Approximate Model for Productivity of Nonconventional Wells in Heterogeneous Reservoirs

Abstract

Summary A semianalytical method is presented for the approximate modeling of the productivity of nonconventional wells in heterogeneous reservoirs. The approach is based on Green's functions and represents an extension of a previous model applicable for homogeneous systems. The new method, referred to as the s-k* approach, models permeability heterogeneity in terms of an effective skin s that varies along the well trajectory and a constant background permeability k*. The skin is computed through local, weighted integrations of the permeability in the near-well region and is then incorporated into the semianalytical solution method. The overall method, which can also model effects due to wellbore hydraulics, is quite efficient in comparison to detailed finite-difference calculations. Results for the performance of nonconventional wells in three-dimensional heterogeneous reservoirs are computed using the s-k* approach and compared to finite-difference calculations resolved on the geostatistical fine grid. The new method is shown to provide an accurate estimate of wellbore pressure and production rate, as a function of position along the wellbore, for various well configurations and heterogeneous permeability fields. The possible use of the overall approach in a simulation while drilling (SWD) tool, in which the well path and trajectory are "optimized" using real-time data, is also discussed. Introduction Nonconventional wells (e.g., horizontal, deviated, or multilateral) have become quite common throughout the oil industry. In designing or optimizing the length and placement of such wells, it is important to estimate accurately the well productivity. One approach for determining this well productivity is to simulate the reservoir performance using a finite-difference simulator. This is the most rigorous approach available, though it is also the most demanding in terms of time and data requirements. An alternate approach for modeling the productivity of nonconventional wells operating under primary production is to employ a semianalytical solution technique. Early work along these lines included single horizontal wells (of infinite conductivity) aligned parallel to one side of a box-shaped reservoir. Solution methods were successive integral transforms1,2 and the use of instantaneous Green's functions,3–6 resulting in infinite series expressions. More complex geometries were considered later7–9 as numerical integration became more feasible. A number of works (see Ouyang,10 and citations therein) include coupling of wellbore hydraulics (i.e., finite-conductivity wells) with reservoir flow. The method we apply in this study9 has one of the most general treatments of wellbore hydraulics. All of the semianalytical techniques mentioned above have the advantage of limited data requirements and high degrees of computational efficiency. These techniques are, however, limited to homogeneous systems or at most strictly layered systems.11,12 This represents a substantial limitation because the productivity of nonconventional wells can be significantly impacted by fine-scale heterogeneities in the near-well region. Fine-scale heterogeneity can be incorporated into detailed simulation models, though the resulting models are complex to build and require substantial computation time to run. The purpose of this paper is to extend an existing semianalytical approach to approximately account for heterogeneity in the near-well region. This will enable us to apply the semianalytical approach to more realistic heterogeneous systems. We accomplish this by introducing an effective skin s into the semianalytical model and then estimating this effective skin as a function of position along the wellbore. The skin is computed via local, weighted integrations of the permeability field in the near-well region. This skin differs significantly from skin in the usual sense, as it is here due to intrinsic heterogeneity in the permeability field rather than from formation damage or stimulation. Away from the wellbore, the reservoir is modeled in terms of the large-scale effective permeability k* The overall method is highly efficient and approximates both near-well effects (through s) and global effects (through k*) with reasonable accuracy. The approach presented here combines and extends formulations developed in two separate earlier studies. These studies addressed the development of a semianalytical well model9 and the approximation of the effects of heterogeneity in the region near a vertical well.13 The semianalytical well model is applicable for very general well configurations and also accounts for pressure drop in the wellbore due to friction, gravitational, and acceleration effects. These can be important in long horizontal wells. The approximate heterogeneity model applied here was developed for the modeling of vertical wells in heterogeneous, two-dimensional areal systems. Both single-well and two-well systems were considered. The basic approach was shown to provide accurate estimates for well productivity, relative to fine-grid simulation results, for many geostatistical realizations over a range of geostatistical parameters. As will be shown below, our new method successfully builds upon both the semianalytical well model and the approximate heterogeneity model. Another technique for approximately modeling the effects of heterogeneity on horizontal wells was previously developed.14 This method, based on a network modeling type of approach, differs considerably from the procedure presented here in that our methodology has as its basis a semianalytical solution technique. The earlier method does, however, display accurate results for a range of problems similar to those considered here. This paper proceeds as follows. We first describe the overall method in some detail. Then, we present numerical results for horizontal and multilateral wells in heterogeneous three-dimensional systems. These results are in many cases compared with detailed finite-difference calculations to assess their level of accuracy. Our new description is shown to provide an accurate estimate of production rate, as a function of position along the wellbore, for a variety of well configurations and for different heterogeneous permeability fields. Finally, we discuss how the overall approach could represent a component of a simulation while drilling (SWD) capability, in which the well path and trajectory are "optimized" using real-time data coupled with our new well model.

Advanced Techniques for Reservoir Simulation and Modeling of Non-Conventional Wells

“…In addition, we present a modified means for approximating the effects of heterogeneity. This new approach provides improved numerical stability over that of our previous approach (Wolfsteiner et al, 2000a(Wolfsteiner et al, , 2000b) and allows the s-k* representation to be used for the highly heterogeneous fluvial reservoir descriptions considered here. Though the models currently used for pressure losses due to wellbore friction, acceleration, inflow, and downhole chokes are relatively simple, the general framework allows for the use of any model or correlation.…”

“…The semi-analytical (Green's function) representation that underlies our overall approach has been described previously (Economides et al, 1996;Maizeret, 1996;Ozkan et al, 1999;Tang et al, 2000;Wolfsteiner et al, 2000aWolfsteiner et al, , 2000b, Our description here will therefore be relatively brief and will emphasize new aspects of the formulation as well as some details required for our description of the residual and Jacobian construction below. Full details can be found in Valvatne (2000).…”

“…Our model for heterogeneity involves the use of an effective skin, introduced to capture the effects of near-well penneability fluctuations (Wolfsteiner et al, 2000a(Wolfsteiner et al, , 2000bDurlofsky, 2000) Wolfsteiner et al (2000a) described previously how skin is actually calculated; here we describe our new procedure for introducing this skin into the formulation.…”

“…As discussed by Wolfsteiner et al (2000a), our approximate s-k* representation of reservoir heterogeneity models the permeability field in terms of a constant, anisotropic background effective permeability k* and a near-well effective skin s. The background permeability is simply the effective large-scale permeability for linear flow, while the skin captures the effect of the "altered" or near-well permeability on the well performance. For the fluvial systems considered here, we found the best accuracy when we took the altered zone to be quite small; i.e., approximately the channel width (Valvatne, 2000).…”

“…The effects of wellbore pressure losses on well performance are included in some of these methodologies (e.g., Ozkan et al, 1999;Tang et al, 2000). In recent work (Wolfsteiner et al, 2000a(Wolfsteiner et al, , 2000bDurlofsky, 2000), our group introduced an approximate method for representing reservoir heterogeneity, which we call the s-k* permeability representation. This representation was implemented into the semianalytical formulation and was shown to provide results in reasonable agreement with finite difference simulations for a variety of problems.…”