Summary. An analytic solution is presented for the pressure response during drawdown and buildup of a horizontal well. This method results from solving the three-dimensional diffusion equation with successive integral transforms. Simplified solutions for short, intermediate, and long times that exhibit straight-line sections when pressure is plotted vs. time are presented. The validity of the method is demonstrated by comparing with results generated numerically by a reservoir simulator and with an analogous analytic solution. Methods for analyzing pressure drawdown and buildup data are presented with examples. The method allows reservoir characteristics, including permeability. skin, and distance to boundaries to be determined. The early-time effects, where the well behaves as if it were in an infinite reservoir, are also discussed. Expressions to determine times to critical events, which are important for well test design, are presented. Introduction The idea of using horizontal wells to increase the area of contacted reservoir dates back to the early 1940's. Until recently, however, very few horizontal wells had been drilled in the noncommunist world. There has been little incentive to spend additional money on a new technique when most reservoirs can be produced by such conventional techniques as stimulation by fracturing. Hydraulic fracturing has been a potential rival to horizontal drilling for a long time, although compared with vertical wells, horizontal wells can increase injection or production rates several times over. The technique was production rates several times over. The technique was considered only when stimulation by hydraulic fracturing from vertical wells was not feasible or practical. The usefulness of horizontal wells has been demonstrated recently in North America and Western Europe. With the current technology, horizontal drainhole distances that are much longer than the fracture lengths achieved by hydraulic fracturing are possible at moderate costs. In 1979, Arco rejuvenated its high-GOR wells through the application of horizontal drainholes. Serious gasconing problems were thus overcome. The danger of extending into the gas problems were thus overcome. The danger of extending into the gas cap had precluded hydraulic fracturing. In 1978, Esso Resources Canada drilled a horizontal well at the Cold Like Leming pilot to field test thermally aided gravity-drainage processes. In 1980, Texaco Canada completed a drilling program to tap an unconsolidated bituminous sand at shallow depths in the Athabasca lease. In Western Europe, between 1979 and 1983. Soc. Natl. Elf-Aquitaine, in association with the Inst. Francais du Petrole, drilled four horizontal wells in oil-bearing reservoirs. Three of these wells are located in Lac Superieur and Castera Lou fields in France. The fourth well is offshore in the karstic reservoir of the Rospo Mare field in the Italian part of the Adriatic sea. The reservoirs consist of thin, soft, vertically fractured, tight formations with fluid interface problems and have been found to be ideal candidates for horizontal wells. When the Rospo Mare pilot was initiated, it was reported that the productivity was 20 times greater in the horizontal well than in neighboring vertical and deviated wells. The horizontal well intersected several voids and was positioned to obtain the greatest possible height above the water/oil contact. Lower near-wellbore drawdown prevented water from coning. Several researchers have studied horizontal well productivity. Efros and later Giger considered the relative economics of the horizontal wells on overall productivity by studying the geometry and spacing of the horizontal wells. Giger investigated the merits of the horizontal wells in preventing water and gas influx during multiphase flow. His calculations show that horizontal wells provide greater sweep efficiencies. Giger also studied the use of horizontal wells to improve oil recovery in formations with fluid interface problems. Laboratory studies have recently been conducted on thermally aided gravity drainage of viscous oil in horizontal wells. Huygen and Black investigated the problem of cyclic steam stimulation through horizontal wells and had encouraging results. Despite the unfavorable mobility ratios associated with heavy oils, horizontal wells provided a more homogeneous steamfront and a much greater injectivity index than vertical wells. Although horizontal drilling activities have been the focus of much attention during recent years, there appears to have been no study in the area of pressure-transient analysis of horizontal wells. The accomplishments so far must be augmented by attempts to understand pressure data from well testing. Cinco et al. presented analytic pressure data from well testing. Cinco et al. presented analytic solutions for unsteady-state pressure distribution created by a directionally drilled well in an anisotropic medium. Gringarten et al. Raghavan et al. and Rodriguez et al. have obtained analytic solutions to the problem of transient flow of fluids toward fully and partially penetrating fractures. These solutions correspond closely to that of a horizontal well. The mathematical method used to solve these problems was based on the use of Green's and source functions. whose usefulness in solving such problems has been demonstrated by Gringarten and Ramey. The derivation of these solutions and the form of the results are complicated. however, and the extension of the methods to the analysis of pressure-transient data from horizontal wells is not immediately apparent. The purpose of this paper is to present a straightforward analytic solution for the pressure drawdown and buildup associated with testing a horizontal well in an undersaturated oil zone. This method is the result of an elegant mathematical procedure using successive integral transforms (Laplace and Fourier). To demonstrate the validity of the derivation, the analytic solutions are compared with the results of numerical studies undertaken as part of this work and an existing analogous analytic solution. Sample calculations are presented to illustrate the determination of reservoir characteristics, including permeability and skin factor. The present state-of-the-art well testing theory does not provide for well test interpretation of horizontal wells. The theory presented in this paper fills this void. SPEFE p. 683

Summary. New analytic solutions are presented in real time and as Laplacetransforms for horizontal wells in reservoirs bounded at the top and bottom byhorizontal planes. Two types of boundary conditions are considered at theseplanes. and the Laplace-transform pressure solutions are used to includewellbore-storage and skin effects. Solutions are based on the uniform-flux, line-source solution, but differ from most existing solutions owing to the use of pressure averaging to approximate the infinite-conductivity wellborecondition and use of the correct equivalent wellbore radius for an anisotropicreservoir. New flow periods (regimes) are identified, and simple equations andexistence criteria are presented for the various flow periods that can occurduring a transient test. Introduction Determination of transient pressure behavior for horizontal wells hasaroused considerable interest over the past 10 years. An extensive literaturesurvey on horizontal wells can be found in Ref. 2. Most work dealing with thehorizontal-well problem uses the instantaneous Green's function techniquedeveloped by Gringarten and Ramey to solve the 3D isotropic diffusivityequation. Goode and Thambynayagam used finite Fourier transforms to solve theanisotropic problem for the line-source case. Because the infinite-conductivityinner-boundary condition (uniform pressure over the sand face) poses a verydifficult boundary-value problem. a uniform-flux condition on the innerboundary is commonly used. The infinite-conductivity solution is thenapproximated with either an equivalent-pressure-point or pressure-averagingtechnique. We prefer the pressure-averaging method because it requires no apriori information. is exact in the limit of a small wellbore radius, and ismore accurate at intermediate times than the equivalent-pressure-point method. These reasons are discussed further in Appendix A. Another feature of thesolutions presented in this paper is the use of the correct equivalent wellboreradius for an anisotropic formation, which guarantees that elliptical-floweffects near the well are treated correctly at late times. At early times it ispreferable to use the elliptical-cylinder solution. Solutions presented in thispaper. however, are sufficient for most practical problems. Using thesetechniques. we extend the work of Goode and Thambynayagam and Clonts and Rameyto obtain new analytic solutions for horizontal wells with and without theeffects of gas cap or aquifer. The wellbore-storage effect is accounted for, and new formulas are presented for the determination of reservoir parametersfrom the characteristics of different flow regimes. Solutions With and Without Gas Cap or Aquifer First we discuss the basic solutions for horizontal wells for theconstant-rate case without wellbore-storage and skin effects. These solutionswill then be combined with constant wellbore storage and/or measured downholeflow rate. The horizontal well shown in Fig. 1 is considered to be completed inan infinite anisotropic medium bounded above and below by horizontal planes. The boundaries of the reservoir in the horizontal directions are considered tobe so far away that they are not seen during the test. The permeabilities inthe principal directions are denoted by kx, ky, and kz. We develop thesolutions for the general case where the three permeabilities are all differentin Appendices A and B, but in the text we consider a transversely isotropicmedium and write k, kv - kH and kz -kv. The flow of a slightly compressiblefluid of constant compressibility and viscosity is assumed throughout themedium. Gravity effects are neglected. Two types of top and bottom boundaryconditions are considered. In the first case. both the top and the bottomboundaries have no-flow conditions. In the second case, one of the boundariesis at constant pressure, while the other is a no-flow boundary as before; thiscase can represent either a gas cap at the top boundary or an active aquifer(in which the water mobility is high compared with the mobility of thereservoir fluid) at the bottom. For convenience, we refer to the first model asthe no-flow-boundary model and to the second as the constant-pressure-boundarymodel. The notation of this paper assumes that in the latter model, theconstant-pressure boundary is at the top (the gas-cap case). but the formulasmay be readily adapted for the case of an aquifer at the bottom. During thelast few years, several solutions for horizontal wells have been presented. Most of these solutions are for the no-flow-boundary model, and apart from thework of Goode and Thambynayagam. none present solutions in the Laplace-transform domain. A solution for the constant-pressure condition atboth the top and bottom boundaries was presented by Daviau et al. This solutionis different from the constant-pressure-boundary solution presented here, inwhich one of the boundaries (top or bottom) is no-flow. This flexibility isimportant because if we have a constant-pressure boundary such as a gas cap, the well is usually drilled close to the other (no-flow) boundary. The solutionmethod is discussed in Appendix A, and the actual solutions are developed in Appendix B. Our solutions differ somewhat from other solutions given in theliterature because we approximate the infinite-conductivity condition byaveraging the pressure along the well length instead of using an equivalentpressure point. A discussion of the pressure-averaging technique is given in Appendix A, together with a derivation of the correct equivalent wellboreradius to be used for an anisotropic formation. We define dimensionless timeand pressure (in field units) by ................(1) ................(2) and other dimensionless parameters ............................(3a) ............................(3b) ............................(3c) In the time domain, the dimensionless pressure response, pD, forconstant-rate drawdown is most conveniently given as a time integral over theinstantaneous Green's function (see Appendix B): ...................(4) .......(5) .........................(6) SPEFE P. 86^

In recent years, pressure transient behavior and inflow performance of horizontal wells have received considerable attention due to the increase in horizontal drilling. In this paper we develop an interpretation methodology for horizontal well pressure transient testing. This methodology is then applied to the interpretation of an actual horizontal well test performed in Prudhoe Bay. The complex geometry associated with horizontal wells makes interpretation of well tests a difficult task. Uniquely determining the system parameters from short time (typical times for vertical well testing) pressure tests is not possible. Combining drawdown and buildup tests, with downhole flowrate measurement is critical for proper interpretation. We also provide a solution for the Inflow Performance of a horizontal well completed in a rectangular drainage volume, where the well can be of arbitrary length and completed at any location within the drainage volume. Introduction Now forward solutions to the diffusivity equation for horizontal well geometry with varied boundary conditions are available in the literature. Moreover, Reiss and Sherrard presented performance and production data from several horizontal wells and mentioned interpretation of well test data. The interpretation of well test data from horizontal wells is a much more difficult task than its vertical counterpart. This difficulty stems from:the main search direction of the parameters usually does not coincide with the depositional environment,the three dimensional nature of the How geometry and lack of radial symmetry, andmore parameters (information) to be obtained. In addition to these difficulties, zonal variation of vertical permeability and shale distribution will make interpretation intricate. A well-defined flow period comparable to that of the infinite acting radial How period (free from storage and boundary effects) of a vertical well is not apparent for horizontal wells. This is largely due to the fact that most horizontal wells will exhibit partial penetration effects, even when they are fully perforated. This fact has already been observed by many authors, and specific methods have been proposed to identify How regimes and their durations under ideal conditions. However, it has not been shown how to extend the identification of How regimes and their usage to the interpretation of real pressure transient tests.

Summary In recent years, pressure-transient behavior of horizontalpressure-transient behavior of horizontal wells has received considerableattention because of the increase in horizontal drilling. This paper presentsan interpretation method for presents an interpretation method forhorizontal-well pressure-transient testing that is applied to a buildup testfrom a horizontal well in the Prudhoe Bay field. The complex flow Prudhoe Bayfield. The complex flow geometry associated with horizontal wells makeswell-test interpretation difficult. Unique determination of the systemparameters from pressure data with a short testing time (typical test times forvertical wells) and/or production time is not possible. We production time isnot possible. We must run drawdown and buildup tests and acquire the downholeflow rate with pressure to estimate the reservoir parameters accurately. Introduction Interpretation of well tests from horizontal wells is much more difficultthan interpretation of those from vertical wells because of a considerablewellbore storage effect, the 3D nature of the flow geometry and lack of radialsymmetry, and strong correlations between certain parameters. Also, zonalvariations of vertical permeability and shale distribution complicateinterpretation. A well-defined flow period, comparable to that of the infinite-acting radialflow period (free from storage and boundary effects) period (free from storageand boundary effects) of a vertical well, is not apparent for horizontal wells, largely because most horizontal wells exhibit partial penetration effects evenwhen they are hilly perforated. Specific methods have been perforated. Specificmethods have been proposed to identify flow regimes and their proposed toidentify flow regimes and their durations under ideal conditions. AlthoughReiss and Sherrard et al. presented performance and production data fromperformance and production data from several horizontal wells and mentionedinterpretation of well-test data, they did not show how to extend theidentification and usage of flow regimes to the interpretation of realpressure-transient tests. This paper presents a method for the interpretation of wells data fromhorizontal wells and analyses of pseudosynthetic and real well-test data. Solutions With and Without Gas Cap or Aquifer In a horizontal well, there is usually considerable wellbore volume (50 to100 bbl) below the measurement point, even if the downhole flow rate ismeasured or a downhole shut-in device is used. The storage effect with thisadditional volume typically lasts longer than that in a vertical well in thesame formation because the anisotropy reduces the effective permeability atearly times to root of kHkV.

Summary This paper presents analytic models to interpret the pressure transientsmeasured by a multiprobe formation tester. The pressure transients measured bya multiprobe formation tester. The multiprobe tester discussed consists ofthree probes. The sink probe generates a pressure pulse by withdrawing fluidfrom the probe generates a pressure pulse by withdrawing fluid from theformation while the resulting pressure response is measured at the sink probeand at each of two observation probes. One observation probe is positioned onthe opposite side of the borehole on the probe is positioned on the oppositeside of the borehole on the same vertical plane as the sink, and the other isdisplaced vertically on the same azimuthal plane as the sink. The effect of thevarious reservoir parameters on the pressure response is discussed. Applicationof the models is demonstrated through an actual field example. Introduction The introduction of the Repeat Formation Tester (RFT SM) tool in themid-1970's coincided with the exploitation of the North Sea. The RFT tool, which is primarily a device for measuring vertical pressure distribution inopen hole, soon gained industry pressure distribution in open hole, soon gainedindustry recognition. While the RFT technique has been successful, it has notreached its full potential because of its inability to give reliablepermeability estimates. The estimated formation parameters and the collectedfluid samples generally are from the parameters and the collected fluid samplesgenerally are from the contaminated zone rather than the native reservoir;consequently, the associated degree of uncertainty is high. With productiondecline of many of the world's large reservoirs, reservoir management isreceiving increasing attention as emphasis shifts from primary to secondaryoperations. As secondary and tertiary schemes become the dominant means ofrecovery, the measurement of permeability in heterogeneous and anisotropicformations will become increasingly important. Because of its effect on manyreservoir displacement processes, vertical permeability often is the singlemost important reservoir parameter. permeability often is the single mostimportant reservoir parameter. Although vertical permeability measurementsroutinely are made from cores, reliable in-situ measurements of verticalpermeability over a large rock volume are much more desirable. The transientwell tests proposed for estimating vertical permeability may be classified asvertical interference testing. permeability may be classified as verticalinterference testing. These techniques have not been widely accepted becausethey usually require cased holes, and the measurements are masked by wellborestorage effects. To eliminate these effects, longer (hence costly) testingtimes are required. For these reasons, industry needs an alternative means ofestimating large-scale, in-situ vertical permeability. permeability. Moran and Finklea first proposed quantitative methods for estimating permeability usingpressure-transient data obtained from the wireline formation tester, thepredecessor to the RFT tool. Their method corresponds to a finite, sphericallyshaped sink of radius r in an infinite, isotropic medium. During drawdown, thespace around the perforation can be divided roughly into three sphericalregions: a steady-state zone near the sink where the total flow rate isindependent of distance r from the sink, the undisturbed formation far from thesink where the flow rate vanishes, and the transition zone between the twowhere the flow rate decreases continuously from the constant steady-state valueto zero at some distance far from the sink. During drawdown, as time increases, the boundaries between successive zones move farther from the sink. Forsufficiently large times, the Moran and Finklea spherical solution yields asteady-state pressure difference, (1) The actual geometry at the RFT probe is not of a spherical source but of adisk source. Therefore, an effective probe radius, r, must be defined. For anisotropic medium, r, was defined as one-half the probe radius. Sharma and Dussan later showed that this definition was not valid and that the correctresult is r =2rp/. The RFT probe enters the formation from the borehole. Because the boreholewall is considered impermeable, the flow pattern cannot be spherical. At thesink probe, in the limit of steady-state flow, Stewart and Wittman accountedfor this wellbore effect in terms of a shape factor by writing Eq. 1 in theform (2) The shape factor, C, is essentially one-half the ratio between the pressuredrop at a specified point with a borehole to the pressure pressure drop at aspecified point with a borehole to the pressure drop without a borehole. Depending on the borehole diameter, C varies between 0.5 for spherical flow and1.0 for hemispherical flow. For an 8-in. borehole in an isotropic formation, Stewart and Wittman defined C as 0.645 when r was taken as one-half the probeaperture radius. Wilkinson and Hammond developed analytical derivations forcalculation of steady-state shape factors as a function of rp /rw andanisotropy. They found that, for isotropic formations with a smallprobe-to-wellbore radius, the effect of the wellbore was small. When rp/rw=0.05, the effective shape factor was found to be 0.96 (it would be unity foran infinite-radius well) when the correct definition of the effective proberadius r =2r / was used. For anisotropic formations with kz less than less thankr, however, the wellbore correction was appreciable. No matter howsophisticated the interpretation, the measurements obtained from the RFT tool(sink probe for the multiprobe configuration) are uncertain for the followingreasons: mudcake blocking the probe, damage to the formation resulting from themechanical setting of the probe, non-Darcy flow near the probe, mud particleshaving migrated into the formation, gas evolution in the near-probe region, andinternal fines migration. The non-Darcy flow results from the high fluidvelocities expected in the near-probe region. According to Muskat, deviationsfrom Darcy's law occur at Reynolds numbers exceeding unity. At high Reynoldsnumbers, the effects of inertial forces and turbulence begin to contributesignificantly to the pressure gradient. It would not be unusual to observe flowin the near-probe region with a Reynolds number greater than 4. Because the Reynolds number is proportional to the fluid velocity and, in spherical flow, the velocity decreases as 1/r, the non-Darcy flow phenomenon is localized tothe sink probe only and has negligible phenomenon is localized to the sinkprobe only and has negligible effect on the observation probes. This paperemphasizes permeability derived from the pressure data recorded away from thepermeability derived from the pressure data recorded away from the sink probe. The multiprobe concept originally was proposed in a U.S. patents in 1956. Fig.1 shows the probe geometry of the multiprobe patents in 1956. Fig. 1 shows theprobe geometry of the multiprobe tool discussed here. In operation, one probe(the sink) withdraws fluid from the formation. The resulting pressure transientis measured simultaneously at the sink and the observation probes. From thesepressure data, both vertical and horizontal permeabilities can be determined. Also, the observation probes are not affected (at least, not to leading order)by the adverse flow effects at the sink probe. Because of the distance betweenthe observation probes and the sink probe, the permeabilities arerepresentative of probes and the sink probe, the permeabilities arerepresentative of a larger length scale than those determined by a single-probetool. SPEFE P. 297