Optimizing Number of Fractures in Horizontal Well
Authors: Guk Vyacheslav , Tuzovskiy Mikhail , Wolcott Don , Mach Joe 
Published: September 2015

Abstract
Numerous papers have been published in recent years on the subject of optimization of multiple transverse fractures in horizontal wells (for instance Saputelli et al., 2014). These studies usually focus on searching for an economical optimum based on multiple runs of 3D or 2D numerical simulator, each for certain fixed properties of hydraulic fractures. What we found missing is a systemized approach to calculate a solution to this problem. The objective of this study is to develop a systemized, rigorous mathematical and unified approach to the design of multiple transverse fractures in horizontal well – an extension of Unified Fracture Design (UFD).
This paper provides a rigorous methodology to optimize the number of fractures (and consequently, fracture geometry) for a given amount of proppant. We follow the UFD concepts and solve our problem in dimensionless variables. For the case of multiple fractures these are: Proppant Number (NP), Penetration Ratio (Ix), Dimensionless Conductivity (CfD) and Aspect Ratio (yeD) for each fracture, which is inversely proportional to the number of fractures. We used the Direct Boundary Element method to generate the Dimensionless Productivity Index (JD) for a given range of these parameters (28,000 runs) for the Pseudo-Steady state case. Finally we plot total JD as a function of the number of fractures for various NP, which allows optimization. In addition, we generate minimum width curves for various proppants, which represent a practical constraint.
Based on our study we found the following:
For a given volume or proppant, NP, total JD for multiple fractures increases to an asymptote as the number of fractures increases. This asymptote represents a technical potential for multiple fractures and for high Proppant Numbers (NP ≥ 100) reaches a technical potential of 3πNP. Below this asymptote, the more fractures that are created for a fixed NP the larger the JD In practice however, there's a minimum fracture width (3 proppant grains), which constrains the fracture geometry and therefore maximum JD. It was shown, that for the case when 20/40 sand is used for multiple hydraulic fracturing of 0.01md formation with square total area, optimal number of factures reduces to approximately Np25. Application of horizontal drilling technology with multiple fractures assumes availability of high Proppant Numbers. We show mathematically that the alternative low Proppant Numbers (NP ≤ 20 for the case in p.3) are impractical for multiple fractures because total JD cannot be significantly higher than JD for optimized single fracture in the same area. In practice this means low formation permeability and/or high proppant volumes are necessary for multiple fracture treatments. Our work shows the methodology to determine optimum geometry and required volume to perform multifracture treatments. Total proppant mass (and hence, NP) used for the fracture design must be selected based on economic considerations. For this purpose we constructed a relationship between total JD and the NP, which accounts for the minimum fracture width requirement.
Our paper presents a mathematically rigorous, systematic and comprehensive approach to the selection of optimal number of transverse hydraulic fractures in a horizontal well. Using the relationship between Proppant Number and maximum practical JD, the proppant mass should be selected for the treatment. Then, based on the formation and proppant permeability, the maximum number of fractures should be calculated for a given NP using the generated type curves and minimum width restriction.