Approximate Model for Productivity of Nonconventional Wells in Heterogeneous Reservoirs
- Christian Wolfsteiner (Stanford U.) | L.J. Durlofsky (Stanford U. and Chevron Petroleum Technology Co.) | Aziz Khalid (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2000
- Document Type
- Journal Paper
- 218 - 226
- 2000. Society of Petroleum Engineers
- 5.1.1 Exploration, Development, Structural Geology, 1.8 Formation Damage, 4.1.5 Processing Equipment, 3.2.8 Well Performance Modeling and Tubular Optimization, 5.3.2 Multiphase Flow, 5.5.3 Scaling Methods, 1.12.2 Logging While Drilling, 5.1.5 Geologic Modeling, 1.6 Drilling Operations, 4.1.2 Separation and Treating, 5.5 Reservoir Simulation, 2.2.2 Perforating, 4.3.4 Scale
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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.
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.
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