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Fig. 3. Distribution of damage zones in the database as a function of fault throw, distinguishing between hanging wall and footwall data. In this study we handle footwall and hanging wall data sepa- rately. Ideally these should be considered together, however, both footwall and hanging wall damage zones are rarely exposed for individual faults. Hence, the expression “damage zone width”, as used here, corresponds to what has been called the “damage zone half width” by some authors (e.g., Hesthammer and Fossen, 2001; Shipton et al., 2006; Fossen et al., 2007; Fossen and Bale, 2007; Lockner et al., 2009; Powers and Jordan, 2010) and thus half the damage zone discussed by others (e.g., Shipton and Cowie, 2001;

Figure 3 Distribution of damage zones in the database as a function of fault throw, distinguishing between hanging wall and footwall data. In this study we handle footwall and hanging wall data sepa- rately. Ideally these should be considered together, however, both footwall and hanging wall damage zones are rarely exposed for individual faults. Hence, the expression “damage zone width”, as used here, corresponds to what has been called the “damage zone half width” by some authors (e.g., Hesthammer and Fossen, 2001; Shipton et al., 2006; Fossen et al., 2007; Fossen and Bale, 2007; Lockner et al., 2009; Powers and Jordan, 2010) and thus half the damage zone discussed by others (e.g., Shipton and Cowie, 2001;

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Fig. 2. Illustration of extensional fault in porous sandstone succession, showing a principal slip surface accommodating nearly all displacement and deformation struc- tures (deformation bands and fractures) in footwall and hanging wall damage zones. Wf = width of the footwall damage zone; Wh = width of the hanging wall damage zone; T = throw. (Aydin and Johnson, 1978; Antonellini et al., 1994; Shipton and Cowie, 2001; Johansen and Fossen, 2008). Fractures that may form in damage zones due to late fault reactivation, typically after substantial lithification and loss of porosity, are not considered in this work.
Fig. 2. Illustration of extensional fault in porous sandstone succession, showing a principal slip surface accommodating nearly all displacement and deformation struc- tures (deformation bands and fractures) in footwall and hanging wall damage zones. Wf = width of the footwall damage zone; Wh = width of the hanging wall damage zone; T = throw. (Aydin and Johnson, 1978; Antonellini et al., 1994; Shipton and Cowie, 2001; Johansen and Fossen, 2008). Fractures that may form in damage zones due to late fault reactivation, typically after substantial lithification and loss of porosity, are not considered in this work.
Fig. 1. Illustrations of a) a damage zone in Utah (from Fossen and Bale, 2007) displaying two distinct deformation band sets, and b) deformation bands in porous sandstone from the Aleria basin of Corsica (France).
Fig. 1. Illustrations of a) a damage zone in Utah (from Fossen and Bale, 2007) displaying two distinct deformation band sets, and b) deformation bands in porous sandstone from the Aleria basin of Corsica (France).
Database overview. The number of scanlines is given for each throw encountered at a certain location. Bold numbers represent the total number of scanlines for the corre sponding location.
Database overview. The number of scanlines is given for each throw encountered at a certain location. Bold numbers represent the total number of scanlines for the corre sponding location.
Fig. 4. Example of outcrop with location of scanline from the quarry and mine system in Wadi el Hommur (Malha Formation, Nubian Sandstone), Sinai, Egypt. Deformation band frequencies along scanlines were recorded only in the footwall in this case (scree-covered slope in the hanging wall).
Fig. 4. Example of outcrop with location of scanline from the quarry and mine system in Wadi el Hommur (Malha Formation, Nubian Sandstone), Sinai, Egypt. Deformation band frequencies along scanlines were recorded only in the footwall in this case (scree-covered slope in the hanging wall).
Fig. 5. Frequency graph illustrating different ways of defining the damage zone width. W5min is the distance from the fault core in which the frequencies are consistently higher than 5 deformation bands per meter. W5max is the distance beyond which the frequencies are consistently lower than 5 deformation bands per meter. W5 is the distance corresponding to an average of 5 deformation bands per meter. W10 and W1 are the distances corresponding to an average of 10 and 1 deformation band(s) per meter, respectively. The example is taken in the footwall side of the fault with a 100 m throw situated in the Wadi el Hommur Mines (Egypt) (same fault as shown in Fig. 4).
Fig. 5. Frequency graph illustrating different ways of defining the damage zone width. W5min is the distance from the fault core in which the frequencies are consistently higher than 5 deformation bands per meter. W5max is the distance beyond which the frequencies are consistently lower than 5 deformation bands per meter. W5 is the distance corresponding to an average of 5 deformation bands per meter. W10 and W1 are the distances corresponding to an average of 10 and 1 deformation band(s) per meter, respectively. The example is taken in the footwall side of the fault with a 100 m throw situated in the Wadi el Hommur Mines (Egypt) (same fault as shown in Fig. 4).
Fig. 6. Damage zone width (W5) as a function of throw. The best fit (black solid line), the confidence and prediction belts have been calculated on the individual data point set. The geometric means have been calculated for different groups of throw using a logarithmic binning along the x-axis. The dotted trend is obtained using only the groups of throw represented by more than 4 damage zones and presenting a lognormal distribution of the widths. Each point is weighted by the variance of its group. In this study, W5 (see above) is used to define damage zone width. A plot of damage zone width versus fault throw reveals a positive but non-linear correlation. The best fit function established is a power-law relationship (Fig. 6). The confidence intervals and prediction belts have been calculated with 95% confidence. The confidence belt around the regression line indicates that, for a given throw, the true value of the width should be located within this belt, for a given data set. The prediction belt defines, for a given throw, the interval within which a new observation of width should be found if this new observation belongs to the same statistical population as the data set used to define the regression line. This belt is larger than the confidence belt because the variance asso- ciated to draw a new point is added to the variance of the existing data set. The high variability of damage zone width is emphasized by the prediction belt spanning more than two orders of magnitude along the y-axis.
Fig. 6. Damage zone width (W5) as a function of throw. The best fit (black solid line), the confidence and prediction belts have been calculated on the individual data point set. The geometric means have been calculated for different groups of throw using a logarithmic binning along the x-axis. The dotted trend is obtained using only the groups of throw represented by more than 4 damage zones and presenting a lognormal distribution of the widths. Each point is weighted by the variance of its group. In this study, W5 (see above) is used to define damage zone width. A plot of damage zone width versus fault throw reveals a positive but non-linear correlation. The best fit function established is a power-law relationship (Fig. 6). The confidence intervals and prediction belts have been calculated with 95% confidence. The confidence belt around the regression line indicates that, for a given throw, the true value of the width should be located within this belt, for a given data set. The prediction belt defines, for a given throw, the interval within which a new observation of width should be found if this new observation belongs to the same statistical population as the data set used to define the regression line. This belt is larger than the confidence belt because the variance asso- ciated to draw a new point is added to the variance of the existing data set. The high variability of damage zone width is emphasized by the prediction belt spanning more than two orders of magnitude along the y-axis.
Fig. 7. Damage zone widths as a function of throw. The represented widths are W10, W5, W1. The trends are calculated using the geometric mean of the widths for the different throw groups.
Fig. 7. Damage zone widths as a function of throw. The represented widths are W10, W5, W1. The trends are calculated using the geometric mean of the widths for the different throw groups.
Fig. 8. A) Average density of deformation bands, D5, in the damage zone as a function of fault throw. B) Maximum density of deformation bands, Dmax, recorded in the damagi zones as a function of throw. In both figures, the gray shading outlines the variability of the density values by considering fluctuation corresponding to the standard deviatio1 around the mean value (the standard deviation for the average density D5 is 9.05 and for the maximum density: 31.49). The best fitting trends are given for information but have n¢ statistical significance since the fitting factors are very low. As opposed to functions that asymptotically approach zero or a constant value, the logarithmic decay function is not valid for an For instance, D5 is the average deformation density calculated up to the distance W5 (Fig. 5 for the definition of W5). The average density is a proxy for the degree of deformation in the damage zone either in the footwall or the hanging wall. Fig. 8A shows the average density D5 calculated for the available damage zones as a function of the fault throw. Plotted data points show no dependence be- tween the average density of deformation bands and the throw. The Decrease of the concentration of subsidiary structures away from a fault is acommon feature generally seen in most lithologies, including sandstones (Knott et al., 1996; Vermilye and Scholz, 1998; Beach et al., 1999; Fossen and Hesthammer, 2000; Hesthammer and Fossen, 2001; Shipton and Cowie, 2001; Du Bernard et al., 2002; Berg and Skar, 2005; Johansen and Fossen, 2008), lime- stones (Micarelli et al., 2003), mudstones (Savage and Brodsky, 2011), and crystalline rocks (Chester et al., 2004). In order to quantify this decrease in our database, different models have been tested: linear, exponential, logarithmic and power-law. Considering fitting factors R* above 0.5, the logarithmic model appears to be the best fitting model for 81% of the damage zones of the database. The logarithmic model is defined as Y=A + Lin(X), with Y representing frequency of deformation bands per meter and X the distance from the fault core (Fig. 9A).
Fig. 8. A) Average density of deformation bands, D5, in the damage zone as a function of fault throw. B) Maximum density of deformation bands, Dmax, recorded in the damagi zones as a function of throw. In both figures, the gray shading outlines the variability of the density values by considering fluctuation corresponding to the standard deviatio1 around the mean value (the standard deviation for the average density D5 is 9.05 and for the maximum density: 31.49). The best fitting trends are given for information but have n¢ statistical significance since the fitting factors are very low. As opposed to functions that asymptotically approach zero or a constant value, the logarithmic decay function is not valid for an For instance, D5 is the average deformation density calculated up to the distance W5 (Fig. 5 for the definition of W5). The average density is a proxy for the degree of deformation in the damage zone either in the footwall or the hanging wall. Fig. 8A shows the average density D5 calculated for the available damage zones as a function of the fault throw. Plotted data points show no dependence be- tween the average density of deformation bands and the throw. The Decrease of the concentration of subsidiary structures away from a fault is acommon feature generally seen in most lithologies, including sandstones (Knott et al., 1996; Vermilye and Scholz, 1998; Beach et al., 1999; Fossen and Hesthammer, 2000; Hesthammer and Fossen, 2001; Shipton and Cowie, 2001; Du Bernard et al., 2002; Berg and Skar, 2005; Johansen and Fossen, 2008), lime- stones (Micarelli et al., 2003), mudstones (Savage and Brodsky, 2011), and crystalline rocks (Chester et al., 2004). In order to quantify this decrease in our database, different models have been tested: linear, exponential, logarithmic and power-law. Considering fitting factors R* above 0.5, the logarithmic model appears to be the best fitting model for 81% of the damage zones of the database. The logarithmic model is defined as Y=A + Lin(X), with Y representing frequency of deformation bands per meter and X the distance from the fault core (Fig. 9A).
Fig. 9. A) Frequency graph for a fault (same fault as in Fig. 4) with 100 m throw. The decrease of the frequency of deformation bands away from the fault core is best represented by a logarithmic fit (Y =A + LIn(X), with Y the number of deformation bands per meter and X the distance from the fault core); B) Logarithmic fit for all the frequency graphs from the database; C) Normalization of all the frequency graphs: The frequency (i.e. number of deformation bands per meter) is divided by D5 and the distance is normalized by W5 (see 3.1 and 3.3 for the definition of W5 and D5); D) Aporma—Lnorma Plot: Anorma = (A + LIn(W5))/D5 and Lnorma = (A/D5 — Anorma)/In(1/W5). Each point corresponds to one damage zone
Fig. 9. A) Frequency graph for a fault (same fault as in Fig. 4) with 100 m throw. The decrease of the frequency of deformation bands away from the fault core is best represented by a logarithmic fit (Y =A + LIn(X), with Y the number of deformation bands per meter and X the distance from the fault core); B) Logarithmic fit for all the frequency graphs from the database; C) Normalization of all the frequency graphs: The frequency (i.e. number of deformation bands per meter) is divided by D5 and the distance is normalized by W5 (see 3.1 and 3.3 for the definition of W5 and D5); D) Aporma—Lnorma Plot: Anorma = (A + LIn(W5))/D5 and Lnorma = (A/D5 — Anorma)/In(1/W5). Each point corresponds to one damage zone
Fig. 11. Correlation dimension Dc as a function of the fault throw for 31 faults from our database. Each point corresponds to a damage zone. Dc appears independent of the throw, so the average Dc is 0.84 with a standard deviation equal to 0.06. Fig. 11 shows the computed correlation dimension Dc as a function of the fault throw. Calculations using the database indicate an average correlation dimension Dc of 0.837 with a standard
Fig. 11. Correlation dimension Dc as a function of the fault throw for 31 faults from our database. Each point corresponds to a damage zone. Dc appears independent of the throw, so the average Dc is 0.84 with a standard deviation equal to 0.06. Fig. 11 shows the computed correlation dimension Dc as a function of the fault throw. Calculations using the database indicate an average correlation dimension Dc of 0.837 with a standard
Fig. 10. A. Correlation analysis with correlation function as a description of the distance r between the deformation bands. If the organization of the deformations bands is fracta the correlation function presents a linear trend in a log—log diagram and its slope is the correlation dimension Dc. In order to define the range over which Dc can be calculated, the | points derivative curve of the correlation function is calculated. B. Theoretical examples of clustering characterized by the correlation dimension Dc and their correspondin: frequency graph. Dc = 0.85 is the distribution closest to the ones observed in nature.
Fig. 10. A. Correlation analysis with correlation function as a description of the distance r between the deformation bands. If the organization of the deformations bands is fracta the correlation function presents a linear trend in a log—log diagram and its slope is the correlation dimension Dc. In order to define the range over which Dc can be calculated, the | points derivative curve of the correlation function is calculated. B. Theoretical examples of clustering characterized by the correlation dimension Dc and their correspondin: frequency graph. Dc = 0.85 is the distribution closest to the ones observed in nature.
Fig. 12. Damage zone width (W5) in hanging wall and footwall as a function of the fault throw. Damage zones in the hanging wall are statistically wider than the damage zones in the footwall.
Fig. 12. Damage zone width (W5) in hanging wall and footwall as a function of the fault throw. Damage zones in the hanging wall are statistically wider than the damage zones in the footwall.
Fig. 13. A) Average density of deformation bands for non-folded and folded (rollover or dragfold) damage zones. Only damage zones associated with faults having a throw of 1-10 m and larger than 100 m have been analyzed. B) Average density D5 and maximum density Dmax as a function of the ratio Tfold/Wfold. C) Illustration of Tfold and Wfold as applied in the analysis of “B”. Tfold is the displacement accommodated by the fold and Wfold is the length other which the layer is considered as folded.
Fig. 13. A) Average density of deformation bands for non-folded and folded (rollover or dragfold) damage zones. Only damage zones associated with faults having a throw of 1-10 m and larger than 100 m have been analyzed. B) Average density D5 and maximum density Dmax as a function of the ratio Tfold/Wfold. C) Illustration of Tfold and Wfold as applied in the analysis of “B”. Tfold is the displacement accommodated by the fold and Wfold is the length other which the layer is considered as folded.
Fig. 14. Damage zone width W5 as a function of throw (T) for only Utah (squares) and Egypt (triangles) damage zones. The table gives the principal characteristics for the two groups of damage zones. Av. D5 is the average damage density of deformation bands in the damage zone calculated up to a width W5. Dc is the average correlation dimension.
Fig. 14. Damage zone width W5 as a function of throw (T) for only Utah (squares) and Egypt (triangles) damage zones. The table gives the principal characteristics for the two groups of damage zones. Av. D5 is the average damage density of deformation bands in the damage zone calculated up to a width W5. Dc is the average correlation dimension.
Fig. 16. Illustration of damage zone evolution with fault throw increase applying the numerical model established for deformation band distribution. At each step, the light colored bars of the frequency graphs correspond to the existing deformation band frequency at the previous step, and the dark colored bars correspond to the newly formed deformation bands. The throw is deduced from the input W5 using the following formula: W5 = 1.86-T°°? (defined for hanging wall damage zone in Fig. 12). The indicated throw corresponds to the damage zone width given on the left graph and AT defines the added throw to reach the damage zone width given in the right graph.
Fig. 16. Illustration of damage zone evolution with fault throw increase applying the numerical model established for deformation band distribution. At each step, the light colored bars of the frequency graphs correspond to the existing deformation band frequency at the previous step, and the dark colored bars correspond to the newly formed deformation bands. The throw is deduced from the input W5 using the following formula: W5 = 1.86-T°°? (defined for hanging wall damage zone in Fig. 12). The indicated throw corresponds to the damage zone width given on the left graph and AT defines the added throw to reach the damage zone width given in the right graph.
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