An efficient semi-analytical solution method appropriate for the approximate modeling of the performance of non-conventional wells operating under primary production is described. The approach, referred to as the s-k*method, accounts approximately for the effects of reservoir heterogeneity and for pressure losses within the wellbore. The accuracy of the method is demon-strated through comparison to detailed finite difference results. Extensive calculations for the performance of non-conventional wells in multiple realizations of log-normally distributed permeability fields are then presented. The effect of wellbore hydraulics on well performance is assessed for several different well geometries and for varying degrees of reservoir heterogeneity. Comparisons between ensemble-averaged quantities (i.e., results averaged over many realizations) and results for a single "average" (homogeneous) reservoir representation are also made. As expected, these calculations differ in general, though for some quantities (e.g., productivity ratio) the homogeneous results give very accurate estimates. The overall approach provides an efficient means for estimating the combined effects of reservoir heterogeneity and wellbore pressure losses on the productivity of non-conventional wells.
The productivity of horizontal, multilateral or other non-conventional wells can be significantly impacted by both reservoir heterogeneity and by pressure losses within the wellbore. Tools capable of modeling these effects are therefore required in order to optimize the design of non-conventional wells. General finite difference reservoir simulators can be used to account for heterogeneity and wellbore flow effects, provided an accurate model for wellbore hydraulics is available within the simulator. The use of conventional reservoir simulators to model the effects of fine scale heterogeneities can be cumbersome, however, because the models can be time consuming to build and may require substantial computation time to run. This is an even more significant issue when many scenarios or geological realizations are to be simulated.
When applicable, alternative procedures, based on semi-analytical solution methods, are simpler and more efficient than finite difference methods. These semi-analytical methods therefore offer significant advantages over general finite difference techniques in some cases. In previous work, we presented a semi-analytical solution technique based on Green's functions for the estimation of the performance of non-conventional wells in heterogeneous reservoirs.1 We refer to the approach as the s-k*method, since heterogeneity is represented in terms of a near-well skin s and a background effective permeability k*. The method extends previous approaches in that it accounts approximately for reservoir heterogeneity and pressure losses in the wellbore due to frictional and accelerational effects. Like other semi-analytical approaches, the s-k*method is applicable in a limited set of cases; e.g., for single phase flow.