Steven R. Pride

Electroseismic wave properties

In a porous material saturated by a fluid electrolyte, mechanical and electromagnetic disturbances are coupled. The coupling is due to an excess of electrolyte ions that exist in a fluid layer near the grain surfaces within the material; i.e., the coupling is electrokinetic in nature. The governing equations controlling the coupled electromagnetic‐seismic (or ‘‘electroseismic’’) wave propagation are presented for a general anisotropic and heterogeneous porous material. Uniqueness is derived as well as the statements of energy conservation and reciprocity. Representation integrals for the various wave fields are derived that require, in general, nine different Green’s tensors. For the special case of an isotropic and homogeneous wholespace, both the plane‐wave and the point‐source responses are obtained. Finally, the boundary conditions that hold at interfaces in the porous material are derived.

Electroseismic waves from point sources in layered media

In a porous material saturated by a fluid electrolyte, mechanical and electromagnetic (EM) disturbances are coupled. The coupling is electrokinetic in nature. The seismic waves generate relative fluid-solid motion that induces an electrical streaming current. When a seismic pulse traverses contrasts in elastic and/or fluid-chemistry properties, the streaming-current imbalance creates dipolar and multipolar charge separations across the interface that, in turn, produce EM disturbances that are measurable at the earths surface. This paper numerically determines a full-waveform electroseismic point-source response in a stratified porous medium. It is shown that the macroscopic governing equations controlling the coupled electromagnetics and acoustics of porous media decouple into two sets corresponding to vertical or horizontal polarization of the transverse wave fields. The frequency content of the converted EM field has the same frequency content (at the generating interface) as the incident seismic pulse. Snapshots in time and converted EM amplitudes versus source to antenna offset are calculated for contrasts in mechanical and/or electrical medium properties. The converted EM radiation pattern away from the interface is similar to having an effective vertical-electric dipole centered right beneath the source on the contrast. The transverse magnetic mode amplitudes fall off rapidly with distance, from the generating interface thus suggesting the importance of a vertical electroseismic profiling geometry to record the converted EM signal at antennas close to an interface of interest.

Deriving the equations of motion for porous isotropic media

The equations of motion and stress/strain relations for the linear dynamics of a two‐phase, fluid/solid, isotropic, porous material have been derived by a direct volume averaging of the equations of motion and stress/strain relations known to apply in each phase. The equations thus obtained are shown to be consistent with Biot’s equations of motion and stress/strain relations; however, the effective fluid density in the equation of relative flow has an unambiguous definition in terms of the tractions acting on the pore walls. The stress/strain relations of the theory correspond to ‘‘quasistatic’’ stressing (i.e., inertial effects are ignored). It is demonstrated that using such quasistatic stress/strain relations in the equations of motion is justified whenever the wavelengths are greater than a length characteristic of the averaging volume size.

Relationships between Seismic and Hydrological Properties

Reflection seismology is capable of producing detailed three-dimensional images of the earth’s interior at the resolution of a seismic wavelength. Such images are obtained by filtering and migrating the seismic data and give geometrical information about where in the earth the elastic moduli and mass densities change. However, information about which specific property has changed and by how much is not contained in the images. Hydrologists can use such migrated images to place geometrical constraints on their possible flow models, but must rely on well data to place constraints on the actual values of the hydrological properties.

Seismoelectric numerical modeling on a grid

Our finite-difference algorithm provides a new method for simulating how seismic waves in arbitrarily heterogeneous porous media generate electric fields through an electrokinetic mechanism called seismoelectric coupling. As the first step in our simulations, we calculate relative pore-fluid/grain-matrix displacement by using existing poroelastic theory. We then calculate the electric current resulting from the grain/fluid displacement by using seismoelectric coupling theory. This electrofiltration current acts as a source term in Poisson’s equation, which then allows us to calculate the electric potential distribution. We can safely neglect induction effects in our simulations because the model area is within the electrostatic near field for the depth of investigation (tens to hundreds of meters) and the frequency ranges ( 10 Hz to 1 kHz ) of interest for shallow seismoelectric surveys.We can independently calculate the electric-potential distribution for each time step in the poroelastic simulation with...

Finite difference modeling of Biot's poroelastic equations at seismic frequencies

[1] Across the seismic band of frequencies (loosely defined as <10 kHz), a seismic wave propagating through a porous material will create flow in the pore space that is laminar; that is, in this low-frequency ‘‘seismic limit,’’ the development of viscous boundary layers in the pores need not be modeled. An explicit time stepping staggered-grid finite difference scheme is presented for solving Biot’s equations of poroelasticity in this low-frequency limit. A key part of this work is the establishment of rigorous stability conditions. It is demonstrated that over a wide range of porous material properties typical of sedimentary rock and despite the presence of fluid pressure diffusion (Biot slow waves), the usual Courant condition governs the stability as if the problem involved purely elastic waves. The accuracy of the method is demonstrated by comparing to exact analytical solutions for both fast compressional waves and slow waves. Additional numerical modeling examples are also presented.

Connecting Theory to Experiment in Poroelasticity

Abstract The variables controlled and measured in elastostatic laboratory experiments (the volume changes, shape changes, confining stresses, and pore pressure) are exactly related to the appropriate variables of poroelastic field theory (the gradients of the volume-averaged displacement fields and the volume-averaged stresses). The relations between the laboratory and volume-averaged strain measures require the introduction of a new porous-material geometrical term. In the anisotropic case, this term is a tensor that is related both to the presence of porosity gradients and to a type of weighted surface porosity. In the isotropic case, the term reduces to a scalar and depends only on the surface-porosity parameter. When this surface-porosity parameter is identical to the usual volume porosity, the relations initially proposed by Biot and Willis are recovered. The exact statement of the poroelastic strain-energy density is derived and is used to define both the laboratory strain measures and the laboratory elastic moduli. Only two restrictions are placed on the materials being treated: (1) the fluid is homogeneous in each sample and (2) the material possesses a rigidity.However, the entire work is limited to linear deformations and long (relative to sample size) wavelengths of applied stress.

Biot slow‐wave effects in stratified rock

The transmission of P‐waves through the stratified layers of a sedimentary basin is modeled numerically using Biot theory. The effects on the transmissivity of frequency, angle of incidence, layer thickness, permeability and elastic compliance of the rocks are all considered. Consistent with previous analytical work, it is found that the equilibration of fluid pressure between the fine layers of a sedimentary sequence can produce significant P‐wave attenuation at low frequencies. For this attenuation mechanism to act within the surface‐seismic band (say, 3–300 Hz), we find that there must be layering present at the scale of centimeters to tens of centimeters. If the layering is restricted to layers of roughly 1 m thickness or greater, then for typical sandstone formations, the attenuation caused by the interlayer flow occurs below the seismic band of interest. Such low‐frequency interlayer flow is called Biot slow‐wave diffusion in the context of Biot theory and is likely to be the dominant source of low‐...

The role of Biot slow waves in electroseismic wave phenomena

The electromagnetic fields that are generated as a spherical seismic wave (either P or S) traverses an interface separating two porous materials are numerically modeled both with and without the generation of Biot slow waves at the interface. In the case of an incident fast-P wave, the predicted electric-field amplitudes when slow waves are neglected can easily be off by as much as an order of magnitude. In the case of an incident S wave, the error is much smaller (typically on the order of 10% or less) because not much S-wave energy gets converted into slow waves. In neglecting the slow waves, only six plane waves (reflected and transmitted fast-P, S, and EM waves) are available with which to match the eight continuity conditions that hold at each interface. This overdetermined problem is solved by placing weights on the eight continuity conditions so that those conditions that are most important for obtaining the proper response are emphasized. It is demonstrated that when slow waves are neglected, it is best to also neglect the continuity of the Darcy flow and fluid pressure across an interface. The principal conclusion of this work is that to properly model the electromagnetic (EM) fields generated at an interface by an incident seismic wave, the full Biot theory that allows for generation of slow waves must be employed.

Seismic stimulation for enhanced oil recovery

The pore-scale effects of seismic stimulation on two-phase flow are modeled numerically in random 2D grain0pack geometries. Seismic stimulation aims to enhance oil production by sending seismic waves across a reservoir to liberate immobile patches of oil. For seismic amplitudes above a well-defined (analytically expressed) dimensionless criterion, the force perturbation associated with the waves indeed can liberate oil trapped on capillary barriers and get it flowing again under the background pressure gradient. Subsequent coalescence of the freed oil droplets acts to enhance oil movement further because longer bubbles overcome capillary barriers more efficiently than shorter bubbles do. Poroelasticity theory defines the effective force that a seismic wave adds to the background fluid-pressure gradient. The lattice-Boltzmann model in two dimensions is used to perform pore-scale numerical simulations. Dimensionless numbers (groups of material and force parameters) involved in seismic stimulation are defined carefully so that numerical simulations can be applied to field-scale conditions. Using the analytical criteria defined in the paper, there is a significant range of reservoir conditions over which seismic stimulation can be expected to enhance oil production.