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By introducing the elastic boundary value condition, the elastic boundary value problem of extended modified Bessel equation is proposed, we can use the following method to solve it. First, two linear independent solutions of extended modified Bessel equation are obtained. Second, the generating function of solution is constructed. Third, the kernel function of solution is constructed using the elastic right value condition. Finally, the solution is obtained by assembling coefficients with the left boundary value condition. As for its application, a fractal homogeneous reservoir seepage model under the elastic outer boundary condition is established, and solution of the model is obtained. Influences of reservoir parameters on characteristic curves corresponding to dimensionless bottom hole pressure and its derivative are analyzed, which provide a new theoretical basis for exploring the flow law of oil. It can be found that seepage model under the elastic outer boundary condition regards three idealized outer boundary conditions (infinite, constant pressure and closed) and homogeneous reservoir seepage model without considering fractal as special cases, so it can reflect real situation of reservoir better and it is helpful to the development of related well test analysis software. Keywords Extended modified Bessel equation • Elasticity • Fractal homogeneous reservoir • The generating function • The kernel function Mathematics Subject Classification 03C98 • 33B99 • 34B60 • 3B340 List of symbols B Oil volume factor (m 3 /m 3) C Well storage coefficient (m 3 /MPa) Communicated by Jorge X. Velasco.

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In the middle and late stages of heavy oil development, formulating a scientific and reasonable mining plan is the key to improving oilfield efficiency. At present, steam stimulation is still the main development method of heavy oil. The determination of its production is not only limited by boiler conditions, surface pipelines, and wellbore conditions but also by the steam absorption capacity of the formation. Therefore, local analysis cannot achieve the best effect in the whole process of steam stimulation. The mechanism model is the most commonly used method to predict heavy oil production, but too many idealized assumptions make the prediction results quite different from the actual production situation. With the rapid development of machine learning, people can achieve rapid prediction of production through field data. However, when the range of the actual parameter is small, the generalization ability of the model is weak and overfitting occurs. Based on the above background, this paper conducts a coupling study on surface steam pipeline flow, steam injection wellbore flow, and formation flow from the perspective of data-driven. Firstly, based on the correlation coefficient and the feature selection of Random Forest, the importance of the characteristics affecting liquid production and water content was ranked. Secondly, through the comparison of five typical machine learning algorithms, we select the optimal prediction model and optimal characteristics suitable for the sample of this paper. Finally, because of the poor generalization ability of the prediction model, we sampled the mechanism model and increased the diversity of steam dryness samples. We find that the accuracy of the optimal prediction model is improved and the generalization ability of the model is improved after the training of new samples. This paper provides a new idea for the production prediction of heavy oil steam stimulation reservoirs, which is helpful for the efficient development of heavy oil reservoirs.

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