**Productivity-Index Behavior for a Horizontal Well Intercepted by Multiple Finite-Conductivity Fractures Considering Nonlinear Flow Mechanisms under Steady-State Condition**

#### **Maojun Cao 1,2,\*, Hong Xiao 1 and Caizhi Wang 2**


Received: 23 March 2020; Accepted: 13 April 2020; Published: 17 April 2020

**Abstract:** In this paper, a mathematical model is proposed to investigate the effect of nonlinear flow mechanisms on productivity-index (PI) behavior in hydraulically fractured reservoirs during steady-state condition. This approach focuses on the fact that PI approaches a constant value at a certain time, indicating the beginning of steady state. In this model, the reservoirs are considered as an elliptical-shaped drainage with constant-pressure boundary, which is depleted by a multiple-fractured horizontal well (MFHW), and various nonlinear flow mechanisms, such as the non-Darcy flow effect and pressure-dependency effect, control flow patterns in the hydraulic fractures. Then, an exact algorithm of solving the resulting nonlinear equations is developed to obtain the PI of MFHW using a semi-analytical approach. Next, type curves are generated to investigate the influences of flow mechanisms and fracture properties on PI. The most interesting points in this study are the following: (1) PI is determined by the properties of MHFW (i.e., dimensions and configuration), the reservoir geometry, and flow mechanism; (2) PI is deteriorated by non-Darcy flow caused by inertial forces; and (3) PI is reduced under the influence of pressure sensitivity caused by the degradation of dynamic conductivity. Generally, this study provides a significant insight into understanding the factors affecting the productivity of a MFHW with nonlinear flow mechanisms.

**Keywords:** horizontal well with multiple finite-conductivity fractures; elliptical-shaped drainage; productivity index; non-Darcy flow; pressure-dependent conductivity
