Frame Moduli Research Articles

Acoustic measures of partial gas saturation in tight sandstones

The acoustic properties of tight sandstones are very sensitive to partial gas/water saturation. Measurements at 5 kHz show that compressional velocity drops sharply with undersaturation. Extensional and shear losses at 5 kHz are large in both fully and partially saturated sands. The shear loss Qs−1 declines linearly with increasing gas content. The extensional loss Qe−1 peaks at 0.80 water saturation, Q−1 is frequency dependent. The attenuation is well explained by hydrodynamic relaxation in the grain contacts. The measurements indicate velocity dispersions consistent with this mechanism. Ultrasonic velocities at 500 kHz are roughly 10–25% higher than acoustic velocities at 5 kHz. The ultrasonic velocities exhibit a weak dependence on saturation. Frame moduli are unrelaxed at these frequencies.

Frame modulus reduction in sedimentary rocks: The effect of adsorption on grain contacts

The adsorption of vapor on the internal surface of a sandstone reduces its frame moduli. A theoretical model has been developed which predicts the weakening as a function of fluid chemistry and effective pressure. The model is based on well established contact theory and surface chemistry. Frame moduli are directly proportional to the stiffness of the grain contacts. The contact stiffness is dependent on surface energy. We show how the adsorption of various chemicals onto quartz surfaces lowers the surface energy, thereby weakening the contact stiffness. The predicted variation of frame moduli with the partial pressure and chemical species is qualitatively consistent with recent experimental results. The calculations suggest that the chemical effect is only significant at low effective pressures. At high pressures, the elastic loading dominates the contribution due to surface energy.

THE INFLUENCE OF PERMEABILITY ON COMPRESSIONAL WAVE VELOCITY IN MARINE SEDIMENTS*

AbstractTo investigate the effect of permeability on the propagation of seismo‐acoustic waves through marine sediments, a theoretical model based on Biot's equations is established which relates the compressional wave velocity measured at a fixed frequency to computed velocities at zero and infinite frequencies in terms of sediment porosity and permeability. The model is examined experimentally in a standard soil mechanics consolidation test (itself dependent, among other things, on sediment porosity and permeability) which has been modified to include measurements of compressional wave velocity at 1 MHz and shear‐wave velocity at 5 kHz. This test allows the elastic modulus of the sediment frame to be assessed under different load conditions simultaneous with the velocity determinations.From a number of tests on different samples, five samples are chosen to typify the range of sediment sizes. The results show that the difference between the measured velocity at 1 MHz and the model‐derived velocity at zero frequency increases with increasing particle size (from clays to fine sand), with decreasing porosity, and with increasing permeability. For sediments coarser than fine sand the simple model breaks down, possibly because of the dominance of scattering/diffraction effects at the high frequency of the experiment. Within this limitation the model seems satisfactory to offer a capability of predicting the permeability of a sea floor sediment to an order of magnitude by the in situ measurement of seismic velocities over a wide range of frequencies; the prediction process requires a good in situ determination of sediment porosity such as that offered by electrical formation factor measurements.

Quantitative methods for microgeometric modeling

The microgeometry or texture of a porous medium determines its physical properties, e.g., the electromagnetic, acoustical, and fluid dynamic properties of sedimentary rock. Unified functional models of texture are clearly needed to test and provide insight for texture-dependent theories relating different properties or explaining empirical observations. A class topological models of sedimentary rock is introduced, together with the necessary topological concepts. Each sample is modeled by two 3-D networks, pore and grain, together with a set of measured parameters such as integranular contact areas or pore/throat ratios (of which any given application may use only a subset). An intuitive example of thin modeling process is provided. A description is given of data acquisition and image analysis procedures used for measurements from reconstructed serial rock sections. Preliminary observations of grain and pore statistics are presented, with comparisons to other related data. The ’’connectivity’’ of the pore (or grain) system of Berea is lower than the connectivities of regular monosized sphere packs with similar porosities and the same mean grain ’’diameters’’. A wide distribution of sizes, aspect ratios, and coordination number were also found for Berea. Illustrative applications of this kind of modeling to nuclear-magnetic resonance, fluid flow, and frame modulus are given.

Geotechnical assessment of superficial marine sediments using in-situ geophysical probes

Abstract A geophysical method is described where electrical resistivity, in terms of formation factor, and compressional wave sound velocity are measured in-situ, on undisturbed sediments, and the measurements then related to porosity using general correlation curves. A survey of a pockmark in the Forties sector of the North Sea detected isolated patches of low geophysical values which showed an increase in inferred porosity of more than 0.10; these patches were assumed to be softer than the surrounding sediments and could represent potential hazards. A resistivity survey in the Irish Sea over a wide variety of sediment types produced a significant correlation between mean size and apparent formation factor (in-situ), which is likely to reflect the natural changes in porosity with mean size. Using values from recent measurements of shear wave velocities in unconsolidated marine sediments, it is suggested that the contribution of the “frame modulus” to the compressional wave velocity is 8–10% whereas that due to the rigidity (as measured by the shear wave velocity) is 2.4%. The propagation of sound and the conduction of electricity in porous media are shown to be controlled by different processes, and as such a combination of these two measurements may uniquely define a third property; cross plots of formation factor and compressional wave travel time showed this to be the case for porosity, and possibly the state of compaction, both properties being assessed directly from the plot.

Elastic wave propagation in fluid-saturated porous media II

Recently, a new method of deriving the coefficients in the strain energy functional of Biot’s theory for elastic waves in fluid-saturated porous media was proposed. The results of this approach have been criticized as being unphysical in the limit K*/Kf → 0, where K* and Kf are the frame and fluid bulk moduli. We show that an error arose due to the literal interpretation of the term ef as the fluid dilatation whereas this term actually differs substantially from the fluid dilatation in some situations. We also show that by lumping part of the solid together with the fluid (conceptually) to form a fictitious fluid suspension, the corresponding fictitious ef may be correctly interpreted as the dilatation of the fluid suspension. This trick leads to a transformation which takes the recently published formulas into standard form. The standard formulas are found to be invariant under the same transformation. New insight into the significance of the standard formulas for the coefficients is obtained.

Biot's second compressional wave: measurement and theory

Recently, Plona reported the experimental observation of Biot's second compressional wave in a fluid-saturated porous medium composed of water and sintered glass beads at frequencies near 1 MHz [T. J. Plona, Appl. Phys. Lett. 36, 259–261 (1980)]. Favorable agreement between the Biot theory and the measured second wave speed (as well as the fast compressional and shear) have been shown by Berryman (to be published to Appl. Phys. Lett.). For these calculations, the moduli of the porous frame were theoretically determined using various existing composite media models together with the moduli and density of the constituent glass and water. In this paper, estimates of the frame moduli are experimentally determined based on ultrasonically measured shear and compressional wave speeds in the air-saturated porous frame. Thus the Biot theory predictions of wave speeds in the water-saturated porous structure are calculated using primarily measured data. Experimental and theoretical results will be shown for a suite of sintered glass bead samples ranging in porosity from 0% to 35%. The validity of using dry velocity measurements as input to the Biot theory will be discussed as well as the dependence of the second wave speed on porosity and permeability.

Confirmation of Biot’s theory

Plona’s recent measurements of elastic-wave speeds in a water-saturated porous structure of sintered glass beads are compared quantitatively to the predictions of Biot’s theory. The theoretical predictions lie within the bounds of experimental error (3%) for the fast compressional wave and for the shear wave in all cases. For the slow compressional wave, the theoretically predicted speeds lie within about 10% of the experimental values and increase with increase in porosity as observed. Our model achieves this agreement with no significant free parameters. The frame moduli are estimated using a recently developed self-consistent theory of composite materials. The induced mass of the frame in a water environment is also estimated theoretically.