Nernst-Planck equation

The efficiency of low salinity waterflooding, particularly during tertiary mode injection, is highly controlled by in situ mixing between the stagnant regions holding high salinity water (HSW) and the flowing regions containing low salinity water (LSW) because it impacts directly the electrokinetics of wettability alteration and the time scale of the low salinity effect. This study aims to address the effects of oil polarity and charged rock surfaces on the time scale of mixing and transport under two-phase flow conditions. A systematic series of micromodel experiments were performed. The micromodels were first saturated with high salinity formation brine and oil (both model and crude oil); thereafter, HSW and LSW were injected sequentially and the mixing time was carefully monitored. Besides, the polarity of model oils was manipulated by adding oleic acid as the representative for the acidic functional groups of crude oil. In addition to the experimental work, conceptual computational fluid dynamics (CFD) simulations were performed to get further insights into the experimental observations and to identify the dominant parameters on the time scale of mixing in stagnant regions. The experimental results show that the time scale of ionic transport in stagnant regions can be slowed down 10−16 times by the increase of the polar fractions of oil, where the time scale cannot be simply described by the Fickian diffusion. It is postulated that the stronger electrical field established in the water film by the increase of oil polarity lowers the solute transport rate according to the Poisson−Nernst−Planck theory. The results were further verified by using crude oil which is highly polar and contains complex types of polar components such as resins/asphaltenes. Experimental results clearly indicated the electrokinetics effect as the time scale was increased even further. Nevertheless, the mixing time does not vary linearly with the polar group fraction and follows a rather logarithmic trend. The CFD simulation confirms that the effective diffusion coefficient (which is influenced by the oil polarity and the induced electric field) is the predominant parameter determining the time scale of mixing. Other parameters such as film thickness/film length and salinity gradient have comparatively a lesser effect on the time scale of mixing in the stagnant regions.

In this study the laws of mechanics for multi-component systems are used to develop a theory for the diffusion of ions in the presence of an electrostatic field. The analysis begins with the governing equation for the species velocity and it leads to the governing equation for the species diffusion velocity. Simplification of this latter result provides a momentum equation containing three dominant forces: (a) the gradient of the partial pressure, (b) the electrostatic force, and (c) the diffusive drag force that is a central feature of the Maxwell-Stefan equations. For ideal gas mixtures we derive the classic Nernst-Planck equation. For liquid-phase diffusion we encounter a situation in which the Nernst-Planck contribution to diffusion differs by several orders of magnitude from that obtained for ideal gases. The study of ion transport in fluids is an important topic with a wide range of applications. Some classic examples are batteries, fuel cells, electroplating, and protection of metal structures against corrosion 1. In addition to the traditional battery, the flow battery or rechargeable fuel cell 2,3 represents an important new technology involving the transport of ions. Ion exchange membranes have a wide range of applications 4 , and the underlying theory has been a matter of concern for several decades 5. Other examples of complex electro-chemical systems are the transport of charged particles in ion channels 6–8 , in protein channels 9 , and during the primordial conversion of light to metabolic energy 10. Often upscaling is necessary for a complete analysis of the transport of electrolytes in charged pores 11,12. Much of ion transport occurs at the nano-scale 13 , and most of the studies use the Nernst-Planck equation 14–16 to describe this type of phenomena. However, there are molecular dynamic simulations indicating that the Nernst-Planck equation does not always provide a complete description 17 , and there are experimental studies that lead to the same conclusion 18. The need to analyze the limits of the Nernst-Planck contribution to diffusion has been emphasized 19 in an exploration of the transport of divalent ions in ionic channels. The authors of this paper have not found a derivation of the Nernst-Planck equation that does not make use of the ideal gas assumption. To be precise we note that ideal gas behavior for mixtures is based on Dalton's laws (see page 114 in ref. 20) that we list as Some care must be taken with the interpretation of Eq. 2 since it applies to all Stokesian fluids and is therefore not limited to ideal gas mixtures (see Appendix B in ref. 21). Caution must also be used with the interpretation of Eq. 3 which is applicable to all ideal fluid mixtures. For example, Eq. 3 should provide reliable results for a liquid mixture of hexane and heptane, but it should not for a mixture of ethanol and water. Here it is important to emphasize that Eq. 1 can only describe a fluid composed of non-interacting particles. To provide a reliable framework in which a fluid is considered to be composed of interacting particles ̶ as in the case of a liquid ̶ it is necessary to understand the process of Nernst-Planck diffusion in a fluid different from an ideal gas. This is the problem under consideration in this paper. In this work we analyze the ion transport process from a fundamental point of view using an axiomatic mechanical perspective. Our analysis is for ideal fluid mixtures, and from the analysis we find that the classic