Semi-Analytic Calculations of Horizontal Well Productivity in Fractured Reservoirs

Abstract

Abstract Simplified analytical relations derived for homogeneous formations are usually applied for the determination of the productivity of horizontal wells regardless of the presence of heterogeneities in the reservoir. Furthermore, complex well architectures and the wealth of completion options currently available can not be properly taken into account, because the well trajectory can only be schematized as a single horizontal wellbore. However, the use of numerical reservoir simulators to reliably forecast the productivity of horizontal wells draining heterogeneous reservoirs may be time-prohibitive or not feasible due to lack of sufficiently detailed information, especially during the appraisal phase or the early stages of production. A new semi-analytic technique is proposed in this paper to solve the inflow equations in an approximate yet reliable manner. The solution to 3-dimensional problems of single-phase flow into a horizontal well, taking into account friction in the wellbore, is provided for both single-layer reservoirs and reservoirs comprising two interfering layers. The method has been extended also to describe the fluid flow when the well intercepts one or more fractures. The presented technique allows very fast calculation of the well productivity in oil and gas reservoirs offering great flexibility in the placement and architecture of the wells. The method has been applied to two field cases for which the well productivity under pseudo steady-state conditions was measured. One of these is a 200-m long horizontal well draining an isotropic carbonatic reservoir and intersected by a natural low-conductivity fracture. The other is a similar well, intercepting a natural high-conductivity fault, but the oil-bearing formation is anisotropic. Good correspondence was found between the actual productivity and the predictions obtained by application of the proposed semi-analytic technique. Introduction Horizontal wells are common practice in the present hydrocarbon industry, and smart wells, including multilateral completions and wells with selective access of different zones, are becoming increasingly commonplace. The modelling of such wells is in many cases not ideal. Areas where improvements are welcome are welltesting, well models in reservoir simulators, and fast models for quick assessment of many field development options. Further, the handling of natural or hydraulic fractures is often sub-optimal. In reservoir simulation, fine grids need to be selected to capture properly the flow behaviour close to the well. Moreover, most reservoir simulators are not equipped with extensive well models, which are required when friction in the well becomes important or when two-phase flow develops in the well. This situation has prompted the development of a number of analytical and semi-analytical tools, some of which are intended for implementation in a reservoir simulator. Most of the first models, and many more recent models as well, assume either constant influx density along the well or infinite well conductivity in a single homogeneous layer [1–7]. Dikken [8] introduced the effect of well conductivity for a single horizontal well in a homogeneous formation. He started with the assumption that the flow is mainly perpendicular to the wellbore, which allowed him to reduce the reservoir to a 2-dimensional flow domain, coupled to a friction model in the well. Others followed this approach [9–12], but 3D models were developed as well [13–17]. A second kind of extension are the multi-layer models [18–20]. Lee and Milliken [18], and Kuchuk and Habashy [19] used a method of reflection and transmission, while Basquet et al [20] used a "quadrupole" method relating the pressures between the various layers. The multi-layer models are also, however, still limited to constant-influx or infinite-conductivity wells. For wells with hydraulic fractures, Prats started working on analytic inflow performance correlations for a single fracture [21]. Van Kruijsdijk [22] used a combination of Laplace transformation and a Boundary Element formulation to model the transient response in fractured reservoirs. His model was later extended to include tight gas reservoirs, where non-Darcy flow in the fracture must be taken into account [23]. Kuppe and Settari [24] have made a number of reservoir simulations to cover multi-fractured reservoir, and to provide engineering correlations for many scenarios. Other contributions relevant to the so far unsolved problem of how to obtain a reliable horizontal well productivity, without being comprehensive, are listed in Refs. [25–28].