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The scaling laws are critically important in researching the characteristics of spontaneous imbibition and predicting oil and gas recovery from fractured reservoirs. This paper focuses on developing a shape factor for scaling the multi-dimensional counter-current imbibition processes. We studied the multi-dimensional imbibition characteristics by establishing numerical simulation models considering nine boundary conditions. On the basis of the capillary diffusion equation of counter-current imbibition, the equations of normalized oil recovery versus imbibition time were obtained under different boundary conditions. Then, we derived a new shape factor by processing and comparing these equations. The results show that the relationship between front distance and imbibition time of 2D and 3D models is different from that of 1D models. The shape factor for the multi-dimensional counter-current imbibition processes is time-dependent and varies with normalized recovery, which is also the reason for the difficulty of using a constant shape factor to scale imbibition data. Numerical simulation results and experimental data validated that the scaling equation could correlate the effect of differences in boundary conditions by introducing the new shape factor. • The difference between one-dimensional and multi-dimensional counter-current imbibition processes is clarified. • A time-dependent shape factor considering different boundary conditions is derived. • The newly proposed shape factor is validated through numerical simulation results and experimental data. • The application conditions of constant shape factor are analyzed. [ABSTRACT FROM AUTHOR]

The initial productivity of directional oil wells is essential to research during the early development stage of offshore oilfields. Since the influence factors of productivity are numerous, the nonlinear relationship among them and productivity is hard to accurately describe by the physical models in practical application. The data-driven model provides an alternative way to deal with this problem, although it neglects the physical correlation between productivity and its influence factors. Therefore, for combining the advantage of the physical model and data-driven model, the reservoir engineering theory was used to constrain the data-driven method in this study. Based on the eXtreme gradient boosting (XGBoost) trees algorithm, this paper proposed a novel physical constrained data-driven model to predict the initial productivity. The reservoir engineering theory, including the productivity formula and the monotonic correlation between productivity and the input feature, was employed as the physical constraints in this model. Moreover, the Spearman Correlation Coefficient and a modified Recursive Feature Elimination were combined to develop a feature selection method for quantitatively selecting the main influence factors of the initial productivity. Based on the production and geological data from 87 wells, the permeability, completed thickness, oil viscosity, clay content, correct factor for interference, choke size, and drawdown were selected from 19 features as the main influence factors of the initial productivity. The prediction accuracy of the model proposed in this paper is 80.18%, which is better than the previous data-driven models. Furthermore, the physical constraints have been proved to improve the forecasting accuracy of the data-driven method by 10.09%. • A novel physical constrained XGBoost model to predict the initial productivity is developed. • Productivity formula and monotonicity relationship among features are used as physical constraints in the data-driven model. • The reservoir engineering theory is considered to modify the RFE method in the feature-selection process. • The accuracy-enhancing effect of physical constraints on the data-driven algorithm is quantitatively evaluated. [ABSTRACT FROM AUTHOR]