Abstract
The flow of fluids in multi-phase porous media results due to many interesting natural phenomena. The counter-current water imbibition phenomena, that occur during oil extraction through a cylindrical well is an interesting problem in petroleum engineering. During the secondary oil recovery process, water is injected into a porous media having heterogenous and homogenous characteristics. Due to the difference in viscosities of fluids in oil wells, the counter-current imbibition phenomenon occurs. At that moment, the imbibition equation V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> =-V <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is satisfied by the viscosities of oil and water. In this article, we have analyzed the governing mathematical model of the imbibition phenomenon occurring during the secondary oil recovery process. A new soft computing algorithm is designed and adapted to analyze the mathematical model of dual-phase flow in detail. Weighted Legendre polynomials based artificial neural networks are hybridized with an efficient global optimizer the Whale Optimization Algorithm (WOA) and a local optimizer the Nelder-Mead algorithm. It is established, that our algorithm LeNN-WOA-NM is efficient and reliable in calculating high-quality solutions in less time. We have compared our experimental outcome with state-of-the-art results. The quality of our solutions is judged based on values of absolute errors, MAD, TIC, and ENSE. It is obvious that LeNN-WOA-NM algorithm can solve real application problems efficiently and accurately.
Highlights
T HIS paper models the phenomena of counter-current imbibition in the multi-phase flow of two immiscible fluids through heterogeneous and homogenous porous mediums that occurs during the secondary oil recovery process or water flooding [1]
Imbibition is one of the interesting phenomena which occurs during oil extraction through a cylindrical well and is an important problem arising in petroleum engineering
The mathematical model for heterogeneous porous media is presented in Eq (29-31), and for homogeneous porous media, the ordinary differential equation together with its conditions is given in Eq (4042)
Summary
T HIS paper models the phenomena of counter-current imbibition in the multi-phase flow of two immiscible fluids through heterogeneous and homogenous porous mediums that occurs during the secondary oil recovery process or water flooding [1]. We have discussed the imbibition phenomenon occurring during the secondary oil recovery (displacement) process through heterogeneous and homogenous porous mediums with capillary pressure. The optimal homotopy analysis method is used to find approximated solutions of imbibition phenomena arising in heterogenous porous media [14] Another interesting problem of spontaneous imbibition is investigated with a multi-layer porous medium [15]. It is worth noting that in this paper the mathematical model is presented to discuss the heterogenous porous media along with homogenous porous media where oil is assumed as the non-wetting fluid and water is considered as wetting fluid Errors in our solutions dictate that LeNN-WOA-NM algorithm calculated the best so far solution for the problem of saturation of injected fluid (water) through heterogeneous and homogenous mediums. The results for saturation of injected fluid are shown through different graphs and tables which dictate the dominance and robustness of the proposed (LeNN-WOA-NM) algorithm
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