The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Dynamics of structures

The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.

/pdf/validity-of-cubic-law-for-fluid-flow-in-a-deformable-rock-2ynq4id3hq.pdf

VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

• Stability of steady frictional sliding, inertia and the quasi-static limit • work on ps 4; see course web site

The Mechanics Of Earthquakes And Faulting

Ultrasonic testing of materials: Josef and Herbert Krautkramer Springer-Verlag Publishing Company, Berlin, Heidelberg, New York (1975) 669 pp, $63.00

https://courses.cit.cornell.edu/mclaskey/pubs/JSV2010.pdf

Beamforming array techniques for acoustic emission monitoring of large concrete structures

This paper describes sensor calibration and signal analysis techniques applicable to the method of acoustic emission (AE) and ultrasonic testing. They are particularly useful for obtaining absolute measurements of AE wave amplitude and shape, which can be used to constrain the physics and mechanics of the AE source. We illustrate how to perform calibration tests on a thick plate and how to implement two different mechanical calibration sources: ball impact and glass capillary fracture. In this way, the instrument response function can be estimated from theory, without the need for a reference transducer. We demonstrate the methodology by comparing calibration results for four different piezoelectric acoustic emission sensors: Physical Acoustics (PAC) PAC R15, PAC NANO30, DigitalWave B1025, and the Glaser-type conical sensor. From the results of these tests, sensor aperture effects are quantified and the accuracy of calibration source models is verified. Finally, this paper describes how the effects of the sensor can be modeled using an autoregressive-moving average (ARMA) model, and how this technique can be used to effectively remove sensor-induced distortion so that a displacement time history can be retrieved from recorded signals.

http://glaser.berkeley.edu/glaserdrupal/pdf/JNDE%20Mclaskey%20Sensor%20Calibration.pdf

Acoustic Emission Sensor Calibration for Absolute Source Measurements

[1] We report on laboratory experiments which investigate interactions between aseismic slip, stress changes, and seismicity on a critically stressed fault during the nucleation of stick-slip instability We monitor quasi-static and dynamic changes in local shear stress and fault slip with arrays of gages deployed along a simulated strike-slip fault (2 m long and 04 m deep) in a saw cut sample of Sierra White granite With 14 piezoelectric sensors, we simultaneously monitor seismic signals produced during the nucleation phase and subsequent dynamic rupture We observe localized aseismic fault slip in an approximately meter-sized zone in the center of the fault, while the ends of the fault remain locked Clusters of high-frequency foreshocks (Mw ~ −65 to −50) can occur in this slowly slipping zone 5–50 ms prior to the initiation of dynamic rupture; their occurrence appears to be dependent on the rate at which local shear stress is applied to the fault The meter-sized nucleation zone is generally consistent with theoretical estimates, but source radii of the foreshocks (2 to 70 mm) are 1 to 2 orders of magnitude smaller than the theoretical minimum length scale over which earthquake nucleation can occur We propose that frictional stability and the transition between seismic and aseismic slip are modulated by local stressing rate and that fault sections, which would typically slip aseismically, may radiate seismic waves if they are rapidly stressed Fault behavior of this type may provide physical insight into the mechanics of foreshocks, tremor, repeating earthquake sequences, and a minimum earthquake source dimension

/pdf/foreshocks-during-the-nucleation-of-stick-slip-instability-3gcoho8u9x.pdf

Foreshocks during the nucleation of stick‐slip instability

Recent modeling studies have explored whether earthquakes begin with a large aseismic nucleation process or initiate dynamically from the rapid growth of a smaller instability in a “cascade-up” process. To explore such a case in the laboratory, we study the initiation of dynamic rupture (stick slip) of a smooth saw-cut fault in a 76 mm diameter cylindrical granite laboratory sample at 40–120 MPa confining pressure. We use a high dynamic range recording system to directly compare the seismic waves radiated during the stick-slip event to those radiated from tiny (M −6) discrete seismic events, commonly known as acoustic emissions (AEs), that occur in the seconds prior to each large stick slip. The seismic moments, focal mechanisms, locations, and timing of the AEs all contribute to our understanding of their mechanics and provide us with information about the stick-slip nucleation process. In a sequence of 10 stick slips, the first few microseconds of the signals recorded from stick-slip instabilities are nearly indistinguishable from those of premonitory AEs. In this sense, it appears that each stick slip begins as an AE event that rapidly (~20 µs) grows about 2 orders of magnitude in linear dimension and ruptures the entire 150 mm length of the simulated fault. We also measure accelerating fault slip in the final seconds before stick slip. We estimate that this slip is at least 98% aseismic and that it both weakens the fault and produces AEs that will eventually cascade-up to initiate the larger dynamic rupture.

/pdf/preslip-and-cascade-processes-initiating-laboratory-stick-rdmae0bogj.pdf

Preslip and cascade processes initiating laboratory stick slip

This paper reviews laboratory observations of earthquake initiation and describes new experiments on a 3‐m rock sample where the nucleation process is imaged in detail. Many of the laboratory observations are consistent with previous work that showed a slow and smoothly accelerating earthquake nucleation process that expands to a critical nucleation length scale Lc, before it rapidly accelerates to dynamic fault rupture. The experiments also highlight complexities not currently considered by most theoretical and numerical models. This includes a loading rate dependency where a “kick” above steady state produces smaller and more abrupt initiation. Heterogeneity of fault strength also causes abrupt initiation when creep fronts coalesce on a stuck patch that is somewhat stronger than the surrounding fault. Taken together, these two mechanisms suggest a rate‐dependent “cascade up”model for earthquake initiation. This model simultaneously accounts for foreshocks that are a by‐product of a larger nucleation process and similarities between initial P wave signatures of small and large earthquakes. A diversity of nucleation conditions are expected in the Earth's crust, ranging from slip limited environments with Lc < 1 m, to ignition‐limited environments with Lc > 10 km. In the latter case, Lc fails to fully characterize the initiation process since earthquakes nucleate not because a slipping patch reaches a critical length but because fault slip rate exceeds a critical power density needed to ignite dynamic rupture. Plain Language Summary In uniquely large‐scale laboratory experiments, a 3‐m rock sample is squeezed until earthquake‐like slip events spontaneously develop on a planar fault cut through the sample. This paper describes the initiation of those slip events—where one part of the fault begins to slip a fraction of a second before the rest of it ruptures (i.e., preslip). The laboratory observations are compared to theoretical models, computer simulations, and field studies of foreshock sequences and other earthquake precursors. Many observations are consistent with previous work that showed slow and smoothly accelerating earthquake initiation—a process termed earthquake nucleation. When the preslip region grows larger than a critical length scale Lc (~1 m), it accelerates unstably and radiates seismic waves like an earthquake. However, some observations show an order of magnitude variation in apparent Lc. The initiation process is sensitive to details such as naturally occurring variation in the strength of the rock/rock fault and perturbations in the rate at which the rock is loaded. Put together, the laboratory work suggests that smoothly accelerating earthquake nucleation is a property of unnaturally smooth and homogenous faults and that Lc is an incomplete metric for characterizing the initiation of earthquakes on realistically rough natural faults.

Gregory C. McLaskey

Papers

Beamforming array techniques for acoustic emission monitoring of large concrete structures

Acoustic Emission Sensor Calibration for Absolute Source Measurements

Foreshocks during the nucleation of stick‐slip instability

Preslip and cascade processes initiating laboratory stick slip

Earthquake Initiation from Laboratory Observations and Implications for Foreshocks