Productivity-Index Optimization for Hydraulically Fractured Vertical Wells in a Circular Reservoir: A Comparative Study With Analytical Solutions

Abstract

Productivity-index (PI) optimization by means of optimal fracture design for a vertical well in a circular reservoir is a canonical problem in performance optimization for hydraulically fractured wells. Recent availability of the exact analytical solution for the pseudosteady-state (PSS) flow of a vertically fractured well with finite fracture conductivity in an elliptical drainage area provides an opportunity to re-examine this fundamental problem in a morerigorous manner. This paper first quantitatively estimates the shape-approximation-induced error in the PI when the exact solution for an elliptical drainage area is applied to a circular drainage area. It is shown that the shape-approximation-induced error in the PSS-flow PI is less than 1% for fracture penetration ratios up to 53%, and this error decreases significantly as the fracture conductivity is increased. PI optimization is then performed with the highly accurate analytical solution for this range of the penetration ratios. The results show that the optimal fracture conductivity increases linearly from 1.39 to 1.71 when the proppant number is increased from 0.0001 to 0.6. PI for the steady-state flow and a popular ad hoc PSS-flow PI are compared with the analytical PSS-flow PI. It is found that both the steady-state and the ad hoc PIs deviate significantly from the analytical PSS-flow PI. In particular, the optimal fracture conductivity for the steady-state flow and the ad hoc PIs decreases with the proppant number, opposite to the trend observed for the optimal fracture conductivity for the PSS flow. It is suggested that the ad hoc PI should be abandoned in favor of the more-rigorous analytical PSS-flow solution.

A New Pseudo Steady-State Constant for a Vertical Well with Finite-Conductivity Fracture

“…In this flow regime, the relationship of dimensionless wellbore pressure and dimensionless average pressure can be expressed as [20][21][22][23][24]:…”

“…where b Dpss is the Pseudo Steady-State (PSS) constant [24]. This PSS constant b Dpss has been widely used to define the appropriate dimensionless decline rate in many currently used production-decline rate analysis models [20,[25][26][27][28].…”

“…In addition, assuming an elliptical boundary, two analytical solutions for a well with an infiniteconductivity fracture and a finite-conductivity fracture were developed by Prats et al [29] and Lu et al [24], respectively.…”