are convenient for analyzing the influence of reservoir parameters on pressure and providing significant convenience to the programming of well-test analysis software.

Transient Rate Decline Analysis for Wells Produced at Constant Pressure

1981 ◽  
Vol 21 (01) ◽  
pp. 98-104 ◽  
Author(s):  
C.A. Ehlig-Economides ◽  
H.J. Ramey
Keyword(s):  
Pressure Drop ◽  
Flow Rate ◽  
Constant Pressure ◽  
Well Test ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Type Curve ◽  
Type Curves ◽  
Constant Rate ◽  
Cumulative Production

Abstract Although constant-rate production is assumed in the development of conventional well test analysis methods, constant-pressure production conditions are not uncommon. Conditions under which constant-pressure flow is maintained at a well include production into a constant-pressure separator or pipeline, steam production into a backpressured turbine, or open flow to the atmosphere.To perform conventional well test analysis on such wells, one common procedure is to flow the well at a constant rate for several days before performing the test. This procedure is not always effective, and often the delay could be avoided by performing transient rate tests instead. Practical methods for transient rate analysis of wells produced at constant pressure are presented in this paper. The most important test is the analysis of the rate response to a step change in producing pressure. This test allows type-curve analysis of the transient rate response without the complication of wellbore storage effects. Reservoir permeability, porosity, and the wellbore skin factor can be determined from the type-curve match. The reservoir limit test is also important. Exponential rate decline can be analyzed to determine the drainage area of a well and the shape factor.The effect of the pressure drop in the wellbore due to flowing friction is investigated. Constant wellhead-pressure flow causes a variable pressure at the sandface because the pressure drop from flowing friction is dependent on the transient rate. Finally, for testing of new wells, the effect of a limited initial flow rate due to critical flow phenomena is examined. Introduction Fundamental considerations suggest that conventional pressure drawdown and buildup analysis methods developed for constant-rate production should not be appropriate for a well produced at a constant pressure. However, a well produced at a constant pressure exhibits a transient rate decline which can be analyzed using techniques analogous to the methods for constant-rate flow. In this paper, analytical solutions for the transient rate decline for wells produced at constant pressure are used to determine practical well test analysis methods.Many of the basic analytical solutions for transient rate decline have been available for some time. The first solutions were published by Moore et al. and Hurst. Results were presented in graphical form for bounded and unbounded reservoirs in which the flow was radial and the single-phase fluid was slightly compressible. Tables of dimensionless flow rate vs. dimensionless time were provided later by Ferris et al. for the unbounded system and by Tsarevich and Kuranov for the closed-boundary circular reservoir. Tsarevich and Kuranov also provided tabulated solutions for the cumulative production from a closed-boundary reservoir. Van Everdingen and Hurst presented solutions and tables of the cumulative production for constant-pressure production. Fetkovich developed log-log type curves for transient rate vs. sine in the closed-boundary circular reservoir. Type curves for rate decline in closed-boundary reservoirs with pressure-sensitive rock and fluid properties were developed by Samaniego and Cinco. A method for determining the skin effect was given by Earlougher. Type curves for analysis of the transient rate response when the well penetrates a fracture were developed by Prats et al. and Locke and Sawyer. SPEJ P. 98^

scholarly journals Characteristic Value Method of Well Test Analysis for Horizontal Gas Well

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Xiao-Ping Li ◽  
Ning-Ping Yan ◽  
Xiao-Hua Tan
Keyword(s):  
Mathematical Models ◽  
Outer Boundary ◽  
Vertical Direction ◽  
Well Test ◽  
Well Test Analysis ◽  
Gas Reservoir ◽  
Test Analysis ◽  
Gas Well ◽  
Characteristic Value ◽  
Wellbore Storage

This paper presents a study of characteristic value method of well test analysis for horizontal gas well. Owing to the complicated seepage flow mechanism in horizontal gas well and the difficulty in the analysis of transient pressure test data, this paper establishes the mathematical models of well test analysis for horizontal gas well with different inner and outer boundary conditions. On the basis of obtaining the solutions of the mathematical models, several type curves are plotted with Stehfest inversion algorithm. For gas reservoir with closed outer boundary in vertical direction and infinite outer boundary in horizontal direction, while considering the effect of wellbore storage and skin effect, the pseudopressure behavior of the horizontal gas well can manifest four characteristic periods: pure wellbore storage period, early vertical radial flow period, early linear flow period, and late horizontal pseudoradial flow period. For gas reservoir with closed outer boundary both in vertical and horizontal directions, the pseudopressure behavior of the horizontal gas well adds the pseudosteady state flow period which appears after the boundary response. For gas reservoir with closed outer boundary in vertical direction and constant pressure outer boundary in horizontal direction, the pseudopressure behavior of the horizontal gas well adds the steady state flow period which appears after the boundary response. According to the characteristic lines which are manifested by pseudopressure derivative curve of each flow period, formulas are developed to obtain horizontal permeability, vertical permeability, skin factor, reservoir pressure, and pore volume of the gas reservoir, and thus the characteristic value method of well test analysis for horizontal gas well is established. Finally, the example study verifies that the new method is reliable. Characteristic value method of well test analysis for horizontal gas well makes the well test analysis process more simple and the results more accurate.

Similar Constructive Method for Solving the Model of the Seepage in Multilayered Reservoir

2013 ◽  
Vol 739 ◽  
pp. 298-302
Author(s):  
Wei Li ◽  
Rong Jun Huang ◽  
Shun Chu Li ◽  
Dong Dong Gui
Keyword(s):  
Structural Equation ◽  
Outer Boundary ◽  
Constructive Method ◽  
Test Model ◽  
Well Test ◽  
Reservoir Model ◽  
Well Test Analysis ◽  
Test Analysis ◽  
The Laplace Transform ◽  
Well Test Model

A well test model analysis that based on the three outer boundary conditions (infinite boundary, closed boundary, constant value out boundary) is established for multilayered reservoir; The solutions to the distribution of reservoir pressure and the bottom-hole pressure are obtained in the Laplace space by the use of the Laplace transform; Though the analysis of solution expressions, the solutions to the reservoir model under the condition of three outer boundaries are found to have the same expression and a new method is obtained to solve the boundary value problem of such models of reservoirsimilar constructive method. The similar structural equation of the solution to the reservoir model ,which is obtained by the similar constructive method, is not only convenient for well test engineer to program the corresponding software for well test analysis but also has an important meaning to the theoretical analysis of the seepage regularity of reservoir.

scholarly journals Bilinear pressure diffusion and termination of bilinear flow in a vertically fractured well injecting at constant pressure

2020 ◽  
Author(s):  
Patricio-Ignacio Pérez D. ◽  
Adrián-Enrique Ortiz R. ◽  
Ernesto Meneses Rioseco
Keyword(s):  
Hydraulic Conductivity ◽  
Flow Regime ◽  
Flow Rate ◽  
Constant Pressure ◽  
Well Test ◽  
High Fracture ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Fracture Length ◽  
Fourth Root

Abstract. This work studies intensively the flow in fractures with finite hydraulic conductivity intersected by a well injecting/producing at constant pressure. Previous investigations showed that for a certain time the reciprocal of flow rate is proportional to the fourth root of time, which is characteristic of the flow regime known as bilinear flow. Using a 2D numerical model, we demonstrated that during the bilinear flow regime the transient propagation of isobars along the fracture is proportional to the fourth root of time. Moreover, we present relations to calculate the termination time of bilinear flow under constant injection or production well pressure, as well as, an expression for the bilinear hydraulic diffusivity of fractures with finite hydraulic conductivity. To determine the termination of bilinear flow regime, two different methods were used: (a) numerically measuring the transient of flow rate in the well and (b) analyzing the propagation of isobars along the fracture. Numerical results show that for low fracture conductivities the transition from bilinear flow to another flow regime occurs before the pressure front reaches the fracture tip and for high fracture conductivities it occurs when the pressure front arrives at the fracture tip. Hence, this work complements and advances previous research on the interpretation and evaluation of well test analysis under different reservoir conditions. Our results aim at improving the understanding of the hydraulic diffusion in fractured geologic media and as a result they can be utilized for the interpretation of hydraulic tests, for example to estimate the fracture length.

Deconvolution of Multiwell Test Data

SPE Journal ◽  
2007 ◽  
Vol 12 (04) ◽  
pp. 420-428 ◽  
Author(s):  
Michael M. Levitan
Keyword(s):  
Test Data ◽  
Rate Variation ◽  
Rate Data ◽  
Well Test ◽  
Well Test Analysis ◽  
Test Analysis ◽  
Deconvolution Analysis ◽  
Deconvolution Algorithm ◽  
Analysis Technique ◽  
Well Tests

Summary The deconvolution analysis technique that evolved with development of the deconvolution algorithms by von Schroeter et al. (2004), Levitan (2005), and Levitan et al. (2006) became a useful addition to the suite of techniques used in well-test analysis. This deconvolution algorithm, however, is limited to the pressure and rate data that originate from a single active well on the structure. It is ideally suited for analysis of the data from exploration and appraisal well tests. The previously mentioned deconvolution algorithm can not be used with the data that are acquired during startup and early field development that normally involve several producing wells. The paper describes a generalization of deconvolution to multiwell pressure and rate data. Several approaches and ideas for multiwell deconvolution are investigated and evaluated. The paper presents the results of this investigation and demonstrates performance of the deconvolution algorithm on synthetic multiwell test data. Introduction Pressure-rate deconvolution is a way of reconstructing the characteristic pressure transient behavior of a reservoir-well system hidden in the test data by well-rate variation during a test. The deconvolution analysis technique that evolved with development of the deconvolution algorithms by von Schroeter et al. (2004), Levitan (2005), and Levitan et al. (2006) became a useful addition to the suite of techniques used in well-test analysis. It has been implemented in commercial well-test analysis software and is routinely used for analysis of well tests. This deconvolution algorithm, however, is applicable only for the case when there is just one active well in the reservoir. It is ideally suited for analysis of exploration and appraisal well tests. The previously described deconvolution algorithm cannot be used for well-test analysis when there are several active wells operating in the field and the bottomhole pressure measured in one well during a well test is affected by the production from other wells operating in the same reservoir. The deconvolution algorithm has to be generalized so that it is possible to remove not only the effects of rate variation of the well itself but also the pressure interferences with other wells in the reservoir. As a result, we would be able to reconstruct the true characteristic well-pressure responses to unit-rate production of each producing well in the reservoir. These responses reflect the reservoir and well properties and could be used for recovering these properties by the techniques of pressure-transient analysis. Multiwell deconvolution thus becomes in a way a general technique for interference well-test analysis. The problem, however, is that the interference pressure signals produced by other wells are small compared to the pressure signal caused by the production of the well itself. These pressure interference signals are delayed in time and the time delay depends on the distance between respective wells. All this makes multiwell deconvolution an extremely difficult problem.

Linearly Supported Radial Flow-A Flow Regime in Layered Reservoirs

2002 ◽  
Vol 5 (02) ◽  
pp. 103-110 ◽  
Author(s):  
Boyun Guo ◽  
George Stewart ◽  
Mario Toro
Keyword(s):  
Radial Flow ◽  
Constant Pressure ◽  
Low Permeability ◽  
Well Test ◽  
High Permeability ◽  
Reservoir Model ◽  
Well Test Analysis ◽  
Pressure Derivative ◽  
Test Analysis ◽  
Diagnostic Plot

Summary This paper discusses pressure responses from a formation with two communicating layers in which a fully penetrated high permeability layer is adjacent to a low-permeability layer. An analytical reservoir model is presented for well-test analysis of the layered systems, with the bottom of the low-permeability layer being a constant-pressure boundary. The strength of the support from the low-permeability layer is characterized with two parameters: layer bond constant and storage capacity. Introduction The log-log plot of pressure derivative vs. time is called a diagnostic plot in well-test analysis. Special slope values of the derivative curve usually are used for identification of reservoir and boundary models. These slopes include 0-slope, 1/4-slope, 1/2-slope, and unity slope. In many cases, however, the derivative curves do not exhibit slopes of these special values, and it is believed that some nonspecial slopes also reflect certain flow patterns in the reservoirs. Layered, thick reservoirs are one such example.1 In a layered reservoir, it is common practice to perforate a high-permeability section intentionally (adjacent sections are known to be less permeable) or unintentionally (adjacent sections are believed to be impermeable). It is expected that the flow in the perforated high-permeability layer will be partially fed by fluids in the adjacent layers. Warren and Root2 classified this type of layered reservoir as one of the dual-porosity systems in which the storage effect of the low-permeability layer is considered while the crossflow between layers is neglected. They presented a model based on the mathematical concept of superposition of the two media, as introduced previously by Barenblatt et al.3 This paper discusses the pressure response from a formation with two communicating layers. The flow in the two-layer system is referred to as Linearly Supported Radial Flow (LSRF) in this study. The reservoir model is depicted in Fig. 1. The LSRF may exist in the drainage area of a vertical well where radial (normally horizontal) flow prevails in a high-permeability layer and linear (normally vertical) flow into the high-permeability layer dominates in a low-permeability layer. The LSRF also may exist in the drainage area of a horizontal well after pseudoradial flow in the high-permeability layer is reached. Two LSRF systems were investigated:an LSRF system with a no-flow boundary at the opposite side of (not adjacent to) the high-permeability layer, andan LSRF system with a constant boundary pressure at the opposite side of (not adjacent to) the high-permeability layer. Model Description LSRF With No-Flow Boundary at Bottom. An LSRF system with a no-flow boundary at the bottom of the low-permeability layer was investigated with a finite-element-based numerical simulator. The simulator was fully tested and commercially available in the market. Model configuration and input data are summarized in Table 1. The model well flowed 1,000 hours at a constant flow rate of 1,000 STB/D. A diagnostic plot of the generated response is shown in Fig. 2. It is seen from the figure that the radial-flow derivative is V-shaped in a certain time period. This is an expected signature of dual-porosity systems. It is concluded that the radial-flow derivative curve is similar to the derivative curve of single-layer double-porosity reservoirs. The signature of the pressure-derivative responses cannot be used for further diagnostic purposes. Other information from fracture/void detections is required. LSRF With Constant-Pressure Boundary at Bottom. Pressure response from an LSRF system with a constant-pressure boundary at the bottom of the low-permeability layer was also investigated with the numerical simulator. Model configuration and input data were kept the same as those in Table 1. The model well flowed 300 hours. A diagnostic plot of the generated response is shown in Fig. 3. It is seen from the figure that pressure derivative drops sharply in the later time. This is an expected signature of reservoirs with bottomwater or gas-cap gas drive. One may use a bottomwater- drive reservoir model to determine horizontal and vertical permeabilities in the perforated layer. However, one cannot be sure whether the derived vertical permeability is the permeability of the perforated layer or the low-permeability layer. Also, one cannot characterize the strength of the waterdrive based on the pressure-transient data. To retrieve true reservoir properties and characterize the strength of the waterdrive based on pressure-transient data, an analytical reservoir model was derived in this study. The mathematical formulation of the model is shown in the Appendix. When U.S. field units are used, the resultant constant-rate solution for oil takes the following form:Equation 1 where pd = p-pwf. The constants B and C are defined asEquations 2 and 3 Noticing that the derivative of Ei (t) is given byEquation 4 the diagnostic derivative of pressure for radial flow becomesEquation 5 Taking the 10-based logarithm of this equation givesEquation 6 This equation indicates that the diagnostic derivative currently used in well-test-analysis practice for radial-flow identification is not a constant during the LSRF (i.e., the radial-flow pressure derivative curve will not have a plateau but will decrease with time). This rate of increase depends on B and C if no other boundary effect exists. Therefore, constants B and C can be used to characterize the strength of the supporting layer.

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