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Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*50 (10): 76–77.

Paper Number: SPE-1098-0076-JPT

Published: 01 October 1998

... type curve graph reservoir permeability GIP Upstream Oil & Gas

**dimensionless****time**liquid solution radial flow Well Production data type curve P R O D U C T I O N T E C H N O L O G Y 76 OCTOBER 1998 Estimation of hydrocarbons in place is required to determine the economic viabil- ity...
Abstract

This article is a synopsis of paper SPE 49222, "Analyzing Well Production Data Using Combined Type-Curve- and Decline-Curve-Analysis Concepts," by R.G. Agarwal, SPE, D.C. Gardner, SPE, S.W. Kleinsteiber, SPE, and D.D. Fussell, SPE, Amoco E&P Co., originally presented at the 1998 SPE Annual Technical Conference and Exhibition, New Orleans, 27-30 September.

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*45 (06): 558–563.

Paper Number: SPE-23591-PA

Published: 01 June 1993

... period wellbore

**dimensionless****time**fracture half-length productivity Summary. This paper describes the use of a single-phase 2D numer- ical model to study the performance of a well intersected by two perpen- dicular vertical fractures assumed to have either infinite or finite conduc- tivities...
Abstract

Summary This paper describes the use of a single-phase 2D numerical model to study the performance of a well intersected by two perpendicular vertical fractures assumed to have either infinite or finite conductivities. Analysis of the simulated drawdown tests at a constant flow rate showed that the transient flow behavior of a well intersected by finite-conductivity (CfD<500) perpendicular fractures does not exhibit the bilinear and formation linear flow periods. However, when fracture conductivities were infinite (CfD =500), the formation linear flow period was observed. This period was used to determine the fracture half length and gave a fracture half-length equal to the sum of the two fracture half-lengths (xf + yf). The beginning of the pseudoradial flow period for any conductivity was found to decrease as yf/xf increased and to increase as the fracture conductivities increased for a given yf/xf. A single hydraulic fracture gave higher productivity than two fractures of the same total length when fracture conductivity was infinite. However, when fracture conductivity was low, the two fractures gave higher productivity than the single fracture. Introduction Hydraulic fracturing is an effective way to increase the productivity of wells producing from low-permeability formations. Considerable research has been conducted to determine the effect of hydraulic fractures on well performance and pressure-transient behavior. The results have been used to improve the design of hydraulic fractures. The increasing use of hydraulic fracturing to improve the productivity of oil and gas wells in low-permeability reservoirs has resulted in many research efforts aimed at increasing fracturing capabilities and evaluating fracture characteristics in the postfracturing period. The massive hydraulic fracturing (MHF) treatment is now a proven technique for developing commercial wells in low-permeability or tight gas formations. The purpose of MHF is to expose a large surface area of the low-permeability formation (in-situ permeability =0.1 md) to flow into the wellbore. Over the past few decades, hydraulic fracturing and explosive shooting of wells have been used to improve the productivity of tight reservoirs. An alternative to these techniques, one that uses certain features of both but appears better for creating multiple fractures, is tailored-pulse loading. Warpinski et al. in 1979 and Schmidt et al. in 1980 demonstrated that propellants can create multiple fractures in the field without damaging the wellbore region. Their results indicated that the long loading time of the pulse allows greater fracture extension and thus gives more opportunity for connecting a wellbore to the reservoir's natural fracture system.

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*39 (09): 1137–1146.

Paper Number: SPE-16183-PA

Published: 01 September 1987

.../, is varied from 10–5 to 5 × 10 -3 by adjusting the wetting-fluid viscosity, A, and the surface tension, a, by adding aqueous surfactants to mixtures of glycerol and water and by altering the bubble velocity, VT. Results show that a

**dimensionless****time**to snap-off depends weakly on the capillary number...
Abstract

Summary. Recent advances in EOR involve generating foam within underground porous media to displace the oil. We investigate the important snap-off mechanism of gas-bubble generation in constricted square capillaries experimentally. The snap-off of smaller bubbles from a larger bubble as it moves through the constriction is recorded on 16-mm movies. The time required for bubbles to snap off once they move past the constriction and the length of the generated bubbles are obtained from viewing the movie frames. The bubble capillary number, uvT/, is varied from 10–5 to 5 × 10 -3 by adjusting the wetting-fluid viscosity, A, and the surface tension, a, by adding aqueous surfactants to mixtures of glycerol and water and by altering the bubble velocity, VT. Results show that a dimensionless time to snap-off depends weakly on the capillary number and that the generated bubble size increases almost linearly with increasing- capillary number. Surfactants create dynamically inmobile interfaces for surfactant solutions of I wt% sodium dodecyl benzene surfactants (SDBS) and Chevron Chaser SD1000. Compared with the surfactant-free solutions, the time to breakup with surfactants increases by a factor of about 3; generated bubble length increases by a factor of at most 3. Introduction Use of steam foams as displacing fluids for EOR has been successful in recent field applications. Foam improves oil recovery by increasing the resistance to flow of the steam through the underground oil-bearing porous medium, thus improving mobility control and decreasing gravity override. Although foam has proved successful in field applications, modeling of foam displacement processes is still at an early stage. To understand foam flow in porous media better, we investigate the mechanisms of foam generation in porous media. Foam ‘veneration processes are important because they control the texture (i.e., the bubble size) of a foam as it flows through the medium. In turn, texture strongly affects the pressure-drop/flow-rate relationship for foam in porous media. Our objective is to quantity in a realistic single-pore model let the time required to snap off a bubble and the size of the generated bubble. The pore space between three touching spheres (an ideal porous medium) is quite angular 4 (see Fig. 1a of Ref. 4). It looks like a constricted triangular capillary. Square capillaries maintain the angularity and also are readily available. Hence, constricted square capillaries are studied in this work. Snap-off occurs when a bubble is disconnected from a larger gas bubble by a growing liquid collar that blocks a pore neck, as shown in Fig. 1. As the gas bubble moves through a constriction, it leaves a channel of liquid in the corners of the capillary. as shown in Steps 1 and 2 of Fig. 1. Liquid then fills in at the pore neck, as depicted in Step 3. Eventually, sufficient liquid collects at the pore neck, and the liquid rearranges to form a lens that blocks the capillary, thus snapping off a bubble in Step 4. Previous Work Roof originally studied snap-off as a mechanism that traps oil drops in porous media. Fried, and in a more careful study, Mast Showed that snap-off is an important mechanism for bubble generation in porous media. Most investigations of snap-off are in cylindrical glass capillaries. Often, a groove is cut along the wall of the capillary to form a channel along which liquid can flow. The advantage of square capillaries is that they naturally have the channels. Arriola et al. studied the trapping of oil drops in constricted square capillaries and determined how small oil drops snap off from a trapped drop. We also study square constricted capillaries. In our case, however, the gas bubbles are always driven through the constriction at a constant volumetric flow' and are not allowed to trap. Accordingly, we observe a mechanism of snap-off different from the one Arriola et al. reported. Roof argued that if the hemispheric front of a nonwetting phase protrudes seven throat radii past the neck of the constriction, the front of the nonwetting phase will snap off. This is a static analysis. It cannot predict dynamic behavior. Strand et al., however, observed that the flow rate of an oil drop through a constriction is important with larger oil drops being snapped off when the oil drop is driven through a capillary at a higher velocity. JPT P. 1137^

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*37 (02): 323–334.

Paper Number: SPE-12177-PA

Published: 01 February 1985

... flow rate straight line storage effect drawdown test upstream oil & gas semilog straight line approximation horner semilog straight line reservoir

**dimensionless****time**flow rate sandface rate formation pressure flow rate data drillstem/well testing artificial intelligence Analysis...
Abstract

Analysis of Simultaneously MeasuredPressure and Sandface Flow Rate in Pressure and Sandface Flow Rate in Transient Well Testing Summary New well test interpretation methods are presented thateliminate wellbore storage (afterflow) effects. These new methods use simultaneously measured sandface flow rateand wellbore pressure data. It is shown that formationbehavior without storage effects (unit response orinfluence function) can be obtained from deconvolution of sandface flow rate and wellbore pressure data. The storage-free formation behavior can be analyzed to identify the system (reservoir flow pattern) that isunder testing and to estimate parameters. Convolution(radial multirate) methods for reservoir parameter estimation and a few synthetic examples for deconvolution and convolution also are presented. Introduction Well testing with measured sandface flow rate can betraced to the beginning of reservoir engineering. The rate must be measured over time to calculate and/or approximate constant rate to obtain even a single reservoir parameter from pressure measurements. This approximate parameter from pressure measurements. This approximate constant rate has been sufficient for estimating permeability, skin, and initial formation pressure during permeability, skin, and initial formation pressure during the radial infinite-acting period. During this period, the well should produce at a constant rate at the sandfaceor at a zero rate if a build up test is conducted. Becauseof compressible fluid in the production string (wellbore storage effects), it takes a long time to reach the radial infinite-acting period. The effect of outer boundaries alsomay start before the end of the wellbore storage effects. In general, the storage capacity of the wellbore, wellbore geometry, near-wellbore complexities, and external boundaries affect transient behavior of a well. During the analysis of pressure-time data, each of these phenomena and its duration must be recognized for the phenomena and its duration must be recognized for the application of semilog and type curve techniques to determine formation flow capacity (kh), damage skin, and average formation pressure. The influence of these phenomena on transient behavior of a well progresses over phenomena on transient behavior of a well progresses over time. For the sake of convenience, the test time can be divided into three periods according to which phenomenonis affecting the pressure. These periods are defined as follows. Early-Time Period The combined effects of wellborestorage, damage skin, and pseudoskin (which includepartial penetration, perforation, acidizing, fractures, partial penetration, perforation, acidizing, fractures, non-Darcy flow, and permeability reduction caused by gassaturation around the wellbore)dominate pressurebehavior. The stratification and dual porosity also mayaffect wellbore pressure during this period. Middle-Time Period During this period, radial flow isestablished. Conventionally, semilog techniques are usedto determine formation, kh and initial pressure and skin. Late-Time Period During this period, outer boundaryeffects start to distort the semilog straight line. For example, the gas cap shows a curve-flattening effect on log-log and Horner plots. Sometimes the separation of these periods from eachother is impossible; particularly, the effects of bottom-water influx and/or gas cap may start during themiddle-time period. Thus, the semilog approach sometimescannot be applied at all. Furthermore, the drawdown or buildup tests as conductedtoday tend to homogenize the reservoir behavior. In other words, most of the reservoirs behave homogeneously duringthe storage-free radial infinite-acting period because mostof the heterogeneous behavior takes place during theearly-time period. The type-curve approaches have been introduced toovercome some of these problems. The theories, applications, and elaborations of the type curve methods, as wellas many references, can be found in Ref. 1. In 1979, Gringarten et al. introduced new type curves that usedifferent parametrization than the earlier ones, namelyRamey, Agarwal et al., McKinley, and Earlougher and Kerschtypes. All the type curves presented by these authors, andmany others, were developed under the assumption that thefluid compressibility (density) in the tubing and annulusremains constant during the test period. During the earlytime, particularly for buildup tests, shut-in pressureincreases very rapidly; thus, the compressibility is usuallyhigher than the compressibility of the fluid in the reservoirfor producing wells. Since the pressure in the wellbore is afunction of the depth, the compressibility of the fluid atthe wellhead can be 10 or even 100 times greater than thecompressibility of the fluid at the bottom. Thus, theassumption that the wellbore storage coefficient is constantduring the drawdown, and particularly during buildup, maynot be correct. JPT p. 323

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*35 (05): 981–990.

Paper Number: SPE-9885-PA

Published: 01 May 1983

... in porous media formation permeability jt graph pressure transient testing fracture conductivity upstream oil & gas fracture correction factor drillstem/well testing square-root-of-time graph hydraulic fracturing fracture length

**dimensionless****time**finite-conductivity fracture drillstem...
Abstract

Introduction Long hydraulic fractures are usually required to optimize recovery from low-permeability gas reservoirs. Since these fractures can be quite expensive to create and since there is still a great deal of "art" associated with fracture design and creation, it is frequently helpful to use pressure-transient tests to determine created fracture properties. In this way, optimal fracture treatments can be developed for a given area. Fast et al. 1 and Kozik and Holditch presented examples showing the potential benefits of such an approach. In addition, Veatch demonstrated the effectiveness of combining the efforts of operations and research personnel to characterize and then to improve stimulation treatments in specific areas. The specific task of the engineer is to estimate propped fracture length and effective fracture conductivity for treatments in a given geological formation, If the reasons for success or failure of previous fracture treatments can be determined, the engineer can then do a better job in the future when designing fracture treatments for the same area. There are currently four basic techniques used to analyze postfracture pressure-transient tests: semilog (pseudoradial flow) analysis, square-root-of-time (linear flow) analysis, type-curve analysis. and reservoir simulator history matching. The strengths and weaknesses of each technique were discussed by Lee and Holditch. No existing technique is without problems or possible ambiguity in some applications; thus, there is a need for still other techniques that may succeed in some situations for which existing techniques are inadequate. The purpose of this paper is to introduce such a new technique for analyzing postfracture pressure-buildup tests. This technique can be particularly helpful when analyzing data from wells in which a finite-conductivity fracture has been created. Proposed Technique The pressure-buildup test analysis technique proposed in this paper requires that the analyst prepare a semilogarithmic graph and a square-root-of-time graph of the test data. The technique also requires use of two correction curves developed from the analytical solutions for finite-conductivity fractures presented by Cinco-Ley et al. The analyst chooses a single straight line on each of the graphs prepared for the test data and solves for fracture length, formation permeability, and fracture conductivity with an iterative procedure. Semilogarithmic Graph As a first step in understanding the basis for the proposed method, consider Fig. 1. JPT P. 981^

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*35 (03): 655–658.

Paper Number: SPE-10856-PA

Published: 01 March 1983

... pipeline segment boundary condition iteration momentum equation

**dimensionless****time**natural gas pipeline noel partial differential equation utah anthony day upstream oil & gas mass flow rate equation tenn continuity inventory Packing and Drafting in Natural Gas Pipelines Noel de Nevers...
Abstract

Summary Increasing and decreasing natural gas pipeline inventory (" packing" and "drafting" ) are examined mathematically. Any line segment's unsteady-state packing or drafting behavior depends on only two dimensionless parameters, packing or drafting behavior depends on only two dimensionless parameters, ( ) and ( ). The influence of ( ) is small, so that for any value of ( ) the behavior of all pipelines can be represented on a single plot; four such plots are shown for four different boundary conditions. Introduction Natural gas dispatchers use increase and decrease of the stored inventory of gas in their pipes as one method of matching time-varying demands with supplies, which generally have less time variation. In pipeline terminology, increasing the inventory (and hence the pressure) is called "line packing," while decreasing it is called "line drafting." This paper examines the limits of this procedure, asking both how much gas can be added to or subtracted from the inventory in a given pipeline segment, and how rapidly this can be accomplished. How Rapidly Can the Inventory Be Depleted or Restored? Although the questions of how large the inventory is and how much can be taken from it for any change in steady-state conditions can be answered very simply, the computation of how rapidly this can be accomplished requires a set of coupled partial differential equations and a numerical solution on a computer. Fortunately, as shown here, the nondimensional results can be summarized in ways that are fairly easy to use. Mathematical Theory It has been known since at least 1951 that the mathematical description of the unsteady-state flow of any gas in a long pipeline is governed by a material-balance equation, ............................(1) and a momentum-balance equation, ............................(2) The mass-balance and momentum equations, expressed with pressure, and mass flow rate, as dependent variables and length, and time, as independent variables and assuming that 's are much smaller than 's and ( / )'s, are ............................(3) and ........................... (4) respectively. The justification for dropping the terms as being much smaller than the others is that changes very slowly with changes in and other variables. JPT p. 655

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*34 (06): 1313–1326.

Paper Number: SPE-6783-PA

Published: 01 June 1982

... 6 1982 1 6 1982 1982. Society of Petroleum Engineers production logging temperature survey heat flow flow rate injection point

**dimensionless****time**injection upstream oil & gas variation well radius conduction conductivity reservoir surveillance fraction temperature...
Abstract

Summary Temperature surveys run during well injection or production can reveal not only intake or production intervals but also the rate of flow. From previous work, the flowrate analysis is based on the "subtangent" or "delta function," which is simply the change in temperature induced by the flow divided by the thermal gradient. This paper investigates accuracy and limitations. of this simplified analysis, and provides design guides for selecting conditions that optimize interpretation. Methods for distinguishing flow-rate changes from changes in formation properties or well radius also are presented. Introduction Diagnostic temperature surveys in wells are at least 60 years old. The earliest use of temperature measurements was primarily for geophysical and geological purposes. For example, determining geothermal gradients, required for regional heat flow studies, was reported by van Orstrand 1 and Heald. 2 Lithological variations were identified by local departures from the prevailing geothermal profile that were caused by thermal property variations of the formations. 3 In the mid-1930's attention turned toward using temperature surveys to diagnose well completion and production problems. 4 Deussen and Guyod 5 described it technique for determining the position of cement emplacement that relied on the temperature increase caused by heat of hydration. Lost circulation zones were identified as warm or cool temperature anomalies, depending on whether the drilling fluid was hotter or colder than the thieving formation. 6 Potential gas production zones were inferred from local anomalies caused by the cooling effect of the depressurized gas as it entered the well. 6 More recently attention has focused on the use of temperature surveys to infer flow rate profiles within wells. The methods can be applied to either injection or production wells. For example, in cased wells positions and magnitudes of casing leaks can be identified. In openhole sections the fluid intake or production strata can be located, and the injectivity or productivity of formations can be profiled. In this manner temperature surveys are similar to flowmeter. surveys (e.g., a spinner survey) except that unlike flowmeter surveys temperature surveys can detect flows behind casing 7 - e.g., water aquifer flows - and are particularly useful for locating hydraulic fractures. 8

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*34 (04): 917–924.

Paper Number: SPE-9889-PA

Published: 01 April 1982

...

**dimensionless****time**planar flow machine learning straight line penneability fracture half-length permeability pseudopressure tech hydraulic fracturing numerical simulation flow period drillstem/well testing approximation drillstem testing reservoir Pressure Buildup From a Vertically Fractured...
Abstract

Summary Buildup theory for a vertically fractured gas well was reexamined assuming planar flow against constants and face pressure. The resultant pressure buildup is an arcsine function of a time group. If this relation is used to analyze buildup, it is possible to calculate the product of permeability and fracture half-length squared. This analysis gives better agreement with a set of field data and numerical simulation as compared with that from constant-rate theories using a Homer plot and atande msquare-root time plot. Introduction Pressure buildup data for a vertically fractured well can be analyzed by the usual Homer method, provided that the drawdown has reached pseudo steady state. For low-permeability reservoirs, the entire well test may lie within the initial flow period. The flow behavior of aportion of this period can be predominantly planar. Millheim and Cichowicz showed that the pressure buildup varies as a tandem square root of time. One ofthe assumptions they used was the constant-ratedrawdown, which usually is not valid for low- permeability reservoirs. In fact, it generally is accepted that as soon as a well is opened to flow in these reservoirs, the sand face pressure drops rapidly to a constant. This boundary condition has been recognized as valid fora large number of well tests. All these papers considered cylindrical radial flow. Agarwal et al. evaluated numerically the drawdown behavior of avertically fractured well under both constant-rate and constant-pressure boundary conditions. However, the rate/time type curves they generated for the drawdown against the constant-pressure case cannot be used to analyze buildup. The appropriate formulation considers the pressure buildup analysis of a vertically fractured well when the drawdown is against a constant sand face pressure. The buildup is derived by use of Duhamel's integrals withthe variable rate being the drawdown rate and the subsequent shut-in rate of zero. The integrand reduces to a simple form at the fracture face. The resultant closed form expression enables the pressure/time data to be plotted into a straight line, the slope of which is characterized by the initial reservoir pressure and the sand face drawdown pressure. Alternately, this slope can be expressed in terms of the last flow rate and the other reservoir/fracture parameters. One of them is the product of the formation permeability and the fracture half-lengthsquared. Without other information, it is possible to calculate only this product and not its individual components. Application of the derived equation to one example gave a result in good agreement with drawdown analysis and numerical simulation. The effects of wellbore storage and fracture damage also were shown to be negligible. Planar flow Period The geometry of the fracture plays an important role in the early-time flow data. Cinco-L. and Samaniego-V. divided the entire flow into four periods. Initially there isa fracture linear flow period in which the flow inside the fracture is the dominant factor. As the fluid from the formation starts its contribution, the flow enters the bilinear flow period. If the fracture is highly conductive, then there will be a period when planar flow from the formation into the fracture is a good approximation. Eventually, pseudo radial flow takes over in any fracture/formation system of sufficient extent. In analyzing a set of well test data using the plan arapproximation, care must be exercised to ensure that these data points fall within the planar flow period. It turns out that the determining factor in each of the early periods is the fracture flow capacity. JPT P. 917^

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*33 (12): 2489–2497.

Paper Number: SPE-8943-PA

Published: 01 December 1981

... drillstem/well testing pressure transient testing gas well elliptical flow equation elliptical system

**dimensionless****time**hydraulic fracturing drawdown elliptical equation reservoir pressure transient analysis finite difference equation radial flow tight gas well pseudopressure drillstem...
Abstract

Summary A set of elliptical equations is developed for use in understanding and evaluating vertically fractured gas wells in low-permeability gas sands. The equations are designed for use with short-term flow tests at either constant-rate or constant-pressure test conditions. Introduction Much new Rocky Mountain-area gas is being produced from tight gas sands. These wells often prove uneconomical unless they are fractured hydraulically. After the wells are fractured, testing generally is required to ensure that sustained production will justify the cost of a pipeline connection. Current practice generally involves a flow test ranging from several hours to several days. A corresponding pressure buildup often ranges from a day to a week or more. Using a well test involving hours to forecast many years' performance can be very misleading if not done with care.A basic understanding of elliptical equations can yield a great deal of useful information from short flow and buildup tests. It also helps set limits on what information can and cannot be expected from tests. Elliptical Equation The elliptical flow equations result from the basic fluid flow equations in porous media with fracture geometry. The fracture is characterized by the distance it extends from the wellbore. It extends from -Xf to +Xf for a total length of 2Xf. Gas flows from the formation into the fracture and is conducted by the fracture to the wellbore. For an infinite conductivity fracture, the inner boundary is one of uniform pressure along the fracture face. Flow may be either at a constant rate, a constant pressure, or a combination of the two.Three points on a given isobar demonstrate steadystate elliptical geometry. Two points are at a distance of +A and -A from the wellbore. The third is at a distance B from the wellbore (Fig. 1). Conformal mapping with the coordinate transformation w = arcsin (Z) is used to transform the X-Y plane into a U - V plane. The transformed system is linear with known solutions for the given conditions. At the external boundary parallel to the fracture, Ve = A′ = B' = arccosh (A) = arcsinh (B). Rearranging this equation yields Ve = ln (A + B) = -ln (A-B) or A2 - B2 = Xf2. This defines an ellipse. The steady-state isobars are concentric ellipses. A is a major semiaxis. B is a minor semiaxis. In this paper, A and B refer to the external ellipse that defines the drainage area. Axf and Bxf refer to the inner ellipse that is the fracture. Fluid flow occurs along the streamlines that are orthogonal to the ellipses. The driving force is the real gas pseudopressure. Steady-State Elliptical Flow Equations Development of steady-state equations will solve few problems but lends to an understanding of unsteady-state problems. The generalized steady-state equation is (1) Expressed in X - Y coordinates, the equation is (2a) Boundary conditions are (2b) JPT P. 2489^

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*33 (02): 370–382.

Paper Number: SPE-8390-PA

Published: 01 February 1981

... transient analysis active well

**dimensionless****time**upstream oil & gas diffusivity ratio 1 2 1981 1 2 1981 1 2 1981 1 2 1981 1981. Society of Petroleum Engineers The Effect of Noncommunicating Layers on Interference Test Data Wei C. Chu, SPE, U. ofTulsa Rajagopal...
Abstract

Summary The characteristics of the pressure vs. time curve at an observation well in a layered reservoir are investigated. The pressure response at the well is dominated by the crossflow effect that results due to interlayer communication and is governed by the properties of the individual layers and the skin regions surrounding flowing and observation wells. It appears that the skin regions may be the dominant influence on the pressure response. Any viable method of analysis should include the effect of the skin regions. Introduction A reliable description of the variations in reservoir properties is needed to design and to ensure the success of an enhanced recovery process. Interference tests commonly are used to obtain information on the character of the porous medium. of the several types of reservoirs that exist, one of the simplest is a reservoir consisting of a number of layers that have distinct values of permeability, porosity, thickness, etc., stacked on top of each other and that are in communication only at the wellbore. A review of the literature indicates that a comprehensive examination of interference test data in a layered reservoir with communication only at the wellbores is yet to appear.The objective of this study is to examine the effect of reservoir parameters (permeability, porosity, thickness, and compressibility) on the pressure response at an observation well in a layered reservoir. The layers are separated by impermeable barriers and are in communication only at the wellbores. Results for two- and three-layer systems are discussed in detail. Ten- and 20-layer systems are examined briefly. The effect of the flow capacity and the diffusivity of the layers on interference data is discussed. The influence of the skin region around the observation and flowing wells is documented. We also document the consequences of communication between the layers at the wellbore and show that the communication at the wellbore has a dominant effect on the observation-well response. In this work, the term "crossflow" is used to refer to the flow between the layers at the observation well.It is emphasized that the intention of this study is to provide a broad, comprehensive understanding of the pressure behavior in reservoirs subject to the assumptions incorporated in our model. This, in turn, should allow a better analysis of data influenced by crossflow and enable the designing of tests to obtain reliable information on the characteristics of a layered reservoir. Procedures used to analyze data are not discussed. Theory and Assumptions The model considered consists of several horizontal, homogeneous layers of differing physical properties, separated by impermeable barriers (see Fig. 1). The top and bottom of the reservoir are sealed by an impermeable boundary. The lateral extent of the reservoir is assumed to be infinite, and each layer is filled with a slightly compressible fluid of constant viscosity. The initial reservoir pressure pi of each layer is the same, and gravity effects are assumed to be negligible. All wells in the reservoir completely penetrate all the layers, and there is communication between the layers at the wellbores. Each layer, denoted by the subscript j, has distinct values of flow capacity kjhj and diffusivity eta j. JPT P. 370^

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*32 (10): 1701–1708.

Paper Number: SPE-6056-PA

Published: 01 October 1980

... Drillstem Testing drawdown

**dimensionless****time**detection match point well pressure time rate boundary infinite reservoir pressure transient testing pressure data pressure behavior well location infinite strip straight line graph procedure flow boundary kh product Detection and Location...
Abstract

A new technique is presented for detecting the existence of and distance to two parallel sealing boundaries surrounding a producing oil or gas well. A log-log plot of time rate of change of well pressure vs. time provides a unique behavior to detect such a condition and allows the use of type-curve matching techniques to determine the kh product. Introduction The idea of detecting and locating a reservoir boundary from transient pressure-time data first appeared in 1951. In that year Horner 1 presented the transient pressure behavior of a constant-rate well located near a linear sealing fault. He also presented a method to calculate the distance to the fault from buidup pressure data. Dolan et al . 2 applied the technique to drillstem tests. Davis and Hawkins 3 gave an equation to determine the distance to the fault from drawdown pressure data. Gray 4 reviewed these methods and discussed their limitations. Bixel et al . 5 considered a more general problem: a well located close to a linear discontinuity across which hydraulic diffusivity changes. They showed a procedure to determine the distance to such a discontinuity. Evrenos and Rejda 6 employed superposition techniques to simulate various combinations of linear boundaries of interest in gas storage systems and showed how a match between the actual test data with various hypothesized model conditions can be used to arrive at a probable configuration of boundary conditions. Overpeck and Holden 7 considered the effect of reservoir anisotropy on fault detection and showed that the distance calculated could differ by as much as 20% with those obtained by assuming isotropic medium. They gave a procedure for imaging when the principal permeability axes are at some angle other than 0 or 90° to the fault boundary. Rodgers and McArthur 8 employed a minimum in standard deviation between observed pressures and a least-squares straight-line fit to determine boundary configuration. Prasad 9 presented a procedure to compute transient pressures for a well between two sealing faults intersecting at any angle. Jones 10 in 1961 drew attention to the possible use of rate change of well pressure with time in detecting reservoir boundaries. Van Poollen 11 presented graphs of time rate of change of well pressure during drawdown for various well locations between faults intersecting at 90 and 36°. Although this approach yields interpretable drawdown behavior, it has not been used widely in the petroleum literature. Witherspoon et al. 12 considered the effect of a linear no-flow or flow boundary and presented the dimensionless pressure behavior caused by a constant-rate producing well at an observation well some distance away. They discussed techniques to analyze drawdown test data.

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*32 (10): 1803–1812.

Paper Number: SPE-8768-PA

Published: 01 October 1980

..., however, wellbore storage effects can be important. JPT P. 1803＾ 1 10 1980 1 10 1980 1 10 1980 1 10 1980 1980. Society of Petroleum Engineers Upstream Oil & Gas

**dimensionless****time**reservoir pressure formation flow capacity outer boundary condition drillstem/well...
Abstract

Transient pressure behavior in the drainage area of a well producing at constant wellbore pressure is discussed. Methods used to analyze flow rate behavior to determine formation flow capacity, porosity-compressibility product, skin factor, and drainage volume are discussed. A new type curve is given for analyzing interference test data obtained by producing (or injecting into) a well at constant pressure. Introduction In a recent study, the analysis of pressure buildup data subsequent to production at a constant wellbore pressure was discussed. It was found that even though the flow period during constant pressure production is discussed in great detail in the literature, some important aspects of well test analysis were never considered. For example, no study examined the transient pressure behavior when the wellbore pressure is constant. The purpose of this paper is to present a comprehensive study of the transient pressure behavior of a reservoir for the constant-wellbore-pressure boundary condition. Specifically, we will document the average reservoir pressure as a function of flow time. This information will be useful in material balance computations. Second, we intend to establish the validity of the effective wellbore radius and the "infinitesimally thin" skin concepts for a well that flows at a constant pressure. Both of these concepts assume that steady-state conditions prevail in the skin region. All studies in the literature implicitly assumed that the "infinitesimally thin" skin concept is applicable when a well produces at a constant pressure. However, it is not readily evident whether this concept can be used when the sandface flow rate is a continuous function of time. Third, the analysis of interference test data when the wellbore pressure is constant will be discussed. This procedure will prove superior to constant-rate testing whenever wellbore storage effects at the active well influence the observation pressure response. If the wellbore pressure is constant, only the reservoir properties affect the pressure response since wellbore storage effects at the flowing well are nonexistent. Consequently, the possibility of conducting an interference test in this manner should be the first consideration whenever the interwell distance is small. Mathematical Model and Assumptions Consider the classic problem of the flow of a slightly compressible fluid in a cylindrical, homogeneous, isotropic reservoir of constant thickness. The outer boundary of the reservoir is either closed or at a constant pressure equal to the initial pressure. The well is located at the center of the cylinder, and fluid is produced at a constant pressure. Initially, the pressure is uniform throughout the reservoir. The skin region in this model is assumed to be an annular region concentric with the wellbore and with a permeability different from the formation permeability. Wellbore storage effects are not considered since the well flows at a constant pressure. (During the buildup period, however, wellbore storage effects can be important. JPT P. 1803＾

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*32 (07): 1251–1261.

Paper Number: SPE-8143-PA

Published: 01 July 1980

... 1980 1 7 1980 1980. Society of Petroleum Engineers Drillstem Testing penetration ratio pressure drop boundary graph vertical permeability pressure behavior flow capacity transient period drillstem/well testing

**dimensionless****time**Gringarten drainage area reservoir early...
Abstract

This paper examines the characteristics of the wellbore pressure drop vs. time curves for a partially penetrating well located at the center of a square drainage region and subject to bottomwater drive. The effects of the penetration ratio and reservoir anisotropy on the transient behavior of the system are investigated. Edgewater- and bottom water-drive systems are compared. Introduction It is a common practice in the petroleum industry to drill wells through a limited portion of the formation or to penetrate the entire thickness and then selectively perforate a limited interval. These two completion techniques are known as partial penetration and restricted entry, respectively. The purpose of partial penetration or limited entry is to avoid or partial penetration or limited entry is to avoid or delay the intrusion of unwanted fluids into the wellbore. Partial penetration is probably the rule in geothermal systems since reservoirs can be extremely thick.Virtually all studies on the transient pressure behavior of both types of wells assume that the top and bottom boundaries are sealed. The objective of these studies has been to determine the horizontal and vertical permeability of the reservoir and/or the productivity loss that results from limiting the productivity loss that results from limiting the interval open to flow. Suprisingly, the pressure transient behavior of wells subject to fluid influx across the bottom boundary has not been investigated until now. The goal of this study was to examine the drawdown and buildup behavior of partially penetrating wells subject to bottomwater partially penetrating wells subject to bottomwater drive and to draw conclusions about pressure transients in these cases. The results obtained in this study also could be used to examine the pressure behavior of partially penetrating wells in a tall steam column supported by a boiling vapor/liquid interface. More specifically, the objectives of this study are to (1) investigate the applicability of conventional techniques for estimating formation flow characteristics, particularly horizontal and vertical permeabilities in systems subject to bottomwater drive, permeabilities in systems subject to bottomwater drive, (2) examine the interaction between the sealed lateral boundaries and the constant-pressure bottom boundary and its implication on well test analysis, and (3) determine the special characteristic features of the shapes of drawdown and buildup curves so that they can be used to identify bottomwater-drive systems. Mathematical Formulation The isometric and top views of the system under study are shown in Fig. 1. The following assumptions are made.1. The reservoir is a parallelepiped with a square drainage area A, uniform thickness h, and porosity phi. It has horizontal and vertical permeabilities k and phi. It has horizontal and vertical permeabilities k and kz, respectively - i.e., an orthotropic (anisotropic) system is considered. The well, which partially penetrates the formation, is at the center of the penetrates the formation, is at the center of the drainage area. It has an infinitesimally small radius rw and length hw.2. A single-phase slightly compressible liquid with compressibility c and viscosity mu flows from the reservoir into the wellbore at a constant reservoir rate qB.3. Initially, the pressure pi is uniform throughout the reservoir. While the upper and the lateral boundaries are sealed, the bottom boundary is kept at a constant pressure equal to the initial pressure. Gravity effects are not included. JPT P. 1251

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*32 (06): 1065–1077.

Paper Number: SPE-4629-PA

Published: 01 June 1980

...

**dimensionless****time**application match point Reservoir Surveillance drillstem/well testing exponential decline rate-time data production forecasting curve analysis rate-time equation inner boundary decline curve analysis equation Trans type curve constant pressure backpressure journal...
Abstract

This paper demonstrates that decline curve analysis not only has a solid fundamental base but also provides a tool with more diagnostic power than has been suspected previously. The type curve approach provides unique solutions on which engineers can agree or shows when a unique solution is not possible with a type curve only. Introduction Rate-time decline curve extrapolation is one of the oldest and most often used tools of the petroleum engineer. The various methods used always have been regarded as strictly empirical and generally not scientific. Results obtained for a well or lease are subject to a wide range of alternate interpretations, mostly as a function of the experience and objectives of the evaluator. Recent efforts in the area of decline curve analysis have been directed toward a purely computerieed statistical approach, its basic objective being to arrive at a unique "unbiased" interpretation. As pointed out in a comprehensive review of the literature by Ramsay, 1 "In the period from 1964 to date (1968), several additional papers were published which contribute to the understanding of decline curves but add little new technology." A new direction for decline curve analysis was given by Slider 2 with his development of an overlay method to analyze rate-time data. Because his method was rapid and easily applied, it was used extensively by Ramsay in his evaluation of some 200 wells to determine the distribution of the decline curve exponent b . Gentry's 3 Fig. 1 displaying the Arps 4 exponential, hypbolic, and harmonic solutions all on one curve also could be used as an overlay to match all of a well's decline data. However, he did not illustrate this in his example application of the curve. The overlay method of Slider is similar in principle to the log-log type curve matching procedure presently being employed to analyze constant-rate pressure buildup and drawdown data. 5–9 The exponential decline, often used in decline curve analysis, readily can be shown to be a long-time solution of the constant-pressure. 10–13 It followed then that a log-log type curve matching procedure could be developed to analyze decline curve data.

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*31 (03): 362–372.

Paper Number: SPE-6838-PA

Published: 01 March 1979

... 1979 1979. Society of Petroleum Engineers hydraulic fracturing drillstem/well testing flow in porous media Upstream Oil & Gas fracture length Drillstem Testing Fluid Dynamics gas well

**dimensionless****time**Formation Permeability fracture flow capacity flow capacity performance...
Abstract

This paper discusses how to analyze past performance and predict futureperformance of tight gas wells stimulated by massive hydraulic fracturing (MHF)using finite fracture flow-capacity type curves. The limitations ofconventional pressure transient analysis and other methods of evaluating MHFtreatment are discussed. A set of constant well-rate and wellbore-pressure typecurves is presented. Introduction Because of the deteriorating gas supply situation in the U.S. and theincreasing demand for energy, the current trend is to consider seriously theexploitation and development of low-permeability gas reservoirs. This has beenpossible because of changes in the economic climate and advances in wellstimulation techniques, such as massive hydraulic fracturing (MHF). It nowappears that MHF is a proven technique for developing commercial wells inlow-permeability or "tight" gas formations. As the name implies, MHF isa hydraulic fracturing treatment applied on a massive scale, which may involvethe use of at least 50,000 to 500,000 gal treating fluid and 100,000 to 1million lb proppant. The purpose of MHF is to expose a large surface area ofthe low-permeability formation to flow into the wellbore. A low-permeabilityformation is defined here as one having an in-situ permeability of 0.1 md orless. Methods for evaluating a conventional (small-volume) fracturing treatmentare available, but the evaluation of an MHF treatment has been a challenge forengineers. To evaluate the success of any type of fracture stimulation, prefracturing rates commonly are compared with postfracturing production rates.These comparisons are valid qualitatively if both pre- and postfracturing ratesare measured under similar conditions (that is, equal production time, samechoke sizes, minimal wellbore effects, etc.). Unfortunately, to evaluate thesuccess of different kinds of fracturing treatments, pre- and postfracturingproduction rates often are measured pre- and postfracturing production ratesoften are measured and compared using not only the same well tested underdissimilar conditions, but also the same kind of comparisons between differentwells that may even have different formation permeabilities. Thus, resultsoften are invalid and may cause misleading conclusions. Moreover, suchcomparisons do not help predict long-term performance. To predict long-termperformance for MHF wells, reliable estimates of fracture length, fracture flowcapacity, and formation permeability are needed. Pressure transient methods for analyzing wells with small-volume fracturingtreatments are based on the concept of infinite or high fracture flow capacityand are used to determine the effectiveness of a stimulation by estimating thefracture length. Our experience indicates that these methods are not adequatefor analyzing wells with finite flow-capacity fractures. Such methods provideunrealistically short fracture lengths for MHF wells provide unrealisticallyshort fracture lengths for MHF wells with finite flow-capacity fractures.Furthermore, fracture flow capacities cannot be determined. Includes associated paper SPE 8145, "Type Curves for Evaluation andPerformance Prediction of Low-Permeability Gas Wells Stimulated by MassiveHydraulic Fracturing."

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*29 (01): 73–80.

Paper Number: SPE-5845-PA

Published: 01 January 1977

... Drillstem Testing pressure data pressure distribution Upstream Oil & Gas

**dimensionless****time**observation well exp fracture orientation hydraulic fracturing orientation Formation Permeability fractured well determination reservoir dimensionless distance journal of petroleum technology...
Abstract

An analytic solution for the pressure distribution around a fractured well in a homogeneous, infinite reservoir is presented in terms of tabulated functions. This solution can be used to determine fracture orientation from interference tests. Introduction The advent of costly tertiary recovery programs has resulted in the need to determine flow patterns created by fractures intersecting wellbores. This requires a knowledge of the compass orientation and length of fractures. Several papers have discussed determination of fracture length by pressure analysis. However, methods for determining the compass orientation using pressure data have not received much attention. This paper pressure data have not received much attention. This paper discusses a method for determining the compass orientation using pressure data obtained at shut-in observation wells due to pressure data obtained at shut-in observation wells due to production or injection at the fractured well. production or injection at the fractured well. In 1960 Elkins and Skov reported a method for determining the orientation of natural fractures. They assumed the system of natural fractures to behave like an anisotropic system and used the classical line-source solution to analyze, pressure behavior. More recently, Pierce et al. demonstrated the use of pulse testing for calculating fracture length and orientation. Other studies on the determination of fracture orientation include (1) use of acoustic signals; (2) measurement of pressure response at observation wells during fracture initiation and propagation; and (3) use of inflation impression packers and television cameras.Even though the problem has been recognized for quite some time, no general method exists for determining pressure distribution in the area drained by a vertically fractured well. The objective of this paper is to present an analytic expression, in terms of simple tabulated functions, for the pressure distribution in a uniform, homogeneous, infinite reservoir drained by a vertically fractured well. This solution may be used to analyze pressure data in adjacent observation wells to determine fracture orientation in a manner analogous to standard interference tests. It also may be used to analyze test data obtained by pulsing a vertically fractured well. pulsing a vertically fractured well. Theory The mathematical model considered in this study is identical to the uniform-flux fracture examined by Gringarten et al. We assume that a single vertical fracture intersects a wellbore located in an infinite, homogeneous, porous medium (Fig. 1). The surface production rate is assumed to be constant and all the production is obtained by means of the fracture. Fluid enters production is obtained by means of the fracture. Fluid enters the fracture at the same rate per unit area of the fracture (uniform-flux fracture). JPT P. 73

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*27 (11): 1392–1400.

Paper Number: SPE-5131-PA

Published: 01 November 1975

...................................(2) JPT P. 1392 1 11 1975 1 11 1975 1 11 1975 1 11 1975 1975. Society of Petroleum Engineers drillstem/well testing pseudoskin factor line source Upstream Oil & Gas

**dimensionless****time**wellbore pressure unsteady-state pressure distribution formation thickness...
Abstract

Analysis of a solution derived to study, the unsteady-state pressure distribution created by a directionally drilled well indicates that the slant of a fully penetrating well creates a negative skin effect that is a function of the angle of slant and the formation thickness. Calculation of this pseudoskin factor permits evaluation of the actual well condition. Introduction Many methods have been developed to analyze transient wellbore pressure data to determine the formation permeability, porosity, average pressure, and well permeability, porosity, average pressure, and well condition. These methods usually are based on solutions of unsteady-state flow problems that consider a fluid flowing toward a fully penetrating well that is perpendicular to the upper and lower formation boundary perpendicular to the upper and lower formation boundary planes. Actually, most wells do not penetrate the planes. Actually, most wells do not penetrate the producing formation perpendicularly. Instead, there is a producing formation perpendicularly. Instead, there is a certain angle between the normal to the formation plane and the well axis, such as when a vertical well penetrates a dipping formation or when a directionally drilled penetrates a dipping formation or when a directionally drilled well penetrates a horizontal formation. These kinds of wells are called "slanted wells." Although such wells are common, there appears to have been only one study of the performance of such completions. Roemershauser and Hawkins studied steady-state flow in a reservoir producing through a fully penetrating, slanted well using an electrical model. They considered a circular reservoir of finite extent and concluded that the slant of a fully penetrating well causes an increase in the well productivity. The increase in well productivity results from the decrease in the resistance productivity results from the decrease in the resistance to flow around the wellbore caused by an increase in the producing-interval area exposed to flow. This increase producing-interval area exposed to flow. This increase in well productivity indicates that a fully penetrating, slanted well creates a negative skin effect. Roemershauser and Hawkins graphed the increase in well productivity vs the angle of slant of the well. There productivity vs the angle of slant of the well. There appears to have been no study of the unsteady-state performance of slanted wells. performance of slanted wells. Mathematical Derivation The unsteady-state laminar flow of a slightly compressible fluid through an anisotropic, homogeneous, porous medium can be described after assuming small porous medium can be described after assuming small pressure gradients everywhere in the reservoir and pressure gradients everywhere in the reservoir and neglecting gravity effects: +++=, ......(1) wherekr= ------- = constant.ct In Eq. 1, it is also assumed that the horizontal permeabilities kx and ky are equal and constant, thus permeabilities kx and ky are equal and constant, thus equaling kr. This assumption is not necessary, and the following results can be generalized to the case of simple anisotropy, where kx, ky, and kr are all constant but are not equal, by redefinition of the horizontal variables x and y. For example, we define z' as z' = z kr/kz................................(2) JPT P. 1392

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*27 (07): 887–892.

Paper Number: SPE-5496-PA

Published: 01 July 1975

... Gringarten graph drillstem/well testing

**dimensionless****time**semilog straight line linear flow period Drillstem Testing Ramey horizontal fracture fractured well fracture Upstream Oil & Gas log-log graph fracture plane application straight line reservoir permeability infinite conductivity...
Abstract

A number of recent studies have resulted in an increased understanding of fractured-well behavior. Two of these studies provide new information on applying log-log type-curve matching procedures to pressure data obtained from fractured wells. This paper compares the applicability of type-curve and conventional semilog methods. Introduction The pressure behavior of fractured wells is of considerable interest because of the large number of wells that intersect fractures. As a result of a number of studies, an increased understanding of fractured-well behavior has been obtained. Although the shape of actual fractures is undoubtedly complicated, most studies assume that real fractures may be ideally visualized as planes intersecting the wellbore. It is generally believed that hydraulic fracturing normally results in one vertical fracture, the plane of which includes the wellbore; however, it is also plane of which includes the wellbore; however, it is also agreed that, if formations are shallow, horizontal fractures can result. The specific orientation of the fracture plane with respect to the wellbore may be subject to debate if the well intersects a natural fracture. Two recent studies provide new information whereby log-log type-curve matching procedures may be applied to pressure data obtained from fractured (vertical or horizontal) wells. These studies also showed that, under conditions that would appear normal, it is likely that horizontal and vertical fractures would affect well behavior sufficiently such that the orientation, vertical vs horizontal, could be determined. The purpose of this paper is to illustrate the applicability of the results paper is to illustrate the applicability of the results obtained in Refs. 1 and 2. Vertically Fractured Wells As mentioned in Ref. 1, new solutions for the transient pressure behavior of a vertically fractured well were pressure behavior of a vertically fractured well were needed because earlier studies were not blended for type-curve analysis. This study examined two boundary conditions on the fracture plane. The first solution, like earlier studies, assumed that the fracture plane is of infinite conductivity. This implies that there is no pressure drop along the fracture plane at any instant in pressure drop along the fracture plane at any instant in time. The second solution, called the uniform-flux solution, gives the appearance of a high, but not infinite, conductivity fracture. (This boundary condition implies that the pressure along the fracture plane varies.) Application of these solutions to field data indicates that the uniform-flux solution usually matches pressure behavior of wells intersecting natural fractures better than does the infinite-conductivity solution. On the other hand, the infinite-conductivity solution often matches the behavior of hydraulically fractured, propped fractured wells better than does the uniform-flux solution. The Infinite-Conductivity Vertical Fracture in A Square Drainage Region Gringarten et al. have presented drawdown data for an infinite-conductivity vertical fracture located at the center of a closed-square drainage region and producing a lightly compressible constant-viscosity fluid at a constant rate. The solution for the producing pressure at time t is kh PwD (tD, Xe/Xf) = (pi - pwf),......(1) 141,2 qB where 0.000264 kt tD = .........................(2) c Xf2 JPT P. 887

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*27 (01): 115–116.

Paper Number: SPE-4954-PA

Published: 01 January 1975

... pressure transient testing Colpitt pressure drawdown test pressure transient analysis Drillstem Testing analysis procedure Test Time production monitoring Upstream Oil & Gas

**dimensionless****time**Reservoir Surveillance drillstem/well testing flow rate dimensionless basis Winestock variable...
Abstract

Use of pressure drawdown tests to characterize gas wells has long been accepted, in principle, to be an excellent formation evaluation technique. In practice, however, the theoretical requirement that flow rates must be maintained strictly constant during the test has led to two major problems: the constant flow rate requirement is frequently ignored, and use of conventional analysis on such tests has led to serious error in many cases, leading to a loss of confidence in drawdown testing; or the constant flow rate is maintained, but only with a great deal of difficulty. Winestock and Colpitts showed that both problems could be avoided if gas-well drawdown tests were run with a fixed choke - usually resulting in a slowly decreasing flow rate during the test - if a simple modification of conventional analysis theory is employed. In fact, Winestock and Colpitts claimed that the method they proposed would be adequate in general for smoothly varying flow rates, even though the total change in flow rate from beginning to end of a test was large. This claim requires justification because the method proposed by Winestock and Colpitts is an approximation, and it clearly must fail when the rate of change in flow rate is sufficiently large. The purpose of our investigations, therefore, was to find the limits of applicability of the Winestock and Colpitts method. First, we reviewed their proposed method and the reason for the method. A frequently used equation describing a pressure drawdown test in a gas well is pressure drawdown test in a gas well is+F (1) In this equation all terms except flowing bottom-hole pressure, Pwf, and test time, t, are assumed to be constant. Winestock Pwf, and test time, t, are assumed to be constant. Winestock and Colpitts point out that use of the equation in this form can lead to serious errors if it is applied to analyze drawdown tests in which flow rates vary even slightly during the test. They propose, however, that for a test with flow rate varying slightly, the test is modeled adequately by a simple rearrangement of Eq. 1: = 712 (2) They propose that the entire group on the left side of Eq. 2, consisting of two variables, bottom-hole flowing pressure, Pwf, and gas production rate qr, both measured pressure, Pwf, and gas production rate qr, both measured during the test, be plotted against the logarithm of test time, t. The permeability thickness product would then be a function of the slope of the resulting line. The turbulence constant, F B, must be obtained before this graph can be prepared. P. 115

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*26 (09): 1035–1043.

Paper Number: SPE-4559-PA

Published: 01 September 1974

... Oil & Gas

**dimensionless****time**reservoir pressure transient period reservoir well-test analysis Well-Test Analysis for Wells Producing From Two Commingled Zones of Unequal Thickness R. Raghavan, SPE-AIME, Amoco Production Co. H. N. Topaloglu, Turkish Petroleum Corp. W. M. Cobb, SPE-AIME...
Abstract

Established methods of analyzing pressure buildup for two-layer no-crossflow systems are extended here to include the effect of the thickness of each zone for a wide range of permeability ratios. In addition, methods are presented for estimating the permeability of the individual layers. Introduction In 1961, Lefkovitz et al. presented solutions describing the pressure behavior of a well producing at a constant rate from a bounded, producing at a constant rate from a bounded, noncommunicating multilayer reservoir. Their study provided a basis for pressure test analysis of wells producing from commingled zones. They recommended a Homer graph for determining average formation flow capacity but found it unsatisfactory for the evaluation of mean or average reservoir pressure. As a result, they suggested that the pressure. As a result, they suggested that the Muskat graph be used to calculate the static reservoir pressure. More recently Cobb et al., using the results of Lefkovitz et al., examined the pressure behavior of a two-layer reservoir for a wide range of producing and shut-in conditions. They assumed that each zone was of equal thickness and they presented results along lines suggested by the general pressure buildup theory for a wide range of permeability ratios. An important conclusion of Ref. 4 was that the permeability and thickness of each individual layer permeability and thickness of each individual layer cannot be evaluated by the conventional semilog techniques. Accordingly, they recommended that an independent effort be made to determine the individual layer characteristics. The primary objective of this paper is to extend the pressure buildup analysis of wells producing from two commingled zones by including the effect of the thickness of each zone for a wide range of permeability ratios. The results presented in this paper correspond to thickness ratios of 2 and 5. It will be shown that the results may also be used to analyze reservoirs with thickness ratios of 1/2 and 1/5, provided the permeability ratio is less than or equal to 10. The permeability ratio is less than or equal to 10. The second objective of this paper is to present methods for estimating the permeability of the individual layers. These methods may also be applied to earlier work related to two-layer commingled fluid production. We consider here a two-layer reservoir that is horizontal and cylindrical; it is enclosed at the top, bottom, and at the external drainage radius by an impermeable boundary. Each layer is homogeneous and is filled with a fluid of small and constant compressibility. The pressure gradients are small, and gravity effects are negligible in the reservoir. The porosity of the layers is assumed to be equal; the porosity of the layers is assumed to be equal; the permeability and thickness of the two zones are the permeability and thickness of the two zones are the parameters under investigation. The initial reservoir parameters under investigation. The initial reservoir pressure is the same in both layers and the surface pressure is the same in both layers and the surface production rate is constant. Finally, it is also assumed production rate is constant. Finally, it is also assumed that the instantaneous sand-face pressure is identical in both layers. Analysis of Dimensionless Pressure And Time Data For convenience of discussion we shall use the dimensionless variables listed below, defined in English engineering units. JPT P. 1035