Abstract

Currently in the oil industry, productivity equations for a single well are erroneously used in all reservoir systems, regardless of the interference effects on flow rates of different wells in the same drainage domain

This paper presents steady state and pseudo-steady state productivity equations of a multiple wells system in an anisotropic circular or rectangular reservoir. Taking fully penetrating vertical wells as uniform line sinks, and solving square matrices of dimension n, where n is the number of wells, simple, accurate multiple wells productivity equations are obtained. The proposed equations which relate the production rate vector to the pressure drawdown vector provide fast analytical tools to evaluate the performance of multiple wells, which are located arbitrarily in an anisotropic circular or rectangular reservoir. This paper also gives equations for calculating skin factors of each well.

It is concluded that, for a given number of wells, well pattern, reservoir boundary conditions, anisotropic permeabilities and skin factor have significant effects on single well productivity and total productivity of the multiple wells system. For a given multiple wells system, if the reservoir is under edge water drive, when producing time is sufficiently long, productivity equation based on the steady-state is required; for a closed-circular or rectangular reservoir, pseudo-steady-state productivity equation is applicable.

Introduction

Well productivity is one of primary concerns in field development and provides the basis for field development strategy. To determine the economical feasibility of drilling a well, petroleum engineers need reliable methods to estimate its expected productivity. Well productivity is often evaluated using the productivity index, which is defined as the production rate per unit pressure drawdown. We often relate the productivity evaluation to the long-time performance behavior of a well, that is, the behavior during pseudo-steady state or steady state flow.

The steady state and pseudo-steady state productivity equations of a fully penetrating vertical well located at the center of a homogeneous, isotropic permeability circular reservoir are sufficiently familiar to petroleum engineers and can be found in any reservoir engineering book[1]. Lu and Tiab extended these well-known productivity equations for a fully penetrating vertical centered well in an isotropic circular reservoir to an off-center partially penetrating vertical well in an anisotropic circular cylinder reservoir[2]. Lu and Tiab also presented steady state and pseudo-steady state productivity equations for a vertical well in an anisotropic box-shaped reservoir[2,3].

The performance of multiple wells system has received attention in the last decade. Camacho-V, Galindo-N, and Prats gave a buildup solution in a system with multiple wells producing at constant wellbore pressures[4]. Umnuayponwiwat, Ozkan, and Raghavan presented equations of pressure transient behavior and inflow performance of multiple vertical and horizontal wells in closed systems[5]. Marhaendrajana and Blasingame presented a solution and associated analysis methodology to evaluate single well performance behavior in a multiple wells reservoir system[6]. Valko, Doublet and Blasingame presented pseudo-steady state productivity index for multiple wells producing from a closed rectangular reservoir[7], however their equation is only applicable to isotropic reservoir, and an infinite series summation is included in their equation.

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