Introduction to Strain and Borehole Strainmeter Data Evelyn Roeloffs U. S. Geological Survey Earthquake Science Center March 28, 2016 Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 1 / 28
Plate Boundary Observatory (PBO) Borehole Strainmeter Network Funded by NSF as part of the Earthscope iniative 78 Gladwin Tensor Strainmeters Installed 2004-2013 Depths 500-800 feet (150-250 m) Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 2 / 28
UNAVCO Engineers Installing B201 (Mount St. Helens) Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 3 / 28
A typical PBO borehole strainmeter installation Strainmeter is grouted into an uncased section of borehole Every PBO borehole also contains a 3-component seismometer 23 of the boreholes also contain pore pressure sensors PBO boreholes at Mount St. Helens and Yellowstone also contain tiltmeters Circular diagram shows BSM gauge orientations Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 4 / 28
Gladwin Tensor Strain Meter Developed in Australia by Michael Gladwin Four gauges measure inner diameter of steel housing Three gauges (CH0, CH1, and CH2) are 120 apart around the borehole axis The fourth gauge (CH3) is perpendicular to CH1 Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 5 / 28
Gladwin Tensor Strain Meter - Capacitive Sensing Element Gauge changes length in response to strain along axis Gap d 1 is fixed Strain changes capacitance across gap d 2 Capacitance changes are measured using a bridge circuit whose other arms are at the surface Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 6 / 28
Strainmeters fill gap between Seismology and GPS Strain = spatial derivative of displacement Seismometer: measures time derivative of displacement; need array to measure strain GPS: measures displacement; need array to measure strain Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 7 / 28
Strainmeter niche is hours to days B201 coldwt201bwa2007 Coldwater Visitor Center Mount St Helens 80 B201 CH0 N 18.0E c 40 c.tb 0-40 7.5 years daily averages nanostrain mm 0-20 -40 0-20 -40 40 20 0-20 -40 12 8 4 0 1008 1004 40 B201 CH1 N138.0E c 20 c.tb 40 B201 CH2 N 78.0E c 20 c.tb B201 CH3 N 48.0E B201 rainfall c c.tb 1000 B201 barometer (hpa) 996 07/18/09 07/25/09 08/01/09 08/08/09 08/15/09 08/22/09 08/29/09 45 days 30-minute samples detrended 4 minutes 20 samples per second Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 8 / 28
Basic GTSM data processing steps 12 8 4 7.5 years 0 1008 1004 1000 daily averages 996 nanostrain mm B201 coldwt201bwa2007 Coldwater Visitor Center Mount St Helens 80 B201 CH0 N 18.0E c 40 c.tb 0-40 40 B201 CH1 N138.0E c 20 c.tb 0-20 -40 40 B201 CH2 N 78.0E c 20 c.tb 0-20 -40 40 20 0-20 -40 B201 CH3 N 48.0E B201 rainfall B201 barometer (hpa) c c.tb 07/18/09 07/25/09 08/01/09 08/08/09 08/15/09 08/22/09 08/29/09 45 days 30-minute samples detrended Applied to figures shown here: Clean the data Remove long-term trends Correct for atmospheric pressure Remove tidal variations To be discussed: Optionally, develop your own calibrations Obtain strain components as linear combinations of gauge elongations Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 9 / 28
What is strain? Strain is a change in one or more dimensions of a solid body, relative to a reference state Size may change Shape may change We assume here that strains are small, so infinitesimal strain theory applies Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 10 / 28
Coordinate Systems Right-handed Cartesian coordinate system Various sets of names for coordinate axes (examples above) Strainmeters do not care about: Curvature of the earth Geodetic reference frames Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 11 / 28
Displacements Material at a point can move in three directions, e.g. (u x, u y, u z ) Various sets of names for components of displacement e.g., 1,2,3 or x, y, z Horizontal axes will not always be East and North Strain is a result of spatially varying displacement Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 12 / 28
Spatial derivatives of displacement Displacements can vary in three coordinate directions 9 partial derivatives u i x j where i, j = 1, 2, 3 Strain components: ɛ ij = 1 2 [ u i x j + u j x i ] Normal strains have i = j: ɛ ii = u i x i (no summation implied) Shear strains have i j, note that ɛ ij = ɛ ji Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 13 / 28
Horizontal strain components +y unstrained material +x ε xx ε yy ε xy contraction in the x-direction (a negative strain) extension in the y-direction (a positive strain) xy shear (a positive strain because y-displacement increases with increasing x) ɛ xx = ux x ɛ yy = uy y ɛ xy = 1 2 [ ux y + uy x ] Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 14 / 28
Units of strain Strain is dimensionless but often referred to as if it had units: ɛ xx = 0.01 might be called 1% strain or 10,000 microstrain or 10,000 ppm strain 1 mm shortening of a 1-km baseline is a strain of 10 6 = -1 microstrain = -1 ppm strain 1 mm lengthening of a 1000-km baseline is a strain of 10 9 = 1 nanostrain = 1 ppb strain Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 15 / 28
Sign conventions The following sign conventions are used here to minimize mathematical confusion: Increasing length ( extension ) is positive strain; decreasing length ( contraction ) is negative strain Increasing area or volume ( expansion ) is positive strain Stresses that produce positive strains are positive (ie., tension is positive) Stresses that produce negative strains are negative (ie., compression is negative)...but note that some publications do not use these sign conventions: In geotechnical literature, contraction and compressional stress are referred to as positive Published work on volumetric strainmeter data describes contraction as positive Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 16 / 28
Example: Locked vs. creeping strike-slip fault Map view Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 17 / 28
GTSM gauge output is proportional to inner diameter change of housing Goal: Measure the strain of the formation Empty Borehole Deforms much more than formation Strainmeter Grouted into Borehole Borehole filled with rigid material Deforms more than formation but less Does not deform at all than empty borehole Deformation depends on relative moduli of strainmeter, grout, and formation Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 18 / 28
Elongation of a single ideal gauge in response to strain Gauge elongation, e i, is a linear combination of strain parallel and perpendicular to the gauge e x = Aɛ xx Bɛ yy A and B are positive scalars with A > B Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 19 / 28
Areal strain, differential extension, engineering shear +y unstrained material +x ε xx +ε yy ε xx ε yy 2ε xy areal contraction (a negative strain) differential extension (a positive strain) engineering shear (a positive strain because y-displacement increases with increasing x) Areal strain ɛ xx + ɛ yy does not change if axes are rotated Differential extension (ɛ xx ɛ yy ) and engineering shear 2ɛ xy are shear strain components Neither shear strain component changes area Shear strains depend on coordinate system Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 20 / 28
Elongation of ideal gauge: Areal strain and differential extension e x = Aɛ xx Bɛ yy can be re-written as linear combination of areal strain and differential extension Rearrange: e x = 0.5(A B)(ɛ xx + ɛ yy ) + 0.5(A + B)(ɛ xx ɛ yy ) Define C = 0.5(A B) and D = 0.5(A + B) e x = C(ɛ xx + ɛ yy ) + D(ɛ xx ɛ yy ) NOTE: ɛ xy does not change length for ideal gauge parallel to x or y Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 21 / 28
PBO 4-component GTSM: Gauge configuration CH1 CH2 x1 =e 1 CH3 =e 3 y 1 60 o 30 o 60 o CH0 =e 2 North φ 0 CH0,CH1, and CH2 are equally spaced CH3 is perpendicular to CH1 θ This is the azimuth on the PBO web page given CW from North CH0 CH2 =e 0 East x 1 and y 1 are Cartesian coordinates parallel and perpendicular to CH1 Azimuths are clockwise from North; Polar coordinate angles are counterclockwise from East Polar coordinates (r, θ) are used for math The polar angle of y 1 is +90 from the polar angle of x 1 Blue dots: end of gauge whose azimuth is given Red circles: ends of CH2 and CH0 that are -120 o and +120 o from CH1 Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 22 / 28
2 strain components from 2 perpendicular ideal gauges x, y = directions of perpendicular gauges CH1 and CH3 e 1 = C(ɛ xx +ɛ yy )+D(ɛ xx ɛ yy ) e 3 = C(ɛ xx +ɛ yy ) D(ɛ xx ɛ yy ) Solve for areal strain and differential extension: (ɛ xx + ɛ yy ) = (e 1 + e 3 )/2C (ɛ xx ɛ yy ) = (e 1 e 3 )/2D Areal strain is proportional to average of gauge elongations Differential extension is proportional to difference between gauge elongations Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 23 / 28
Transforming horizontal strains to rotated coordinates Horizontal strain tensor can be expressed in a coordinate system rotated about the vertical axis ɛ x x + ɛ y y 1 0 0 ɛ xx + ɛ yy ɛ x x ɛ y y = 0 cos 2θ sin 2θ ɛ xx ɛ yy 2ɛ x y 0 sin 2θ cos 2θ 2ɛ xy Areal strain is invariant under rotation: ɛ x x + ɛ y y = ɛ xx + ɛ yy for any value of θ Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 24 / 28
Expressing gauge elongations in rotated coordinates Each gauge has its own gauge-parallel coordinates, e.g.: e 1 = C(ɛ xx + ɛ yy ) + D(ɛ xx ɛ yy ) e 0 = C(ɛ x x +ɛ y y )+D(ɛ x x ɛ y y ) Express CH2 elongation in CH1-parallel coordinates: ɛ x x + ɛ y y = ɛ xx + ɛ yy ɛ x x ɛ y y = cos 2θ(ɛ xx ɛ yy ) + sin 2θ(2ɛ xy ) e 0 = C(ɛ xx + ɛ yy ) + D cos 2θ(ɛ xx ɛ yy ) + D sin 2θ(2ɛ xy ) e 0 = C(ɛ xx + ɛ yy ) 0.5D(ɛ xx ɛ yy ) + 0.866D(2ɛ xy ) for θ = 60 NOTE: e 0 does not respond to 2ɛ x y, but does respond to 2ɛ xy Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 25 / 28
3 gauge elongations to 3 strain components x, y are parallel and perpendicular to CH1 = e 1 3 identical gauges 120 apart (CH2, CH1, CH0) = (e 0, e 1, e 2 ) Express elongations in CH1-parallel coordinates: e 0 = C(ɛ xx + ɛ yy ) + Dcos( 240 )(ɛ xx ɛ yy ) + Dsin( 240 )(2ɛ xy ) e 1 = C(ɛ xx + ɛ yy ) + D(ɛ xx ɛ yy ) e 2 = C(ɛ xx + ɛ yy ) + Dcos(240 )(ɛ xx ɛ yy ) + Dsin(240 )(2ɛ xy ) Solve for strain components: (ɛ xx + ɛ yy ) = (e 0 + e 1 + e 2 )/3C (ɛ xx ɛ yy ) = [(e 1 e 0 ) + (e 1 e 2 )]/3D 2ɛ xy = (e 0 e 2 )/[2(0.866D)] Areal strain is proportional to average of outputs from equally spaced gauges Shear strains are proportional to differences among gauge outputs Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 26 / 28
Gauge elongations to strain components: Example Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 27 / 28
Any questions? Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016 28 / 28