- Examination of a Theoretical Model of Streaming Potential Coupling Coefficient, D. T. Luong and R. Sprik

International Journal of Geophysics

Research Article (12 pages), Article ID 471819, Volume 2014 (2014)

Published 27 May 2014

International Journal of Geophysics

Volume 2015, Article ID 941246, 8 pages

http://dx.doi.org/10.1155/2015/941246

## Comment on “Examination of a Theoretical Model of Streaming Potential Coupling Coefficient”

School of Earth and Environment, University of Leeds, Leeds LS2 9JT, UK

Received 15 May 2015; Accepted 1 October 2015

Academic Editor: Rudolf A. Treumann

Copyright © 2015 P. W. J. Glover. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Recently, Luong and Sprik published an article that compared measurements that had been made on 20 samples of saturated rock with a number of empirical models and Glover et al.’s 2012 theoretical model for zeta potential and streaming potential coefficient. They found that none of the empirical models could reproduce the streaming potential coefficient measurements which had been made in the presence of low pore fluid salinities, and the theoretical method could only do so if a constant zeta potential was invoked. This contribution in the form of a comment (i) indicates at least three possible errors in modelling that contribute to the mismatch between the theoretical model and the data at low salinities and (ii) carries out individual modelling on all of samples of Luong and Sprik’s 2014 dataset, showing that Glover et al.’s 2012 theoretical model matches the data well when the zeta potential is allowed to vary and good match can only be obtained with a constant zeta potential if an unrealistic value of zeta potential offset is used.

#### 1. Introduction

Recently, Luong and Sprik [1] published an examination of Glover et al.’s theoretical approach to zeta potential and streaming potential modelling of porous media [2, 3]. Luong and Sprik [1] recognized that although the comparison of the model with data in Glover et al. [3] was carried out with a large database which was extracted from the literature and which, at the time, was believed to represent a large portion of available data, there were problems with the test data. The problems included lack of reliable knowledge with regard to the pore fluid salinities, pH, and temperatures at which the experiments were carried out, as well as lack of the microstructural parameters for each sample. Microstructural rock parameters are necessary in order to carry out modeling for individual rock samples rather than generic rock types.

We have also recognized the same limitations in the published data and have set out independently to generate a high quality database in order to resolve the deficiency. The new approach to measurement which we developed was published in Walker et al. [4], and we now have 1253 measurements in our database.

Luong and Sprik [1] have followed the same approach. They have carried out high quality measurements on 20 rock samples over a restricted salinity range (136 measurements). They compared their data with the available empirical models [5–8] for streaming potential coefficient. They also implemented Glover et al.’s [3] theoretical model for individual rock samples using the cementation exponent, formation factor, and permeability to characterize the rock microstructure. In this process, they effectively use the fluid permeability as a proxy for the rock’s modal grain size. Their conclusion was that none of the empirical models could account for the low salinity behaviour of the streaming potential coefficient, and neither could Glover et al.’s [3] model if the zeta potential was calculated with the approach of Revil et al. [9]. They repeated the model restricting the zeta potential to be constant following the assumption of Allègre et al. [10], whence they found that Glover et al.’s [3] model fitted the data well over the whole salinity range.

We have carried out theoretical modelling of Luong and Sprik’s [1] data and come to different conclusions. In particular, Glover et al.’s [3] theory models Luong and Sprik’s [1] data well at all salinities and with variable zeta potential within the limitations that we only know the limits of pore fluid pH under which the measurements were made.

This comment restricts itself to (i) making the results of our implementation of the modelling of Luong and Sprik’s [1] data available in the literature and (ii) pointing out a number of reasons why it differs from that published by Luong and Sprik [1].

#### 2. Modelling Difficulties

Luong and Sprik [1] retained most of the parameters used in Glover et al.’s [3] modelling in order to facilitate intercomparison. However, and significantly, they modified the equation for modelling the streaming potential coefficient. Glover et al. [3] had implemented the solution for the streaming potential coefficient using the equationwhich is an approximation of for . In these equations, is the streaming potential pressure coefficient relative to pore fluid pressure, in V/Pa [11], is the dynamic viscosity of the pore fluid (in Pa·s), is the relative dielectric permittivity of the pore fluid (no units), is the dielectric permittivity of vacuum (in F/m), is the zeta potential (in V), is the modal grain size of the rock (in m), is the specific surface conductivity (surface conductance, in S), is the electrical conductivity of the pore fluid (in S/m), and is the formation factor (no units).

We will refer to (2) as the exact solution and (1) as the RGPZ solution. In these equations, the streaming potential coefficient depends upon a group of pore fluid properties (, , and ), two interfacial properties , and two properties describing the microstructure of the rock ( and ).

Luong and Sprik [1], however, considered that the modal grain diameter of the rock was not convenient, opting to rewrite (1) in terms of fluid permeability according towhere is the steady-state fluid permeability of the rock. They do not provide a derivation or published reference for this equation.

There are three aspects of Luong and Sprik’s [1] modelling that are of concern, each of which is discussed in one of the following subsections.

##### 2.1. Modelling Equations

The first is that we could not generate (3) from either (1) or (2) using accepted relationships between and the characteristic pore size scale length defined by Johnson et al. [12] and between and the steady-state permeability [13] or by using the RGPZ equation [14] or by invoking the well-known relationship in the basic Helmholtz-Smoluchowski relationship [11]. In all cases, we obtain (3) but with the number 4 replacing 2 in the second term of the denominator.

Figure 1 shows an intercomparison of the use of (1), (2), and (3) for one of the samples of Luong and Sprik [1]. The sample chosen was DP50. This sample has a well-known permeability from which (3) may be implemented. It and all the DP labelled samples have the added advantage of also having a measured characteristic pore diameter range. We have taken the central value of this pore diameter range as the modal pore diameter and used the Theta Transformation [15] to calculate the modal grain size. Hence, we have an independent value of grain diameter for use in (1) and (2). Both figures are for a pH of 6.7, which is the same as that used by Luong and Sprik [1]. The left-hand panel (Figure 1(a)) shows the models with zeta potential offset of −15 mV, which we find fits the data best and is consistent with values used by other authors. The right-hand panel (Figure 1(b)) shows the models with zeta potential offset of −45 mV, which is that used by Luong and Sprik [1] and is unrealistically high. Between them, these two diagrams incorporate all of the parameter values used by Luong and Sprik [1] and can hence be used to draw direct conclusions about the effectiveness of Luong and Sprik’s [1] implementations of the streaming potential coefficient modelling.