A fractal model for streaming potential coefficient in porous media

Abstract

Streaming potential is the result of coupling between a fluid flow and an electric current in porous rocks. The modified Helmholtz–Smoluchowski equation derived for capillary tubes is mostly used to determine the streaming potential coefficient of porous media. However, to the best of our knowledge, the fractal geometry theory is not yet applied to analyse the streaming potential in porous media. In this article, a fractal model for the streaming potential coefficient in porous media is developed based on the fractal theory of porous media and on the streaming potential in a capillary. The proposed model is expressed in terms of the zeta potential at the solid−liquid interface, the minimum and maximum pore/capillary radii, the fractal dimension, and the porosity of porous media. The model is also examined by using another capillary size distribution available in published articles. The results obtained from the model using two different capillary size distributions are in good agreement with each other. The model predictions are then compared with experimental data in the literature and those based on the modified Helmholtz–Smoluchowski equation. It is shown that the predictions from the proposed fractal model are in good agreement with experimental data. In addition, the proposed model is able to reproduce the same result as the Helmholtz–Smoluchowski equation, particularly for high fluid conductivity or large grain diameters. Other factors influencing the streaming potential coefficient in porous media are also analysed.

A Physically Based Analytical Model to Describe Effective Excess Charge for Streaming Potential Generation in Water Saturated Porous Media

“…When it is not the case, alternative formula have been proposed by several researchers (e.g., Glover & Déry, 2010;Revil et al, 1999). Various model using capillaries to predict the coupling coefficient under saturated and partially saturated conditions can be found in the literature (e.g., Jackson, 2010;Jackson & Leinov, 2012;Rice & Whitehead, 1965;Ishido & Mizutani, 1981;Thanh et al, 2017). The second approach to simulate streaming current generation is more recent and focuses on the excess charge that is dragged by the water flow.…”

“…The classical approach relies on the use of the coupling coefficient, which is a rock-dependent property that relates the difference of hydraulic pressure to the difference in electrical potential. Various model using capillaries to predict the coupling coefficient under saturated and partially saturated conditions can be found in the literature (e.g., Jackson, 2010;Jackson & Leinov, 2012;Rice & Whitehead, 1965;Ishido & Mizutani, 1981;Thanh et al, 2017). The so-called Helmholtz-Smoluchowski coupling coefficient has been developed from a capillary tube model, and its final expression does not depend on the geometrical properties of the porous medium.…”

A physically based model for the electrical conductivity of water-saturated porous media

“…Existing electrical models with surface conductivity based on different approaches such as the two-resistors in parallel approach [e.g., Waxman & Smits, 1968, Brovelli et al, 2005, the effective medium [e.g., Hanai, 1961, Bussian, 1983, the volume averaging theorem [e.g., Pride, 1994, Linde et al, 2006 have also been presented in literature. Recently, fractal models have been proven to be an alternative and useful means for studying the transport phenomenon and analyzing the macroscopic transport properties of porous media [e.g., Mandelbrot, 1982, Thompson et al, 1987, Feder & Aharony, 1989, Thompson, 1991, Sahimi, 1993, Ghanbarian-Alavijeh et al, 2011, Xu, 2015, Thanh et al, 2018, Guarracino & Jougnot, 2018. Fractal theory in porous media has been applied to derive theoretical electrical conductivity models [e.g., Katz & Thompson, 1985, Pape et al, 1987, Roy & Tarafdar, 1997, Coleman & Vassilicos, 2008a,b, Wei et al, 2015.…”

Examination of the fractal model for streaming potential coefficient in porous media

“…Fractal models on porous media have attracted increasing interests from many different disciplines [e.g., [15][16][17][18][19][20][21][22]. Recently, Luong et al [23] have presented a fractal model for the streaming potential coefficient in porous media based on the fractal theory of porous media and on the streaming potential in a capillary. The proposed model has been applied to explain the dependence of the streaming potential coefficient on the grain size.…”

“…In this work, the fractal model for the streaming potential coefficient in porous media presented in [23] is examined by calculating the zeta potential that is normally determined by a conventional Helmholtz-Smoluchowski (HS) equation. Obtained values are then compared with experimental data available in literature.…”