Roy J. Greenfield

Factors controlling joint spacing in interbedded sedimentary rocks: integrating numerical models with field observations from the Monterey Formation, USA

Abstract Local tensile stress normal to a joint is reduced in the vicinity of the joint because such stresses are not transmitted across free surfaces. This stress reduction prevents the formation of new joints in the vicinity of existing joints, and thus influences joint spacing. Lateral extent of this stress reduction shadow increases with joint height, which corresponds to bed thickness for many sedimentary rocks. The linear correlation between joint spacing and bed thickness commonly observed in outcrop is a direct result of this relationship. However, other factors in addition to bed thickness influence joint spacing. We evaluate these factors through both a review of the Hobbs model for joint spacing and a 2D finite element simulation of a crack confined to a lithology-controlled mechanical unit. The stress reduction shadow increases in length with incresaing Young’s modulus of the jointing bed, though fracture stress, flaw size, flaw distribution and extensional strain all interact with bed thickness and elastic properties ultimately to control joint spacing. One explanation for the observed decrease in joint spacing with increasing Young’s modulus in outcrops of the Monterey Formation is that beds with higher Young’s moduli fail at lower magnitudes of extensional strain.

Numerical solutions of the response of a two-dimensional earth to an oscillating magnetic dipole source

A finite difference formulation is developed for computing the frequency domain electromagnetic fields due to a point source in the presence of two‐dimensional conductivity structures. Computing costs are minimized by reducing the full three‐dimensional problem to a series of two‐dimensional problems. This is accomplished by Fourier transforming the problem into the x-wavenumber (kx) domain; here the x-direction is parallel to the structural strike. In the kx domain, two coupled partial differential equations for H⁁x(kx,y,z) and E⁁x(kx,y,z) are obtained. These equations resemble those of two coupled transmission sheets. For a requisite number of kx values these equations are solved by the finite difference method on a rectangular grid on the y-z plane. Application of the inverse Fourier transform to the solutions thus obtained gives the electric and magnetic fields in the space domain. The formulation is general; complex two‐dimensional structures containing either magnetic or electric dipole sources can ...

Finite-element analysis of the stress distribution around a pressurized crack in a layered elastic medium: implications for the spacing of fluid-driven joints in bedded sedimentary rock

Bedding-perpendicular joints confined to individual beds in interbedded sedimentary rocks commonly exhibit spacings which are proportional to the thickness of the jointed bed, and which vary according to lithology or structural position. The mechanical explanation for this relationship is well understood when the joints are driven by far-field crack-normal tensile stresses, but poorly understood for cracks driven by elevated fluid pressures, where the crack-driving stress is the difference between the crack-normal compression and the fluid pressure in the crack. Through a series of finite-element numerical models, we investigate how various parameters influence the driving-stress distribution around pressurized cracks in layered media, and thereby identify factors influencing the spacing of fluid-driven joints. For the situation we modeled, we observe that: (1) crack-driving stress is reduced in the vicinity of pressurized joints, and that the extent of the stress reduction depends on the contrast in elastic properties between the layers; and (2) crack-driving stress distribution depends on the ambient pore pressure during jointing. These results indicate the spacing of fluid-driven joints should depend on lithology and pore pressure.

Multidimensional GPR array processing using Kirchhoff migration

We compare the ability of several practical ground-penetrating radar (GPR) array processing methods to improve signal-to-noise ratio (SNR), increase depth of signal penetration, and suppress out-of-plane arrivals for data with SNR of roughly 1. The methods include two-dimensional (2-D) monostatic, three-dimensional (3-D) monostatic, and 3-D bistatic Kirchhoff migration. The migration algorithm is modified to include the radiation pattern for interfacial dipoles. Results are discussed for synthetic and field data. The synthetic data model includes spatially coherent noise sources that yield nonstationary signal statistics like those observed in high noise GPR settings. Array results from the model data clearly indicate that resolution and noise suppression performance increases as array dimensionality increases. Using 50-MHz array data collected on a temperate glacier (Gulkana Glacier, AK), we compare 2-D and 3-D monostatic migration results. The data have low SNR and contain reflections from a complex, steeply dipping bed. We demonstrate that the glacier bed can only be accurately localized with the 3-D array. In addition, we show that the 3-D array increases SNR (relative to a 2-D array) by a factor of three.

Radar signature of a 2. 5-D tunnel

The effects of an infinitely long cylindrical void on short‐pulse cross‐borehole radar waveforms are modeled and analyzed. Pulsed electromagnetic sensing system (PEMSS) data are of particular interest. The PEMSS system developed by the Southwest Research Institute uses a vertically oriented electric dipole that emits a short electromagnetic pulse with peak power output centered around 30 MHz, which gives wavelengths of roughly 1.5 cavity diameters. The transmitter and receiver are typically located in boreholes separated by approximately 30 m. The model is based on field solutions for a vertically oriented point‐source electric dipole. A three‐dimensional (3-D) analytical frequency domain derivation of the Green’s function is found using a spatial Fourier transform over the cylinder axis. The resulting wavenumber integral is evaluated by a numerical integration over wavenumber. Time‐domain waveforms are produced by applying a Fourier transform to a 7-80 MHz band of frequencies in the Green’s function spec...

Application of a simple relation for describing wave velocity as a function of pressure in rocks containing microcracks

The occurrence of microcracks and pore space reduces the P and S wave velocities in rocks from those expected for the intrinsic mineral matrix. In the low-pressure regime, before microcracks close, the elastic properties and wave velocities are characteristically nonlinear. This nonlinear behavior reflects the effects of the crack closure spectra on the P and S wave velocities. Several efforts have been made in the past to describe or model this behavior using phenomenological relations. However, success has been limited to a rather restricted pressure range. In the present study, two simple relations are considered that have a clear physical basis and are convenient to use and interpret. Both are tied to the functional form V2 = A + BP + Ce−P/τ. The fitting parameters A and B can be identified directly as second- and third-order elastic properties of the intrinsic mineral matrix of the rock specimen, and C and τ describe the nature of the microcrack distribution. The results from fitting a large number of published igneous rock data sets indicate that the relations studied yield rms errors that are (1) always within the stated accuracy of the data, (2) generally within the stated precision of the data, (3) comparable in “goodness of fit” to previously published but more complex relations over limited pressure range data sets, and (4) clearly superior over extended pressure ranges. Because of their comparative simplicity, demonstrated accuracy, and clearly defined fitting parameters, they are well suited for systematic studies of large numbers of igneous rock data sets.

Seismic source model for moving vehicles

We develop a method for the loading of ground by moving vehicles in large finite-difference time-domain simulations of seismic wave propagation. The objective is to realistically produce two distinct types of ground loading for either wheeled or tracked vehicles in our propagation models: lower frequency loading associated with suspension dynamics and higher frequency impulsive loading associated with tire treads or wheels rolling over individual track blocks. These loading characteristics are important because field measurements show that vehicle ground forcing in both frequency bands produces seismic surface waves that networked sensors can remotely process for security applications. The method utilizes a vehicle-dynamics model to calculate a response to vehicle acceleration and ground features such as bumps; calculates forces transmitted to the ground; distributes these forces to staggered points of a finite-difference model; and simulates seismic wave propagation away from the vehicle. We demonstrate the method using bounce-and-pitch models of wheeled and tracked vehicles. We show that by carefully preprocessing force inputs, we can accurately simulate wave propagation and seismic signatures in finite-difference analyses of vehicles moving continuously over terrain.

Seismic radiation from a point source on the surface of a cylindrical cavity

The presence of a void or cavity in the vicinity of a seismic source will modify the radiated signal from the classic solution for a point force in an infinite medium. To study this effect, solutions were obtained for the seismic fields from a point force applied to the surface of a cylindrical cavity in an elastic medium. The solutions were evaluated to give P‐ and S‐wave frequency domain radiation patterns. For P wavelengths less than about 3 times the cavity diameter, the cavity acts to decrease the P‐wave amplitude going outward in the direction opposite the source. Data taken in two coal mines show this shielding effect. High‐frequency energy was observed, with surface seismometers, for signals generated by hitting the mine roof, whereas the high‐frequency energy was much smaller on signals generated by hitting the floor. Time domain calculations show that the P‐wave signal is delayed by approximately the time it takes an S‐wave to propagate around the cylinder.

Quick and accurate Q parameterization in viscoelastic wave modeling

We introduce a procedure for including the attenuation factor Q in a consistent manner in seismic modeling and show 3D examples. The Q fitting over a chosen frequency band involves two algorithms: The first creates starting values of relaxation times, and the second does nonlinear inversion using the results of the first as initial values. The resulting Q function gives a good approximation to a constant Q over the chosen frequency band. The algorithm is combined with a finite-difference (F-D) code that includes topographies in 3D seismic media. The velocity-stress formulation for viscoelastic wave modeling is used with an arbitrary number of relaxation mechanisms to model a desired Q behavior. These equations are discretized by high-order F-Ds in the interior of the medium, and we gradually reduce the F-D order to two at the stress-free surface, where we implement our free-surface boundary conditions. The seismic F-D algorithm is applied to a marine seismic experiment, with and without viscoelasticity, to emphasize the importance of including physical attenuation and dispersion in seismic modeling. Their inclusion, even for marine surveys, is clearly important for lossy ocean bottoms. Our procedure for more accurate modeling of physical dispersion and attenuation may increase future motivation to include viscoelasticity in seismic inversion.

Electromagnetic wave propagation in disrupted coal seams

It is of economic importance to locate zones of geological disturbance in longwall coal panels prior to the start of mining operations. This can be done with electromagnetic (EM) longwall attenuation tomography in which the amplitude of a single frequency wave in the 300 kHz range is measured over a number of raypaths. The main component of the signal is a fundamental waveguide mode. Zones with high attenuation rates have been observed to be zones of geological disturbance. A two‐dimensional TM (magnetic field transverse to the plane of propagation) mode, time domain finite difference solution was developed and used to determine the excess attenuation (EA) that occurs in disturbed zones. EA is defined as the actual attenuation going through a disturbed zone less the attenuation for an equal length of the normal, undisturbed waveguide. A typical model has a coal seam bordered above and below by rock with a disturbed zone; various types of disturbances such as sand channels and faults were modeled. Syntheti...