Well Configurations in Anisotropic Reservoirs

Abstract

Summary Horizontal wells have emerged as a new means for well productivity enhancement. Simultaneously, they have brought forward the need to recognize and account for permeability anisotropies, including vertical-to-horizontal and horizontal-to-horizontal directions. In addition, there is the possibility of multiple horizontal drainholes emanating from the same vertical well. Performance relationships for the most interesting well configurations are presented including both early-time and late-time differences rather than only bounded flow regimes. Solutions for arbitrarily oriented single or multiple horizontal wells are introduced along with a discussion of well known existing relationships. Introduction It is a foregone conclusion that horizontal wells will capture an ever increasing share of all petroleum wells drilled. The performance of these wells depends greatly on appropriate reservoir selection, substantial predrilling formation evaluation and optimized completion and stimulation practices. There have been several attempts to describe and estimate horizontal well productivity and for injectivity indexes and several models have been employed for this purpose. Following the tradition of vertical well productivity models, analogous well and reservoir geometries have been considered. A widely used approximation for the well drainage is, conveniently, a parallelepiped model with no-flow or constant-pressure boundaries at the top or bottom and either no flow or infinite-acting boundaries at the sides. One of the earliest models was introduced first by Borisov (1964) assuming a constant pressure drainage ellipse whose dimensions depend on the well length. This configuration evolved into a widely used equation presented by Joshi (1988) accounting for vertical-to-horizontal permeability anisotropy and, adjusted by Economides et al. (1991) for a wellbore in elliptical coordinates. This model, while useful for first approximations and comparisons with vertical well productivity indexes, does not account for either early-time or late-time phenomena nor, more importantly, realistic well and reservoir configurations. Babu and Odeh (1989) used an expression for the pressure drop at any point by integrating appropriate point source (Green's) functions in space and time. Their solutions for various no-flow boundary positions include infinite-sum expressions, accounting for individual pseudosteady-state pressure drops. These forms are rather complicated and cumbersome to calculate. Using vertical well analogs, Babu and Odeh (1989) grouped their solutions into reservoir/well configuration shape factors and a (horizontal) partial penetration skin effect.

Hyperbolic-elliptic models for well-reservoir flow

“…The integral representation above is flexible and may be applied to reservoir problems with complex well configurations. Successful applications of this approach have been reported by Economides et al [6] and Ouyang et al [27], see also references therein.…”

“…As a consequence, there is at most one entropy weak solution to (6). The entropy weak solution u satisfies the following estimates for any t ∈ (0, T ):…”

“…In the remaining part of this paper we focus exclusively on the model problem (4)- (6). We are interested in general existence and uniqueness results that apply for our model problem, which might be considered as a simplest possible approximation to the more general well-reservoir model (1).…”

“…Then there exists an entropy weak solution to (6) in the sense of Definition 3.2. Moreover, for any (fixed) T > 0, let u, v : (0, T ) × R → R be two entropy weak solutions with…”

Productivity and injectivity of horizontal wells. Quarterly report, January 1, 1996--March 31, 1996

“…A) First, the objective was to reproduce the results obtained by Ekonomides et al [2]. We considered one of his examples: a 200 ft long horizontal well in a box.…”