A Simple and Accurate Mathematical Model for Predicting Productivity of Multifractured Horizontal Wells

Abstract

Abstract Horizontal wells drilled in the direction of the minimum horizontal stress allow for multiple transverse fractures be hydraulically created for enhancing well productivity in low-permeability oil and gas reservoirs. Dominating factors affecting the productivity of the multi-fractured horizontal wells vary with reservoir and fracture properties, as well as well trajectories. A simple and accurate mathematical model for evaluating and optimizing productivity of this type of wells is not available and is highly desirable to reservoir engineers. This paper fills the gap. After a case study with existing analytical models that were derived for multi-fractured wells we found that these models cannot describe the performance of oil wells with acceptable accuracy. We then formulated a simple analytical model that better describes the productivity of multi-fractured horizontal wells. The new model couples the radial flow in the non-fractured region of reservoir, the linear flow toward the fractures in the fractured region, the linear flow in the fracture, and the radial flow in the fracture toward the horizontal wellbore. It can model pseudo-steady state flow of reservoir fluids in sections reservoir of any shape with the fractured region being located at any area in the reservoir. The difference between the production rate given by the new model and the actual production rates was found to be less than 5% in the two cases studied. This paper provides reservoir engineers a simple and accurate tool for predicting, evaluating, and optimizing the performance of multi-fractured horizontal oil and gas wells. Introduction The use of horizontal wells for producing oil and gas from low-permeability reservoirs is now firmly established within the energy industry. Stimulation of a horizontal well in a low-permeability reservoir can further increase well productivity. Multi-fractured horizontal wells have been successfully applied in a number of oil and gas fields. However, there is lack of an analytical method that can be used for evaluating and optimizing the performance of this type of wells. Giger et al. (1984) presented the first mathematical model for analyzing productivity of horizontal wells intersecting fractures. Giger et al.'s model does not rigorously couple the flow in the reservoir and in the fracture. Karcher et al. (1986) and Soliman et al. (1990) adapted Giger's model in their numerical simulators for productivity improvement studies. Joshi (1988) derived an analytical model on the basis of infinite fracture conductivity. Mukherjee and Economides (1991) developed a simplified model on the basis of Joshi's (1988) work and Prats' correlation of dimensionless wellbore radius. Economides et al. (1989) performed a numerical stimulation of horizontal well production and confirmed Joshi's (1988) equation. Raghavan and Joshi (1993) presented a mathematical model for predicting productivities of horizontal wells with multiple transverse fractures. The model uses the effective wellbore radius (in radial flow) to simulate the fluid flow toward the fractured well. Flow within the fracture was not considered. Wan and Aziz (2002) presented a semi-analytical well model for horizontal wells with multiple hydraulics fractures. More practical models were presented by Li et al. (1996) and Wei and Economides (2005) that couple the reservoir linear flow, fracture linear flow, and fracture radial flow. Guo and Schechter (1997) presented a rigorous mathematical model coupling the reservoir linear flow and fracture linear flow. But the fracture radial flow was not considered. None of the existing mathematical models couples the reservoir radial flow, reservoir linear flow, fracture linear flow, and fracture radial flow. Such an analytical model is highly desirable for evaluation and optimization of multi-fractured horizontal well.