Seismic Wave Attenuation in Cracks and Pores of Partially Saturated Rock

Attenuation results in loss of seismic wave intensity over distance and time. Attenuation via viscous dissipation results from relative fluid-solid motion and pressure gradients within the fluid, which is contained in a pore or crack. Here, a classic derivation of dissipative attenuation is summarized and its drawbacks and need for derivations specific to the microscopic scale are explained. Two derivations of viscous dissipation at the microscopic scale of a pore or crack are presented and compared. One derivation involves a highly elongate partially fluid-filled crack. The last derivation is based upon a moderately elongate fully saturated pore. The derivation for the partially filled crack yields maximum attenuation at seismic frequencies, but the approach utilized for the fully saturated elongate pore is most practical. Introduction and Relevance Attenuation is an all-encompassing term for the loss of intensity of seismic waves during passage through the Earth. Attenuation is grouped into two categories, scattering and dissipation. Scattering is elastic and preserves energy, and will not be addressed here. Dissipation is inelastic and exothermic, where the propagating wave (system) loses energy as heat to the rock body (surroundings) [Mavko et al., 1979]. In place of attenuation measurements at locations along the wave, many analytical models have been derived to represent attenuation and yield specifics for the general representation of attenuation Q [Shearer, 2009, p. 164]: € 1 Q(ω) = − ΔE 2πE (1) Here, I will focus on viscous dissipation at the microscopic scale of pores and/or cracks containing fluid, after giving a brief introduction to the subject of dissipative attenuation. There are many causes of dissipative attenuation. Anelastic behavior of rocks and friction at grain boundaries; phase change and thermal relaxation; point defect hysteresis friction; and viscous fluid flow or viscous dissipation [Jackson and Anderson, 1970; Mavko et al., 1979] are just a few examples of causes of dissipative attenuation. The focus of this paper is viscous fluid flow, which involves a porous and/or cracked medium that is partially or fully saturated. Pressure gradients and relative fluid-solid motion are key aspects of viscous dissipation, which includes shear relaxation via fluid-enabled sliding, pressure changes, and re-equilibration of pressure between or within cracks and/or pores [Mavko et al., 1979; Walsh, 1995]. There are three general scales at which attenuation is observed or analyzed (Figure 1). The macroscopic scale is on the order of wavelengths and is essentially the scale of observation in most situations. At the mesoscopic scale, heterogeneities in the rock body at a scale Figure 1. Scales of attenuation [Muller et al., 2010]. Top: Macroscopic scale of attenuation, on the scale of wavelengths, or observation scale. Middle: Mesoscopic scale, between pore size and wavelength. Bottom: Microscopic scale, at the level of individual pores in the rock. intermediate between the lower limit of pore size and upper limit of wavelength are addressed [Muller et al., 2010]. The microscopic scale is on the order of micrometers, where the focus is a single fluid-bearing pore or crack and its response to a seismic wave. The fluid involved may be a liquid or gas, and the crack or pore may be fully or partially saturated. The physical response is quite interesting under partial saturation, when the surface contact area to volume ratio of the liquid is relatively high, leading to more viscous flow per quantity fluid. There are many reasons to determine ways to accurately represent attenuation, other than the attraction of a fascinating intellectual problem. Interest in fluid-related dissipative attenuation was triggered due to laboratory experiments on previously dry rocks that were slightly wetted and found to have greatly increased attenuation [Mavko et al., 1979]. Applications of attenuation exist in several earth science fields: in hydrogeology to determine aquifer properties; in industry to distinguish between oil, gas, and water; and in consulting or development of geothermal power to determine if liquid or gaseous water is present [Mavko and Nur, 1979; Walsh, 1995; Pride et al., 2004; Gurevich et al., 2010; Muller et al., 2010]. The primary focus of this paper shall be on key aspects of derivations of fluid-related attenuation. The derivation of Biot [1956] is regarded as classic and may be appropriate over the mesoscopic scale, but has some drawbacks. The derivations of Mavko and Nur [1979] and Gurevich et al. [2010] specifically address fluid in a crack or pore at the microscopic scale. Mavko and Nur [1979] focus on flow to obtain attenuation due to a partially fluid-filled crack. Gurevich et al. [2010] primarily utilize bulk moduli to arrive at attenuation due to a saturated elongate pore. Each approach has strengths and weaknesses that will be discussed. Biot’s approach Prior to addressing the more modern approaches specific to the microscopic scale, a classic approach to seismic wave attenuation and its drawbacks will be addressed. The derivation begins with the motion equation, adapted to simply equal total force. In the ideal case with no dissipation, the forces on the fluid, Q, and on the solid, q, are [Biot, 1956]: