A model is presented to describe the attenuation of seismic waves in rocks with partially liquid‐saturated flat cracks or pores. The presence of at least a small fraction of a free gaseous phase permits the fluid to flow freely when the pore is compressed under wave excitation. The resulting attenuation is much higher than with complete saturation as treated by Biot. In general, the attenuation increases with increasing liquid concentration, but is much more sensitive to the aspect ratios of the pores and the liquid droplets occupying the pores, with flatter pores resulting in higher attenuation. Details of pore shape other than aspect ratio appear to have little effect on the general behavior provided the crack width is slowly varying over the length of the liquid drop.

Interpretation of seismic velocity and attenuation in partially molten rocks has been limited, with few exceptions, to models that assume the melt to be distributed either as spheres or as thin films. However, other melt phase geometries, such as interconnected tubes along grain edges, might equally well account for seismic observations if there is a much larger fraction of melt. Seismic velocity and attenuation are estimated in rocks in which the melt phase has the tube geometry, and the results are compared with resuits expected for the more familiar film model under similar conditions. For a given melt fraction, tubes are found to give moduli intermediate between moduli for rigid spherical inclusions and compliant films. For example, in polycrystalline olivine at 20 kbar the model predicts a decrease in V s of 10% and a decrease in I/•, of 5% at 0.05 melt fraction, without considering inelastic relaxation. Shear attenuation appears to be dominated by viscous flow of melt between the tubes and/or films. For olivine the tube model predicts the increment of relaxation due to melt, •//x, to be 0.01 at 0.05 melt fraction. Relaxation of the bulk modulus is dominated by flow between melt pockets of different shape, heat flow, and solid-melt phase change. If melt is present, considerable bulk attenuation is expected, although the relaxation may be observable only at long periods, outside the seismic body wave band. Paper number 80B0266. 5173 5174 MAVKO: STANFORD Q CONFERENCE Thermodynamic considerations [Bulau and Waft, 1977; Bulau et al., 1979] and experimental results [Waft and Bulau, 1977, 1979] suggest that tubes, rather than films, may be the expected equilibrium melt geometry under some conditions. Frank [1968] postulated the tube geometry to model melt percolation and to explain certain aspects of heat and mass transport in a convecting upper mantle. Walker et al. [1978] refined Frank's tube model to calculate melt mobility and melt-solid segregation. The effect of solid tubular inclusions (with circular cross section) on the effective elastic properties of a composite has been treated by Wu [1966], Boucher [1974], and Walpale [1969].In the tube model suggested by Smith [1964], shown schematically in Figure 1, each tube is roughly triangular in cross section with sharp edges. In the film geometry the flat grain faces are coated. Details of the shape (e.g., tube versus film) depend on the relative solid-solid and solid-melt surface energies, as well as whether or not the system is in equilibrium. The factors affecting shape are discussed by Smith

The sudden appearance of a dislocation, representing an earthquake, in an elastic layer (the lithosphere) overriding a viscoelastic half space (the asthenosphere) is followed by time-dependent surface deformation, which is very similar to in situ postseismic deformation. The spectacular postseismic deformation following the large Nankaido earthquake of 1946 yields for the asthenosphere a viscosity of 5 x 10(19) poise and a 50 percent relaxation of the shear modulus. Large thrust type earthquakes may provide, in the future, a new method for exploring the rheology of the earth's upper mantle.

The interpretation of the seismic low‐velocity zone as a region of partially molten rock is extended to explain the transient displacements following the 1946 Nankaido earthquake. Three partial melt models are considered to account for the observed time constant of 3–5 years: large‐scale diffusion of melt through a porous matrix can decay over thousands to billions of years and is much too slow. Simple shearing in ‘penny‐shaped’ cracks happens on a seismic time scale and is much too rapid. Interconnected penny‐shaped cracks at different orientations with respect to the principal stresses respond on an intermediate time scale by short‐range melt squirt from one crack to another, providing a reasonable mechanism to account for the transient deformation at Nankaido, while components of shear parallel to each individual crack relax quickly according to the better‐known mechanism for seismic attenuation in the low‐velocity zone.

Dislocation theory is used to study the deformation of nonelliptical thin cracks in a loaded elastic material. The cracks considered are two-dimensional with nonblunted, tapered ends such that opposite faces are tangent to each other at points of contact. Under compression the cracks shorten by closing near the crack tips, the proportion of crack surface area in contact becoming gradually larger. Some cracks make contact between the crack tips, becoming multiple cracks. Normal stresses on the crack surface vary rapidly over the closed portions from zero near the open surfaces to a peak value at the original crack tip. Stresses remain finite everywhere. At a given load the effective rock compressibility due to arbitrarily shaped, tapered cracks depends only on crack length, giving results identical to a distribution of elliptical cracks of the same lengths. However, at different loads the varying length causes the modulus to vary. As a result, interpretation of features like porosity and modulus under varying applied stress will depend on the specific crack model chosen. In particular, a single aspect ratio of a simply-tapered crack yields the same nonlinear effect as a flat distribution of elliptical cracks. Consequently, estimates of crack spectra from nonlinear strain data are totally nonunique.