In boreholes, temperatures vary and to extract hydrocarbon saturation from conductivity measurements, the influence of temperature on water and rock conductivities must be accounted for. The mobility [Formula: see text] of the counter‐ions due to clays and the electrical conductivity of pore‐filling brine show large changes with variation in temperature, whereas the microgeometry of the pore space exhibits negligible change. Using this idea, the temperature dependence of [Formula: see text] is extracted using data on dc electrical conductivity of shaly sands (σ) containing varying amounts of clay. The mobility of [Formula: see text] counter‐ions is found to vary approximately linearly with temperature. This explicit relationship is tested by comparing the predicted temperature dependence against the measured temperature dependence of conductivity of a set of rocks with high and low clay content. While the rock conductivity shows a large temperature dependence, the resistivity index is less sensitive to temperature. An approximate formula, which is superior to Arps’s formula, for water conductivity as a function of temperature is obtained.
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
We find the formula σ=φm[σω+AQv/ (1+CQv/σw)] that embodies the fact that the rock conductivity σ is a nonlinear function of water conductivity σw, can fit data on 140 cores rather well with m≊2, A=1.93×m (mho/m)(l/mol) and CQv=0.7 (mho/m). The observed curvature at low salinity is due to an interplay of tortuosity and water conductivity. Empirical correlation shows that m increases with the clay content, as the tortuosity increases with the clay content. Thus, the conductivity of a fully water saturated clay bearing (shaly) sand is completely determined from porosity φ, charge density Qv, and water conductivity σw.
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^
Summary When a horizontal well is selectively completed, the productive length maynot be the entire drilled length of the horizontal section. The productivity ofthe well will be affected by the total length and productivity of the well willbe affected by the total length and distribution of the open intervals. Thispaper presents a method to calculate the inflow performance of a horizontalwell that is selectively completed. Introduction Completion of horizontal wells as open holes or with slotted (Figs. 1a and 1b) leaves operators with little or no opportunity to perform diagnostic orremedial work. perform diagnostic or remedial work. Many horizontal wells thathave been producing for several years are now producing for several years arenow experiencing production problems that can be attributed to the lack ofcompletion control. This is particularly true in areas where the targets arethin and coning n a potential problem. It is inevitable that certain portionsproblem. It is inevitable that certain portions of the well will be closer tothe fluid/fluid contact than others and that hydraulic isolation of thoseportions would significantly improve long-term performance. Also, when drillingcontrol or sufficient geological knowledge is absent, portions of the well maynot even lie within the reservoir. Considerable knowledge has been developed onhow to complete horizontal wells successfully; what is missing is a method forpredicting the performance of different completion strategies. It may be thatit is not practical or cost effective to open the entire practical or costeffective to open the entire length of the well within the reservoir. Figs. 1c and 1d are schematics of a selectively completed well. The well in Fig. 1c was completed with external casing packers (ECP's) that had alternate slotted andunslotted sections. Fig. 1d shows that the same effective completion wasachieved with a cemented liner that was subsequently perforated. For thepurpose of inflow perforated. For the purpose of inflow calculations, thesecompletions are identical. As horizontal well technology has developed, severalinflow performance formulas for horizontal wells have been discussed in theliterature. In this paper, we expand the work of Goode and Kuchuk to includethe effects of having only a portion of the well open. They present a solutionfor the inflow performance of a horizontal well producing from a reservoir ofuniform producing from a reservoir of uniform thickness within a closed, rectangular drainage region. The well can be placed arbitrarily within thedrainage volume, provided that the distance from any part of the well (open toflow) to a lateral boundary is large compared with the scaled of the reservoir. In practice this is not an unduly restrictive assumption, unless the verticalpermeability is extremely low. It is much less permeability is extremely low. It is much less restrictive than the geometry required by Giger el al. and Karcher el al, where the well must be short enough, compared with the boundedregion, to permit the development of radial flow before the effect of thelateral boundaries is felt. The formulas presented by Babu and Odeh and Goodeand Kuchuk are for a well placed inside a drainage volume with no flux crossingany external boundary, while the formulas presented in Refs. 3 through 5 assumea constant pressure at the external lateral boundaries and no-flux conditionsat the top and bottom. The no-flux condition on all external boundaries is themost relevant for practical purposes and is the boundary condition used in thiswork. Mathematical Model We consider a horizontal well of length 2L 1/2 centered at (x w, y w, z w)and producing from a rectangular region of dimensions Lx and Ly, through n popen intervals, with Segment i of length 2L i centered at x i (Fig. 2). Thedistribution and number of open interviews is arbitrary, as is the position ofthe well within the drainage area, provided that the distance from any openinterval to a lateral is large compared will the scaled reservoir thickness h(kx/k z) 1/2. With this restriction, the pressure will be vertically equilibratedbefore the influence of any lateral boundaries is felt, and we may derive theinflow formula by writing the pressure drop as a sum of two terms. We firstpressure drop as a sum of two terms. We first consider a 2D fracture problemthat is analogous to the horizontal well problem at long times. We then accountfor the pressure drop in the third dimension, z, by including a pseudoskin, SzD, determined by solution of pseudoskin, S zD, determined by solution of a 3Dproblem (not 2D as in Refs. 3 through 5) that excludes the lateral boundaries. The inflow performance of a well is related to the long-time behavior of theconstant-rate pressure. At long times, when no flux is permitted to cross theexternal boundaries, the difference between the average pressure in thereservoir and wellbore pressure pressure in the reservoir and wellbore pressureapproaches a constant value that we call the inflow pressure. JPT P. 983
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