water Article Spontaneous Imbibition in a Fractal Network Model with Different Wettabilities Shaobin Cai 1,2 , Li Zhang 3 , Lixin Kang 1,2 , Yongfei Yang 1,2, * , Wenlong Jing 1,2 , Lei Zhang 1,2, *, Chao Xu 1,2 , Hai Sun 1,2 and Mozhdeh Sajjadi 4   Citation: Cai, S.; Zhang, L.; Kang, L.; Yang, Y.; Jing,W.; Zhang, L.; Xu, C.; Sun, H.; Sajjadi, M. Spontaneous Imbibition in a Fractal Network Model with Different Wettabilities. Water 2021, 13, 2370. https:// doi.org/10.3390/w13172370 Academic Editors: Jianchao Cai and Steffen Berg Received: 28 June 2021 Accepted: 26 August 2021 Published: 29 August 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Key Laboratory of Unconventional Oil & Gas Development, Ministry of Education, China University of Petroleum (East China), Qingdao 266580, China; S19020157@s.upc.edu.cn (S.C.); kanglixinupc@163.com (L.K.); jingwenlongupc@163.com (W.J.); Z19020079@s.upc.edu.cn (C.X.); sunhai@upc.edu.cn (H.S.) 2 Research Center of Multiphase Flow in Porous Media, School of Petroleum Engineering, China University of Petroleum (East China), Qingdao 266580, China 3 Geological Exploration & Development Research Institute, CNPC Chuanqing Drilling Engineering Company Limited, Chengdu 610051, China; zhangl-sc@cnpc.com.cn 4 College of Chemical Engineering, University of Tehran, Tehran 1417466191, Iran; sajjadi.mozhdeh@ut.ac.ir * Correspondence: yangyongfei@upc.edu.cn (Y.Y.); zhlei84@163.com (L.Z.) Abstract: In this work, we derived a mathematical model for spontaneous imbibition in a Y-shaped branching network model. The classic Lucas–Washburn equation was used for modeling the im- bibition process occurring in the Y-shape model. Then, a mathematical model for the Newtonian fluid’s imbibition was derived to reveal the relationship between dimensionless imbibition time and length ratio, radius ratio, and wetting strength. The dimensionless imbibition time in the model was adopted to compare with that of the capillary bundle model. Different length and radius ratios were considered in the adjacent two-stage channels, and different wettabilities were considered in the different branches. The optimal radius ratio, length ratio, and wetting strength were calculated under the condition of the shortest imbibition time. In addition, the shortest dimensionless imbibition time of the three-stage Y-shaped branching network model was calculated when the wettability changes randomly. The results indicate that the imbibition time changed mostly when the wettability of the second branch changed, and the second branch was the most sensitive to wettability in the model. Keywords: porous media; capillary force; imbibition; fractal; L–W equation 1. Introduction Research on percolation theory is of great significance in various disciplines, such as soil physics [1], enhancing oil recovery [2–5], rock physics [6,7], fluid flows in porous media [8–10], and growth of branched structures [11]. Among this research in porous media, much of the literature concerns drainage processes rather than imbibition processes. However, imbibition processes take control of most fluid flows in tight porous media rather than drainage processes. The pressure difference opposed in the tight porous media [12–14] is most likely to fail in mobilizing hydrocarbon due to low connectivity of pores. The imbibition processes controlled by capillary force [15,16] gained from the extremely small size of the pores would be the dominant force. The wetting phase fluid enters the porous medium spontaneously and replaces the nonwetting phase fluid originally existing in the porous medium under the action of capillary force. This process is often referred to as spontaneous imbibition [17,18]. The existing literature on spontaneous imbibition theory is extensive and focuses particularly on the capillary bundle model. The existing theoretical models for the study of spontaneous imbibition mainly include the Lucas–Washburn model [19,20], Terzaghi model [21], Handy model [22], and dimensionless time scale model [23–28]. Lucas [19] simplified porous media into capillary bundles and proposed a capillary osmosis model. Based on Lucas’s Water 2021, 13, 2370. https://doi.org/10.3390/w13172370 https://www.mdpi.com/journal/water