A B S T R A C T An over-deterministic method has been employed for calculating the stress intensity factors (SIFs) as well as the coefficients of the higher-order terms in the Williams series expansions in cracked bodies, using the conventional finite element analysis. For a large number of nodes around the crack tip, an over-determined set of simultaneous linear equations is obtained, and using the fundamental concepts of the least-squares method, the coefficients of the Williams expansion can be calculated for pure mode I, pure mode II and mixed mode I/II conditions. A convergence study has been conducted to examine the effects of the number of nodes used, the number of terms in Williams expansion and the distance of the selected nodes from the crack tip, on the accuracy of the results. It is shown that the simple method presented in this paper, yields accurate results even for coarse finite element meshes or in the absence of singular elements. The accuracy of SIFs and the coefficients of higher-order terms are validated by using the available results in the literature.Keywords coefficients of the higher-order terms; crack; finite element analysis; overdeterministic method; stress intensity factor (SIF); T-stress. N O M E N C L A T U R Ea = crack length or semi-crack length A 0 , B 0 = parameters related to the crack tip translation components A n (n = 1, 2, . . .) = mode I coefficients of the crack tip asymptotic field B 2 = parameter related to rigid body rotation of the crack B n (n = 1, 3, 4, . . .) = mode II coefficients of the crack tip asymptotic field[C] = matrix of nodal positions E = Young's modulus F 0 , g 0 = functions of material constants f I n (r, θ), f II n (r, q ), g I n (r, θ), g II n (r, θ) = functions of the coordinates r and θ h = semi-height of the specimens k = number of utilized nodes K I , K II = mode I and mode II stress intensity factors N , M = number of mode I and mode II terms in the truncated displacement series r, θ = polar coordinate components T = T-stress [U] = column matrix of the nodal displacement componentsCorrespondence: M. R. Ayatollahi.

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Abstract. To characterize the stress field at the Grimsel Test Site (GTS) underground rock laboratory, a series of hydrofracturing and overcoring tests were performed. Hydrofracturing was accompanied by seismic monitoring using a network of highly sensitive piezosensors and accelerometers that were able to record small seismic events associated with metre-sized fractures. Due to potential discrepancies between the hydrofracture orientation and stress field estimates from overcoring, it was essential to obtain highprecision hypocentre locations that reliably illuminate fracture growth. Absolute locations were improved using a transverse isotropic P-wave velocity model and by applying joint hypocentre determination that allowed for the computation of station corrections. We further exploited the high degree of waveform similarity of events by applying cluster analysis and relative relocation. Resulting clouds of absolute and relative located seismicity showed a consistent east-west strike and 70 • dip for all hydrofractures. The fracture growth direction from microseismicity is consistent with the principal stress orientations from the overcoring stress tests, provided that an anisotropic elastic model for the rock mass is used in the data inversions. The σ 1 stress is significantly larger than the other two principal stresses and has a reasonably welldefined orientation that is subparallel to the fracture plane; σ 2 and σ 3 are almost equal in magnitude and thus lie on a circle defined by the standard errors of the solutions. The poles of the microseismicity planes also lie on this circle towards the north. Analysis of P-wave polarizations suggested double-couple focal mechanisms with both thrust and normal faulting mechanisms present, whereas strike-slip and thrust mechanisms would be expected from the overcoring-derived stress solution. The reasons for these discrepancies can be explained by pressure leak-off, but possibly may also involve stress field rotation around the propagating hydrofracture. Our study demonstrates that microseismicity monitoring along with high-resolution event locations provides valuable information for interpreting stress characterization measurements.

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Fluid pressure within the Earth's crust is a key driver for triggering natural and human-induced seismicity. Measuring fluid pressure evolution would be highly beneficial for understanding the underlying driving mechanisms and supporting seismic hazard assessment. Here we show that seismic velocities monitored on the 20-m scale respond directly to changes in fluid pressure. Our data show that volumetric strain resulting from effective stress changes is sensed by seismic velocity, while shear dislocation is not. We are able to calibrate seismic velocity evolution against fluid pressure and strain with in situ measurements during a decameter-scale fluid injection experiment in crystalline rock. Thus, our 4-D seismic tomograms enable tracking of fluid pressure and strain evolution. Our findings demonstrate a strong potential toward monitoring transient fluid pressure variations and stress changes for well-instrumented field sites and could be extended to monitoring hydraulic stimulations in deep reservoirs. Plain Language Summary The pressure of fluids in the subsurface is generally a function of depth as well as the regional geological history. Changes to the subsurface fluid pressure-be it natural or human induced-disturb the stress field and are known to drive volcanic eruptions, as well as to trigger earthquakes. For example, pressure increase by fluid injection for hydraulic stimulation and wastewater disposal has been linked to earthquake activity. Unfortunately, pressure measurements need direct access through boreholes, so that pressure data are only available for few locations. A method for estimating the spatial distribution of fluid pressure remotely would thus be highly beneficial. From measurements in a 20-m-scale experiment in granite, we find that fluid pressure propagation can be predicted from observed seismic velocity variations, based on a strong correlation between observed changes in seismic velocities and fluid pressure measured within the rock. As seismic velocities can be readily measured on the reservoir scale, our results demonstrate a strong potential of seismic velocity monitoring for remotely estimating fluid pressure changes in deep reservoirs, along faults, or in volcanic systems. The estimated pressure and stress changes could be an important input to real-time risk analysis of fault reactivation and volcanic eruptions.