Study of the pressure transient behavior of directional wells considering the effect of non-uniform flux distribution

During the production, the fluid in the vicinity of the directional well enters the wellbore with different rates, leading to non-uniform flux distribution along the directional well. However, in all the existing studies, it is oversimplified to a uniform flux distribution, which can result in inaccurate results for field applications. Therefore, this paper proposes a semi-analytical model of a directional well based on the assumption of non-uniform flux distribution. Specifically, the direction well is discretized into a carefully chosen series of linear sources, such that the complex well trajectory can be captured and the non-uniform flux distribution along the wellbore can be considered to model the three-dimensional flow behavior. By using the finite difference method, we can obtain the numerical solutions of the transient flow within the wellbore. With the aid of Green's function method, we can obtain the analytical solutions of the transient flow from the matrix to the wellbore. The complete flow behavior of a directional well is perfectly represented by coupling the above two types of transient flow. Subsequently, on the basis of the proposed model, we conduct a comprehensive analysis of the pressure transient behavior of a directional well. The computation results show that the flux variation along the direction well has a significant effect on pressure responses. In addition, the directional well in an infinite reservoir may exhibit the following flow regimes: wellbore afterflow, transition flow, inclined radial flow, elliptical flow, horizontal linear flow, and horizontal radial flow. The horizontal linear flow can be observed only if the formation thickness is much smaller than the well length. Furthermore, a dip region that appears on the pressure derivative curve indicates the three-dimensional flow behavior near the wellbore.