Effect of Thermal Pressurization on Radiation Efficiency

Transcription

1 Bulletin of the Seismological Society of America, Vol. 99, No. 4, pp , August 2009, doi: / Effect of Thermal Pressurization on Radiation Efficiency by Jeen-Hwa Wang Abstract The radiation efficiency, η R, is an important parameter showing the source property. It is strongly affected by the variation in shear stress with slip. Thermal pressurization is considered to be a significant mechanism in controlling such a variation, thus influencing η R. In this study, the formula of η R as a function of slip, δ, on the basis of two end-member models of thermal pressurization, that is, the adiabatic-undrained-deformation (AUD) model and slip-on-a-plane (SOP) model proposed by Rice (2006), is derived. The controlling parameters of the AUD and SOP models are, respectively, δ c and L, which are dependent on thermal, mechanical, and hydraulic parameters of fault rocks. Modeled results suggest that thermal pressurization controls the variation in shear stress with slip and thus influences the radiation efficiency. Results show that η R increases with δ. The increasing rate of η R with δ is high at small δ and low at large δ. This indicates that η R varies very much with δ for small earthquakes and only slightly depends on δ for large events. For the two endmember models, η R increases with decreasing δ c (or L ). When δ c L, η R is higher for the AUD model than for the SOP model. The thermal pressurization model is also applied to investigate the shear stress slip function in a 5 5 km square covering a drilled site on the fault plane of the 1999 Chi-Chi, Taiwan, earthquake inferred from seismograms. Results show that the AOD model is more appropriate to describe the inferred shear stress slip function than the SOP model, and the proposed model is a modified one from the AUD model by including a small amount of loss of frictional heat from the slip zone during faulting. Introduction After an earthquake ruptures, the shear stress, τ, on the fault plane will decrease from an initial τ o to a dynamic τ d and finally becomes τ f. The weakening mechanism is not simple (see Bizzarri and Cocco, 2005) and could be either slip-dependent or rate- and state-dependent (see Bizzarri and Cocco, 2006a. It is noted that slip, δ, and the sliding velocity, v, are each a function of time.) For a long time, there were debates concerning the comparison between the slipdependent and rate- and state-dependent constitutive laws. Some researchers (e.g., Ohnaka, 2004) are in favor of the slip-dependent law, while others (e.g., Bizzarri and Cocco, 2005) prefer the rate- and state-dependent law. Bizzarri and Cocco (2005) found that the rate-weakening law can also yield slip-weakening behavior of shear stress evolution. To simplify the problem, only the function of shear stress versus slip, which might be caused by different mechanisms, is used and displayed in Figure 1. In general τ d is equal to or smaller than τ f (Kanamori and Heaton, 2000). According to the slip- and rate-weakening frictional law, the frictional stress changes from τ o to τ d during a characteristic slip displacement, D c (see Ida, 1972; Marone, 1998; Wang, 2002). The static stress drop Δσ s τ o τ f and the dynamic stress drop Δσ d τ o τ d are usually used to specify the change of stresses on a fault. The strain energy, ΔE, can be approximated by the area of a trapezoid underneath the linearly decreasing function of stress versus slip (Fig. 1). ΔE is almost transferred into three kinds of energy, that is, the seismic radiation energy (E s ), fracture energy (E g ), and frictional energy (E f ), that is, ΔE E s E g E f. E s is the energy radiated through seismic waves. E g is the energy used to extend a fault plane. E f results from the dynamic frictional stress and can generate heat. E s is measured directly from seismograms with some corrections. From Figure 1, wehavee g Δσ d AD=2 E s (see Kanamori and Brodsky, 2004) where A and D are, respectively, the fault plane and an average displacement. Because the three parameters, that is, Δσ d, A, and D, can be evaluated from the source rupture processes inversed from seismograms, E g can be estimated indirectly from seismograms. However, it is difficult to accurately evaluate ΔE and E f due to incomplete data. Wang (2004, 2006) proposed ways of evaluating ΔE and E f when Global Positioning System data are available. The seismic efficiency, η, which is defined to be the ratio of E s to ΔE, that is, E s =ΔE, has long been taken to present the level of seismic-wave radiation generated from an 2293

2 2294 J.-H. Wang Figure 1. The curve AD represents the slip-decreasing function of shear stress. The shear stress slip function, lines AC and CD, represent linear slip-weakening friction, D c equals the characteristic slip displacement, D max equals the maximum slip, τ o equals initial stress (or static frictional stress), τ d equals dynamic frictional stress, and τ f equals final stress. The strain energy, ΔE,is the area of a trapezoid below line AD, E s equals seismic radiation energy, E g equals fracture energy, and E f equals frictional energy. earthquake source. The uncertainty of evaluating η is high due to the difficulty of accurately measuring ΔE. Hence, Kanamori and Heaton (2000) defined a new parameter, that is, the radiation efficiency, η R, which is η R E s = E s E g. This parameter can be evaluated directly from seismograms. Venkataraman and Kanamori (2004) observed η R 0:25 1 for most earthquakes. To decrease the effective frictional strength for initiating faulting or to reduce dynamic friction for maintaining faulting, different mechanisms, including hydrodynamic lubrication (Brodsky and Kanamori, 2001), thermal pressurization (Bizzarri and Cocco, 2006b,c; Fialko, 2004), flash faulting (Rice, 2006), gel formation (Goldsby and Tullis, 2002), and melting lubrication (Spray, 1993) have been proposed. A summary of the mechanisms can be found in Rice (2006). Except for flash heating and melting lubrication, the pore fluid pressure plays a significant role on reducing the frictional stress. Sibson (1973) first proposed that interaction between heat and pore fluid pressures results in thermal pressurization. This mechanism assumes that fluids are present within the fault zone and the shear stress τ during seismic slip can be represented by τ f σ n p, where f is the frictional coefficient, and σ n and p are the normal stress and pore fluid pressure, respectively, and the pore fluid pressure can be affected by heating. Figure 1 shows that the radiation efficiency is mainly controlled by the shear stress slip function. This makes the radiation efficiency a good constraint to specify the shear stress slip function. In this study, an attempt is made to theoretically derive the slip-dependent function of the radiation efficiency from the shear stress slip function (denoted by the τ-δ function hereafter) on the basis of two end-member models of one-dimensional thermal pressurization proposed by Rice (2006). Moreover, the τ-δ function and radiation efficiency on the basis of a controlling parameter for the individual end-member model will be performed from the theoretical formula. Modeled results will be used to examine the effect of thermal pressurization on the radiation efficiency. In addition, the τ-δ function within the square of 5 5 km around a drilled site on the fault plane of the 1999 M s 7.6 Chi-Chi, Taiwan, earthquake inferred from seismograms is taken as an example to describe how to apply the thermal pressurization model to interpret such a function. In order to model the τ-δ function, two ways are suggested to evaluate η R : one from the physical quantities of fault rocks measured from the core samples obtained from the drilled site and the other directly from the inferred τ-δ function of a 5 5 km square from seismograms. Brief Description of Theory One-Dimensional Thermal Pressurization On a fault plane with an area of A and an average displacement D, the frictional energy caused by the dynamic friction stress, τ d,ise f τ d DA, which could result in a temperature rise, ΔT. Frictional heat can conduct outward from the slipping zone to wall rocks. Theoretical analyses (Bizzarri and Cocco, 2006b; Fialko, 2004) show that ΔT is described by an error function of distance and decays outward from the fault plane. Considering a 1D fault plane, the x-and y-axes denote the directions along and normal to the fault plane, respectively. Under thermal pressurization, the energy and fluid mass conservation equations can be written as (see Rice, 2006) T= t σγ t =ρc 1=ρc ρcα t T= y = y; (1) p= t Λ T= t n p = t =β 1=ρ f β ρ f βα h p= y = y; (2) where t is the time, T is the temperature, p is the pore fluid pressure, γ is the shear strain, γ t dγ=dt is the shear rate, ρ is the density of fault zone, ρ f is the density of fluids, c is the heat capacity, α t is the thermal diffusivity, α h is the hydraulic diffusivity, and n p is the inelastic (or plastic) porosity after deformation. The parameter β is the volumetric pore fluid storage coefficient and equal to n β f β n where n is the starting porosity before deformation, β f is the isothermal compressibility of the pore fluid (dρ f =ρ f β f dp), and β n is the isothermal compressibility of the pore space (dn=n β f dp). Λ is the undrained pressurization factor and can be written as λ f λ n = β f β n where

3 Effect of Thermal Pressurization on Radiation Efficiency 2295 λ f dρ f =ρ f λ f dt and λ n dn=n λ n dt are, respectively, the isobaric, volumetric thermal expansion coefficients for the pore fluid and pore space. The parameters α t and α h are, respectively, equal to K=ρc and κ=η f β where K is the thermal conductivity, κ is the permeability, and η f is the fluid viscosity. Rice (2006) proposed two end-member models for thermal pressurization: the adiabatic-undrained-deformation (AUD) model and slip-on-a-plane (SOP) model. The detailed description about the two end-member models can be found in his paper. Only a brief description is given here. The first model corresponds to a homogeneous simple shear strain γ at a constant normal stress σ n on a spatial scale of the sheared layer that is broad enough to effectively preclude heat or fluid transfer. The second model shows that all sliding is on the plane with τ 0 f σ n p o where p o is the pore fluid pressure on the sliding plane (y 0). For this second model, heat is transferred outward from the fault plane. The τ-δ functions, τ δ, caused by thermal pressurization are (Rice, 2006): 1. For the AUD model, τ AUD δ f σ n p o exp δ=δ c : (3) In equation (3), δ c ρch=fλ where h and f are, respectively, the thickness and friction coefficients of the slip zone. 2. For the SOP model, τ SOP δ f σ n p o exp δ=l erfc δ=l 1=2 : (4) In order to derive equation (4), the sliding velocity, V, is set to be constant. In equation (4), erfc z is the complementary error function, and L 4 ρc=λ 2 αt 1=2 α 1=2 h 2 =f 2 V where V is the slip rate, and δ Vt is the displacement. The two parameters δ c and L control the τ-δ functions of the individual end-member models. A detailed description of how to drive equations (3) and (4) can be found in Rice (2006). In equation (3), δ c, which is associated with the thickness h of the slip zone, is a characteristic displacement of the τ-δ function. For equation (4) Rice (2006) addressed that there is not a characteristic displacement for the τ-δ function. The two end-member models will be applied in the following section to study the effect of thermal pressurization on the radiation efficiency as well as the τ-δ function. It is noted that the SOP model with a constant sliding velocity is far away from reality because the source rupture process in general is specified with varying sliding velocities. In fact, the AUD model is not a real one either. However, the two models represent the two end members of thermal pressurization, and they have analytic solutions. This makes us easily explore the problem. If the two end-member models cannot interpret the observations, we must give up thermal pressurization as a mechanism in controlling earthquake ruptures. On the other hand, if the two end-member models can partly interpret the observations, thermal pressurization would be a significant mechanism in controlling earthquake rupture and would thus affect the radiation efficiency. The acceptable model of thermal pressurization should be in between them. To Model the Radiation Efficiency Unlike the previous notations, E s and E g, respectively, represent the seismic radiation and fracture energies per unit area in this section. Rice (2006) used an expression to describe E g (denoted by G in his article) under thermal pressurization. The expression is E g δ f σ n p o 1 1 δ=δ c exp δ=δ c δ c (5) for the AUD model and E g δ f σ n p o exp δ=l erfc δ=l 1=2 1 δ=l 1 2 δ=πl 1=2 L (6) for the SOP model. From Figure 1, we have E s E g τ 0 τ δ δ=2, where δ is the (final) displacement, and τ 0 and τ δ are, respectively, the shear strength and shear stress at δ. Hence, E s E g can be written as E s E g f σ n p o 1 exp δ=δ c δ=2 (7) for the AUD model with equation (3) and E s E g f σ n p o 1 exp δ=l erfc δ=l 1=2 δ=2 (8) for the SOP model with equation (4). The radiation efficiency is defined to be η R E s = E s E g. E s can be written as E s E g E g. Inserting equations (5) (8) into E s E g E g = E s E g leads to η RAUD δ=δ c exp δ=δ c =f 1 exp δ=δ c δ=δ c g (9) for the AUD model and η RSOP δ=l exp δ=l erfc δ=l 1=2 1 2 δ=πl 1=2 =f 1 exp δ=l erfc δ=l 1=2 δ=l g (10) for the SOP model. Obviously, the radiation efficiency is mainly controlled by δ=δ c for the AUD model and by δ=l for the SOP model. Equations (9) and (10) show that

4 2296 J.-H. Wang η RAUD and η RSOP are zero when δ 0 and 1 when δ approaches infinity. Modeled Results From equations (9) and (10), δ c and L control the radiation efficiency. The two parameters are a function of thermal and hydraulic parameters. Rice (2006) compiled the values of related thermal and hydraulic parameters under σ n p o MPa from different sources. He also calculated the values of those parameters at ambient p and T and their averages on p-t path in the intact elastic walls and highly damaged walls when σ n 196 MPa, p o 70 MPa, and T 210 C for a mature fault plane at 7 km. The values of related parameters are ρc 2:7 MPa= C, Λ 0:31 MPa= C, α t 0: m 2 =sec, α h 6: m 2 =sec, and V 1 m=sec. Field surveys suggests that slip in an individual event occurs mainly within a thin shear zone, <1 5 mm, within a finely granulated, ultracataclastic fault core (see Rice, 2006). Hence, we have δ c 2: =f when h m and L 3: =f 2. Rice (2006) took f 0:25, leading to δ c 0:108 m and L 0:049 m. In this study, we consider five values of δ c and L, that is, 0.3, 0.5, 1.0, 3.0, and 5.0 m. These values are larger than those used by Rice (2006). Hence, the related values of f are about 0.090, 0.054, 0.027, 0.009, and for δ c and 0.102, 0.079, 0.056, 0.032, and for L. The shear strength τ o f σ n p o 126 MPa is shown in Figure 2. The plots of shear stress versus slip for the five values of δ c of the AUD model and those for the five values of L of the SOP model are displayed in Figures 2a and 2b, respectively. Figure 3 shows the plots of radiation efficiency versus slip: (a) for the AUD model with δ c 0:3, 0.5, 1.0, 2.0, and 5.0 m and (b) for the SOP model with L 0:3, 0.5, 1.0, 3.0, and 5.0 m. Discussion There is a characteristic slip distance, that is, δ c, in equation (3) yet not in equation (4). This is due to the fact that a finite thickness of the slip zone, that is, h, is defined for the AUD model yet not for the SOP model. Considering this characteristic slip distance, the AUD model is somewhat similar to, and the SOP model is different from the rate- and statedependent friction law (Ruina, 1983). Figure 2 shows that τ decreases with increasing δ. The decreasing rate of τ with δ is larger at small δ than at large δ and also larger for the AUD model than for the SOP model. At a certain δ, τ increases with δ c and L. The difference in τ between two values of δ c (or L ) first increases and then decreases with increasing δ for the AUD model and first increases with δ and then becomes slightly dependent on δ for the SOP model. As mentioned previously, for the two end-member models an increase in δ at small δ results in a large decrease in τ; while an increase in δ at large δ leads to a small decrease in τ. This indicates that after the initiation of an earthquake, τ drops mainly in the early stage with small δ. For the same slip, the stress drop is higher for the AUD model than for the SOP model and also higher at small δ c (or L ) than at large δ c Figure 2. The plots of shear stress versus slip for different values of δ c and L : (a) for the adiabatic-undrained-deformation model and (b) for the slip-on-a-plane model.

5 Effect of Thermal Pressurization on Radiation Efficiency 2297 Figure 3. The plots of radiation efficiency, η R, versus slip for different values of δ c and L : (a) for the adiabatic-undrained-deformation model and (b) for the slip-on-a-plane model. (or L ). This indicates that for the AUD model, a thin slip zone (with small h) yields a higher stress drop than a thick slip zone (with large h) when other parameters do not change. This is in agreement with the results obtained by Bizzarri and Cocco (2006c). For the two end-member models, the stress drop becomes constant when δ is larger than a certain value. When δ c L, τ is larger for the AUD model than for the SOP model when δ is very small, while τ is smaller for the AUD model than for the SOP model when δ is large. This might imply that for a general thermal pressurization process, the AUD model dominates in the early stage with small δ, and the SOP model plays the main role in the latter stage with large δ. When a fault breaks, slip generates heat. In the early stage, the temperature abruptly increases within the slip zone due to an adiabatic process of heat. This makes the fault zone melt, thus yielding melting lubrication. This could accelerate slip. When time increases, heat propagates outward from the slip zone, and temperature decreases. This makes the SOP model become predominant. Figure 3 shows that the radiation efficiency, η R, increases with δ. At a certain δ, the range of η R remarkably varying with δ c is larger for the AUD model than that varying with L for the SOP model. The range of δ, within which η R remarkably increases with δ c, increases with δ c for the AUD model, but it does not change too much with δ c for the SOP model. When δ c L, η R is higher for the AUD model than for the SOP model. For the two end-member models, η R increases with decreasing δ c (or L ). When δ c is small, the increasing rate of η R with δ is high at small δ and low at large δ, and the increasing rate is larger for small δ c than for large δ c. This indicates that η R varies very much with the size of an earthquake for small earthquakes and is almost constant for large events. The values of η R for large earthquakes (M w 7 9) evaluated by Venkataraman and Kanamori (2004) seem to be independent on M w. This provides indirect evidence to confirm the previous assumption. Wang and Ou (1998) found that the average displacement of an earthquake increases with fault length (or size) of the earthquake. Results suggest that for a large earthquake with a long displacement (or large size), seismic radiation could be much stronger in the early stage of rupture with small δ than in the latter one with large δ. The curves with individual values of δ c and L as shown in Figure 2 (associated with equations 3 and 4) and Figure 3 (related to equations 9 and 10) demonstrate two kinds of physical meaning. On one hand, a curve and the related equation show the scaling law of shear stress or radiation efficiency versus slip for the assembly of events with different final displacements whose fault planes are specified with a particular value of δ c or L. On the other hand because slip is a function of time during the rupture of a single earthquake whose fault plane is characterized by a particular value of δ c or L, time is an implicit variable of the equations. Hence, a curve can also represent the evolution of shear stress or radiation efficiency with slip during the event. In other words, Figures 2 and 3 implicitly display the temporal variations in shear stress and radiation efficiency, respectively. In this study, the second kind of physical meaning is considered for exploring the inferred shear stress slip function. An Example for the Modeled and Inferred Shear Stress Slip Functions of an Earthquake On 20 September 1999, the M s 7.6 Chi-Chi earthquake ruptured the Chelungpu fault, which is a 100 km long and east-dipping thrust fault with a dip angle of 30, in central

6 2298 J.-H. Wang Figure 4. The epicenter (in a solid star), the Chelungpu fault (in a solid line), and the borehole site of TCDP (in solid circles). Figure 5. The shear stress slip function (shown by a solid line) inferred by Ma and Mikumo (unpublished manuscript, 2008) from seismic data. The dashed line represents the static stress-slip function. The two vertical thin dotted lines denote δ 10:7 and 12.3 m, respectively. The two horizontal thin dotted lines denote τ m 5:5 MPa and τ f 4:6 MPa, respectively. τ m and τ f are explained in the text. Taiwan (Ma et al., 1999; Shin, 2000). The epicenter and the surface trace of the fault are displayed in Figure 4. Ma and Mikumo (unpublished manuscript, 2008) inversed the τ-δ functions on the fault plane of the Chi-Chi earthquake from local seismograms. The spatial discretization of the fault plane was made by the adoption of identical squares of 5 5 km. The inferred τ-δ function of the square covering the drilled site on the fault plane as described in the following section is displayed with a thin solid line in Figure 5 where the minimum value is null because only the change of shear stress can be evaluated from seismograms. This figure shows that the static stress decreases linearly from τ o 11:2 MPa at δ 0 to τ f 4:6 MPa at δ 12:3 m, and the (dynamic) shear stress decreases from τ o at δ 0 to 0 at δ 10:7 m and then increases from 0 at 10.7 m to τ f 4:6 MPa at 12.3 m. The static stress related to δ 10:7 m is denoted by τ m and equals 5.5 MPa. Of course, this τ-δ function is an average of the 5 5 km square rather than that at a particular site. In addition, Guatteri and Spudich (2000) assumed that the τ-δ function cannot be exactly determined from seismograms due to inclusion of artifacts caused by low resolution using longer-period waveforms. Nevertheless, the inferred τ-δ function is still taken as an example to describe how to apply the thermal pressurization model to interpret the τ-δ function. From the inferred τ-δ function, I assume that thermal pressurization was the predominant mechanism in controlling the variation in shear stress and thus affecting faulting in the range of δ from 0 to 10.7 m, and other mechanisms could play the main role when δ > 10:7 m. In 2005, the Taiwan Chelungpu fault Drilling Project (TCDP) was conducted, and two deep holes cutting the fault plane were drilled (Ma et al., 2006; Song et al., 2007). The two deep holes are 40 m apart: hole A with a depth of 2000 m and hole B with a depth of 1300 m. The solid circle in Figure 4 displays the localities of the two holes. The fault zone at depths m is considered to be associated with the 1999 Chi-Chi earthquake rupture and denoted by the FZA1111 (Hung et al., 2007). Continuously coring and geophysical well loggings were made at the two holes. Ma et al. (2006) called a 12 cm thick zone at depths of m the primary slip zone (PSZ) and named the bottom 2 cm thick subzone the major slip zone (MSZ), which is least deformed and regarded as the slip zone of this earthquake. The traced fissures of PSZ are shown in Figure 6. In order to apply the thermal pressurization model to explore the inferred τ-δ function as that displayed in Figure 5, we must first have the value of η R. Two ways are suggested to evaluate η R. Equations (9) and (10) are essentially defined on the basis of local physical quantities at a site, for example, the drilled site of the TCDP. Hence, the first way is that η R is evaluated from the physical quantities of fault rocks measured from the core samples obtained at the drilled site. Because there is a lack of the inferred τ-δ function at such a small drilled site, the inferred one like that in Figure 5 is taken for comparison. For the second way, it is

7 Effect of Thermal Pressurization on Radiation Efficiency 2299 Figure 6. The traced fractures in a 12 cm thick core sample of the primary slip zone (PSZ) drilled from hole A. The arrow points to the upward direction. The major slip zone (denoted by MSZ) is at the bottom 2 cm. (This figure is reproduced from Ma et al., 2006). assumed that equations (9) and (10) can be used to evaluate η R not only from local physical quantities at a site but also from the global quantities that are the averages of local ones at all sites in an area; for example, the 5 5 km square covering the drilled site. The inferred τ-δ function represents the effects on ruptures caused by global quantities of fault rocks in the study area. Hence, the second way is that η R is evaluated directly from the inferred τ-δ function. First, the first way is considered. As mentioned previously, it is not easy to evaluate η R at the drilled site because E s and the stress drop cannot be measured from seismograms at such a small area. An alternative approach must be taken into account. Kanamori (2004) approximated the radiation efficiency using the following formula on the basis of grain size and physical properties at a site of the slip zone: η 0 R 1= 1 6λG c T=D =μe R d ; (11) where λ is the correction for grain roughness, G c is specific fracture energy, T=D is the ratio of the slip thickness (T) to the total displacement (D), e R is the scaled energy, and d is the average grain size. Ma et al. (2006) applied equation (11) to estimate the radiation efficiency at the drilled site. The values of G c, T=D, and d measured from the TCDP by Ma et al. (2006) are G c 1 J=m 2, T=D (due to T 12 cm and D 300 m), and d 1: m. The common value of μ for crustal rocks is Pa. The value of e R for the Chi-Chi earthquake evaluated by Venkataraman and Kanamori (2004) is The value of λ ranges in general from 5 to 22 (see Wilson et al., 2005). Ma et al. (2006) selected λ 6:6 for calculations. Consequently, the value of η 0 R estimated by them is This value is close to η R 0:8 for the whole fault plane of the Chi-Chi earthquake estimated by Venkataraman and Kanamori (2004) from teleseismic data and larger than η R 0:67 for the northern fault plane evaluated by Wang (2006) from local seismograms. For comparison, in this study the upper bound of λ, that is, 22, is also taken into account. This leads to η 0 R 0:68, which is close to Wang s η R 0:67. Although the maximum displacement, δ max, at the study site reported by Ma et al. (2006) is 8.3 m, Figure 5 shows δ max 10:7 m below which thermal pressurization plays a significant role on controlling rupture as mentioned previously. Hence, δ max 10:7 m is taken in this study. In Figure 3, η R 0:88 and 0.68 are displayed with two dotted lines, and δ max 10:7 m and 12.3 m are shown by dashed lines. It can be seen from Figure 3 that there are two intersecting points for each end-member model. The functions of η R versus δ, which can pass through the individual intersecting points are depicted with broad solid curves. The values of δ c are 1.2 m for η R 0:88 and 2.5 m for η R 0:68, and those of L are 0.5 m for η R 0:88 and 10.0 m for η R 0:68. The τ-δ functions associated with these parameters are plotted in Figure 7a for λ 6:6 and Figure 7b for λ 22. In Figure 7, the solid and dashed curves represent the AUD and SOP models, respectively. The value of τ is larger for the AUD model than for the SOP model at small δ and opposite at large δ. For the two end-member models, τ is smaller for λ 6:6 than for λ 22. The plots of the two end-member models for λ 6:6 cannot explain the inferred τ-δ function. On the other hand, the two end-member models with λ 22 are more capable of describing the inferred τ-δ function, at least for its pattern of variation, than those with λ 6:6. Considering the pattern of the function as shown in Figure 7b, the plot of the AUD model is more able to describe the general trend of the inferred τ-δ function than the SOP model. In addition to the pattern, we can also consider the value of apparent characteristic slip distance D c. In Figure 7, the inferred τ-δ function leads to D c 10 m. Figure 7b shows that the SOP model has a larger value of D c 10 m than the AUD model ( 8 m). This indicates that the SOP model provides a more acceptable value of D c than the AUD model. However, the inferred value of τ at δ 10:7 m is close to the modeled value from the AUD model and much smaller than that from the SOP model. This means that the stress drop can be more reliably estimated from the AUD model than from the SOP model. Hence, the comparisons for the three properties between the two end-member models suggest that the AUD model is better than the SOP model. However, the modeled value of τ is smaller than the inferred one, and for the inferred τ-δ function, τ does not remarkably drop at small δ as expected by the two end-member models. In addition, in the latter stage the inferred τ-δ function is far away from the modeled function from the SOP model. There are five possible reasons to result in the differences. First, the inferred τ-δ function could not be completely correctly inversed from seismograms due to some uncertainties caused by data and inversion technique. Second, Guatteri and Spudich (2000) found that two rupture models having different strength excesses and slip-weakening distances were inversed from the same waveforms at periods that were one third of the rise time. This indicates that the wavelengths of seismograms control the resolution of

8 2300 J.-H. Wang Figure 7. The plots of shear stress versus slip: (a) for δ c 1:2 m and L 0:5 m when λ 6:6 and (b) for δ c 0:5 m and L 10:0 m when λ 22:0 (Thin solid line for the shear stress slip function at 5 5 km square around the drilled site inferred by Ma and Mikumo (unpublished manuscript, 2008) from seismic data. Solid line for the adiabatic-undrained-deformation model. Dashed line for the slip-on-a-plane model). detecting the earthquake rupture processes and source parameters. The wavelengths of seismograms used by Ma and Mikumo (unpublished manuscript, 2008) to inverse the τ-δ function of a 5 5 km square are much longer than the dimension of the drilled site. Hence, the resolution using their results to detect the physical quantities at the drilled site is very low, thus causing the difference between the inferred and modeled τ-δ functions. Third, the inferred τ-δ function was obtained for a 5 5 km square, while the modeled function is made only for the small drilled site. The values of physical and chemical parameters measured from the 12 cm core samples can only show the properties at the drilled site whose area is a very small. The τ-δ function modeled on the basis of those measured values represents merely the behavior at the drilled site. On the other hand, the τ-δ function inferred from seismograms shows the average behavior over the 5 5 km square that covers the drilled site and is much larger. The drilled site made a contribution to the average behavior of the 5 5 km square. However, there is a difference between the behavior of the square and that of the drilled site because the physical properties on a fault are spatially inhomogeneous. Hence, the modeled τ-δ function cannot be the representative of the square, thus causing the difference. Fourth, now only the one-dimensional thermal pressurization process is considered, while the real fault zone is in fact two-dimensional. Fifth, thermal pressurization might not be the unique mechanism in controlling the rupture of the Chi-Chi earthquake, and others must be taken into account. Although the two end-member models can only partly describe the observations, results still suggest that thermal pressurization would be a significant mechanism in controlling the variation in shear stress versus slip and would thus affect the radiation efficiency. Considering the second way, the values of E s, E g, and η R are evaluated from the inferred τ-δ function displayed in Figure 5. Because τ increases with δ when δ > 10:7 m, E s and E g are evaluated for two cases. For the first case, the values of E s and E g are evaluated from the function in the range of δ from 0 to 10.7 m, excluding the increasing part, and E s E g is 10:7 τ o τ m =2. The results are E s E g 89:1 J=m 2, E s 41:1 J=m 2, and E g 48:0 J=m 2, thus leading to η R 0:46. For the second case, the values of E s and E g are evaluated from the function in the range of δ from 0 to 12.3 m, including the increasing part, and E s E g is 12:3 τ o τ f =2. The results are E s E g 97:6 J=m 2, E s 47:8 J=m 2, and E g 49:8 J=m 2, thus leading to η R 0:49. The value of η R evaluated for the second case is only slightly higher than that for the first case. This indicates that the energies generated when δ > 10:7 m were much smaller than those for the whole rupture processes. The present two values of η R are both smaller than those evaluated from the first way. This again suggests spatial inhomogeneity of physical quantities on the square. Based on the two end-member models of thermal pressurization, the modeled curves of η R versus δ associated with the two cases are displayed in Figure 3. The results

9 Effect of Thermal Pressurization on Radiation Efficiency 2301 are for the AUD model, δ c 3:3 m for δ max 10:7 m and δ c 3:5 m for δ max 12:3 m and for the SOP model, L 40:0 m for δ max 10:7 m and L 30:0 m for δ max 12:3 m. The values of δ c and L evaluated from the second way are higher than those from the first way. The value of δ c for δ max 10:7 m is slightly smaller than that for δ max 12:3 m. The value of L for δ max 10:7 m is larger than that for δ max 12:3 m. The modeled τ-δ functions for the two cases are displayed in Figure 8: (a) for δ max 10:7 m and (b) for δ max 12:3 m. The solid and dashed lines represent the modeled τ-δ functions from the AUD and SOP models, respectively. Figure 8a shows that the AUD model can interpret the inferred τ-δ function due to the following reasons: (1) the solid line does not depart too much from the thin solid line, (2) the patterns of the two lines are similar, and (3) the inferred and modeled stress drops are almost equal. Clearly, the modeled τ-δ function for the two cases cannot interpret the increasing part of the inferred τ-δ function when δ > 10:7 m. Nevertheless, the difference between the two modeled τ-δ functions for δ > 10:7 m and δ max 12:3 m from the AUD model is very small. This is mainly due to a small difference between the values of η R for the two cases. Although other mechanisms could play a significant role in controlling earthquake rupture when δ > 10:7 m as suggested previously, they only made a small effect on the whole rupture processes. On the other hand, Figure 8b displays that most of the modeled values of shear stresses from the SOP model are larger than the inferred ones and the dashed line is far away from the thin solid line. The stress drop calculated from the SOP model is smaller than one half of the inferred one. The values of D c of the modeled τ-δ functions for the AUD and SOP models are the same and almost equal to δ max. Comparisons between the results from the two end-member models lead to a conclusion that the AUD model is more appropriate than the SOP model to control the τ-δ function and thus affect the radiation efficiency. In addition, a comparison between Figures 7 and 8 suggests that the second way of evaluating η R is better than the first one. Figure 8 also shows that the values of τ calculated from the AUD model are almost lower than the inferred ones, and those from the SOP model are almost higher than the inferred ones. This means that during faulting the inferred shear stresses on the fault plane within the 5 5 km square were slightly higher than those predicted by the AUD model and much lower than those expected from the SOP model. The first and second reasons to cause the difference between the inferred and modeled τ-δ functions for the first way as mentioned previously would also be valid for the second way. However, there might be one more reason. For the AUD model, frictional heat generated during faulting is totally kept in the slip zone due to the request of the adiabatic process for this model. This could cause melting or partial melting of fault rocks. Spray (1993) proposed that such melting or partial melting could lubricate the fault plane and thus reduce the shear stresses. On the other hand, for the SOP model frictional heat is totally transferred outward from the slip zone after the fault breaks. This is unable to yield Figure 8. The plots of shear stress versus slip: (a) for δ c 3:3 m and L 40:0 m when δ max 10:7 m and (b) for δ c 3:5 m and L 30:0 m when δ max 12:3 m. (Thin solid line for the shear stress slip function at the 5 5 km square around the drilled site inferred by Ma and Mikumo (unpublished manuscript, 2008) from seismic data. Solid line for the AUD model. Dashed line for the SOP model).

10 2302 J.-H. Wang melting or partial melting of fault rocks and thus cannot remarkably decrease the shear stress. Figure 8 shows that a small percentage of frictional heat would be transferred outwards from the study square during faulting, and most of the frictional heat was kept in the slip zone. Although this reduced the shear stresses within the 5 5 km square on the fault plane, the amount of reduction is smaller than that predicted from the AUD model. In spite of the existence of possible uncertainties of the inferred τ-δ function, I propose that a model modified from the AUD model by including a small amount of loss of frictional heat and/or pore fluids from the slip zone during faulting could elucidate the function. Although it is not the main goal of this study to examine whether the inferred τ-δ function is acceptable or not, I discuss this issue briefly. As mentioned by Guatteri and Spudich (2000), different rupture models with different τ-δ functions that have different strength excesses and slip-weakening distances D c (denoted by d c in their paper) can lead to the same waveforms. This means that the inversion solution of the τ-δ function is not unique due to low resolution when longerperiod waveforms are used. The stress drop and D c are two important parameters in controlling the τ-δ function. From the fifth and sixth figures in Guatteri and Spudich (2000), we can see that they used short D c and long D c models to simulate waveforms. Similar simulated waveforms lead to almost equal values of E s generated from the two rupture models. Under a fixed stress drop, the rupture model or the τ-δ function can be uniquely determined when D c is definitely evaluated. Estimates of D c could be affected by the source time function (Piatanesi et al., 2004; Tinti et al., 2005) and filtration (Spudich and Guatteri, 2004). Numerous ways (e.g., Bizzarri and Cocco, 2003; Wang, 2006) have been suggested for reliably estimating D c. Reliable determination of D c is out of the scope of this study and will not described further. The fifth figure in Guatteri and Spudich (2000) shows that the two rupture models with different values of D c result in different values of E g. This makes the values of η R different for the two models even though they have the same stress drop. This result is consistent with my basic ideal as shown in Figures 7 and 8 that η R is a significant parameter in controlling the τ-δ functions. The rupture model or the τ-δ function can be uniquely determined when η R is definitely evaluated. In principle, the value of η R can be evaluated from an independent way for which the τ-δ function is not given. For example, for a mode-3 crack η R is a function of v R =β (see Venkataraman and Kanamori, 2004) η R 1 1 v R =β = 1 v R =β 1=2 ; (12) where v R and β respectively, are the rupture and S-wave velocities. From equation (12), η R can be evaluated when v R =β is given. On the other hand, v R =β can also be evaluated when η R is known. Hence, on the fault plane at the drilled site the values of v R =β are 0.97 and 0.96, respectively, from η R 0:88 and 0.86, while over the 5 5 km square covering the drilled site v R =β is 0.55 from η R 0:46. Ma et al. (2001) inferred that average v R =β on the northern segment of the Chelungpu fault plane is The values of v R =β evaluated from core data are higher than the average while that from the inferred τ-δ function is lower than the average. The value of v R =β at a site or on a small square is not necessarily equal to the average over a large fault plane because the physical properties of materials are not homogeneous. However, at present it is hard to say which is better than the other. Because there is only an inferred τ-δ function on the 5 5 km square covering the drilled site, we cannot directly confirm if such a function is completely correct or not. Nevertheless, indirect evidence can be found to help us to examine this problem. The parameter δ c of the AUD model is similar to D c of the one-variable, rate- and state-dependent friction law because δ c and D c are the characteristic slip displacements of individual models. But, the parameter L of the SOP model is dissimilar to D c because the SOP model has no characteristic slip displacement (Rice, 2006). Wang (2006) inferred that average D cn on the northern segment of the Chelungpu fault must be between 1.8 and 3.7 m. From near-fault seismograms Mori (2005) estimated D cn 2:3 m. As mentioned previously, δ c is 3.3 m for δ max 10:7 m, and 3.5 m for δ max 12:3 m. The two values of δ c are within the range of D c inferred by Wang (2006) and slightly larger than that estimated by Mori (2005). This suggests that the inferred τ-δ function is acceptable. In addition, a model that is modified from the AUD model by including a small amount of loss of frictional heat and/ or pore fluids from the slip zone during faulting is assumed to interpret the τ-δ function. From modeling of history of frictional heating on the fault plane at the drilled site on the basis of observed data from core samples, I (unpublished manuscript, 2008) found that during faulting most of the heat was retained within the slip zone, and only little heat was conducted outside the zone. My result is consistent with the assumption of this study. This is also indirect evidence to suggest that the inferred τ-δ function is acceptable. Conclusions From two end-member models, the shear stress, τ, decreases with increasing slip, δ. The decreasing rate of τ with δ is first high and then low, and it is higher for the AUD model than for the SOP model. At a certain δ, τ increases with δ c and L. The radiation efficiency, η R, is strongly affected by the variation of shear stress with slip. Thermal pressurization is considered to be a significant mechanism in controlling such a variation, thus influencing the radiation efficiency. In this study, the formula of radiation efficiency as a function of slip on the basis of two end-member models of thermal pressurization are derived. The parameters δ c and L control the shear stress slip function, thus affecting the radiation efficiency. Modeled results show an increase in η R with slip. When δ c L, η R is higher for the AUD model than for

11 Effect of Thermal Pressurization on Radiation Efficiency 2303 the SOP model. For the two end-member models, η R increases with decreasing δ c (or L ). The increasing rate of η R with δ is high at small δ and low at large δ. This indicates that η R varies very much with the size of an earthquake for small events and is almost constant for large events. For a large earthquake, seismic radiation could be stronger in the early stage of rupture than in the latter stage. For the 1999 M s 7.6 Chi-Chi, Taiwan, earthquake, two ways are proposed to evaluate η R : one from the physical quantities of fault rocks measured from the core samples obtained from the TCDP and the other directly from the τ-δ function of a 5 5 km square covering the drilled site inferred by Ma and Mikumo (unpublished manuscript, 2008) from seismograms. From the value of η R, the modeled τ-δ functions are calculated on the basis of the AUD and SOP models. For the first way, the modeled τ-δ functions for the two end-member models with λ 6:6 cannot well explain the inferred τ-δ function. When λ 22, the AUD model is more appropriate than the SOP model in controlling the τ-δ function because the result from the AUD model is able to interpret the pattern of variation in inferred shear stress with slip. For the second way, the modeled τ-δ function calculated from the AUD model can explain the inferred τ-δ function well yet not from the SOP model. Results suggest that thermal pressurization is a significant mechanism in controlling the variation in shear stress and thus in affecting the radiation efficiency. In spite of the existence of possible uncertainties of the inferred τ-δ function, the proposed model to interpret such a function should be a modified one from the AUD model by including a small amount of loss of frictional heat and/ or pore fluids from the slip zone during faulting. In addition, the second way of evaluating η R is better than the first one. Data and Resources All data used in this article came from published sources listed in the references, core data from Ma et al. (2006), and the inferred shear stress slip function from Ma and Mikumo (unpublished manuscript, 2008). Acknowledgments The author would like to thank associate editor of BSSA Michel Bouchon, Andrea Bizzarri, and an anonymous reviewer for their valuable comments and suggestions to improve this article. K.-F. Ma provided the unpublished shear stress slip function at a subfault around the drilled site inferred from seismograms. This study was financially supported by Academia Sinica and the National Sciences Council under Grant No. NSC M MY3. References Bizzarri, A., and M. Cocco (2003). Slip-weakening behavior during the propagation of dynamic ruptures obeying rate-and state-dependent friction laws, J. Geophys. Res. 108, no. B8, 2373, doi /2002JB Bizzarri, A., and M. Cocco (2005). 3D dynamic simulations of spontaneous rupture propagation governed by different constitutive laws with rake rotation allowed, Ann. Geophys. 48, Bizzarri, A., and M. Cocco (2006a). Comment on Earthquake cycles and physical modeling of the process leading up to a large earthquake, Earth Planets Space 58, Bizzarri, A., and M. Cocco (2006b). A thermal pressurization model for the spontaneous dynamic rupture propagation on a three-dimensional fault: 1. Methodological approach, J. Geophys. Res. 111, B05303, doi /2005JB Bizzarri, A., and M. Cocco (2006c). A thermal pressurization model for the spontaneous dynamic rupture propagation on a three-dimensional fault: 2. Traction evolution and dynamic parameters, J. Geophys. Res. 111, B05304, doi /2005JB Brodsky, E. E., and H. Kanamori (2001). Elastohydrodynamic lubrication of faults, J. Geophys. Res. 106, Fialko, Y. A. (2004). Temperature fields generated by the elastodynamic propagation of shear cracks in the Earth, J. Geophys. Res. 109, B01303, doi /2003JB Goldsby, D. L., and T. E. Tullis (2002). Low frictional strength of quartz rocks at subseismic slip rates, Geophys. Res. Lett. 29, no. 17, 1844, doi /2002GL Guatteri, M., and P. Spudich (2000). What can strong-motion data tell us about slip-weakening fault-friction laws? Bull. Seismol. Soc. Am. 90, no. 1, Hung, J.-H., Y.-H. Wu, E.-C. Yeh,, and TCDP Scientific Party (2007). Subsurface structure, physical properties, and fault zone characteristics in the scientific drill holes of Taiwan Chelungpu-fault Drilling Project, Terr. Atmos. Ocean. Sci. 18, Ida, Y. (1972). Cohesive force across the tip of a longitudinal-shear crack and Griffith s specific surface energy, J. Geophys. Res. 77, no. 20, Kanamori, H. (2004). The diversity of the physics of earthquakes, Proc. Jpn. Acad. 80, Kanamori, H., and E. E. Brodsky (2004). The physics of earthquakes, Rept. Prog. Phys. 67, Kanamori, H., and T. H. Heaton (2000). Microscopic and macroscopic physics of earthquakes, in Geocomplexity and Physics of Earthquakes, J. B. Rundle, D. L. Turcotte, and W. Klein (Editors), American Geophysical Monograph 120,, Ma, K.-F., C.-T. Lee, Y.-B. Tsai, T.-C. Shin, and J. Mori (1999). The Chi- Chi, Taiwan earthquake: Large surface displacements on an inland thrust fault, in Eos, Transaction, American Geophysical Monograph 80, Ma, K.-F., J. Mori, S.-J. Lee, and S.-B. Yu (2001). Spatial and temporal distribution of slip for the 1999, Chi-Chi, Taiwan, earthquake, Bull. Seismol. Soc. Am. 91, Ma, K.-F., S.-R. Song, H. Tanaka, C.-Y. Wang, J.-H. Hung, Y.-B. Tsai, J. Mori, Y.-F. Song, E.-C. Yeh, H. Sone, L.-W. Kuo, and H.-Y. Wu (2006). Slip zone and energetics of a large earthquake from the Taiwan Chelungpu-fault Drilling Project (TCDP), Nature 444, Marone, C. (1998). Laboratory-derived friction laws and their application to seismic faulting, Annu. Rev. Earth Planet. Sci. 26, Mori, J. (2005). Using near-field seismograms to estimate the slip weakening distance, Eos, Trans. AGU, Suppl. 86, no. 52, F1420. Ohnaka, M. (2004). Earthquake cycles and physical modeling of the process leading up to a large earthquake, Earth Planets Space 56, Piatanesi, A., E. Tinti, M. Cocco, and E. Fukuyama (2004). The dependence of traction evolution on the earthquake source time function adopted in kinematic rupture models, Geophys. Res. Lett. 31, doi / 2003GL Rice, J. R. (2006). Heating and weakening of faults during earthquake slip, J. Geophys. Res. 111, B05311, doi /2005JB Ruina, A. (1983). Slip instability and state variable friction laws, J. Geophys. Res. 88, Shin, T.-C. (2000). Some seismological aspects of the 1999 Chi-Chi earthquake in Taiwan, Terr. Atmos. Ocean. Sci. 11,