Pressure Drawdown and Buildup Analysis of Horizontal Wells in Anisotropic Media

Abstract

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

Parametric analysis of diffusivity equation in oil reservoirs

“…Goode offered an analytic solution for three dimensional diffusivity equations during drawdown and buildup of a horizontal model to get some specifications of reservoir. He applied numerical solution for validation of his procedure (Goode 1987). In some cases, the researchers face with heterogeneous reservoirs instead of homogeneous, therefore, they use some synthetic methods to solve the problem (Dake 1983).…”

Tunnel Inflow Monitoring in Permeable Rocks Nearby Cannero River, Piedmont (Northern Italy)

“…For this reason it needs to propose a forecasting method in order to predict, in approximate way, the interference of a tunnel on the shallow aquifers. Kawecki (2000), Odeh and Babu (1993), Goode and Thambynayagam (1987), Dunning et al (2004) obtained analytical solutions for the transient state for horizontal drainage system similar to the tunnels but in an infinite domain without boundary conditions. Therefore, the relationships are not useful for direct application in reality.…”

“…For the first case, the Goode et al (1987)'s relationship, valid for a tunnel located at a distance z w from the bedrock partially penetrating a confined aquifer ( Figure 3) has been used. Figure 3.…”

“…Figure 3. Partially penetrating tunnel modeling in a confined aquifer (Goode & Thambynayagam, 1987). The piezometric level is represented in a blue line…”

“…This relationship (1.1) depends also on the location of tunnel due to the different depth (Goode & Thambynayagam, 1987) 1. -]is the skin tunnel factor due to the loss of the head pressure in the vertical flux.…”