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### NARROW

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H.J. Ramey

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Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*44 (06): 650–659.

Paper Number: SPE-20592-PA

Published: 01 June 1992

Abstract

Distinguished Author Series articles are general, descriptiverepresentations that summarize the state of the art in an area of technology bydescribing recent developments for readers who are not specialists in thetopics discussed. Written by individuals recognized as experts in the area, these articles provide key references to more definitive work and presentspecific details only to illustrate the technology. Purpose: to informthe general readership of recent advances in various areas of petroleumengineering. Abstract The decade of the eighties produced important findings in well testanalysis. However, the large number of new and sometimes competitive methodsalso has produced confusion. The main objective of this paper is to considerthe current state of practical well test analysis methods. Often new studiesproduce conclusions that time proves incomplete or partly untrue. The storagelog type curve was initially presented as a method to analyze shorttime data. This was later found to be impossible. But the diagnose tic value of thelog-log curves was far more important than shorttime analysis. Then later thederivative was added to the type curve and one conclusion was that Hornercurved were no longer necessary and that short time analysis was now possible. Neither conclusion is entirely correct, yet the diagnostic value of thederivative remains. Another major development was computer aidedinterpretation. The computer was necessary to differentiate data, and of greathelp in preparing the large number of graph required for modern interpretation. An important breakthrough resulted with development of nonlinear regression forspecific models and the ability to consider rate variation. Results of aninterpretation could be used to simulate the test data and a comparison ofsimulation and field data made. The regression coefficient or confidence limitprovided a quantitative measure of the agreement of field data with the modelchosen. Results may also be used to determine where a correct straightlinecould be on a Homer buildup graph. This procedure proven that it is verydifficult to find a Horner straightline slope with the precision previuslythought possible. Widespread use of electronic pressure gauges and computerdata acquisition has created net problems for a well test analyst. A newproblem revealed by type curve analysis is presented. Introduction A 1976 study reviewed important findings of the previous decade and stressedpractical applications of new methods not described in the originalpublications. At that time. three log-log versions of the storage and skinproblem had appeared, and several type curves for fractured wells had beenpresented. Many well test analysts threw up their hands at the proliferation oftype curves. The 1976 study concluded that a major advantage of the log-logtype curve was that it was usually possible to identify the flow model and findthe start of a semilog straightline for the appropriate model. It wasrecommended that a Horner straightline should still be used as the final basisof analysis where possible. The 1976 study pointed out problems with fracturetype curves (short apparent fracture lengths for large jobs), and otherexisting worries on selection of an appropriate type curve. One major advantageof the understanding that new methods brought was that it was possible tocorrect bad test data and fill in missing data in many cases. Since 1976, problems identified in that study have been solved. and true breakthroughspresented. The purpose of this study is to present useful practical methods forwell test analysis and design. Important new information includes selection ofan industry standard storage type curve. development of derivative methods, solution of the finite fracture conductivity problem. and development ofcomputer aided interpretation and design. STORAGE TYPE CURVE The traditional engineering dilemma is two methods to solve a problem whichyield different answers. Well test analysis is replete with this problem. Papers often say something like the data was analyzed by the Smith methodyielding 20 md, the Jones method yielding 2 darcies, and the von Schultz typecurve yielding 0.15 md. The average is …" Remarkably, the three differentmethods are usually different graphs of the same solution. The second well testanalysis monograph by R. C. Earlougher, Jr. was published in 1977, but thelog-log type curves described for the first time in an SPE monograph wereconsidered controversial by the SPE Board of Directors. An early SPE Boarddecision was not to publish full-scale log-log type curves as this wouldindicate SPE approval of log-log type curves, or a particular type curve. Fortunately, this decision was reversed. Gringarten et al. ended thecontroversy over the best form of the wellbore storage and skin effect typecurve in 1978. Their type curve of the log of dimensionless pressure vs the logof tD/CD with CD exp 2s as a parameter has become the industry standard. Theiroriginal type curve combined radial flow and fracture flow results andindicated the effect of producing time on buildings. It was a remarkableimprovement which caught immediate acceptance. So one problem identified inref. 1 was solved.

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*34 (11): 2656–2666.

Paper Number: SPE-10178-PA

Published: 01 November 1982

Abstract

Summary The use of type-curve matching in pressure-transientanalysis has become an essential method for estimatingreservoir parameters with short-time data. Although theoriginal type-curve matching techniques were based onsolutions for pressure drawdown problems, thesesolutions have been extended, and justifiably so, to pressurebuildup problems. However, there has been anindiscriminate use of these solutions interchangeablywithout proper justification for each application. As thispaper points out, the drawdown solution is not the same paper points out, the drawdown solution is not the same as the buildup solution for a fractured well producing athigh flow rates before shut-in. The bilinear period, however, is observed in the buildup solution. In this paper the effects of producing time on pressurebuildup analysis also are investigated for fractured wellsat low and high flow rates. An example is presented foranalyzing short-time data for a fractured well producingat high flow rates. Introduction Pressure buildup testing is probably the best known and Pressure buildup testing is probably the best known and most extensively used of all transient well-testingtechniques. During a pressure buildup test, a producing wellis shut in, and the pressure/time performance of the wellis recorded and analyzed for reservoir properties andwellbore conditions. The latter are analyzed on the basisof the early-time period of the test. In general, pressure buildup analysis can becomplicated by many factors such as producing time andwellbore and boundary conditions. Recently, Agarwalpresented a study showing that the effects of producing presented a study showing that the effects of producing time on pressure buildup analysis can be eliminatedcompletely, by replacing the dimensionless shut-in time by anequivalent dimensionless term, teD. Such a procedurenormalizes the effects of producing times, and henceconventional drawdown type curves can he used. Raghavan made a similar attempt to account forproducing time effects by presenting new type curves for producing time effects by presenting new type curves for dimensionless pressure rise vs. dimensionless shut-intime, with producing time as a parameter. We havefound this concept a useful one and have incorporatedthe dimensionless pressure rise concept with theAgarwal method and have included them in this study. Analyzing early-time pressure buildup data forfractured wells is similar to techniques used for drawdowndata. The use of type-curve matching has improved thequality of this early-time analysis greatly. One of thefirst papers to introduce type-curve matching forfractured wells was by Gringarten et al. Their studypresented new values of dimensionless pressure drop vs. presented new values of dimensionless pressure drop vs. dimensionless time for a fracture in an infinite reservoir.Several cases were studied involvinginfinite-conductivity vertical fractures, uniform-flux verticalfractures, and horizontal fractures. In 1976, Cinco-Ley et al. used a semianalyticalmethod to simulate the transient-pressure behavior of awell with a finite-conductivity vertical fracture in aninfinite slab reservoir. Type curves as well assemilogarithmic analysis methods were presented foranalyzing the short-time data. Agarwal et al. presentedsimilar results for the constant-rate case and introducedtype curves for the constant-pressure case. This analysiswas designed mainly for massive hydraulically fracturedwells. JPT P. 2656

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*34 (05): 976–988.

Paper Number: SPE-9272-PA

Published: 01 May 1982

Abstract

Summary Well test methods established in petroleum engineering have been applied successfully to geothermal wells. Modifications have been necessary because of property differences and distinctive geometries of geothermal fields. This paper presents a comprehensive state of the art in pressure transient analysis of geothermal steam wells. The techniques encompass drawdown and conventional buildup as well as the newer fractured parallelepiped models. The latter have been used successfully in the analysis of field data from Larderello, Italy, and The Geysers in California. Field examples follow the presentation of each technique. Introduction Two publications - by Homer and Miller et al. - have formed what generally is recognized as the basis of modem well test analysis. Significant contributions to the understanding of fundamental concepts were made by van Everdingen and Hursts and Matthews et al. The methods described in these papers have been applied successfully to geothermal wells. Notable are the publications by Ramey, Ramey and Gringarten, and Barelli et al. Recently, considerable efforts have been made to describe geothermal pressure transient analysis assuming distinct geometries penetrated by fractures. These configurations have been used successfully to describe the geothermal reservoirs at Larderello and The Geysers. The results were presented by Barelli et al., Cinco-Ley et al., and Economides et al. Drawdown testing also has generated attention. New techniques have been developed to contend with the frequent shut-ins and flow rate fluctuations. Methods that use effluent chemistry may offer an assessment of the liquid reserves. Methods outlined in this paper assume a vapor-dominated zone within the reservoir. "Immobile" liquid water throughout the reservoir has been proved experimentally by Hsieh. He discovered that absorbed water may account for as much as 15 times the amount of fluid in the vapor phase for typical conditions found in geothermal steam reservoirs. Work is now under way at the U. of Alaska on the effect of a "source term" in the solution of the diffusivity equation. Developement of Fundamental Concepts Well test analysis techniques and basic equations were derived from the solutions of partial differential equations describing fluid flow through porous media. The most familiar form of the flow equation is (1) This equation-the result of the continuity principle, Darcy's law, and an equation of state-presumes radial flow, homogeneous (constant beta and isotropic (constant k) medium, uniform thickness (constant h), fluid of small and constant compressibility (c), constant viscosity (mu), no gravity effects, and single-phase flow. Although some of these assumptions often are violated, they have been proved flexible and several solutions have represented real cases sufficiently. Traditionally, three major categories of "drainage area" have been considered: infinitely acting, no flow outer boundary, and constant-pressure outer boundary. The solution most commonly encountered in the classic references of Carslaw and Jaeger and Muskat is for an infinite reservoir. The "line-source solution" in particular describes the pressure history at the wellbore. JPT P. 976^

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*28 (09): 1097–1106.

Paper Number: SPE-5596-PA

Published: 01 September 1976

Abstract

Pressure buildup, interference, and pulse tests in a naturally fractured Pressure buildup, interference, and pulse tests in a naturally fractured dry gas reservoir are influenced by reservoir limits. Type curves are matched to test data to estimate drainage area and to compute porosity and permeability. Calculated porosity and permeability values compare well permeability. Calculated porosity and permeability values compare well with published data for natural fracture systems. Introduction The case studied is a dry gas reservoir in which three wells are completed. The wells are spaced 2 and 8 miles apart in a 10-mile line along the crest of an anticline with about 100 sq miles of closure (Fig. 1). The dashed contour in Fig. 1 is the drainage boundary that was initially estimated from geologic and production test data assuming a uniform gas-water contact. This drainage area is about 18 miles long and 3 miles wide. Only one productive stratigraphic unit is common to all three productive stratigraphic unit is common to all three wells. This is a naturally fractured zone of thinly bedded, clean orthoquartzites that accounts for 90 percent of deliverability at Well 1, 95 percent at Well 2, and 100 percent at Well 3. Type of completion, fractured zone thickness, and other reservoir data are presented in Table 1. No cores were taken directly from the naturally fractured orthoquartzite zone, but cores from other orthoquartzites had 2.5-percent average porosity and less. than 0.1-md permeability to air. Test data studied in this field case history have two chronological groupings: data recorded when Well 2 was completed, consisting of one pressure drawdown and four pressure buildup tests at Well 2; and data obtained 4 years later, consisting of pressure interference at Wells 3 and 1 caused by flowing Well 2 for 450 hours, pressure buildup at Well 2 immediately following the interference test, and pulse response at Well 3 caused by pulsing Well 2. The field was never on production except to conduct pressure transient and production except to conduct pressure transient and deliverability tests. Analysis of the field tests data is organized into four sections: (1) general discussion of the pressure drawdown and buildup behavior in light of recently published well-test theory; (2) computation of porosity and published well-test theory; (2) computation of porosity and estimation of drainage area by matching the buildup data to type curves; (3) computation of porosity, permeability, and drainage area by matching the permeability, and drainage area by matching the interference data to type curves; and (4) analysis of pulse behavior in the presence of reservoir limits. General Pressure-Buildup Behavior Buildup Tests 1 through 4, recorded at completion of Well 2, are presented in Tables 2 through 5. The pressure drawdown corresponding to Buildup Test 4 is also pressure drawdown corresponding to Buildup Test 4 is also shown in Table 5. Fig. 2 is a graph of pressure as a function of the logarithm of time for the drawdown test. All four buildup tests are plotted in Fig. 3, using the technique of Horner. Pressure buildup during Test 1 becomes a linear function of the logarithm of the Horner time ratio, and extrapolates to initial pressure at infinite shut-in time. Each of the other tests plotted in Fig. 3 has an early period in which pressure is a linear function of the period in which pressure is a linear function of the logarithm of the Horner time ratio and a late period in which pressure bends upward. JPT P. 1097

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

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 (10): 1290–1298.

Paper Number: SPE-5319-PA

Published: 01 October 1975

Abstract

Many formations, such as channel sand, appear to exhibit simple ky-kxanisotropy. Directional permeability has an important effect onplanning fluid-injection oil recovery. The method discussed in thispaper uses data obtained during water injection to determine major andminor permeability axes and the orientation of the unknown permeability matrix. Introduction It has long been known that many formations appear toexhibit simple ky-kx anisotropy. This model also may beapplicable for formations containing trending fracturepatterns. Knowledge of directional permeabilityobviously would have an important effect on planningreservoir development, particularly for fluid injectionoperations. A recent study by Papadopulos outlines methods for interference analysis in anisotropic formations. Themethod is used here to analyze a field water-injection testto determine major and minor permeabilities and theorientation of the unknown permeability matrix. The method also may be extended to pressure falloff (orbuildup) interference analysis. The theory of flow of either fluid or heat through ananisotropic medium is well established. A brief reviewof this information is sufficient here. We consider that awell is produced at a constant volumetric rate in aninfinite, anisotropic medium. The formation has aconstant thickness and porosity, and the total systemeffective compressibility is constant. Collins has presentedperhaps the best-known solution to this problem in thearea of oil production technology. In the mid-1960's, aseries of papers appeared in the groundwater hydrologyliterature dealing with the problem of well testanalysis for anisotropic aquifers. Studies by Hantush, Papadopulos, and Walton are particularly notable. ThePapadopulos study appears to be well-suited for applicationto analysis of oil and gas well test data and serves asthe basis for this study. The purpose of this paper is topresent analysis of actual field data with the Papadopulosmethod in a form readily useable by petroleum engineers.The method is extended to falloff data. Theory The Collins solution for the pressure field caused by awell producing from an anisotropic reservoir is correct, but it assumes the directions of the major and minorpermeability axes are known and are aligned with thewell-location coordinate system. In the general well testanalysis situation, the direction of the major permeabilityaxis would be unknown. Fig. 1 shows the known welllocation x-y coordinate system with the unknown permeability axes, X-Y, oriented at some unknown angle 0.The pressure at (x, y, t) caused by a line-source well at theorigin was presented by Papadopulos: 2 h (pi - px, y, t)kxxkyy - kxy ----------------- =141.2 q B 1- Ei .....(1)2 kxx = 1/2{(kxx+kyy) + [(kxx-kyy)2+4kxy2] 1/2}....(2) kyy = 1/2{(kxx+kyy) − [(kxx-Kyy)2+4kxy2]1/2}.....(3) kxx − kxx= arctan ......................(4)kxy JPT P. 1290^

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

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*26 (09): 1035–1043.

Paper Number: SPE-4559-PA

Published: 01 September 1974

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

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*26 (01): 85–92.

Paper Number: SPE-3913-PA

Published: 01 January 1974

Abstract

A somewhat neglected method of calculating bottom-hole pressures, that of Sukkar and Cornell, does not involve trial and error and is very fast for hand calculation; but it does not allow for the severe conditions of modern gas wells. Here the method is refined to extend the pressures and temperatures, to allow for sour gas, and to make results readily accessible for hand calculation. Introduction Modem trends in gas well drilling have been toward deep, hot, high-pressure completions. Currently, gas wells are being drilled to depths of about 25,000 ft in the southwestern U. S. As would be expected, bottom-hole pressures for these wells are high (about 16,000 psi), and bottom-hole temperatures are in excess of 400 deg. F. The gas produced from many of these deep wells is sour. To run bottom-hole pressure bombs is costly. and because the sour gas rapidly corrodes wirelines it may also be disastrous.There is often no indication of liquid in these wells, so it is possible to compute bottom-hole pressures from measured wellhead data. Of the many methods of computing bottom-hole pressure, perhaps the best known are the method for static and flowing gas columns outlined in the State of Texas backpressure manual, and the static and flowing gas column method described by Cullender and Smith. The static method described in Ref. 1 is the Rzasa and Katz Method II, related to the older Rawlins and Schellhardt method. The flowing method in Ref. 1 is a modification of the static column method, which is based on estimating flowing friction from the Weymouth gas flow equation. Although the static method in Ref. 1 can be used with several depth increments to provide reasonable answers for deep wells, the flowing method is not recommended for deep, high-rate wells. The Cullender and Smith method is based upon a mechanical energy balance and is generally reliable for both static and flowing gas columns. But methods in both Ref. 1 and Ref. 2 involve tedious trial-and-error solutions. The Cullender and Smith method is best solved by computer if many determinations are required. There is a method that does not involve trial and error, however, that is very fast for hand calculations; but surprisingly, this method has received very little attention and use. This paper is based on that technique.A method presented by Fowler, involves integrated values of the gas law deviation factor, Z, with pressure, and is a direct method of calculating static pressure, and is a direct method of calculating static bottom-hole pressure, assuming a constant average temperature. Sukkar and Cornell extended Fowler's analysis and presented a general approach to calculating both static and flowing bottom-hole pressures for pure natural gas. They derived a pressure integral for perfectly vertical pipe by assuming negligible kinetic energy change, steady-state isothermal flow, and no work done by the gas in flow. Sukkar and Cornell evaluated the integral generally in terms of pseudoreduced pressures. The integral contains the pseudoreduced pressures. The integral contains the gas law deviation factor as a function of pressure, but it is assumed that temperature can be treated as an average over the depth range of interest. This is not an important weakness in the method because a depth of interest can be broken into several intervals to provide accuracy. The integral is based on a mechanical provide accuracy. The integral is based on a mechanical energy balance for a flowing column and is essentially identical with the Cullender and Smith result. JPT P. 85

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*25 (11): 1244–1250.

Paper Number: SPE-4371-PA

Published: 01 November 1973

Abstract

A case history is used to illustrate that wellbore storage can act differently in injection well falloff and injectivity tests. This can cause test curves to have shapes characteristic of mobility banks and thus render some falloff tests useless. Injectivity tests, however, may be interpretable. Introduction Injectivity or pressure falloff tests in injection wells are commonly used to investigate formation properties. Ideally, such tests provide information about formation permeability, skin factor, distance to fluid banks, and permeability, skin factor, distance to fluid banks, and distance to boundaries. Wellbore storage 9 and its effects on transient testing have been described in the literature. The storage of fluids in the wellbore due to compression or changing liquid level causes transient tests to act differently at short times than they would in the absence of wellbore storage effects; long-time behavior is essentially unaffected. Wellbore storage is a general term encompassing the more specific terms, "afterflow" (pressure buildup) and "unloading" (pressure falloff and drawdown); we use only the general term in this paper. The specific term commonly, applied to injectivity tests is wellbore storage.We show here that under certain circumstances wellbore storage effects, in particular changing wellbore storage, can make test interpretation for formation characteristics practically impossible. We include field data illustrating this problem, provide an explanation of the behavior of these data, make suggestions for injection well testing and test analysis, and illustrate the analysis technique. The problem of wellbore storage effects in injection well testing is much too broad and complex to be treated exhaustively in this paper, but we feel that the material presented is detailed enough to identify, illustrate, and at least partially solve the problem. partially solve the problem.We first encountered and finally recognized this problem as a result of a series of injectivity and problem as a result of a series of injectivity and falloff tests on several wells. The purpose of the testing was to locate and estimate the distance to fluid banks. Fig. 1 shows data from one of these falloff tests, from a 1,000-ft-deep water injection well. Pressure data are from a permanently installed surface-recording down-hole gauge. Fig. 1 has several bends that might be interpreted as banks, boundaries, interference from adjacent injection or production wells, commingled zones, etc. It is not difficult to pick four, five, or even six different slopes from this particular test. As a result of the multiple slope changes, there were great differences of opinion about how to interpret these data; test interpretation was never satisfactory to all involved.Fig. 2 shows injectivity test data for the same well. This figure also has several bends that might be interpreted as banks, boundaries, interference, etc. (The straight line in Fig. 2 is used later in an example calculation.) Detailed analysis of these two figures shows that they do not have the same sequence of slopes and that the slopes do not change at the same time. There are one or two places where the slopes do appear to be the same, but these slopes do not occur at equivalent times. Furthermore, we expect the general appearance of the injectivity and the falloff tests to be the same; clearly, the shapes of these two curves are quite different. JPT P. 1244

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*24 (01): 27–37.

Paper Number: SPE-3014-PA

Published: 01 January 1972

Abstract

Pressure buildup for two-layer no-crossflow systems has been carefully Pressure buildup for two-layer no-crossflow systems has been carefully studied to determine the proper application of conventional analysis methods. Results of using single-layer buildup plotting forms suggested by Muskat, Miller-Dyes-Hutchinson, and Horner indicate that---under well defined conditions-all three methods can also be applied to two-layer systems. Introduction The three most common graphical techniques used to interpret buildup behavior are the methods of Muskat, Miller-Dyes-Hutchinson, and Homer. Initially, all three methods were developed for a well producing from a reservoir consisting of a single homogeneous layer. (They were lately reviewed by Ramey and Cobb.) In recent years, however, investigators have conducted studies on wells with commingled fluid production from two or more noncommunicating zones. In those cases, fluid is produced into the wellbore from two or more separate layers and is carried to the surface through a common wellbore. The layers are hydraulically connected only at the wellbore. Lefkovits et al., and Duvaut have presented identical rigorous solutions that describe the pressure behavior of a constant-terminal-rate well producing from a bounded, noncommunicating, multilayer reservoir with contrasting properties. Both Lefkovits et al. and Papadopulos have properties. Both Lefkovits et al. and Papadopulos have presented pressure behavior for the infinitely large presented pressure behavior for the infinitely large mulitlayer case. Although much has been published on the behavior of noncommunicating layered systems, the knowledge of well-test applications must be considered elementary. Consequently, pressure buildup for two-layer, no-crossflow systems has been carefully analyzed to determine the proper application of conventional analysis methods to this class of reservoir system. It is remarkable that the duration of transients is often orders of magnitude longer for multilayer systems than for a single layer. The only existing method for determining fully-static pressure for layered systems requires that pseudosteady state be assumed, so one principal objective of this study was to arrive at improved methods of estimating fully static pressure. As a practical step, we decided to limit our attention to systems of only two commingled zones. Finally, it should be emphasized that certain of the following results were first presented by Lefkovits et al. in a pioneering study of pressure buildup in such systems. For example, Ref. 6 clearly describes the extended duration of transients in multilayered systems and thoroughly presents the pressure behavior during drawdown. But only scanty pressure behavior during drawdown. But only scanty information was presented for analysis of pressure buildup by means of the Horner plot, and the method recommended for determining static pressure was the Muskat trial-and-error plot. The Muskat method requires the assumption that the well had been produced long enough to reach pseudosteady state - a very long time as shown by Lefkovits et al. Determination of static pressure for such systems can be exceedingly important. We wish to emphasize that we believe our contribution through this study is a clear definition of the applicability of conventional pressure buildup analysis methods to this important class of problems. JPT P. 27

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*23 (12): 1493–1505.

Paper Number: SPE-3012-PA

Published: 01 December 1971

Abstract

Pressure buildup analysis, when properly done, yields the same results nomatter which of the conventional methods is used. As illustrated here with aclosed square, general interpretation equations may be derived that are correctfor closed drainage regions of any shape. Introduction In 1935, Theis 1 showed that buildup pressures in a shut-in waterwell should be a linear function of the logarithm of the time ratio( t +? t )/? t , and that the slope of the line is inverselyproportional to the mean formation effective permeability. Muskat discussedpressure buildup in oil wells in 1937, and proposed determination of staticpressure by a semilog trial-and-error plot that has been found to be applicableto a variety of buildup cases. In the late 1940's, van Everdingen presented aseries of lectures on well test analysis in the U.S. that related to aclassical study of unsteady flow by van Everdingen and Hurst. 3 In1951, Horner 4 presented a study of pressure buildup that appears tohave summarized fundamental efforts of a number of pioneering researchers inthe Shell companies. Horner also recommended a semilog buildup curve identicalwith the Theis curve, and presented a method for extrapolation to fully builtup static pressure for a closed circular reservoir. This sort of semilogpressure buildup plot is often referred to in the oil industry as a Hornerplot. About the same time, Miller-Dyes-Hutchinson 5 presented ananalysis for buildup when the well had been produced long enough to reachpseudosteady, or true steady state prior to shut-in. Their work indicated thatbuildup pressures should plot as a linear function of the logarithm of shut-intime. As in the Horner plot, the slope of the straight line was shown to beinversely proportional to the permeability to the flowing fluid. It isinteresting that about the same time Jacob 6 presented a similarapproach to determine aquifer transmissibility for a water well productiontest. Miller-Dyes-Hutchinson presented several important extensions of builduptheory. One was an initial attempt to apply buildup theory to multiphase flow. This problem was later resolved by Perrine. 7 More important to thisstudy, an approach to extrapolation to fully built-up static pressure waspresented for outer boundary conditions of either no flow (closed), or constantpressure (water drive). Thus, by the early 1950's, both Horner and Miller-Dyes-Hutchinson hadpresented methods for determining permeability and static pressure from buildupdata. Although there were similaries - both involved semilog plotting - therewere confusing differences between the methods. Perrine7 presented an excellentreview of this theory in 1956 that clearly indicates the state of understandingat that time. He stated that Horner-type plot (which he referred to as the vanEverdingen-Hurst method) was valid for a new well in a large reservoir, butthat the Miller-Dyes-Hutchinson approach was best for older wells in fullydeveloped fields. The latter is a widely held misconception.

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*22 (07): 837–838.

Paper Number: SPE-2871-PA

Published: 01 July 1970

Abstract

A materials balance for a volumetric gas reservoir leads to the well known conclusion that p/z should be a linear function of the cumulative gas produced. Data needed to prepare this balance usually take the form of tabulations of mean reservoir pressure and corresponding cumulative gas production in chronological sequence. It is then necessary to find corresponding gas law deviation factors, z's, compute p/z, and make the plot of p/z vs Gp. If the data form a straight line, it is possible to forecast pressures after any future cumulative gas production by finding the p/z for some future value of Gp. Thus performance p/z for some future value of Gp. Thus performance matching and production forecasting require finding p/z corresponding to some p, or finding the value of p p/z corresponding to some p, or finding the value of p that corresponds to a specific p/z. To aid these calculations, a large-scale plot of p/z vs p is often constructed. But if a plot of z vs p is available (as it usually is), this step is not necessary, and it is also not necessary to calculate p/z. This can be seen with the aid of Fig. 1. Fig. 1 is a conventional plot of z vs p that has been rotated 90 degrees to illustrate the basis of a simple graphical interpretation. The heavy line presents the relationship between z and p for a particular reservoir temperature and gas composition. First, assume that we wish to know the value of pressure that corresponds to a specific value of p/z, say 5,000 psia. The statement p/z = 5,000, or p = 5,000z represents the equation of a straight line on a graph of p vs z. The line passes through the point (0, 0), and also through the point where p = 5,000 and z = 1. This line is shown as the light line on Fig. 1. The intersection of the light line and the heavy line provides both the p and z that correspond to a p/z value provides both the p and z that correspond to a p/z value of 5,000. Now let us assume we want the p/z value corresponding to a given pressure p. Locate the pressure on the heavy line. Pass a straight line from the pressure on the heavy line. Pass a straight line from the origin (0, 0) through the point, and read p/z from the pressure scale where the line intersects the line z = 1. Thus it is not necessary to determine z to find p from p/z, or p/z from p - if a graph such as Fig. 1 p from p/z, or p/z from p - if a graph such as Fig. 1 is available. To summarize, any straight line through the origin on Fig. 1 will represent some constant value of p/z. The intersection of such a line with an appropriate z curve provides a point (p, z) corresponding to the particular value of p/z. This is a graphical solution particular value of p/z. This is a graphical solution of two simultaneous equations. This method could be used to find an empirical pressure coordinate scale for a plot of p vs Gp that would yield a straight line satisfying a gas materials balance. Often, z vs p plots prepared for field work have limited z and p scales such that the graphs can be read with good accuracy. An example is given in Fig. 2. Conventional alignment of the coordinate axes is used in this case. The point at the origin (0, 0) is not on the graph in this case. The graph can still be used readily to find values of p corresponding to p/z values. Again, assume we wish the pressure p corresponding to a p/z value of 5,000 psia. Plot the point p = 5,000 on the line z = 1. P. 837

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*22 (01): 97–104.

Paper Number: SPE-2336-PA

Published: 01 January 1970

Abstract

The log-log type-curve described here shows clearly the presence and duration of wellbore storage as well as the presence and duration of wellbore storage as well as the presence of linear flow due to fracturing. It can be used to presence of linear flow due to fracturing. It can be used to obtain quantitatively the information normally obtained from pressure buildup analyses and to identify the proper straight pressure buildup analyses and to identify the proper straight line in pseudo-radial flow for a fractured well. Introduction Specifications for modem well testing (drawdown or buildup) are usually written in such a way that a well will be tested for a period of time long enough to reach and define a proper "straight line" when test data are plotted in conventional manners. Pressure data obtained before the straight line is reached are not often analyzed, despite the fact that a number of publications have advanced methods for doing so. publications have advanced methods for doing so. One reason for this situation is that many factors are known to affect the short-time data. "Short-time data" signify data obtained before a conventional straight line is reached. Some of the factors are the effects of wellbore storage, perforations, partial penetration, and well stimulation such as fracturing or acidizing. Although the effects of such factors are generally known, the duration and importance have not been clearly defined in all cases particularly when these effects are combined in a well test. However, recent studies have revealed a great deal of potentially useful information concerning the analysis of short-time well test data. Our purpose here is to illustrate the interpretation of short-time well test data through presentation of field examples. Factors to be considered presentation of field examples. Factors to be considered will include wellbore storage, well damage, and fractured wells. Wellbore Storage and Skin Effect The effect of wellbore storage or unloading was originally considered by van Everdingen and Hurst. These studies called attention to the fact that the storage or unloading of fluid contained within the wellbore could cause a significant difference between the surface production rate and the sand-face flow rate in a well immediately following sudden changes in production rate. Gladfelter et al. presented a method production rate. Gladfelter et al. presented a method for correcting pressure buildup data for the changing sand-face flow rate. In another publication a method was presented for estimating the duration of the storage effect, and the Gladfelter et al. correction was generalized to apply to drawdown data. Fundamentally, wellbore storage can occur in several ways. Fluid can be stored by compression of the fluid in a completely filled wellbore, or by movement of a gas-liquid interface. Russell presented a method for analysis of the latter case, pointing out that the virtue of the method was that it was not necessary to know the sand-face flow rate as was the case in the Gladfelter et al. method. One problem with Russell's method was that only a portion of the short-time data was used, yet no criterion for selection of the data was presented. presented. Recently, Agarwal et al. re-examined this problem and presented dimensionless pressure-dimensionless time plots for the case of a well in an infinitely large reservoir, producing at constant surface production rate, and having wellbore storage and a skin effect. Fig. 1 presents a portion of their results. The usual definitions of dimensionless groups were employed. JPT P. 97

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*20 (10): 1187–1194.

Paper Number: SPE-1839-PA

Published: 01 October 1968

Abstract

During interpretation of pressure buildup tests on gas wells in a tight dolomite gas reservoir, peculiar behavior was noticed. Two straight lines were apparent. Effective permeability to gas taken from either straight line was about the same, and the Miller-Dyes-Hutchinson dimensionless time check for the straight line was proper for both straight lines. Geological data indicated the likelihood of scattered trending fractures in the reservoirs. Since the first straight line yielded permeability values close to the geometric mean permeability from core analyses, it was postulated that the reservoir model was that of an acidized well completed in the tight dolomite, but that widely scattered hairline fractures caused the mean permeability of the reservoir distant from the well to be higher than the matrix permeability. Because all other studies of fractured reservoirs to the authors' knowledge assumed that the fracture matrix was dense enough to communicate directly with the well, no interpretative methods were available. The Hurst line-source solution for a radial change in permeability for interference between oil reservoirs was adapted to pressure buildup testing. The result indicated that the first straight line should yield the proper matrix permeability and wellbore skin effect. The second straight line may be extrapolated to obtain static pressure. The time of bend between the straight lines was used to estimate distance to a fracture. Application to field test data is shown. It is believed that the methods developed and the case history presented will add to present tools available for pressure buildup interpretation. Introduction Since the pioneer studies by Miller, Dyes, and Hutchinson and Horner in 1950 and 1951, well test analysis has become recognized as one of the most powerful tools available to both production and reservoir engineers. Well test analysis serves as a logical basis for well stimulation and completion analysis, and for long-term reservoir engineering. Since the early 1950's, much effort has been placed on the development of well-test analytical methods. Reservoir and well conditions of increasing complexity have been considered systematically to provide the analyst with a catalog of causes and effects. Matthews and Russell state that some 200 papers dealing with this subject have been published in the last 35 years. Developments in well test analysis appear to have originated in one of two ways. Either a physically realistic field condition was anticipated and analytical solutions for the condition achieved, or anomalous field test behavior was recognized and interpretative methods sought for the anomaly. In recent years, it has appeared that the latter has inspired an increasing number of studies. The analyst today finds an increasing number of known cause and effect studies available for well test analysis, the classic of which is that of finding the specific flow problem that generated the answer the well behavior. Although it may be impossible to achieve this goal uniquely, the analyst often is able to select a useful interpretation that combines all known performance and geologic dataor to show that various logical alternatives would not significantly affect the interpretation. During a recent reservoir study, we observed gas well test behavior that did not appear to fit behavior described previously. Although it cannot be said that we have found a unique interpretation, we shall present in this paper the peculiar behavior observed, and describe the reservoir and interpretative methods developed. Reservoir Description The subject gas reservoir is a 9-mile-long, narrow dolomite reservoir lying within a limestone of Ordovician age. (See Fig. 1.) The dolomitized rock in the field consists of dark brown to buff, dense to coarsely crystalline, vugular dolomite containing numerous hairline fractures, many of which may have been closed in the reservoir and parted when cores were brought to the surface. Larger fractures are also apparent in core, but usually are filled and sealed with euhedral dolomite crystals. Portions of the north flank of the reservoir are known to be cut by a sealing fault downthrown to the north. Gas wells located near the fault have higher open flow potentials than those more distant from the fault. This is believed to be a result of higher permeability near the fault due to more extensive and open fractures. Detailed coring and core analysis have been performed on several of the wells in this reservoir. JPT P. 1187ˆ

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*20 (08): 877–887.

Paper Number: SPE-1835-PA

Published: 01 August 1968

Abstract

A systematic study has been made of the application of the real gas pseudo-pressure m(p) to short-time gas well testing. The m(p) function can be used in real gas flow problems to account for the variation of viscosity and z-factor with pressure. A mathematical model was solved numerically to generate solutions of various real gas flow problems. The erects of turbulence, formation damage and wellbore storage were included in the model. The analysis of simulated well tests showed that the interpretation methods normally used for liquid flow are generally accurate when the m(p) is used. For practical rates without turbulence, the solutions are not rate-sensitive, and the flow capacity kh and skin effect can be determined accurately from either a buildup or a drawdown test. However, the kh calculated from a drawdown test can be significantly low when turbulence is present. A case was simulated in which this error was 36 percent. Turbulence does not affect the determination of kh from buildup tests. The proper determination of kh from p and p2 buildup and drawdown plots is developed by analyzing their relationship to the m(p) method. A simple equation is given that can be used for long-range gas well performance forecasting. This expression is compared with a method presented by Russell et al. Introduction The formation flow capacity and wellbore damage condition can be determined for liquid-producing tests by means of buildup and drawdown tests. These tests make use of short-time pressure transient data rather than stabilized flow tests that are often used for gas well testing. Tracy presented a method of gas well testing that was based on ideal gas equations and utilized p buildup plots. This method was shown to be good for low pressure wells. Matthews suggested plotting p and using an average slope and average gas properties. This method was more successful on high pressure wells. Al-Hussainy, Ramey and Crawford showed that variation in gas properties could be simplified by using the real gas pseudo-pressure m(p). Some cases showed that the use of the m(p) function provided an accurate method of interpreting buildup and drawdown tests. Because the work of Al-Hussainy et al. included only a few actual cases, there was a need to explore the use of the m(p) method for a greater variety of flow conditions. The occurrence of turbulent flow around the wellbore often is an important factor in gas well testing. Swift and Kiel, and Carter et al. treated turbulence in gas well testing, but did not consider the variation of viscosity and z-factor with pressure. This paper presents the results of an investigation of the application of the m(p) method to buildup and drawdown testing. The investigation included the effects of turbulence, formation damage and wellbore storage. Gas Flow Equations To formulate the mathematical model, many of the assumptions usually used in well testing theory are applied. The system has radial geometry with a closed outer boundary and is composed of a horizontal porous formation that has uniform and isotropic rock properties and uniform thickness. Allowance is made, however, for a radial region of reduced permeability near the wellbore. This region represents formation damage. The geometry of the system is shown in Fig. 1. Darcy's law does not always apply to gas flow. JPT P. 877ˆ

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*20 (02): 119–120.

Paper Number: SPE-2112-PA

Published: 01 February 1968

Abstract

In 1954 Matthews, Brons and Hazebroek presented a method for correcting extrapolated buildup pressures on a Theis or Horner-type plot to static pressure. Matthews, Brons and Hazebroek considered buildup in a large number of geometric reservoir shapes with wells at various locations within a particular shape. In 1961 Brons and Miller pointed out that the presence of pseudo-steady state was indicated for the various shapes at long producing times, and that simple flow equations involving a producing times, and that simple flow equations involving a shape factor could be written for this condition. In 1965 Dietz extended this idea to show that static pressure could be found on the extension of the buildup straight line at a dimensionless time of where A = drainage shape area, CA = shape factor and other symbols are SPE standard. Dietz work generalized the older Miller-Dyes-Hutchinson buildup method to geometric shapes other than circular. As stressed by Dietz, the method assumed that the well had been produced to pseudo-steady state prior to shut in. Neither pseudo-steady state prior to shut in. Neither Matthews-Brons-Hazebroek nor Dietz cataloged the actual buildup curves involved in the various shapes, although a few cases were shown as examples. The purpose of this article is to present the Miller-Dyes-Hutchinson type of pressure buildup curves for most of the Matthews-Brons-Hazebroek shapes. As would be expected, assymetric location of a well in a particular drainage shape may lead to unusual bends in the pressure buildup curves. Because it is currently becoming practice to infer reservoir heterogeneities from the shape of the buildup curve, it appears useful to be aware of anomalies which may be a result of well location and drainage shape. Figs. 1 through 4 present Miller-Dyes-Hutchinson pressure buildup curves for the rectangular shapes considered. pressure buildup curves for the rectangular shapes considered. Symbols used are SPE standard; It refers to elapsed time after shut-in. Fig. 1 shows buildup curves for a square reservoir shape with three different well locations. The heavy lines are the buildup curves. Curves 1 and 2 are similar; it would be impossible to detect the effect of movement of the well from an actual buildup curve. Curve 3, however, has a peculiar bend at a dimensionless buildup time of about 0.02. This bend could be misinterpreted as the result of a general fracturing in the reservoir as studied by Warren and Root. Another interesting result is indicated by the dashed lines on Fig. 1. These lines represent the extrapolation of the initial straight line, and have the usual slope of 1.151. The intersection of the dashed lines with the ordinate at zero occurs at a dimensionless buildup time equal to the reciprocal of the shape factor Ca, for each shape. Note that the shape factors shown are the modified ones reported by Earlougher et al. P. 119

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*20 (02): 199–208.

Paper Number: SPE-1956-PA

Published: 01 February 1968

Abstract

EARLOUGHER JR., ROBERT C., MARATHON OIL CO., LITTLETON, COLO. JUNIOR MEMBER AIME RAMEY JR., H.J., STANFORD U., STANFORD, CALIF. MILLER, F.G., STANDARD OIL CO. OF CALIFORNIA, SAN FRANCISCO, CALIF. MUELLER, T.D., STANDARD OIL CO. OF CALIFORNIA, SAN FRANCISCO, CALIF. MEMBERSAIME Abstract There are many studies of flow in radial systems that can be used to interpret unsteady reservoir flow problems. Although solutions for systems of infinite extent can be used to generate solutions for finite flow systems by super-position, application is tedious. In this paper a step is made toward simplifying calculations of such solutions for finite flow systems. Superposition is used to produce a tabulation of the dimensionless pressure drop function at several locations within a bounded square that has a well at its center. The square system provides a useful building block that may be used to generate flow behavior for any rectangular shape whose sides are in integral ratios. Values of the tabulated dimensionless pressure drop function are simply added to obtain the dimensionless pressure drop function for the desired rectangular system. The rectangular system may contain any number of wells producing at any rates. Furthermore, the outer boundaries of the rectangular system may be closed (no-flow) or they may be at constant pressure. Mixed conditions also may be considered. Tables of the dimensionless pressure drop function for the square system are prevented and various applications of the technique are illustrated. Introduction In 1949 van Everdingen and Hurst published solutions for the problem of water influx into a cylindrical reservoir. Since this problem is mathematically identical with the depletion of a cylindrical reservoir with a well at the origin, the van Everdingen-Burst solution may be used to study the depletion problem. In their analysis, they assumed that the fluid had a small, constant compressibility such that flow was governed by the diffusivity equation (1) For a constant production rate q starting at time zero, van Everdingen and Hurst showed that the unsteady pressure distribution for both finite and infinite systems could be expressed in terms of a dimensionless pressure (2) (3) (4) where rw = wellbore radius (reservoir radius for influx)pD = dimensionless pressure at rD at tD. Tabulations of the dimensionless pressure drop for a unit value of rD were provided by van Everdingen and Hurst, and later by Chatas. Others also presented values in graphical or tabular form. If the radius of the well becomes vanishingly small, rW 0, the line source solution may be used for Eq. 2 when infinite systems are considered. (5) where -Ei(-x) is the well known exponential integral. If the argument of the exponential integral is small enough, (6) Eqs. 5 and 6 are excellent approximations for Eq. 2 under certain conditions. In 1954, Matthews, Brons and Hazebroek demonstrated that solutions such as Eq. 5 can be superposed to generate the behavior of bounded geometric shapes; i.e., the behavior of a bounded single-well system can be calculated by adding together the pressure disturbances caused by the appropriate array of an infinite number of wells producing from an infinite system. These wells are referred to as image wells. Matthews, Brons and Hazebroek considered systems containing a single well producing at a constant rate. JPT P. 199ˆ

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*19 (11): 1500–1506.

Paper Number: SPE-1645-PA

Published: 01 November 1967

Abstract

This paper presents the results of applying the Buckley-Leverett' displacement theory to petroleum reservoirs consisting of a finite number of layers. The layers are assumed to communicate only in the wellbores, and the reservoir may be represented as a linear system. Most previous investigations of this nature were limited by assumptions and by inconsistent calculation techniques. This study improves on previous work by applying the Buckley-Leverett displacement theory to a noncommunicating layered system where permeability, porosity, initial saturation, residual saturation and relative permeability vary from layer to layer in a logical and consistent manner. Gravity and capillary-pressure effects are neglected. A modification of the Higgins-Leighton calculation method was used in this study. Waterflood predictions were made with all properties varying, and then with only permeability varying using several mobility ratios. These results were compared with the Stiles and Dykstra-Parsons predictions. It is shown that the latter methods generally give poor values for the breakthrough recovery and pessimistic predictions for the performance after breakthrough. Similar results were obtained for a gas-displacement case. Introduction Field experience with immiscible displacement usually shows constant producing conditions until breakthrough of the displacing fluid. Then oil production continues at increasing displacing-to-displaced fluid ratios until the economic limit is reached. Three different ideal mechanisms are known that will produce this behavior: relative permeability effects as described by Buckley-Leverett frontal advance theory, vertical stratification as considered by Stiles, Dykstra and Parsons and others and different path lengths involved in areal (two-dimensional) flow between wells as described by Dyes et al. Without question, a combination of these factors modified by formation heterogeneity and other known and unknown factors actually does control the behavior of real systems, This paper presents results of an investigation of certain factors that should affect performance but which have received little attention to date. In 1944, Law demonstrated that porosity and permeability are often found to have normal and logarithmic-normal distributions, respectively, throughout cored intervals in natural formations. This led to the concept of the noncommunicating, multilayered reservoir model for immiscible displacement. This model assumes that the reservoir is composed of a number of layers that communicate only at the wellbores. Each layer is individually homogeneous, but may be different from every other layer. Stiles presented one of the earliest applications of this model to waterflood performance. In addition, Stiles assumed that the initial saturations and relative permeabilities were the same for each layer, porosity was the same. displacement was piston-like, fluids were incompressible and injection into each layer was proportional to that layer's permeability capacity (permeability-thickness product). The last assumption would be true if the mobility ratio for the displacement were unity." Dykstra and Parson's used the same model as Stiles, but rigorously included mobility ratios other than unity for piston-like displacement. Dykstra and Parsons used their general result to produce charts for log-normal permeability distributions between layers. Similarly, Muskat published analytical solutions for linear and exponential permeability distributionsIn 1959, Roberts described a scheme for calculating water-drive performance for the noncommunicating, layered reservoir model which considered two-phase flow in the displaced region. Roberts used the same model and assumed that the injection rate into a layer was proportional to that layer's permeability capacity, but that flood front locations could be evaluated from the Dykstra-Parsons results. These assumptions are inconsistent, and a material balance cannot be maintained except for a mobility ratio of unity. At the same time, Kufus and Lynch coupled Buckley-Leverett displacement theory with the layered model to provide an improvement of the Dykstra-Parsons method that was consistent. In 1960, Higgins and Leighton presented a numerical method for calculating waterflood performance also considering two-phase flow in the displaced region. The result was used to investigate variation in absolute permeability and oil viscosity. An excellent, detailed history of using the noncommunicating, layered reservoir model was presented by Nielsen. The preceding techniques (and many related ones) were similar in that differences in initial saturations, residual saturations and relative permeabilities from layer to layer were neglected. It is well known that the irreducible water saturation is an important function of absolute permeability. JPT P. 1500ˆ

Journal Articles

Journal:
Journal of Petroleum Technology

Publisher: Society of Petroleum Engineers (SPE)

*J Pet Technol*18 (05): 624–636.

Paper Number: SPE-1243-A-PA

Published: 01 May 1966

Abstract

The effect of variations of pressure-dependent viscosity and gas lawdeviation factor on the flow of real gasses through porous media has beenconsidered. A rigorous gas flow equation was developed which is a second order, non-linear partial differential equation with variation coefficients. Thisequation was reduced by a change of variable to a form similar to thediffusivity equation, but with potential-dependent diffusivity. The change ofvariable can be used as a new pseudo-pressure for gas flow which replacespressure or pressure-squared as currently applied to gas flow. Substitution of the real gas pseudo-pressure has a number of importantconsequences. First, second degree pressure gradient terms which have commonlybeen neglected under the assumption that the pressure gradient is smalleverywhere in the flow system, are rigorously handled. Omission of seconddegree terms leads to serious errors in estimated pressure distributions fortight formations. Second, flow equations in terms of the real gaspseudo-pressure do not contain viscosity or gas law deviation factors, and thusavoid the need for selection of an average pressure to evaluate physicalproperties. Third, the real gas pseudo-pressure can be determined numericallyin terms of pseudo-reduced pressures and temperatures from existing physicalproperty correlations to provide generally useful information. The real gaspseudo-pressure was determined by numerical integration and is presented inboth tabular and graphical form in this paper. Finally, production of real gascan be correlated in terms of the real gas pseudo-pressure and shown to besimilar to liquid flow as described by diffusivity equation solutions. Application of the real gas pseudo-pressure to radial flow systems undertransient, steady-state or approximate pseudo-steady-state injection orproduction have been considered. Superposition of the linearized real gas flowsolutions to generate variable rate performance was investigated and foundsatisfactory. This provides justification for pressure build-up testing. It isbelieved that the concept of the real gas pseudo-pressure will lead to improvedinterpretation of results of current gas well testing procedures, both steadyand unsteady-state in nature, and improved forecasting of gas production. Introduction In recent years a considerable effort has been directed to the theory ofisothermal flow of gases through porous media. The present state of knowledgeis far from being fully developed. The difficulty lies in the non-linearity ofpartial differential equations which describe both real and ideal gas flow. Solutions which are available are approximate analytical solutions, graphicalsolutions, analogue solutions and numerical solutions. The earliest attempt to solve this problem involved the method ofsuccessions of steady states proposed by Muskat. 1 Approximateanalytical solutions 2 were obtained by linearizing the flow equationfor ideal gas to yield a diffusivity-type equation. Such solutions, thoughwidely used and easy to apply to engineering problems, are of limited valuebecause of idealized assumptions and restrictions imposed upon the flowequation. The validity of linearized equations and the conditions under whichtheir solutions apply have not been fully discussed in the literature. Approximate solutions are those of Heatherington et al ., 2 MacRoberts 4 and Janicek and Katz. 5 A graphical solutionof the linearized equation was given by Cornell and Katz. 6 Also, byusing the mean value of the time derivative in the flow equation, Rowan andClegg 7 gave several simple approximate solutions. All the solutionswere obtained assuming small pressure gradients and constant gas properties. Variation of gas properties with pressure has been neglected because ofanalytic difficulties, even in approximate analytic solutions. Green and Wilts 8 used an electrical network for simulatingone-dimensional flow of an ideal gas. Numerical methods using finite differenceequations and digital computing techniques have been used extensively forsolving both ideal and real gas equations. Aronofsky and Jenkins 9,10 and Bruce et al . 11 gave numerical solutions for linear andradial gas flow. Douglas et al . 12 gave a solution for asquare drainage area. Aronofsky 13 included the effect of slippage onideal gas flow. The most important contribution to the theory of flow of idealgases through porous media was the conclusion reached by Aronofsky andJenkins 14 that solutions for the liquid flow case 15 couldbe used to generate approximate solutions for constant rate production of idealgases.