Production Forecasting with Logistic Growth Models
- Aaron James Clark (Petrobras) | Larry Wayne Lake (U. of Texas at Austin) | Tadeusz Wiktor Patzek (U. of Texas at Austin)
- Document ID
- Society of Petroleum Engineers
- SPE Annual Technical Conference and Exhibition, 30 October-2 November, Denver, Colorado, USA
- Publication Date
- Document Type
- Conference Paper
- 2011. Society of Petroleum Engineers
- 1.6.6 Directional Drilling, 5.6.9 Production Forecasting, 5.7 Reserves Evaluation, 1.6 Drilling Operations, 3.2.3 Hydraulic Fracturing Design, Implementation and Optimisation, 2 Well Completion
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With the commercial development of extremely low permeability oil and gas reservoirs, new challenges have arisen both from operational and reservoir standpoints. Reservoir models, which previously yielded reasonable results for reserves estimates and production forecasts, no longer do so. Various new models and techniques have been proposed to improve the accuracy and reliability of reserves estimates; however, none have gained widespread industry acceptance. This paper will propose a new empirical model for production forecasting in extremely low permeability oil and gas reservoirs based on logistic growth models. The new model incorporates known physical volumetric quantities of oil and gas into the forecast to constrain the reserve estimate to a reasonable quantity. The new model is easy to use, and it is very capable of trending existing production data and providing reasonable forecasts of future production. The logistic growth model does not extrapolate to non-physical values.
One source of production to meet the demand for oil and gas has come from extremely low permeability oil and gas reservoirs, often referred to as unconventional resources. These unconventional resources typically exhibit permeabilities in the nanodarcy to microdarcy range. The production from these wells is characterized by very long and extended periods of transient flow before reaching the reservoir boundary, and entering into boundary-dominated flow. Thanks to drastic improvements in both horizontal drilling and hydraulic fracturing, these resources have been able to be successfully exploited in North America. Along with providing many new challenges in successfully drilling and completing unconventional wells, they have also provided new challenges in accurately forecasting the reserves.
The traditional empirical Arps' equation used to forecast reserves in conventional reservoirs, often gives over estimates of reserves when used in these extremely low permeability formations. Commonly the b values obtained are greater than 1, which results in an unrealistic production rate that never approaches 0. The purpose of this paper, however, is not to discuss the errors encountered when using the Arps' equation, as the problem has already been thoroughly discussed in the past. Lee (2010) and Ilk et al. (2008) have both provided good explanations of the problems encountered with the traditional models as have numerous other authors over the years. Various new models for use in production forecasting in tight reservoirs have been proposed including Maley (1985), Ilk et al. (2010), Kupchenko et al. (2008), and Valko (2009). The industry has been slow to adopt the new methods for forecasting reserves despite the apparent over estimation of reserves encountered when using the traditional decline curve models. This paper presents a new method for empirically forecasting production based on the logistic growth model.
Logistic Growth Models
Logistic growth curves are a family of mathematical models used to forecast growth in numerous applications. Originally developed by the Belgian mathematician Pierre Verhulst in the 1830s, logistic growth curves were used to model population growth. Verhulst based his ideas on the works of Malthus who believed that the population of a particular country or region would only be able to grow to a certain size before competition for resources would cause the growth to stabilize. Verhulst took this idea and by adding a multiplicative factor to the equation for exponential growth, created the logistic growth model.
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