David Linton Johnson

Acoustic slow waves and the consolidation transition

We have investigated the ultrasonic properties of unconsolidated (loose) glass beads and of lightly fused (consolidated) glass beads when the pore space is saturated with water. At a frequency of 500 kHz we have observed a single compressional wave in the former whose speed is 1.79 km/s and two distinct compressional waves with speeds 2.81 km/s and 0.96 km/s in the latter. The Biot theory is shown to give an accurate description of this phenomenon. We also analyze the acoustics of low temperature Heu2009ii in packed powder superleaks; either the fast wave for unconsolidated systems or the slow wave in a highly consolidated (fused) frame may be considered to be the 4th sound mode. In all such systems, the acoustic properties can be very simply understood by considering the velocities of propagation as continuous functions of the elastic moduli of the solid skeletal frames.

Permeability and borehole Stoneley waves: Comparison between experiment and theory

We performed laboratory experiments to evaluate theoretical models of borehole Stoneley wave propagation in permeable materials. A Berea sandstone and synthetic samples made of cemented glass beads were saturated with silicone oils. We measured both velocity and attenuation over a frequency band from 10 kHz to 90 kHz. Our theoretical modeling incorporated Biot theory and Deresiewicz-Skalak boundary conditions into a cylindrical geometry and included frequency-dependent permeability. By varying the viscosity of the saturating pore fluid, we were able to study both low-frequency and high-frequency regions of Biot theory, as well as the intermediate transition zone. In both low-frequency and high-frequency regions of the theory, we obtained excellent agreement between experimental observations and theoretical predictions. Velocity and attenuation (1/Q) are frequency-dependent, especially at low frequencies. Also at low frequencies, velocity decreases and attenuation increases with increasing fluid mobility (permeability/viscosity). More complicated behavior is observed at high frequencies. These results support recent observations from the oil field suggesting that Stoneley wave velocity and attenuation may be indicative of formation permeability.

High‐frequency acoustic properties of a fluid/porous solid interface. I. New surface mode

We use Biot’s theory to search numerically for the velocities of the various surface waves, both the true slow surface wave and the pseudosurface waves, at an interface between a fluid half‐space and a half‐space of a fluid‐saturated porous medium. We focus on the high‐frequency range where the Biot–Plona slow wave is propagatory. We find that for an open‐pore surface situation, the true surface mode exists for a limited range of material parameters and changes continuously into a slightly leaky pseudo‐Stoneley mode as the mode speed increases past the slowest bulk wave speed; for the sealed pore situation there exist simultaneously a true surface mode (for all values of material parameters) and a pseudo‐Stoneley mode. The pseudo‐Rayleigh mode has features similar to those of the pseudo‐Rayleigh mode for a fluid/nonporous solid case.

Probing porous media with first and second sound. II. Acoustic properties of water‐saturated porous media

The ultrasonic properties (reflection/transmission and bulk attenuation/speed) of porous and permeable media saturated with a Newtonian fluid, namely water, are considered. The frequency dependence of the transmission amplitudes of pulses is measured through a slab of thickness d1, repeated for another slab of thickness d2 for a given material. With these two measurements on two different thicknesses, it is possible in principle to separate bulk losses from reflection/transmission losses for compressional waves in these materials. The bulk properties are calculated from the Biot theory for which all of the input parameters have been measured separately; the attenuations are particularly sensitive to the values of Λ, determined from second‐sound attenuation measurements reported in the companion article. There is excellent quantitative agreement between the theoretical and experimental values in the cases considered; there are no adjustable parameters involved. The reflection and transmission coefficients ...

Nonlinear elasticity of granular media

The finite and incremental elasticity of a random packing of identical spheres is derived using energy methods. We consider different models for the contact forces between spheres, all of which are based upon or related to the fundamental Hertz theory; we consider only the special cases of perfect friction (no tangential slip) or no tangential friction. The existence of a strain energy function for the medium depends critically upon the type of contact. If the tangential contact stiffness is independent of the normal force, then the energy is well defined for all values of the macroscopic strain. Otherwise, the strain energy of the system is path dependent, in general. However, the concept of a quadratic strain energy function is always well defined for incremental motion superimposed on large confining stress and strain. For all models considered, we derive the changes in wave speeds due to incremental strains. For the models based upon an energy function we derive expressions for the third-order elastic constants as a function of confining pressure.

High‐frequency acoustic properties of a fluid/porous solid interface. II. The 2D reflection Green’s function

A 2‐D space–time reflection Green’s function is developed using the Cagniard–de Hoop technique for a fluid/lossless porous media planar interface configuration. It is then used to study the effects of the various surface waves whose velocities were calculated in the previous article. The time‐dependent pressure response to a modeling source signal (Blackman–Harris pulse) for certain simulated experimental configurations is also presented. It is concluded that the amplitudes with which the new surface modes are generated are large enough to enable their detection.

Capillary forces in the acoustics of patchy-saturated porous media

A linearized theory of the acoustics of porous elastic formations, such as rocks, saturated with two different viscous fluids is generalized to take into account a pressure discontinuity across the fluid boundaries. The latter can arise due to the surface tension of the membrane separating the fluids. We show that the frequency-dependent bulk modulus K(omega) for wavelengths longer than the characteristic structural dimensions of the fluid patches has a similar analytic behavior to the case of a vanishing membrane stiffness and depends on the same parameters of the fluid-distribution topology. The effect of the capillary stiffness can be accounted for by renormalizing the coefficients of the leading terms in the low-frequency limit of K(omega).

Linear and Nonlinear Elasticity of Granular Media: Stress-Induced Anisotropy of a Random Sphere Pack

We develop an effective medium theory of the nonlinear elasticity of a random sphere pack based upon the underlying Hertz-Mindlin theory of grain-grain contacts. We compare our predictions for the stress-dependent sound speeds against new experimental data taken on samples with stress-induced uniaxial anistropy. We show that the second-order elastic moduli, C ijkl , and therefore the sound speeds, can be calculated as unique path-independent junctions of an arbitrary strain environment, {∈ kl }, thus generalizing earlier results due to Walton. However, the elements of the stress tensor, σ ij , are not unique functions of {∈ kl } and their values depend on the strain path. Consequently, the sound speeds, considered as functions of the applied stresses, are path dependent. Illustrative calculations for three cases of combined hydrostatic and uniaxial strain are presented. We show further, that, even when the additional applied uniaxial strain is small, these equations are not consistent with the usual equations of third-order hyperelasticity. Nor should they be, for the simple reason that there does not exist an underlying energy function which is simply a function of the current state of the strain. Our theory provides a good understanding of our new data on sound speeds as a function of uniaxial stress.

Can one hear the shape of a saturation patch

[1]xa0The theory of the acoustics of patchy-saturation in porous media is used to analyze experimental data on wave velocity and attenuation in partially water saturated limestones. It is demonstrated that the theory can be used to deduce the value of V/A, the ratio of the volume to area of the water patch, and lf, the Poisson size of the water patch. One can “hear” the shape of a patch if the properties of the rock and the measurement frequencies are such as to satisfy the specific requirements for the validity of the theory.

A New Modular Sonic Tool Provides Complete Acoustic Formation Characterization

An improved estimation of sonic slownesses and a comprehensive mechanical characterization of the wellbore rock rely on a complete characterization of the compressional and shear slowness in terms of their radial, azimuthal, and axial variations. The new modular sonic tool accomplishes this by incorporating improved monopole and cross-dipole transmitter technology while featuring an extensive receiver array incorporating 13 axial levels of 8 azimuthal sensors each. Each receiver is individually digitized resulting in 104 waveforms per transmitter firing leading to an extremely reliable and accurate slowness estimation. This comes about through improved borehole mode extraction/rejection and enhanced wavenumber resolution at all frequencies. Formations exhibit wide, and sometimes complex, acoustical behaviors ranging from isotropic, anisotropic with its various mechanisms and significant radial slowness gradients. Radial rock property variations arise because of non-uniform stress distributions and mechanical or chemical near-wellbore alteration due to the drilling process. Anisotropy can be caused by intrinsic shale properties or external differential stresses. The critical data required to invert for these rock parameters underlying these acoustic behaviors are derived from the new tool through the use of broadband dispersion curves associated with propagating borehole acoustic modes. In this paper, we highlight tool features that have an important impact on seismic, borehole seismic, and sonic applications. The acquired high quality waveforms and advanced processing techniques lead to improved compressional and shear slowness estimates, radial profiling of shear and compressional slowness, enhanced anisotropy detection and mechanism identification, and reliable through casing slowness measurements. Examples are shown from several wells in Norway and Mexico.