A Semianalytic Model for the Productivity Testing of Multiple Wells
- Peter A. Fokker (TNO Inst. Appl. Geosciences) | Francesca Verga (Politecnico di Torino)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- June 2008
- Document Type
- Journal Paper
- 466 - 477
- 2008. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 1.2.2 Geomechanics, 5.5.3 Scaling Methods, 5.3.1 Flow in Porous Media, 5.5.7 Streamline Simulation, 3 Production and Well Operations, 5.6.4 Drillstem/Well Testing, 5.1.2 Faults and Fracture Characterisation, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 5.5 Reservoir Simulation, 4.6 Natural Gas, 5.5.8 History Matching
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We present a semianalytic method for modeling the productivity testing of vertical, horizontal, slanted, or multilateral wells. The method is applicable to both oil and gas reservoirs and automatically accounts for well interference. The use of analytic expressions ensures that short-time transient behavior and long-time semisteady-state behavior are handled appropriately, whether close to the well or further into the reservoir. Calculation times are still very limited—on the order of a few minutes to a few seconds when all wells are vertical. This makes the tool suitable for evaluating well testing and determining well productivity.
We based the approach on an earlier derived productivity prediction tool, in which the steady-state equations were solved. It has now been extended to solve the time-dependent diffusion equation. In our current method, the equations have first been transformed using the Laplace transformation. The expressions for the producing wells are combined with auxiliary sources outside the reservoir. The crux of the semianalytic method involves an adjustment of the positions and strengths of these sources in order to approximate the boundary conditions at the reservoir boundaries. The solution obtained is transformed back into the time domain by use of a Stehfest algorithm.
The new approach has been validated with numeric tools, including both reservoir simulators and well-test interpretation software. Validations were performed with artificial cases and with field production data, using both single-well and multiple-well production tests. The results of these tests were excellent.
In a previous paper, we presented a novel semianalytic method to calculate productivities of complex wells in steady-state or pseudosteady-state conditions (Fokker et al. 2005b). The method is very flexible in that it is able to combine finite-conductivity wells, well interference, nonhomogeneous reservoirs, and finite-conductivity natural or hydraulic fractures. Innovative well geometries and completion, including variably spaced producing intervals, can be taken into account to estimate the well productivity under any flow conditions. Despite this, calculations with the new approach are fast.
The methodology that we propose can be classified as an example of the method of fundamental solutions, as reviewed by Fairweather and Karageorghis (1998). A similar method has been developed to predict subsidence caused by exploiting hydrocarbon reservoirs in a layered subsurface (Fokker and Orlic 2006). For a review of other available fast models for predicting productivity, see Fokker et al. (2005b) and the references provided therein.
Our new method is suitable for incorporation into a reservoir simulator. It can also be applied as a fast model to evaluate completion scenarios for a given reservoir (e.g., in an automated decision support environment). In this paper, we present a refinement of the method, for describing the pressure—and, thus, the evolution—of well productivity under transient conditions.
The available literature on well testing is extensive, and any summary within the context of this paper is bound to be incomplete. The interested reader is therefore referred to Matthews and Russell (1967), Earlougher (1977), Lee (1982), Bourdet (2002), Horne (1995), Zheng and Corbett (2005), and Gringarten (2008) for an overview of the currently available interpretation methodology. For the testing of horizontal wells or multilaterals (including interference between wells or well branches), we also refer the reader to select papers: Ozkan et al. (1989), Kuchuk (1995), Basquet et al. (1998), Ding (1999), Al-Khamis et al. (2001, 2005), Kuchuk and Onur (2003), Yildiz (2003), Shimamoto (2006), and Medeiros et al. (2006). Most of the approaches in this field use expressions that are, in principle, exact solutions of the boundary-value problem. Shimamoto (2006) uses a totally different approach, by combining a streamline simulation and a mapping of the physical space on a radially symmetric domain to obtain simple, approximate "S-functions" that represent the transient response of the well. Our approach is more like the approach of Furman and Neuman (2003), who use "analytic elements" (i.e., analytic functions), in such a way that the boundary or interface conditions are approximated.
The method we propose can be applied to determine the well productivity of any type of well in the short or long term; in the latter case, the method's reliability can be improved by fine-tuning the model with the transient pressure response obtained from a well test.
The diffusion equation is transformed with the Laplace transform. In Laplace space, the solution is obtained similarly to the original, pseudosteady-state solution. Because no analytic solution to the transformed diffusion equation of a finite well segment in 3D is available, a numeric integration has to be used. A back transformation with the Stehfest algorithm completes the approach.
We will start with a brief mathematical description of the new approach. Then we will present a validation of the method for oil and gas wells of simple geometry by comparison with the results obtained with a reservoir simulator and with well-testing software. The usefulness of the code will be demonstrated with some applications to data obtained in real well tests and a comparison with a well-testing software package. We complete the paper with a discussion and conclusions.
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