1
The Microscale Origin of the Seismoelectric Effect: Electrokinetics, Streaming Potential

Niels Grobbe1,2 and André Revil3

1 Hawai'i Institute of Geophysics and Planetology, University of Hawai'i at Manoa, HI, USA

2 Water Resources Research Center, University of Hawai'i at Manoa, HI, USA; Formerly at: Delft University of Technology, The Netherlands; Massachusetts Institute of Technology, MA, USA

3 CNRS, Université Savoie Mont-Blanc, France; Department of Geophysics, Université Savoie Mont-Blanc, Chambéry, France

ABSTRACT

In this chapter, we discuss the microscale origin of coupled seismo-electromagnetic fields. We introduce the electrical double layer (EDL) and important electrokinetic processes that take place in porous materials that have a certain degree of pore-fluid saturation. The electrical double layer at the microscale plays a crucial role in the seismoelectric and electroseismic coupling processes. As a consequence of the presence of an EDL, several electrokinetic phenomena can occur, including the electro-osmotic phenomenon, where an electric field drives fluid flow, and the streaming potential phenomenon, where a hydraulic gradient creates an electric field. The streaming potential mechanism lies at the heart of the seismoelectric effect (mechanical to electromagnetic coupling). Electro-osmosis results in the so-called electroseismic effect (electromagnetic to mechanical coupling). For seismoelectric (and electroseismic) phenomena, we can distinguish three types of coupling: coseismic fields, interface response fields (or seismoelectric conversion), and source-converted fields. We introduce the so-called seismoelectric coupling coefficient, which describes the amount of coupling that naturally occurs between mechanical and electromagnetic fields in porous media.

1.1 INTRODUCTION

HThe use of seismo-electromagnetic and electromagneto-seismic coupling effects are quite novel in geophysics. These types of coupling are one class of so-called electrokinetic effects in porous media in a broad sense (including colloidal suspensions). These coupling effects in the constitutive equations at the macroscale (i.e., at a representative elementary volume) are intrinsically associated with the existence of an electrical double layer at the surface-interface between solid particles and pore water. These phenomena have been extensively studied throughout the years (e.g., Lyklema, 2002, 2003). They have a long history outside the realm of geophysics. For instance, Reuss (1809) was the first to prove the existence of such an electrokinetic effect. He performed a U-tube experiment demonstrating that the application of an electric current to a clay-sand-water mixture could cause the water to rise (the so-called electro-osmosis effect).

To understand electrokinetic phenomena in the context of geophysical applications, we consider a porous material such as a rock or soil. For the sake of simplicity, we first consider this material to be fully water saturated by an electrolyte. The surface of the grains (assumed to be insulating) is covered by reactive sites, such as silanol and aluminol sites for aluminosilicates. For some clays such as smectite, the mineral can also bear a charge through isomorphic substitutions in the crystalline framework (Davis et al., 1978; Avena and De Pauli; 1998, Revil et al., 2012).

The chemical reactions between the mineral surface and the pore water are collectively assembled in a so-called speciation model for a given mineral in contact with a specific pore water solution. These chemical reactions generate a net charge on the mineral surface depending on the presence and concentration of so-called potential-determining ions (e.g., Avena & De Pauli (1998) for clays, Heberling et al. (2014) for calcite). For alumino-silicates, these potential-determining ions are essentially H+ and OH-, because the surface sites are basically weak acids. Some ions from the pore water adsorb as inner- and outer-sphere complexes on the mineral surface, forming a so-called Stern layer (Stern, 1924; Davis et al., 1978). In this way, the Stern layer partially compensates the surface charge.

The surface charge that is not compensated in the Stern layer is compensated in the so-called diffuse layer. In this diffuse layer, the counterions and co-ions obey Boltzmann statistics. In other words, at equilibrium, the flux of ions due to the Coulombic field created by the surface charge counterbalances the diffusive flux of ions (Gouy, 1910; Chapman, 1913; Revil & Jardani, 2013). As a result, the potential of the diffuse layer more or less exponentially decreases away from the grain surface toward the bulk of the pore space. As a side note, an exact exponential decay for the electrical potential in the diffuse layer (in absence of a macroscopic electrical field) is the result of linearizing the Poisson-Boltzmann equation, obtained by combining Gauss' law with the Boltzmann model of concentration in the Coulombic field created by the mineral surface charge (including the charge density of the Stern layer).

When the grains and the pore water are moving, the place of zero relative velocity is called the shear plane. This shear plane occurs somewhere in the diffuse layer. For the sake of simplicity, it is often considered to be at the interface between the Stern and diffuse layer. The electrostatic potential on this plane is referred to as the zeta potential. The Stern and diffuse layers together form the so-called electrical double layer (EDL). The characteristic length scale corresponding to half of the diffuse layer thickness is referred to as the Debye length, and is for typical reservoir rocks on the order of a few tens of nanometers (Schoemaker et al., 2012). The above is schematically displayed in Figure 1.1.

Figure 1.1 Sketch explaining the electrical double layer (EDL), which forms around the grain surfaces in a porous medium saturated with a fluid. The chemical speciation between the mineral and the pore water generates here a negative charge on the mineral surface. This negative charge is partially compensated by an excess of charge in the Stern layer (we do not distinguish here between inner and outer sphere complexes). Charge balance is obtained by the formation of a diffuse layer around the grain in which the charge concentration obeys Boltzmann statistics. The d-Plane is defined at the plane separating the diffuse layer from the Stern layer and is, therefore, the inner plane of the diffuse layer. The shear plane is assumed to be the same as this d-Plane. The o-Plane is the surface plane.

Revil and Mahardika (2013) further distinguish two end-member scenarios, based on the size of the EDL with respect to the size of the pores:

1. A thick double layer, where the counterions of the diffuse layer are uniformly distributed in the pore space,
2. A thin double layer, where the thickness of the diffuse layer is much smaller than the size of the pores.

These end-member scenarios have an impact on the theoretical formulations and expressions for important petrophysical parameters such as the seismoelectric coupling coefficient, the permeability, and the electrical conductivity. These implications are further discussed in section 1.2. Note that throughout this book, we use the following convention

for the definition for the forward temporal Fourier transform and

for the forward spatial Fourier transform. Here, ? denotes the angular frequency in [rad s-1], t is time, j indicates the imaginary unit, xi indicates the three spatial directions of the right-handed Cartesian coordinate system (with the positive x3 direction pointing downward [depth]), and ka, with a = 1 or 2, denotes the horizontal wavenumber in those respective directions. The hat indicates a quantity in the space-frequency domain, the tilde sign denotes a quantity in the horizontal wavenumber-frequency domain. Throughout this book, Latin subscripts can take the values 1, 2, and 3, corresponding to the three spatial coordinate principal directions, whereas Greek subscripts can take the values 1 and 2. The Einstein summation convention holds for repeated indices (unless indicated otherwise).

As a consequence of the presence of an EDL, several electrokinetic phenomena can occur, including the electro-osmotic phenomenon, where an electric field drives fluid flow, and the streaming potential phenomenon, where a hydraulic gradient creates an electric field.

Helmholtz (1879) derived a mathematical description for electro-osmosis and electro-phoresis. The Helmholtz-Smoluchowski equation is one of the models used to describe such phenomena, and relates the streaming potential to controlling parameters such as the zeta-potential and fluid viscosity:

where Cs0 denotes the steady-state streaming potential coefficient and ?? is the induced electrical potential difference (streaming potential) as a consequence of the pore-pressure difference...