Annual Report 2000 - WIT
Annual Report 2000 - WIT Annual Report 2000 - WIT
and » ˆ Ö Ö is the scalar hydraulic diffusivity. If a time-harmonic perturbation 1äåjæÄçèšF¬´ºŠétˆ=Ÿ ê ×^í)î ë ˆ Ö Ö é “)»òñ 132 Thus, »P×]Ú is assumed to be pressure independent in eq. (1). To introduce the concept of triggering fronts let us firstly recall the form of the solution of (1) in the case of a homogeneous poroelastic medium. In the case of anisotropic homogeneous medium equation (1) takes the following form Ö Ö ¦È (2) ‘»P×]Ú If the medium is also isotropic » (i.e., then Ö Ö Ç7Ç ‘» Ò7Ò ‘» Õ7Õ , and »½×…Úß‘Î , if ºáà ‘uÝ ), ‘» ‘»ãâP¦• (3) Ê1× ÊÛÚ of the pore-pressure perturbation is given on a small spherical surface of the ê radius with the center at the injection point, then the solution of equations (3) is • (4) é where is the angular frequency ëá‘ôó õtó and is the distance from the injection point to the point, where the solution is looked for. From equation (4) we note that the solution can be considered as a spherical wave (it corresponds to the slow compressional wave in the Biot theory) with the attenuation coefficient equal ö é “)» to and the slowness equal to ̨$÷ é “(» . !š{릕.ˆ=Ÿt‘Á1äÆì)Å åjæÄçÃ漢{º¦¬Į̈Ÿš{ël¬·êÛŸjð In reality the pore pressure at the injection point is not a harmonic function. Let us roughly approximate the pore pressure perturbation at the injection point by a step function 蚈=Ÿø‘¿1ä , if ˆPùúÎ and 蚈=Ÿø‘ÓÎ if ˆ½ûôÎ . For instance, this can be a rough approximation in some cases of a borehole fluid injection (e.g. for a hydraulic fracturing or other fluid tests). In a given elementary volume of the medium located at the distance ëü‘ýó õ ó from the injection source, the triggering of microseismic events starts just before the substantial relaxation of the pore pressure has been reached. For a particular seismic event at the time ˆFä the time evolution of the injection signal for the time ˆÿþψFä is of no relevance for this event anymore. Thus, this event is triggered by the rectangular signal 蚈=ŸP‘1ä if Ρ ˆ¢ ˆFä and 蚈=ŸP‘ Î if ˆ û Î or ˆ þ ˆFä . The power spectrum of this signal has the dominant part in the frequency range below “¤£è¦ˆFä (note that the choice of this frequency is of partially heuristic character; see the related discussions in Shapiro et al., 1997 and 1999). Thus, the probability, that this event was triggered by signal components from the frequency range é¥ ¦£!0ˆFä is high. This probability for the lower energetic high frequency components is small. However, the propagation velocity of high-frequency components is higher than those of the low frequency components (see eq. 4). Thus, to a given time ˆFä it is probable
‘Î–È Ì©¨)¢ËÒ? ¨ 133 that events will occur at distances, which are smaller than the travel distance of the slow-wave signal with the dominant frequency “¤£è0ˆFä . The events are characterized by a significantly lower probability for larger distances. The spatial surface which separates these two spatial domains we call the triggering front. Triggering fronts in homogeneous anisotropic media Let us firstly assume, that the medium is homogeneous and isotropic. Then the slowness of the slow wave (see eq. 4) can be used to estimate the above mentioned size of the spatial domain, where microseismic events are characterized by a high probability. We obtain (see also Shapiro et al. 1997) ëß‘ ÷ § £¦»¤ˆÆÈ (5) This is the equation for the triggering front in an effective isotropic homogeneous poroelastic medium with the scalar hydraulic diffusivity » . With a correctly selected value of the hydraulic diffusivity, equation (5) corresponds to the upper bound of the cloud of events in the plot of their spatio-temporal distribution (i.e., the plot of ë versus ˆ ). In Figure 1a such a spatio-temporal distribution of the microseismicity is shown for the the microseismic data collected in December 1983 during the hydraulic injection into crystalline rock at a depth of 3463 meters at the Fenton Hill (USA) geothermal energy site (see for details and further references Fehler et al., 1998). We see a good agreement between the theoretical curve » with and the data. distance from injection source [m] 1000 900 800 700 600 500 400 300 200 100 Events Fenton Hill D= 0.17 m 2 /s a) distance from injection source [m] 1000 900 800 700 600 500 400 300 200 100 Events Soultz ’93 D= 0.05 m 2 /s b) 0 0 10 20 30 40 50 time from beginning of injection [h] 0 0 50 100 150 200 250 300 350 400 time from beginning of injection [h] Figure 1: Distances of events from the injection source versus their occurrence time for a) the Fenton Hill experiment, 1983 and b) the Soultz-sous-Forets experiment, 1993 Such a good agreement supporting the above concept of the triggering of microseismicity can be observed in many other cases. For example Figure 1b shows
- Page 93 and 94: ê 81 ing was realized by an implem
- Page 95 and 96: ˆ 83 -0.05 -0.1 Amplitude -0.15 -0
- Page 97: 85 on a second-order approximation.
- Page 100 and 101: 88 kinematic (related to traveltime
- Page 102 and 103: ¨ º Ý È · § À · Á · Á ¼
- Page 104 and 105: ° ´ ´ ã ° ¼ ´ ´ þ Ò Ò ¥
- Page 106 and 107: 94 1 taper value ksi1 0 ksi2 Figure
- Page 108 and 109: 96 0 1 2 3 4 5 Depth [km] CMP [km]
- Page 110 and 111: 98 CONCLUSION As a generalization o
- Page 112 and 113: 100 weight during the stacking proc
- Page 114 and 115: 102 are not illuminated for every o
- Page 116 and 117: 104 obtained from Zoeppritz' equati
- Page 118 and 119: 106 PreSDM of Porous layer 0.358 re
- Page 120 and 121: 108 REFERENCES Gassmann, F., 1951,
- Page 122 and 123: 110 et al., 1992), Hanitzsch (1997)
- Page 124 and 125: 112 consecutive wavefronts). Howeve
- Page 126 and 127: 114 Numerical example We test the W
- Page 128 and 129: 116 Versteeg, R., and Grau, G., 199
- Page 130 and 131: 118 indicator. To extract elastic p
- Page 132 and 133: € b b 120 S where is the position
- Page 134 and 135: É É É É 122 The À constant (da
- Page 136 and 137: 124 0 Distance [ m ] 1000 2000 3000
- Page 138 and 139: 126 Kosloff, D., Sherwood, J., Kore
- Page 140 and 141: 128
- Page 142 and 143: 130 penetration distances and a pot
- Page 146 and 147: Å éÙó ó ».-.‹œì/-j쨋
- Page 148 and 149: 136 TRIGGERING FRONTS IN HETEROGENE
- Page 150 and 151: Ö ˆ Ö “P£'ˆ é ˆ ó,Lˆ¨
- Page 152 and 153: Ö Ö Ê » Ø Ö » Ö Ê Ê Ê Ö
- Page 154 and 155: 142 300 250 200 a) eikonal equation
- Page 156 and 157: 144 Shapiro, S. A., and Müller, T.
- Page 158 and 159: » i » m ×]Ú ‘Œƒšj'Ÿlk hM
- Page 160 and 161: 148 RESULTS Soultz-sous-Forêts Inv
- Page 162 and 163: …„ƒ 150 for the smaller ellips
- Page 164 and 165: 152 Figure 5: The cloud of events f
- Page 166 and 167: …„ƒ Î Î ÎÄÈ‹•¨ Î-È
- Page 168 and 169: 156 straightforward. The new approa
- Page 170 and 171: 158
- Page 172 and 173: 160 1982). There is also the so-cal
- Page 174 and 175: { ³ u ´ ´ ´ ´ -y ›-^œ ´ ´
- Page 176 and 177: Ï u u ¬ 164 To summarize, we deri
- Page 178 and 179: Ý ì á ä é å¤æDçzè ä é
- Page 180 and 181: ø M ã = ã Ù ø ä Ú ã Ù ø
- Page 182 and 183: ã Q c ã ä 170 a=10m; std.dev.=8%
- Page 184 and 185: 172 Frankel, A., and Clayton, R. W.
- Page 186 and 187: 174 It is well-known that inhomogen
- Page 188 and 189: ã ä ä ŸSŸ S ¡S¡ ©©¨¨ æ
- Page 190 and 191: Kneib, 1995 285-6000 superposition
- Page 192 and 193: 180 parameter: Hurst coefficient pa
‘Î–È Ì©¨)¢ËÒ? ¨<br />
133<br />
that events will occur at distances, which are smaller than the travel distance of the<br />
slow-wave signal with the dominant frequency “¤£è0ˆFä . The events are characterized<br />
by a significantly lower probability for larger distances. The spatial surface which<br />
separates these two spatial domains we call the triggering front.<br />
Triggering fronts in homogeneous anisotropic media<br />
Let us firstly assume, that the medium is homogeneous and isotropic. Then the slowness<br />
of the slow wave (see eq. 4) can be used to estimate the above mentioned size of<br />
the spatial domain, where microseismic events are characterized by a high probability.<br />
We obtain (see also Shapiro et al. 1997)<br />
ëß‘ ÷ § £¦»¤ˆÆÈ (5)<br />
This is the equation for the triggering front in an effective isotropic homogeneous<br />
poroelastic medium with the scalar hydraulic diffusivity » .<br />
With a correctly selected value of the hydraulic diffusivity, equation (5) corresponds<br />
to the upper bound of the cloud of events in the plot of their spatio-temporal<br />
distribution (i.e., the plot of ë versus ˆ ). In Figure 1a such a spatio-temporal distribution<br />
of the microseismicity is shown for the the microseismic data collected in<br />
December 1983 during the hydraulic injection into crystalline rock at a depth of 3463<br />
meters at the Fenton Hill (USA) geothermal energy site (see for details and further<br />
references Fehler et al., 1998). We see a good agreement between the theoretical<br />
curve » with and the data.<br />
distance from injection source [m]<br />
1000<br />
900<br />
800<br />
700<br />
600<br />
500<br />
400<br />
300<br />
200<br />
100<br />
Events Fenton Hill<br />
D= 0.17 m 2 /s<br />
a)<br />
distance from injection source [m]<br />
1000<br />
900<br />
800<br />
700<br />
600<br />
500<br />
400<br />
300<br />
200<br />
100<br />
Events Soultz ’93<br />
D= 0.05 m 2 /s<br />
b)<br />
0<br />
0 10 20 30 40 50<br />
time from beginning of injection [h]<br />
0<br />
0 50 100 150 200 250 300 350 400<br />
time from beginning of injection [h]<br />
Figure 1: Distances of events from the injection source versus their occurrence time for<br />
a) the Fenton Hill experiment, 1983 and b) the Soultz-sous-Forets experiment, 1993<br />
Such a good agreement supporting the above concept of the triggering of microseismicity<br />
can be observed in many other cases. For example Figure 1b shows
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