Coupled Electro-hydrodynamic Transport in Geological Fractures,
Abstract
Fractures are very common features in subsurface crystalline rocks, where they are organized in
networks of interconnected elements [1]. A number of essential mechanical properties of the rock
formations, such as their mechanical strength and their transport properties (hydraulic and electri-
cal conductivities), are dictated by the behavior of the fracture networks. Within these networks,
individual geological fractures are the basic structural unit controlling the
ow of
uids and the trans-
port of solute chemical species. Their length is distributed over a very large range, which strongly
constrains the connectivity and hydraulic behavior of the network [2]. Fracture wall roughness is
responsible for
ow channeling (and therefore, heterogeneity) within the fracture plane, which, at
the fracture scale, impacts the fracture's transmissivitty [3, 4]. The characteristic length scale Lc
at which the two fracture walls are matched [5, 6], plays a crucial role as it is the upper limit scale
for
ow heterogeneities [7]. When Lc is suciently large with respect to the distance between two
intersections with other fractures, fracture wall roughness also impacts the distribution of
uxes
in-between fractures of the network [8].
The most prevalent way of computing the transport properties and transmissivitty of a rough
fracture in an ecient way and without resorting to a full three-dimensional
ow simulation, is to
use the lubrication approximation, which leads to a Darcy
ow type equation for the pressure, the
Reynolds equation [3]. This method has been used extensively to simulate the
ow [3, 9], as well as
the electric current (without
ow) through a rough fracture [10]. However, the e ect of the electrical
properties of the fracture walls on the transport properties of a fracture still remains an open question,
to the best of our knowledge. Since dissolved minerals and salts are ever present in the
uids inside
the fracture, Electrical Double Layers (EDL) almost inevitably form at the
uid-solid interface [11],
and their strength depends on the chemical properties of the rock and ionic strength of the
uid.
Therefore, the ocurrence, at the fracture scale, of externally-imposed or naturally-occurring gradients
in electrical potential and/or ionic concentration, can lead to signi cant changes in the
uid motion
through the fracture, as compared to
ows driven primarily by hydraulic head di erences.
In this work, we attempt to explore the
ow dynamics that result from such coupled electro-
hydrodynamic forcings. To this end, we generalize the standard lubrication theory for
ow, to
include a description of the coupled transport of
uid massn, solutes, and electrical current under
application of xed di erences in hydraulic head (or pressure), electrical potential and concentration
across the fracture. By invoking the requirement of conservation of volumetric
ow rate, ions and
electrical charge
uxes, a coupled system of equations can be derived, which governs the spatial
distribution of electrical potential, pressure and concentration in the bulk
uid within the fracture.
This system of equations is the generalization of the Reynolds equation to the coupled transport
of
uid mass, solutes, and electrical charges. It is solved using an iterative Finite Volume Method
to gain insight into the dynamics of the coupled transport processes, in geological fractures with a
realistic aperture field. We investigate in particular the role of the characteristic length scale Lc.