Stephen R. Brown

Rupture mechanism and interface separation in foam rubber models of earthquakes: a possible solution to the heat flow paradox and the paradox of large overthrusts

Abstract Spontaneous stick-slip along the interface between stressed foam rubber blocks is a simple analog of earthquake rupture and stick-slip. Results from this model are used to elucidate the role of normal stress variations along the interface in the stick-slip process. Observations indicate significant normal interface vibrations and separation during slip, suggesting that dynamic changes in normal stress (rather than a drop in the coefficient of friction) may control stick-slip, as suggested, for example, by Tolstoi, Oden and Martins and Brune and co-workers. Observations of particle trajectories indicate that stick-slip shear motion is associated with various degrees of fault separation. For an asymmetric model, the motion is consistent with slipping motion of the type suggested by Schallamach and Price. For a symmetric model, the motion is similar to that suggested by Comninou and Dundurs. If interface waves of this type, involving separation during slip, occur in earthquakes, they may be a solution to the heat flow paradox, since a major part of the slip occurs during separation and during low normal stress. Thus frictional heat generation is reduced. Normal interface vibrations during stick-slip may explain the high corner frequency of P wave spectra and the generally high levels of P wave spectra beyond the corner frequency. Schallamach-Comninou type waves are consistent with the partial stress drop-abrupt locking-self healing models of Brune and Heaton.

Applicability of the Reynolds Equation for modeling fluid flow between rough surfaces

Predictions of the Reynolds equation for flow between rough-walled surfaces have been compared to a more exact calculation of Navier-Stokes flow based on a lattice-gas automaton method. Two-dimensional channels were constructed with an idealized sinusoidal roughness on each wall. Flow in the channels was studied by both methods for various amplitude to wavelength ratios of the roughness, surface separations, relative alignment or phase of the sinusoids, and Reynolds numbers. The Reynolds equation overestimates fluid velocity as the surfaces are placed together or the amplitude of the roughness increases relative to its wavelength.

A simplified spring‐block model of earthquakes

The time interval between earthquakes is much larger than the actual time involved during slip in an individual event. The authors have used this fact to construct a cellular automaton model of earthquakes. This model describes the time evolution of a 2-D system of coupled masses and springs sliding on a frictional surface. The model exhibits power law frequency-size relations and can exhibit large earthquakes with the same scatter in the recurrence time observed for actual earthquakes.

Fluid permeability of deformable fracture networks

We consider the problem of defining the fracture permeability tensor for each grid block in a rock mass from maps of natural fractures. For this purpose we implement a statistical model of cracked rock developed by M. Oda, where the permeability tensor is related to the crack geometry via a volume average of the contribution from each crack in the population. In this model, tectonic stress is implicitly coupled to fluid flow through an assumed relationship between crack aperture and normal stress across the crack. We have included three enhancements to the basic model. (1) A realistic model of crack closure under stress has been added along with the provision to apply tectonic stresses to the fracture system in any orientation. The application of compressive stress results in fracture closure, and consequently, a reduction in permeability. (2) The fracture permeability can be linearly superimposed onto an arbitrary anisotropic matrix permeability. (3) The fracture surfaces are allowed to slide under the application of shear stress, causing fractures to dilate and result in a permeability increase. Through two examples we demonstrate that significant changes in permeability magnitudes and orientations are possible when tectonic stress is applied to fracture systems.

Measuring fracture apertures: A comparison of methods

Two methods for measuring aperture distributions within rough-walled fractures are compared; magnetic resonance imaging (MRI) and spectrophotometric analysis (SA) of epoxy replicas. Comparisons with observations of flow through the fracture for Reynolds numbers less than one indicate that the resolution and accuracy of SA is sufficient to predict the effective transmissivity of the fracture using the locally applied cubic law. Observed effective transmissivities for higher Reynolds number flows are lower than predicted using the local cubic law. MRI apertures are generally consistent with SA, except for those less than 100 microns which were not detectable. The lower resolution of the MRI data results in a poor estimate of the effective transmissivity, indicating that despite their low transmissivities, the smallest apertures (<100 microns in this fracture) must still be accurately measured to predict flow through the fracture.

Formation of voids and veins during faulting

Abstract We have developed a quantitative model of void and vein development in fault zones based on observations of fault roughness and the contact characteristics of rough, directionally anisotropic fractal surfaces. This model includes progressive dilation of the fault during slip events and the elastic deformation of the surfaces normal to the fault plane (closure) as new void space develops. The model predicts vein geometries that are qualitatively similar to those observed in fault-controlled mineral deposits. The statistics of the vein system are described and strategies for sampling of such structures by drilling are developed. The results have significant implications for evaluation of ore reserves and evaluating the fluid transport properties of faults.

Origin of rate dependence in frictional sliding

Experiments indicate that frictional resistance to sliding between macroscopic, clean, dry surfaces depends on the average rateV at which the surfaces are translated relative to each other. Using a new lattice automaton, we obtain results suggesting that rate-dependent macroscopic dynamics may arise from microscopic interactions between contact points which decay from a metastable state with a finite lifetimeΓ. Sliding is accommodated by clusters, or avalanches, of failed lattice contact points, and corresponds to successive quenches into the metastable state by an electromechanical loading system with a finite response timeΔ. Under the quasistatic assumptionΔ ≫Γ, rate dependence is a consequence of the increase in correlation length ξd of clusters of failed lattice points as quench rate increases. Special cases of the model are isomorphic to the selforganized criticality model for sandpiles, and to block-spring models of the type first studied by Burridge and Knopoff for earthquakes.

Mixing at fracture intersections: Influence of channel geometry and the Reynolds and Peclet Numbers

The 3D lattice Boltzmann (LB) method was used to model mixing at three types of continuous fracture intersections: planar, fluted (containing parallel grooves), and rough-walled. Peclet number (Pe) varied from 3 to 400, and Reynolds number (Re) varied from 0.5 to 100. In both planar- and rough-walled intersections, the mixing ratio (Mr) decreases with increasing Pe, though the decrease is less dramatic for the rough-walled geometry. In planar-walled intersections, the Mr decreases with increasing Re; however, the fluted and rough-walled intersections show the opposite trend. Overall, the impact of inertial effects is slight for Re ≤ 10. The effects of channel length are also small; the calculated Mr varies little for LB simulations with length/width ≥ 1.

Effective media theory with spatial correlation for flow in a fracture

Standard effective media theory, real-space renormalization, and first-order perturbation theory all predict that a percolation threshold, where the electrical and hydraulic conductivity become zero, is reached when the area of contact is half of the total area. However, calculations of the flow properties of joints, whether simulated by continuous fields or discretely as a network of resistors, show that flow is maintained at appreciably larger contact areas. This behavior has been termed “channeling” in the literature. We have modified the standard effective media theory by introducing a rudimentary sort of nonrandomness in an attempt to simulate features like topographic hills and valleys. Using this new model, we calculate overestimates and underestimates of electrical and hydraulic conductivity as a function of the separation between the fracture surfaces. The underestimate is found to be very close to the standard effective media theory, real-space renormalization, and first-order perturbation theory results. The mean of the overestimate and underestimate is a good approximation to the published values found from calculations of electrical and hydraulic conductivities for simulations of real joints.

Microscopic analysis of macroscopic transport properties of single natural fractures using graph theory algorithms

Given an aperture distribution for a single fracture, a graph theory model has been employed to simulate the transport properties of the fracture. In the processes of imbibition and drainage, the connectivity of each phase at a particular capillary pressure can be sought out by the priority-first search algorithm of graph theory. Then, the flow path in each phase can be identified, and flow equations can be established and solved in an optimal manner. It was observed that in a simulated rough fracture, a few flow “channels” can dominate the permeability of the whole system.