A Hybrid Numerical-Analytical Model of Finite-Conductivity Vertical Fractures Intercepted by a Horizontal Well

Abstract

Abstract This paper presents a hybrid, numerical-analytical model for the pressure-transient response of a finite-conductivity fracture intercepted by a horizontal well. The model dynamically couples a numerical fracture model with an analytical reservoir model. This approach allows us to include finer details of the fracture characteristics while keeping the computational work manageable. For example, the fracture may have irregular shape, nonuniform width, and variable conductivity, and the well may not intersect the fracture at its geometric center. This model can be used as a pressure-transient analysis tool to interpret complex pressure-transient characteristics or as a diagnostic tool to investigate the influence of various fracture properties on the production performance of fractured horizontal wells. We discuss the effect of fracture characteristics on the existence of different flow regimes. We present the distribution of flow rate into the fracture and flow convergence toward the wellbore. The model helps to understand the productivity loss due to flow choking and nonDarcy flow at the intersection of the horizontal well. Introduction Fracturing horizontal wells is a common practice in tight formations.1,2 Choking of flow around the horizontal well-fracture intersection, however, strongly influences the flow characteristics and reduces the productivity of the fracture. Fig. 1 shows the pressure surface on the fracture plane for a square hydraulic fracture that is intercepted by a horizontal well. The apex of the surface indicates the location of the well intersection and the increased pressure gradients around the well highlights the choking effect. This aspect of transverse hydraulic fractures emanating from horizontal wells is different from vertical wells. Raghavan et al.3 and Chen and Raghavan4 used vertical-well fracture models5,6 to approximate the pressure-transient responses of fractured horizontal wells (Fig. 2). These models assume uniform-flux or infinite-conductivity rectangular fractures with uniform width. The fracture communicates with the wellbore over its entire height. Therefore, flow is linear within the fracture and the choking of flow around the well intersection is ignored. In more rigorous fractured horizontal well models,7,8 a circular, fully penetrating fracture has been assumed (Fig. 3). In these models, the fracture has uniform width and conductivity and the well pierces the fracture plane at the center. Choking of flow around the wellbore intersection is taken into account but the circular geometry of the fracture imposes radial flow within the fracture. In general, different fracture geometries may lead to different fracture flow regimes (Fig. 4). If the fracture is a long rectangle, for example, instead of radial flow, linear flow dominates in the fracture. Because the estimation of fracture conductivity and half-length from pressure-transient analysis is strongly influenced by the specific characteristics of certain flow regimes (Fig. 5), the pressure-transient model should take into account the correct fracture geometry. This study presents a hybrid numerical-analytical model for the pressure-transient performance of fractured horizontal wells. In this model, the fracture flow is numerically simulated and dynamically coupled with an analytical solution for the reservoir flow. Compared with a fully numerical model, using an analytical solution for the reservoir reduces the computational work and allows concentrating on the details of the fracture flow. It also provides great flexibility to model and analyze different fractured horizontal well scenarios. For example, the fracture can have an irregular shape due to geological complexities, conductivity can be variable within the fracture because of non-uniform gel and proppant placement or non-planar fracture profile, and the fracture width may be variable because of an elliptical cross-section. It is also possible to incorporate nonDarcy flow within the fracture. The model presented in this paper is not limited to transverse horizontal-well fractures. Because the wellbore is represented as a source term in the fracture grid, several grids may include the wellbore source terms to simulate the appropriate intersection of the wellbore and the fracture plane (Fig. 6).