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A correlation radius estimate between in-panel faults and high-stress areas using Monte Carlo simulation and point process statistics
International Journal of Coal Geology 175 (2017) 51–62
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International Journal of Coal Geology journal homepage: www.elsevier.com/locate/coal
A correlation radius estimate between in-panel faults and high-stress areas using Monte Carlo simulation and point process statistics
MARK
Tongjun Chena,b,⁎, Tapan Mukerjic,d, Linming Doue a Key Laboratory of Coalbed Methane Resources and Reservoir Formation Process of the Ministry of Education, China University of Mining and Technology, Xuzhou, Jiangsu, China b School of Resources and Geosciences, China University of Mining and Technology, Xuzhou, Jiangsu, China c Department of Energy Resources Engineering, Stanford University, Stanford, CA, USA d Department of Geophysics, Stanford University, Stanford, CA, USA e School of Mines, China University of Mining and Technology, Xuzhou, Jiangsu, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Correlation association Quantitative estimate In-panel fault High-stress area Monte Carlo simulation Point process statistics
Since quantitative correlation association between in-panel faults and high-stress areas have not been well understood, we propose a workflow to quantitatively estimate this spatial association through Monte Carlo simulations and point process statistics using measured fault traces and tomographic seismic velocities as inputs. According to different distribution scenarios of fault traces and high-stress areas based on in situ characteristics, we build three different spatial statistical models: a no spatial correlation model, an anti-correlation model and a correlation model to analyze and compare with the observed data. By estimating and cross plotting RHA (Ratio of High-stress Areas over total area) and RFL (Ratio of included Fault-trace Length over total fault-trace length) pairs for Monte Carlo realizations of those models, we generate a template to estimate the correlation association between in-panel faults and high-stress areas for the study panel. After comparing the observed cross plots of RHA vs. RFL pairs with the template, we find that the in-panel faults and high-stress areas have positive correlation association and yield an estimate of correlation radius for the study panel. This result is in accordance with previous geological analysis. However, the estimated correlation radius can be affected by velocity artifacts and inaccurate interpreted faults. Considering the influence of velocity artifacts, we achieve a calibrated template to better estimate the correlation radius between in-panel faults and high-stress areas. This estimate could be a practical parameter to optimize mining methods and to minimize stress related rock failures.
1. Introduction With the massive mining of underground coal in China, dynamic failures such as gas outbursts and rock bursts have caused losses of several million dollars and several hundred lives. According to present researches, the presence of faults is a primary factor affecting gas outbursts and rock failures (Cao et al., 2001; Chen et al., 2015; Dou et al., 2012; Hanson et al., 2002; Wold et al., 2008; Zhai et al., 2016). As the results revealed in China and around the world, normal faults, strike slip faults as well as reverse faults can induce severe gas outbursts and rock failures (Cao et al., 2001; Shepherd et al., 1981; Wold et al., 2008; Zhai et al., 2016). Among all outbursts, most of the severe outbursts have been located in strongly deformed zones along the axes of faults. Because of the tectonic deformation and structural heterogeneity, stress and gas are concentrated in the narrow zone (Shepherd et al., 1981;
Wold et al., 2008). Apart from natural factors, some human factors, such as CO2 injection and underground mining activity, can also affect the reactivation of pre-existing faults and cause local stress field perturbations (Cappa and Rutqvist, 2011; Faulkner et al., 2010, 2006; Jeanne et al., 2014; Rinaldi et al., 2014). Nevertheless, fault zones and fault systems control the mechanics and flow properties of the crust. If pre-existing stress in a mining zone is high, the perturbation related to underground mining activities may have more chance to reactivate preexisting faults or generate new rock failures than the same perturbations in a less critically-stressed zone. In general, geomechanics, pore pressure, constitutive laws, and stress/strain boundary conditions govern the association between faults and local stress (Brady and Brown, 2013; Faulkner et al., 2010, 2006; Heap et al., 2010). Traditionally, researchers use Boundary Element Method, Finite Element Method and other numerical methods to
⁎ Corresponding author at: Key Laboratory of Coalbed Methane Resources and Reservoir Formation Process of the Ministry of Education, China University of Mining and Technology, Xuzhou, Jiangsu, China. E-mail addresses: [email protected] (T. Chen), [email protected] (T. Mukerji), [email protected] (L. Dou).
http://dx.doi.org/10.1016/j.coal.2017.04.001 Received 10 January 2017; Received in revised form 10 April 2017; Accepted 10 April 2017 Available online 11 April 2017 0166-5162/ © 2017 Elsevier B.V. All rights reserved.
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simulate the effect of faults on local stress field and to identify the correlation radius between faults and their associated high-stress zones. Those methods need some preconditions such as obtaining exact information of fault geometry, geomechanical properties, pore pressure and stress/strain boundary condition (Brady and Brown, 2013; Kattenhorn et al., 2000; Li et al., 2011). In many practical situations, those preconditions are incompletely accessible before coal extraction in underground coalmines. Instead of geomechanical modeling, active seismic survey and passive seismic monitoring are popular methods to delineate the fault distribution and estimate the velocity heterogeneity in underground coalmine panels (Chen et al., 2015; Dou et al., 2012; Ge, 2005; Hatherly, 2013; Mason, 1981; Maxwell and Young, 1995; Si et al., 2015; Zuo et al., 2009). The delineations of high-velocity areas and detected or measured in-panel faults are the most accessible data. Because stress and elastic wave velocity have positive correlation in the shallow crust, the monitored velocities can be used to locate high-stress areas and to estimate a relatively accurate stress level (Chen et al., 2015; Dou et al., 2012; Mason, 1981). If one can use those monitored velocities and detected/measured in-panel faults to better understand the spatial correlation and association between high-stress areas and inpanel faults, it could help coalmine operators to minimize stress related rock failures and gas outbursts. Stochastic simulation as a popular modeling tool has been used in geosciences for many years. In the coal industry, researchers use it to estimate coal-bed methane resources, to map the heterogeneities of coal quality, and to quantify the geological uncertainties and risks (Olea and Luppens, 2015; Tercan and Sohrabian, 2013; Zhou et al., 2012). Geologists use it to simulate 2D and 3D fracture and fault distributions, to analyze the association between faults and fractures, and to model the association between stress field and fractures (Kattenhorn et al., 2000; Noroozi et al., 2015; Viruete et al., 2001, 2003). Point process statistics (also called Boolean spatial process) have been widely used in mining, geology, forestry, and environmental sciences to analyze the geometrical patterns formed by objects that are distributed randomly in one-, two- or three-dimensional space (Connor and Hill, 1995; Illian et al., 2008; Moller and Waagepetersen, 2004; Russell et al., 2016). In order to estimate the association between in-panel faults and high-stress areas under incomplete information of underground coalmine panel, stochastic simulation and point process statistics are useful options. This paper, following the qualitative analysis of Chen et al. (2015), proposes a quantitative analysis workflow to estimate the spatial correlation association between in-panel faults and high-stress areas in an underground coalmine panel using monitored tomographic velocities from active seismic data, in situ measured faults, Monte Carlo simulation and point process statistics. The focus is on statistical spatial modeling of observed data, with the purpose of getting some practical estimates of spatial correlation association.
In the Yanshanian Orogeny (208 Ma) and the Himalayan Orogeny (65 Ma), the coal field of this panel experienced a series of tectonic movements. During roadway excavation and in-panel coal extraction, the operators of this mine identified in-panel faults and measured their characteristics daily. As a result, the 163L02C panel shows fifteen mapped faults (Fig. 3). Their strikes are shown in Fig. 4. In general, those faults belong to two main fault sets. One set consists of normal faults with NNW-SSE trend, and the other consists of normal faults with NNE-SSW trend. Because most faults are along NNW-SSE trend, this trend dominates the faulting characteristics of this panel. Considering the fault trends, most of them are near north direction with up to ± 30° deviation. This characteristic is a key input factor during fault simulation. The fault throws are small (0.7–1.6 m). We ignore the influence of fault throw during fault simulation. The fault lengths differ largely from fault to fault (Chen et al., 2015). The longest one is F3 (209 m), and the shortest one is F14 (17 m). We consider the influence of fault length during fault simulation. Chen et al. (2015) carried out active seismic tomography in this underground panel to measure the velocity distribution before coal extraction. Lithologically the formations are quite homogeneous and lithological variations are not a source of P-velocity heterogeneity in this case, as analyzed in Chen et al. (2015). Since P-velocity has a positive correlation with stress in the shallow crust, imaged P-velocities will indirectly depict the stress distribution of the panel (Chen et al., 2015; Dou et al., 2012; Mason, 1981; Maxwell and Young, 1995; Young and Maxwell, 1992). In order to achieve a categorical assessment, Chen et al. (2015) classified the imaged P-velocities into three stress levels (High, Medium and Low) using a threshold method as shown in Fig. 5. Almost all of the high-stress area are near faults except for area B. The spatial distribution of high-stress areas shows qualitative correlation with local stress field, previous mining activity and local fault distributions (Chen et al., 2015). After previous mining activities, the west and east sides of 163L02C panel have formed two mined out voids. Because of the existence of those voids, the coal bed and its main roof in this island panel have been detached from its adjacent strata. Therefore, the maximum horizontal principal stress (with near E-W direction) has been relieved from this panel (Chen et al., 2015). This may cause perturbations of the local stress field and the concentration of high stress around faults. Instead of a qualitative analysis as in Chen et al. (2015), we focus on the fault zone to analyze quantitatively the correlation association between measured faults and categorized highstress areas as shown in Fig. 5. The fault zone is defined as the rectangle with dashed lines as most of the in-panel fault traces and high-stress areas are within this rectangle.
2. General geology
Geologically, the existence of faults will be accompanied with the heterogeneity distribution of stress as shown in Fig. 5. A deterministic estimate between in-panel faults and their related high-stress areas will be convenient and practical for coalmine operators to optimize mining methods and minimize stress related rock failures. In this section, we propose a feasible workflow to quantitatively analyze the correlation association with Monte Carlo simulation and point process statistics. The input data are simulated fault traces and high-stress areas referenced from the measured in situ characteristics. Because the coalbed is relatively flat and all faults found in the study panel are normal faults with limited dip angle and throw variations (Chen et al., 2015), we ignore the influences of fluctuations in the coalbed floor and small variations in fault's dip angle and throw. We use a 2D-panel model to approximate the true 3D panel. In addition, we ignore the damage zones around faults during simulation because the true damage zones in the 163L02C panel are nearly invisible. The concerned variables are fault strikes, fault lengths, and tomographic seismic velocities.
3. Estimates of correlation association
The panel used in this study is a longwall island panel of 163L02C from Jining3 coalmine. This mine is a key production colliery in Southwest Shandong coalfield that is located in Shandong province, eastern China as shown in Fig. 1. The target of this panel is No.3 coal (about 675 m in depth). Its lithology is anthracite with glass luster, and its thickness is 3.0–6.0 m. Except for the faulted areas, the coal-bed thickness is relatively uniform. Both the coal bed and its roof and floor belong to Lower Permian formation as shown in Fig. 2. The direct roof is thin and non-uniform silt sandstone; while the direct floor is thin and non-uniform mudstone. In contrast with the thin and non-uniform direct roof and floor, the main roof and floor are thick and uniform sandstone as shown in Fig. 2. Since they are more stiff, uniform and thick than the coal bed, generated seismic waves will likely refract from them for a source deployed in the coal bed. Therefore, this panel is suitable for the imaging using refraction tomography (Chen et al., 2015). 52
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Fig. 1. The location of Jining3 coalmine.
Fig. 4. Rose map of faults of 163L02C panel. The radius indicates fault lengths, and the angles imply fault strikes. Every solid dot corresponds to an individual fault.
simulated fault lengths by a uniform distribution between 30 m and 200 m, and restrict the fault strikes between − 30° and 30°. We also simulate spatially expanding zones of high-stress areas, starting with 1% of the grid cells. The locations of the initial seed cells with respect to the location of the simulated faults are selected according to three different spatial models described in the next section: a) random, b) anti-correlation, and c) correlation. The initial seed cells are dilated with a radius r at each iteration. The Monte Carlo approach can give stochastic results in the form of distributions. It is more convenient and practical for mine operators to have a useable deterministic estimate to optimize mining methods and to minimize the stress related underground rock failures and gas outbursts. Hence we assimilate and synthesize the Monte Carlo results
Fig. 2. Stratigraphic column around the No. 3 coal, edited from Chen et al. (2015). In contrast to the roof and floor, the No. 3 coal has a characteristically low seismic velocity.
3.1. Simulation and statistical procedures The Monte Carlo procedure includes three main steps: generating in-panel faults, generating high-stress areas, and statistically analyzing the correlation pattern of in-panel faults and high-stress areas. In order to analyze digitally, we discretize the studied fault zone into 1 m × 1 m grids with a total of 400 × 135 grid cells. Because the true number of mapped in-panel faults in the fault zone is eleven, we uniformly draw a random number of faults within the range of 11 ± 6. Referring to the true measured characteristics of in-panel faults (Fig. 4), we limit the
Fig. 3. Plan view map of 163L02C panel, edited from Chen et al. (2015). The length and width of this panel is 756 m and 135 m respectively. The red solid curves are fault traces and the gray labels are fault names and fault throws.
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Fig. 5. Categorical assessment of stress levels in 163L02C panel, edited from Chen et al. (2015). The gray areas are high-stress class (H), and the other areas are low-and-medium stress class (L & M). All high-stress areas are labeled as A, B, C, D and E respectively. The rectangle with green dashed lines defines the fault zone used in the simulations.
3.3. Anti-correlation model
to give statistically calibrated estimates. The spatial statistical analysis has five sub steps. As the high-stress zones are expanded spatially, at each dilation iteration (1) we calculate the Ratio of included Fault-trace Length over total fault-trace length (RFL). If a fault trace totally or partly overlaps with high-stress areas, the trace or the part of the trace will be treated as included fault trace. (2) Then, we count the Ratio of High-stress Areas over total area (RHA) of the fault zone. (3) After that, we dilate the high-stress areas with a given radius r and compute RHA(r) and RFL(r) with the dilated areas. (4) We repeat steps (1) ~ (3), dilating and expanding the high stress areas, until all fault traces are included in the dilated high-stress areas. (5) Finally, we plot RFL(r) against RHA(r) and fit a trend curve. In this way, one can quantify the spatial correlation pattern of in-panel faults and high-stress areas. As shown below, the RFL versus RHA plots show different patterns depending on the spatial correlation or non-correlation between faults and high-stress areas.
In comparison with the randomly uncorrelated model, we build an anti-correlation model to analyze the anti-correlation association between in-panel fault traces and high-stress areas. In this scenario, the high-stress areas tend to avoid fault traces in the fault zone. In general, the simulation procedure of this model is similar to the procedure of no spatial correlation model. At first, we randomly generate fault traces with a uniform distribution function as in the previous model. Then, we randomly generate high-stress areas with a uniform distribution function and avoid generating high-stress areas within the range of a certain anti-correlation radius from any fault trace. (In point process statistics, models similar to this are termed hard-core point process, e.g. Illian et al., 2008). The anti-correlation model needs an additional parameter of anti-correlation radius to control the minimum distance between fault traces and high-stress areas. Because no measured high-stress areas appear around the tips of the fault traces (Fig. 5), we ignore the tip influence during the simulation of high-stress areas, and only simulate high-stress cells over the fault zone avoiding the simulated fault traces by a distance given by the range of anti-correlation radius. The generating method of highstress cells follow the sequential procedure as below: (1) randomly generate a cell location in the fault zone with a uniform distribution function; reject if it falls within the anti-correlation radius of any fault trace; (2) assign the corresponding cell as high-stress cell if it is not a previously assigned high-stress cell; (3) repeat step (1) and (2) until the high-stress cells account for 1% fault zone cells. As an example, Fig. 8 shows one realization. Finally, as before, we dilate the high-stress areas, count and cross-plot RHA and RFL. Simulations are done for a set of anti-correlation radii (10 m, 20 m, 30 m, 40 m and 50 m) with 100 realizations for each anti-correlation radius to obtain the RFL-RHA cross plot as shown in Fig. 9. Comparing with Figs. 7 and 9 is obviously different. Almost all cross-plotted points in this figure move off the diagonal and bend toward the bottom-right corner. This phenomenon is a distinguishable characteristic of the anticorrelation model from the uncorrelated model. If in-panel fault traces and high-stress areas have anti-correlation association, the cross plot of RHA vs. RFL pairs will be located in the figure's lower right half. For a given anti-correlation radius, simulated points are monotonically increasing, beginning at (0, 0), bent in the middle, and ending at (1, 1). With increasing anti-correlation radius, the bent curvature in the middle increases gradually. In order to quantitatively estimate an anti-correlation radius, we fit every set of points for a fixed anti-correlation radius with different functional forms:a Weibull function,
3.2. No spatial correlation model In general, geological factors will have independent spatial distributions in underground coalmine if they do not have any association at all. If in-panel faults and high-stress areas do not have any spatial association, they will be independently distributed in the study panel. In order to analyze the pattern of cross plots between RFL and RHA under independent random condition, we build a no spatial correlation model to simulate in-panel faults and high-stress areas and to count their RFL and RHA. In the first step, we randomly generate in-panel fault traces with a uniform distribution function with the parameters described in the previous section. In the second step, we randomly generate initial high-stress areas with a uniform distribution function too, without any spatial correlation to the previously simulated fault traces. The method of generating high-stress cells follows the sequential procedure as below: (1) randomly generate a cell location in the fault zone with a uniform distribution function and assign it as high-stress cell; (2) randomly generate the next high-stress cell with a uniform distribution function avoiding the previously assigned high-stress cells; (3) repeat step (2) until the proportion of high-stress cells accounts for 1% of the total panel cells. Fig. 6(a) shows one example of this initial stage. Then we gradually dilate the high-stress areas with increasing radii (5 m, 10 m, 15 m…) until all fault traces have been included in the dilated high-stress areas. Fig. 6(b) shows one iteration of the dilation. With increasing dilation radius, the included fault-trace length will increase correspondingly until all fault traces fall into high-stress areas. At every dilation step, RHA and RFL are computed. Finally, Fig. 7 cross plots all pairs of RHA and RFL for 100 simulations. In this figure, almost all cross-plotted points are near and around the diagonal with very small deviation. The near diagonal characteristics of RHA vs. RFL pairs indicate that the in-panel faults and high-stress areas are distributed randomly with no spatial correlation in the study panel. In this study, we treat those RFL-versus-RHA characteristics as indicators of a spatially uncorrelated model.
f (x ) = abx (b −1) e−ax
b
(1)
a power-law function,
f (x ) = ax b a Gaussian function, 54
(2)
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Fig. 6. A realization of in-panel fault traces and simulated high-stress areas for the no spatial correlation model. Black lines are fault traces, and gray areas are high-stress areas. Highstress areas in (b) are dilated from (a) with a 5 m radius.
f (x ) = ae−(
x−b 2 c )
template to estimate the anti-correlation radius between in-panel fault traces and high-stress areas for the study panel.
(3)
and an exponential function x c
f (x ) = 1 − e−( b )
3.4. Correlation model (4) Instead of uncorrelated or anti-correlated association, fault traces and high-stress areas may have a positive spatial correlation association in the underground panel. In this scenario, high-stress areas will tend to appear preferentially near the fault traces rather than off the fault traces. We build a correlation model to simulate the characteristics of this scenario. At first, we randomly generate fault traces with a uniform
respectively. Among the equations, the coefficients a, b and c are used to adjust the shape of fitted curves. Since the Weibull function has the best R-square goodness of fit (> 0.93) with 95% confidence bounds, we use its fit result as the fitted trend of anti-correlation model as shown in Fig. 10(a). The fitted curves as shown in Fig. 10(b) can be treated as a
Fig. 7. Cross plot of RHA vs. RFL pairs for no spatial correlation model, for 100 simulations.
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Fig. 8. A realization of simulated in-panel fault traces and high-stress areas of anti-correlation model. Black lines are fault traces, and gray areas are high-stress areas. The anti-correlation radius used in this model is 20 m.
left half. With the decreasing of correlation radius, the symbols will bend toward the top-left corner gradually. Because all cross plots are monotonically increasing, beginning at (0, 0), bent in the middle, and ending at (1, 1), this phenomenon is much more obvious in the middle range. In order to quantitatively estimate a correlation radius, we fit every set of points for a fixed correlation radius with different functional forms:an inverse Weibull function,
distribution function as we did in the previous two cases. Then, we randomly generate initial high-stress seed areas with a uniform distribution function and keep only those high-stress areas within the range of correlation radius around the previously simulated fault traces. Similar to the anti-correlation model, the correlation model needs a correlation radius to control the maximum distance between in-panel fault traces and high-stress areas. The generating method of high-stress cells follow the sequential procedure as below: (1) randomly generate a cell location in the fault zone with a uniform distribution function; accept if it is at least within the correlation radius of any one fault trace; (2) assign the corresponding cell as high-stress cell if it is not a previously assigned high-stress cell; (3) repeat step (1) and (2) until the high-stress cells account for 1% fault zone cells. As an example, Fig. 11 shows a realization. Finally, as before, we dilate the initial highstress areas, count and cross-plot RHA and RFL, for different correlation radii (10 m, 20 m, 30 m, 40 m and 50 m), with 100 simulations for each correlation radius (Fig. 12a). Fig. 12(a) differs obviously from Figs. 7 and 9. All cross-plotted symbols in this figure move off the diagonal and bend toward the topleft corner. This phenomenon is a distinguishable characteristic of correlation model from uncorrelated model and anti-correlation model. If in-panel fault traces and high-stress areas have correlation association, the cross plot of RHA vs. RFL pairs will be located in figure's upper
f (x ) = 1 − abx (b −1) e−ax
b
(5)
an inverse Gaussian function,
f (x ) = 1 − ae−(
x−b 2 c )
(6)
a power-law function (Eq. (2)),and an exponential function (Eq. (4)).respectively. Since the exponential function has the best R-square goodness of fit (> 0.96) with 95% confidence bounds, we use its fit result as the fitted trend for the correlation model. The fitted curves as shown in Fig. 12(b) can be treated as a template to estimate the correlation radius between in-panel fault traces and high-stress areas for the study panel.
Fig. 9. Cross plot of RHA vs. RFL pairs for anti-correlation model. The larger the anti-correlation radius, the further are the points from the diagonal.
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Fig. 10. Fitted RHA-RFL cross plots with Weibull function. In (a), the anti-correlation radius of scattered points and fitted curve is 30 m; in (b), the labels on the fitted curves indicate the anti-correlation radii.
Fig. 11. A realization of in-panel fault traces and initial high-stress seed areas for the correlation model. Black lines are fault traces, and gray areas are high-stress areas. The correlation radius used in this model is 20 m.
artifacts. Those artifacts may come from the inaccurate picking of first breaks, limited seismic bandwidth, and the azimuthal limitations of observation. Because of the existence of those artifacts, the estimate of correlation radius may be different from the previous estimate. Consequently, we build a noisy correlation model to calibrate the influence of velocity artifacts. The modeling procedures of the noisy correlation model are similar to the procedures of correlation model. At first, we randomly generate fault traces with a uniform distribution function as we did for all scenarios. As in the correlation model, we generate high-stress areas distributed near the faults traces within the correlation radius. But in addition, to simulate noise artifacts a small fraction of ‘false’ high-stress areas are uniformly sprinkled throughout the fault zone irrespective of their distance from the fault traces. In the following simulations we assume this fraction to be 30% of the high-stress areas. As an example, Fig. 13 shows one realization. We carry out 100 simulations for each set of correlation radii (10 m, 20 m, 30 m, 40 m and 50 m) and fit every set of symbols for a fixed correlation radius with an exponential function (Eq. (4)) as we did in Section 3.4. The RHA-RFL cross plots and the fitted curves are shown in Fig. 14. The R-squares, goodness of fit, are > 0.96 with 95% confidence bounds. In order to analyze the influence of random artifacts on the estimate of correlation radius, we overlay the observed scatter points of the fault zone on the trend curves estimated from ‘noisy’ simulations. The solid points are along and around the trend curve of 20 m, indicating the correlation radius is estimated to be 20 m if tomographic velocities contain 30% random artifacts. In comparison with Fig. 12(b), the solid points and the 20 m trend curve have closer association. In sum, the actual correlation radius will be smaller than the estimate of correlation model from high-stress areas inferred from tomographic velocities containing random artifacts. The larger the percentage of random artifacts, greater is the overestimate because the random artifacts make the observed curve move closer to the diagonal. In the limit (when the tomographically interpreted high stress areas are completely unreliable
3.5. In situ estimates of correlation radius Through the simulations of Sections 3.2–3.4, we achieved three basic observations about the cross plot of RHA vs. RFL pairs. (1) If the cross-plotted points are below the diagonal, fault traces and high-stress areas will have anti-correlation spatial association. (2) If the crossplotted points are along the diagonal with a small deviation, fault traces and high-stress areas will have no spatial correlation association. (3) If the cross-plotted points are above the diagonal, fault traces and highstress areas will have positive spatial correlation association. If other factors remain unchanged, a further distance from the diagonal will imply a smaller correlation radius. Since we already have the templates to estimate the correlation radius or anti-correlation radius between in-panel fault traces and highstress areas for the study panel, we first count and cross plot the RHA and RFL of the fault zone in163L02C panel using the measured faults and estimated high-stress areas and overlay them over the template as shown in Fig. 12(b). Because the black points are far above the diagonal, this implies that the fault traces and high-stress areas in the fault zone have positive correlation association. This outcome is in accordance with the geological analysis of Chen et al. (2015). To estimate the correlation radius quantitatively, we see the observed black points being scattered between the 20 m curve and the 30 m curve. This implies that the fault traces and high-stress areas in the studied fault zone have a correlation radius between 20–30 m. 4. The influence of model inputs 4.1. Velocity artifacts The data we used to estimate the correlation association between high-stress areas and faults are the measured in-panel faults and subsurface tomographic seismic velocities as shown in Fig. 5. As an indirect measurement, the tomographic velocities may contain some 57
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Fig. 12. Cross plot of RHA vs. RFL pairs for correlation model (a) and fitted curves for RHA-RFL cross plots with an exponential function (b). In (b), every labeled curve is a fitted result of a set of single-colour symbols. The labels on the curves are the corresponding correlation radii. The solid points in (b) are observed RHA-RFL cross plot of the fault zone in Fig. 5.
4.2. Interpreted faults
and are just randomly distributed) the actual spatial correlation is lost and it appears (incorrectly) to be uncorrelated. Thus the estimate with velocity artifacts (provided it is not too large) provides an upper bound on the correlation radius. This maximum estimate can be a useful parameter to design the plan for the mine and to prevent occurrences of stress related rock failures and gas outbursts.
In comparison with the underground measured in-panel faults, interpreted faults from 3D seismic and/or underground seismic data are often more easily accessible for most practical applications. In such cases, the interpreted faults cannot be 100% correct. In this section, we analyze the influence of uncertainties in interpreted faults on the estimate of correlation radius between in-panel faults and high-stress areas.
Fig. 13. A realization of the noisy correlation model with a correlation radius of 20 m.
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Fig. 14. Cross plot of RHA vs. RFL pairs for noisy correlation model (a) and fitted curves for RHA-RFL cross plots with an exponential function (b). In (b), every labeled curve is a fitted result of a set of single-colour symbols. The labels on the curves are the corresponding correlation radii. The solid points in (b) are observed RHA-RFL cross plot of the fault zone in Fig. 5.
Fig. 15. A realization of in-panel fault traces and initial high-stress seed areas for the correlation model with − 90° ~ 90° fault strike distribution. Black lines are fault traces, and gray areas are high-stress areas. The correlation radius used in this model is 30 m.
30 m, 40 m and 50 m). As an example, Fig. 15 shows one realization. Unlike the previously simulated fault traces, the simulated fault traces in Fig. 15 have a much wider range of strikes as expected. The cross plot of RHA vs. RFL pairs for this new model are shown in Fig. 16. In comparison with the correlation model, the scatter points of this new model have a much larger deviation range. Fitting the scatter points of each set of correlation radii (10 m, 20 m, 30 m, 40 m and 50 m) with an exponential function (Eq. (4)) as we did in Section 3.4, the fitted R-
First of all, we analyze the influence of deviation in interpreted fault strike on the estimate. The modeling procedures are similar to the procedures of the correlation model. The only difference is the deviation range of simulated fault strikes. In this experiment, we expand the limit of the fault strikes to a radical range of − 90° ~ 90° rather than the previous range of − 30° ~ 30°, implying a large uncertainty in the fault interpretation. As we did in Section 3.4, we carry out 100 simulations for each set of correlation radii (10 m, 20 m, 59
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Fig. 16. Cross plot of RHA vs. RFL pairs for correlation model with different range of fault strikes.
smaller influence on the accuracy of the estimate compared to the uncertainties in fault strikes. In sum, both inaccurate fault strikes and lengths will affect the accuracy of the estimated correlation radius. In comparison with inaccurate fault lengths, inaccurate fault strikes may have a larger influence. If one hopes to estimate the correlation radius between inpanel faults and high-stress areas, it is crucial to provide good fault strike distribution.
squares of this new model are < 0.86 with 95% confidence bounds. This result is much worse than the correlation model, indicating inaccurate inputs of fault strikes have strong influence on the accuracy of the estimate. Fortunately, few true panel have this wide (− 90° to + 90°) of an uncertainty in fault strike distribution. Then, we analyze the influence of fault length deviation on the estimate. The modeling procedures are similar to the procedures of correlation model. The only difference is the deviation range of simulated fault lengths. In this experiment, we limit the fault lengths to a range of 100–200 m rather than the previous range of 30–200 m because small faults with short lengths are not interpretable from seismic data. As we did in Section 3.4, we carry out 100 simulations for each set of correlation radii (10 m, 20 m, 30 m, 40 m and 50 m). As an example, Fig. 17 shows one realization. Unlike the previous simulated fault traces, the simulated fault traces in Fig. 17 have a longer averaged fault trace. The cross plot of RHA vs. RFL pairs for this new model is shown in Fig. 18. In comparison with the correlation model, the scatter points of this new model have a little bit larger deviation range. By fitting the scatter points of each set of correlation radii (10 m, 20 m, 30 m, 40 m and 50 m) with an exponential function (Eq. (4)) as we did in Section 3.4, the fitted R-squares of this new model are about 0.94 with 95% confidence bounds. This result is only a little bit worse than the correlation model, indicating inaccurate inputs of fault lengths have
5. Conclusions In this paper, we presented a workflow to quantitatively estimate the correlation association between in-panel faults and high-stress areas in an underground coalmine panel using true measured in-panel faults, tomographic seismic velocities, Monte Carlo simulations and point process statistics. Through modeling, simulation and comparison, we can conclude: (1) The proposed workflow of this paper can produce a deterministic estimate of correlation radius rather than a qualitative description of correlation association between in-panel faults and high-stress areas. This estimate could be a practicable parameter for coalmine operators to optimize mining methods and to minimize stress
Fig. 17. A realization of in-panel fault traces and initial high-stress seed areas for the correlation model with 100–200 m fault lengths. Black lines are fault traces, and gray areas are highstress areas. The correlation radius used in this model is 30 m.
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Fig. 18. Cross plot of RHA vs. RFL pairs for correlation model with different range of fault lengths.
consortia sponsors.
related rock failures and gas outbursts. (2) In the fault zone of the 163L02C panel, Monte Carlo simulations show that high-stress areas have positive correlation with in-panel faults. This result is in accordance with the geological analysis of this panel (Chen et al., 2015). In this specific example, the estimated correlation radius is 20–30 m. If ones hope to estimate a template for their own application, ones should follow the workflow and generate their own template calibrated to the geology of the specific application area. (3) Velocity artifacts affect the estimate of correlation radius between in-panel faults and high-stress areas. If those artifacts are uniformly random, the correlation radius estimate using a template constructed with no noise will be an overestimation with increasing percentage of artifacts. Therefore, the estimate of correlation model is an upper bound estimate of correlation radius between in-panel faults and high-stress areas. (4) In comparison with the variation of fault lengths, too large a variation of fault strikes will decrease the reliability of fitted correlation radius, although this situation does not often appear in true underground panels. (5) The preconditions of this proposed workflow are a relative flat panel, small fault throws and small fault-throw deviations, and small lithologic variability. If these preconditions are not satisfied, the applicability of this proposed workflow is unknown. Conceptually the workflow can be easily expanded from a 2-D flat panel to a full 3-D volume. But incorporating large fault throws, complex faulting regimes and very heterogeneous lithologies are much more challenging. A further research is a compulsive option for more complex faulting regimes.
References Brady, B.H., Brown, E.T., 2013. Rock Mechanics: For Underground Mining. Springer Science & Business Media. Cao, Y.X., He, D.D., Glick, D.C., 2001. Coal and gas outbursts in footwalls of reverse faults. Int. J. Coal Geol. 48 (1–2), 47–63. Cappa, F., Rutqvist, J., 2011. Modeling of coupled deformation and permeability evolution during fault reactivation induced by deep underground injection of CO2. Int. J. Greenh. Gas Control 5 (2), 336–346. Chen, T., Wang, X., Mukerji, T., 2015. In situ identification of high vertical stress areas in an underground coal mine panel using seismic refraction tomography. Int. J. Coal Geol. 149, 55–66. Connor, C.B., Hill, B.E., 1995. Three nonhomogeneous Poisson models for the probability of basaltic volcanism: application to the Yucca Mountain region, Nevada. J. Geophys. Res. Solid Earth 100 (B6), 10107–10125. Dou, L., Chen, T., Gong, S., He, H., Zhang, S., 2012. Rockburst hazard determination by using computed tomography technology in deep workface. Saf. Sci. 50 (4), 736–740. Faulkner, D.R., Mitchell, T.M., Healy, D., Heap, M.J., 2006. Slip on ‘weak’ faults by the rotation of regional stress in the fracture damage zone. Nature 444 (7121), 922–925. Faulkner, D.R., Jackson, C.A.L., Lunn, R.J., Schlische, R.W., Shipton, Z.K., Wibberley, C.A.J., Withjack, M.O., 2010. A review of recent developments concerning the structure, mechanics and fluid flow properties of fault zones. J. Struct. Geol. 32 (11), 1557–1575. Ge, M., 2005. Efficient mine microseismic monitoring. Int. J. Coal Geol. 64 (1), 44–56. Hanson, D.R., Vandergrift, T.L., DeMarco, M.J., Hanna, K., 2002. Advanced techniques in site characterization and mining hazard detection for the underground coal industry. Int. J. Coal Geol. 50 (1), 275–301. Hatherly, P., 2013. Overview on the application of geophysics in coal mining. Int. J. Coal Geol. 114, 74–84. Heap, M.J., Faulkner, D.R., Meredith, P.G., Vinciguerra, S., 2010. Elastic moduli evolution and accompanying stress changes with increasing crack damage: implications for stress changes around fault zones and volcanoes during deformation. Geophys. J. Int. 183 (1), 225–236. Illian, J., Penttinen, A., Stoyan, H., Stoyan, D., 2008. Statistical Analysis and Modelling of Spatial Point Patterns. 534 John Wiley & Sons. Jeanne, P., Guglielmi, Y., Cappa, F., Rinaldi, A.P., Rutqvist, J., 2014. The effects of lateral property variations on fault-zone reactivation by fluid pressurization: application to CO2 pressurization effects within major and undetected fault zones. J. Struct. Geol. 62, 97–108. Kattenhorn, S.A., Aydin, A., Pollard, D.D., 2000. Joints at high angles to normal fault strike: an explanation using 3-D numerical models of fault-perturbed stress fields. J. Struct. Geol. 22 (1), 1–23. Li, L., Yang, T., Liang, Z., Zhu, W., Tang, C., 2011. Numerical investigation of groundwater outbursts near faults in underground coal mines. Int. J. Coal Geol. 85 (3–4), 276–288. Mason, I., 1981. Algebraic reconstruction of a two-dimensional velocity inhomogeneity in the High Hazles seam of Thoresby colliery. Geophysics 46 (3), 298–308. Maxwell, S., Young, R., 1995. A controlled in-situ investigation of the relationship
Acknowledgement Financial support for this work, provided by the National Natural Science Foundation of China (No. 41374140 and No. 41430317), Natural Science Foundation of Jiangsu Province (BK20130175) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), are gratefully acknowledged. We also acknowledge Stanford Center for Reservoir Forecasting (SCRF) and Stanford Rock Physics and Borehole Geophysics (SRB) 61
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Viruete, J.E., Carbonell, R., Jurado, M.J., Marti, D., Perez-Estaun, A., 2001. Twodimensional geostatistical modeling and prediction of the fracture system in the Albala Granitic Pluton, SW Iberian Massif, Spain. J. Struct. Geol. 23 (12), 2011–2023. Viruete, J.E., Carbonell, R., Marti, D., Perez-Estaun, A., 2003. 3-D stochastic modeling and simulation of fault zones in the Albala granitic pluton, SW Iberian Variscan massif. J. Struct. Geol. 25 (9), 1487–1506. Wold, M.B., Connell, L.D., Choi, S.K., 2008. The role of spatial variability in coal seam parameters on gas outburst behaviour during coal mining. Int. J. Coal Geol. 75 (1), 1–14. Young, R., Maxwell, S., 1992. Seismic characterization of a highly stressed rock mass using tomographic imaging and induced seismicity. J. Geophys. Res. Solid Earth 97 (B9), 12361–12373. Zhai, C., Xiang, X., Xu, J., Wu, S., 2016. The characteristics and main influencing factors affecting coal and gas outbursts in Chinese Pingdingshan mining region. Nat. Hazards 1–24. Zhou, F., Allinson, G., Wang, J., Sun, Q., Xiong, D., Cinar, Y., 2012. Stochastic modelling of coalbed methane resources: a case study in Southeast Qinshui Basin, China. Int. J. Coal Geol. 99, 16–26. Zuo, J.P., Peng, S.P., Li, Y.J., Chen, Z.H., Xie, H.P., 2009. Investigation of karst collapse based on 3-D seismic technique and DDA method at Xieqiao coal mine, China. Int. J. Coal Geol. 78 (4), 276–287.
between stress, velocity and induced seismicity. Geophys. Res. Lett. 22 (9), 1049–1052. Moller, J., Waagepetersen, R.P., 2004. Statistical Inference and Simulation for Spatial Point Processes. 298 Chapman and Hall/CRC. Noroozi, M., Kakaie, R., Jalali, S.M., 2015. 3D stochastic rock fracture modeling related to strike-slip faults. J. Min. Environ. 6 (2), 169–181. Olea, R.A., Luppens, J.A., 2015. Mapping of coal quality using stochastic simulation and isometric logratio transformation with an application to a Texas lignite. Int. J. Coal Geol. 152, 80–93. Rinaldi, A.P., Jeanne, P., Rutqvist, J., Cappa, F., Guglielmi, Y., 2014. Effects of fault-zone architecture on earthquake magnitude and gas leakage related to CO2 injection in a multi-layered sedimentary system. Greenh. Gases Sci. Technol. 4 (1), 99–120. Russell, J.C., Hanks, E.M., Haran, M., 2016. Dynamic models of animal movement with spatial point process interactions. J. Agric. Biol. Environ. Stat. 21 (1), 22–40. Shepherd, J., Rixon, L., Griffiths, L., 1981. Outbursts and geological structures in coal mines: a review. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 267–283 (Elsevier). Si, G., Durucan, S., Jamnikar, S., Lazar, J., Abraham, K., Korre, A., Shi, J.-Q., Zavšek, S., Mutke, G., Lurka, A., 2015. Seismic monitoring and analysis of excessive gas emissions in heterogeneous coal seams. Int. J. Coal Geol. 149, 41–54. Tercan, A.E., Sohrabian, B., 2013. Multivariate geostatistical simulation of coal quality data by independent components. Int. J. Coal Geol. 112, 53–66.
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International Journal of Coal Geology journal homepage: www.elsevier.com/locate/coal
A correlation radius estimate between in-panel faults and high-stress areas using Monte Carlo simulation and point process statistics
MARK
Tongjun Chena,b,⁎, Tapan Mukerjic,d, Linming Doue a Key Laboratory of Coalbed Methane Resources and Reservoir Formation Process of the Ministry of Education, China University of Mining and Technology, Xuzhou, Jiangsu, China b School of Resources and Geosciences, China University of Mining and Technology, Xuzhou, Jiangsu, China c Department of Energy Resources Engineering, Stanford University, Stanford, CA, USA d Department of Geophysics, Stanford University, Stanford, CA, USA e School of Mines, China University of Mining and Technology, Xuzhou, Jiangsu, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Correlation association Quantitative estimate In-panel fault High-stress area Monte Carlo simulation Point process statistics
Since quantitative correlation association between in-panel faults and high-stress areas have not been well understood, we propose a workflow to quantitatively estimate this spatial association through Monte Carlo simulations and point process statistics using measured fault traces and tomographic seismic velocities as inputs. According to different distribution scenarios of fault traces and high-stress areas based on in situ characteristics, we build three different spatial statistical models: a no spatial correlation model, an anti-correlation model and a correlation model to analyze and compare with the observed data. By estimating and cross plotting RHA (Ratio of High-stress Areas over total area) and RFL (Ratio of included Fault-trace Length over total fault-trace length) pairs for Monte Carlo realizations of those models, we generate a template to estimate the correlation association between in-panel faults and high-stress areas for the study panel. After comparing the observed cross plots of RHA vs. RFL pairs with the template, we find that the in-panel faults and high-stress areas have positive correlation association and yield an estimate of correlation radius for the study panel. This result is in accordance with previous geological analysis. However, the estimated correlation radius can be affected by velocity artifacts and inaccurate interpreted faults. Considering the influence of velocity artifacts, we achieve a calibrated template to better estimate the correlation radius between in-panel faults and high-stress areas. This estimate could be a practical parameter to optimize mining methods and to minimize stress related rock failures.
1. Introduction With the massive mining of underground coal in China, dynamic failures such as gas outbursts and rock bursts have caused losses of several million dollars and several hundred lives. According to present researches, the presence of faults is a primary factor affecting gas outbursts and rock failures (Cao et al., 2001; Chen et al., 2015; Dou et al., 2012; Hanson et al., 2002; Wold et al., 2008; Zhai et al., 2016). As the results revealed in China and around the world, normal faults, strike slip faults as well as reverse faults can induce severe gas outbursts and rock failures (Cao et al., 2001; Shepherd et al., 1981; Wold et al., 2008; Zhai et al., 2016). Among all outbursts, most of the severe outbursts have been located in strongly deformed zones along the axes of faults. Because of the tectonic deformation and structural heterogeneity, stress and gas are concentrated in the narrow zone (Shepherd et al., 1981;
Wold et al., 2008). Apart from natural factors, some human factors, such as CO2 injection and underground mining activity, can also affect the reactivation of pre-existing faults and cause local stress field perturbations (Cappa and Rutqvist, 2011; Faulkner et al., 2010, 2006; Jeanne et al., 2014; Rinaldi et al., 2014). Nevertheless, fault zones and fault systems control the mechanics and flow properties of the crust. If pre-existing stress in a mining zone is high, the perturbation related to underground mining activities may have more chance to reactivate preexisting faults or generate new rock failures than the same perturbations in a less critically-stressed zone. In general, geomechanics, pore pressure, constitutive laws, and stress/strain boundary conditions govern the association between faults and local stress (Brady and Brown, 2013; Faulkner et al., 2010, 2006; Heap et al., 2010). Traditionally, researchers use Boundary Element Method, Finite Element Method and other numerical methods to
⁎ Corresponding author at: Key Laboratory of Coalbed Methane Resources and Reservoir Formation Process of the Ministry of Education, China University of Mining and Technology, Xuzhou, Jiangsu, China. E-mail addresses: [email protected] (T. Chen), [email protected] (T. Mukerji), [email protected] (L. Dou).
http://dx.doi.org/10.1016/j.coal.2017.04.001 Received 10 January 2017; Received in revised form 10 April 2017; Accepted 10 April 2017 Available online 11 April 2017 0166-5162/ © 2017 Elsevier B.V. All rights reserved.
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simulate the effect of faults on local stress field and to identify the correlation radius between faults and their associated high-stress zones. Those methods need some preconditions such as obtaining exact information of fault geometry, geomechanical properties, pore pressure and stress/strain boundary condition (Brady and Brown, 2013; Kattenhorn et al., 2000; Li et al., 2011). In many practical situations, those preconditions are incompletely accessible before coal extraction in underground coalmines. Instead of geomechanical modeling, active seismic survey and passive seismic monitoring are popular methods to delineate the fault distribution and estimate the velocity heterogeneity in underground coalmine panels (Chen et al., 2015; Dou et al., 2012; Ge, 2005; Hatherly, 2013; Mason, 1981; Maxwell and Young, 1995; Si et al., 2015; Zuo et al., 2009). The delineations of high-velocity areas and detected or measured in-panel faults are the most accessible data. Because stress and elastic wave velocity have positive correlation in the shallow crust, the monitored velocities can be used to locate high-stress areas and to estimate a relatively accurate stress level (Chen et al., 2015; Dou et al., 2012; Mason, 1981). If one can use those monitored velocities and detected/measured in-panel faults to better understand the spatial correlation and association between high-stress areas and inpanel faults, it could help coalmine operators to minimize stress related rock failures and gas outbursts. Stochastic simulation as a popular modeling tool has been used in geosciences for many years. In the coal industry, researchers use it to estimate coal-bed methane resources, to map the heterogeneities of coal quality, and to quantify the geological uncertainties and risks (Olea and Luppens, 2015; Tercan and Sohrabian, 2013; Zhou et al., 2012). Geologists use it to simulate 2D and 3D fracture and fault distributions, to analyze the association between faults and fractures, and to model the association between stress field and fractures (Kattenhorn et al., 2000; Noroozi et al., 2015; Viruete et al., 2001, 2003). Point process statistics (also called Boolean spatial process) have been widely used in mining, geology, forestry, and environmental sciences to analyze the geometrical patterns formed by objects that are distributed randomly in one-, two- or three-dimensional space (Connor and Hill, 1995; Illian et al., 2008; Moller and Waagepetersen, 2004; Russell et al., 2016). In order to estimate the association between in-panel faults and high-stress areas under incomplete information of underground coalmine panel, stochastic simulation and point process statistics are useful options. This paper, following the qualitative analysis of Chen et al. (2015), proposes a quantitative analysis workflow to estimate the spatial correlation association between in-panel faults and high-stress areas in an underground coalmine panel using monitored tomographic velocities from active seismic data, in situ measured faults, Monte Carlo simulation and point process statistics. The focus is on statistical spatial modeling of observed data, with the purpose of getting some practical estimates of spatial correlation association.
In the Yanshanian Orogeny (208 Ma) and the Himalayan Orogeny (65 Ma), the coal field of this panel experienced a series of tectonic movements. During roadway excavation and in-panel coal extraction, the operators of this mine identified in-panel faults and measured their characteristics daily. As a result, the 163L02C panel shows fifteen mapped faults (Fig. 3). Their strikes are shown in Fig. 4. In general, those faults belong to two main fault sets. One set consists of normal faults with NNW-SSE trend, and the other consists of normal faults with NNE-SSW trend. Because most faults are along NNW-SSE trend, this trend dominates the faulting characteristics of this panel. Considering the fault trends, most of them are near north direction with up to ± 30° deviation. This characteristic is a key input factor during fault simulation. The fault throws are small (0.7–1.6 m). We ignore the influence of fault throw during fault simulation. The fault lengths differ largely from fault to fault (Chen et al., 2015). The longest one is F3 (209 m), and the shortest one is F14 (17 m). We consider the influence of fault length during fault simulation. Chen et al. (2015) carried out active seismic tomography in this underground panel to measure the velocity distribution before coal extraction. Lithologically the formations are quite homogeneous and lithological variations are not a source of P-velocity heterogeneity in this case, as analyzed in Chen et al. (2015). Since P-velocity has a positive correlation with stress in the shallow crust, imaged P-velocities will indirectly depict the stress distribution of the panel (Chen et al., 2015; Dou et al., 2012; Mason, 1981; Maxwell and Young, 1995; Young and Maxwell, 1992). In order to achieve a categorical assessment, Chen et al. (2015) classified the imaged P-velocities into three stress levels (High, Medium and Low) using a threshold method as shown in Fig. 5. Almost all of the high-stress area are near faults except for area B. The spatial distribution of high-stress areas shows qualitative correlation with local stress field, previous mining activity and local fault distributions (Chen et al., 2015). After previous mining activities, the west and east sides of 163L02C panel have formed two mined out voids. Because of the existence of those voids, the coal bed and its main roof in this island panel have been detached from its adjacent strata. Therefore, the maximum horizontal principal stress (with near E-W direction) has been relieved from this panel (Chen et al., 2015). This may cause perturbations of the local stress field and the concentration of high stress around faults. Instead of a qualitative analysis as in Chen et al. (2015), we focus on the fault zone to analyze quantitatively the correlation association between measured faults and categorized highstress areas as shown in Fig. 5. The fault zone is defined as the rectangle with dashed lines as most of the in-panel fault traces and high-stress areas are within this rectangle.
2. General geology
Geologically, the existence of faults will be accompanied with the heterogeneity distribution of stress as shown in Fig. 5. A deterministic estimate between in-panel faults and their related high-stress areas will be convenient and practical for coalmine operators to optimize mining methods and minimize stress related rock failures. In this section, we propose a feasible workflow to quantitatively analyze the correlation association with Monte Carlo simulation and point process statistics. The input data are simulated fault traces and high-stress areas referenced from the measured in situ characteristics. Because the coalbed is relatively flat and all faults found in the study panel are normal faults with limited dip angle and throw variations (Chen et al., 2015), we ignore the influences of fluctuations in the coalbed floor and small variations in fault's dip angle and throw. We use a 2D-panel model to approximate the true 3D panel. In addition, we ignore the damage zones around faults during simulation because the true damage zones in the 163L02C panel are nearly invisible. The concerned variables are fault strikes, fault lengths, and tomographic seismic velocities.
3. Estimates of correlation association
The panel used in this study is a longwall island panel of 163L02C from Jining3 coalmine. This mine is a key production colliery in Southwest Shandong coalfield that is located in Shandong province, eastern China as shown in Fig. 1. The target of this panel is No.3 coal (about 675 m in depth). Its lithology is anthracite with glass luster, and its thickness is 3.0–6.0 m. Except for the faulted areas, the coal-bed thickness is relatively uniform. Both the coal bed and its roof and floor belong to Lower Permian formation as shown in Fig. 2. The direct roof is thin and non-uniform silt sandstone; while the direct floor is thin and non-uniform mudstone. In contrast with the thin and non-uniform direct roof and floor, the main roof and floor are thick and uniform sandstone as shown in Fig. 2. Since they are more stiff, uniform and thick than the coal bed, generated seismic waves will likely refract from them for a source deployed in the coal bed. Therefore, this panel is suitable for the imaging using refraction tomography (Chen et al., 2015). 52
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Fig. 1. The location of Jining3 coalmine.
Fig. 4. Rose map of faults of 163L02C panel. The radius indicates fault lengths, and the angles imply fault strikes. Every solid dot corresponds to an individual fault.
simulated fault lengths by a uniform distribution between 30 m and 200 m, and restrict the fault strikes between − 30° and 30°. We also simulate spatially expanding zones of high-stress areas, starting with 1% of the grid cells. The locations of the initial seed cells with respect to the location of the simulated faults are selected according to three different spatial models described in the next section: a) random, b) anti-correlation, and c) correlation. The initial seed cells are dilated with a radius r at each iteration. The Monte Carlo approach can give stochastic results in the form of distributions. It is more convenient and practical for mine operators to have a useable deterministic estimate to optimize mining methods and to minimize the stress related underground rock failures and gas outbursts. Hence we assimilate and synthesize the Monte Carlo results
Fig. 2. Stratigraphic column around the No. 3 coal, edited from Chen et al. (2015). In contrast to the roof and floor, the No. 3 coal has a characteristically low seismic velocity.
3.1. Simulation and statistical procedures The Monte Carlo procedure includes three main steps: generating in-panel faults, generating high-stress areas, and statistically analyzing the correlation pattern of in-panel faults and high-stress areas. In order to analyze digitally, we discretize the studied fault zone into 1 m × 1 m grids with a total of 400 × 135 grid cells. Because the true number of mapped in-panel faults in the fault zone is eleven, we uniformly draw a random number of faults within the range of 11 ± 6. Referring to the true measured characteristics of in-panel faults (Fig. 4), we limit the
Fig. 3. Plan view map of 163L02C panel, edited from Chen et al. (2015). The length and width of this panel is 756 m and 135 m respectively. The red solid curves are fault traces and the gray labels are fault names and fault throws.
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Fig. 5. Categorical assessment of stress levels in 163L02C panel, edited from Chen et al. (2015). The gray areas are high-stress class (H), and the other areas are low-and-medium stress class (L & M). All high-stress areas are labeled as A, B, C, D and E respectively. The rectangle with green dashed lines defines the fault zone used in the simulations.
3.3. Anti-correlation model
to give statistically calibrated estimates. The spatial statistical analysis has five sub steps. As the high-stress zones are expanded spatially, at each dilation iteration (1) we calculate the Ratio of included Fault-trace Length over total fault-trace length (RFL). If a fault trace totally or partly overlaps with high-stress areas, the trace or the part of the trace will be treated as included fault trace. (2) Then, we count the Ratio of High-stress Areas over total area (RHA) of the fault zone. (3) After that, we dilate the high-stress areas with a given radius r and compute RHA(r) and RFL(r) with the dilated areas. (4) We repeat steps (1) ~ (3), dilating and expanding the high stress areas, until all fault traces are included in the dilated high-stress areas. (5) Finally, we plot RFL(r) against RHA(r) and fit a trend curve. In this way, one can quantify the spatial correlation pattern of in-panel faults and high-stress areas. As shown below, the RFL versus RHA plots show different patterns depending on the spatial correlation or non-correlation between faults and high-stress areas.
In comparison with the randomly uncorrelated model, we build an anti-correlation model to analyze the anti-correlation association between in-panel fault traces and high-stress areas. In this scenario, the high-stress areas tend to avoid fault traces in the fault zone. In general, the simulation procedure of this model is similar to the procedure of no spatial correlation model. At first, we randomly generate fault traces with a uniform distribution function as in the previous model. Then, we randomly generate high-stress areas with a uniform distribution function and avoid generating high-stress areas within the range of a certain anti-correlation radius from any fault trace. (In point process statistics, models similar to this are termed hard-core point process, e.g. Illian et al., 2008). The anti-correlation model needs an additional parameter of anti-correlation radius to control the minimum distance between fault traces and high-stress areas. Because no measured high-stress areas appear around the tips of the fault traces (Fig. 5), we ignore the tip influence during the simulation of high-stress areas, and only simulate high-stress cells over the fault zone avoiding the simulated fault traces by a distance given by the range of anti-correlation radius. The generating method of highstress cells follow the sequential procedure as below: (1) randomly generate a cell location in the fault zone with a uniform distribution function; reject if it falls within the anti-correlation radius of any fault trace; (2) assign the corresponding cell as high-stress cell if it is not a previously assigned high-stress cell; (3) repeat step (1) and (2) until the high-stress cells account for 1% fault zone cells. As an example, Fig. 8 shows one realization. Finally, as before, we dilate the high-stress areas, count and cross-plot RHA and RFL. Simulations are done for a set of anti-correlation radii (10 m, 20 m, 30 m, 40 m and 50 m) with 100 realizations for each anti-correlation radius to obtain the RFL-RHA cross plot as shown in Fig. 9. Comparing with Figs. 7 and 9 is obviously different. Almost all cross-plotted points in this figure move off the diagonal and bend toward the bottom-right corner. This phenomenon is a distinguishable characteristic of the anticorrelation model from the uncorrelated model. If in-panel fault traces and high-stress areas have anti-correlation association, the cross plot of RHA vs. RFL pairs will be located in the figure's lower right half. For a given anti-correlation radius, simulated points are monotonically increasing, beginning at (0, 0), bent in the middle, and ending at (1, 1). With increasing anti-correlation radius, the bent curvature in the middle increases gradually. In order to quantitatively estimate an anti-correlation radius, we fit every set of points for a fixed anti-correlation radius with different functional forms:a Weibull function,
3.2. No spatial correlation model In general, geological factors will have independent spatial distributions in underground coalmine if they do not have any association at all. If in-panel faults and high-stress areas do not have any spatial association, they will be independently distributed in the study panel. In order to analyze the pattern of cross plots between RFL and RHA under independent random condition, we build a no spatial correlation model to simulate in-panel faults and high-stress areas and to count their RFL and RHA. In the first step, we randomly generate in-panel fault traces with a uniform distribution function with the parameters described in the previous section. In the second step, we randomly generate initial high-stress areas with a uniform distribution function too, without any spatial correlation to the previously simulated fault traces. The method of generating high-stress cells follows the sequential procedure as below: (1) randomly generate a cell location in the fault zone with a uniform distribution function and assign it as high-stress cell; (2) randomly generate the next high-stress cell with a uniform distribution function avoiding the previously assigned high-stress cells; (3) repeat step (2) until the proportion of high-stress cells accounts for 1% of the total panel cells. Fig. 6(a) shows one example of this initial stage. Then we gradually dilate the high-stress areas with increasing radii (5 m, 10 m, 15 m…) until all fault traces have been included in the dilated high-stress areas. Fig. 6(b) shows one iteration of the dilation. With increasing dilation radius, the included fault-trace length will increase correspondingly until all fault traces fall into high-stress areas. At every dilation step, RHA and RFL are computed. Finally, Fig. 7 cross plots all pairs of RHA and RFL for 100 simulations. In this figure, almost all cross-plotted points are near and around the diagonal with very small deviation. The near diagonal characteristics of RHA vs. RFL pairs indicate that the in-panel faults and high-stress areas are distributed randomly with no spatial correlation in the study panel. In this study, we treat those RFL-versus-RHA characteristics as indicators of a spatially uncorrelated model.
f (x ) = abx (b −1) e−ax
b
(1)
a power-law function,
f (x ) = ax b a Gaussian function, 54
(2)
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Fig. 6. A realization of in-panel fault traces and simulated high-stress areas for the no spatial correlation model. Black lines are fault traces, and gray areas are high-stress areas. Highstress areas in (b) are dilated from (a) with a 5 m radius.
f (x ) = ae−(
x−b 2 c )
template to estimate the anti-correlation radius between in-panel fault traces and high-stress areas for the study panel.
(3)
and an exponential function x c
f (x ) = 1 − e−( b )
3.4. Correlation model (4) Instead of uncorrelated or anti-correlated association, fault traces and high-stress areas may have a positive spatial correlation association in the underground panel. In this scenario, high-stress areas will tend to appear preferentially near the fault traces rather than off the fault traces. We build a correlation model to simulate the characteristics of this scenario. At first, we randomly generate fault traces with a uniform
respectively. Among the equations, the coefficients a, b and c are used to adjust the shape of fitted curves. Since the Weibull function has the best R-square goodness of fit (> 0.93) with 95% confidence bounds, we use its fit result as the fitted trend of anti-correlation model as shown in Fig. 10(a). The fitted curves as shown in Fig. 10(b) can be treated as a
Fig. 7. Cross plot of RHA vs. RFL pairs for no spatial correlation model, for 100 simulations.
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Fig. 8. A realization of simulated in-panel fault traces and high-stress areas of anti-correlation model. Black lines are fault traces, and gray areas are high-stress areas. The anti-correlation radius used in this model is 20 m.
left half. With the decreasing of correlation radius, the symbols will bend toward the top-left corner gradually. Because all cross plots are monotonically increasing, beginning at (0, 0), bent in the middle, and ending at (1, 1), this phenomenon is much more obvious in the middle range. In order to quantitatively estimate a correlation radius, we fit every set of points for a fixed correlation radius with different functional forms:an inverse Weibull function,
distribution function as we did in the previous two cases. Then, we randomly generate initial high-stress seed areas with a uniform distribution function and keep only those high-stress areas within the range of correlation radius around the previously simulated fault traces. Similar to the anti-correlation model, the correlation model needs a correlation radius to control the maximum distance between in-panel fault traces and high-stress areas. The generating method of high-stress cells follow the sequential procedure as below: (1) randomly generate a cell location in the fault zone with a uniform distribution function; accept if it is at least within the correlation radius of any one fault trace; (2) assign the corresponding cell as high-stress cell if it is not a previously assigned high-stress cell; (3) repeat step (1) and (2) until the high-stress cells account for 1% fault zone cells. As an example, Fig. 11 shows a realization. Finally, as before, we dilate the initial highstress areas, count and cross-plot RHA and RFL, for different correlation radii (10 m, 20 m, 30 m, 40 m and 50 m), with 100 simulations for each correlation radius (Fig. 12a). Fig. 12(a) differs obviously from Figs. 7 and 9. All cross-plotted symbols in this figure move off the diagonal and bend toward the topleft corner. This phenomenon is a distinguishable characteristic of correlation model from uncorrelated model and anti-correlation model. If in-panel fault traces and high-stress areas have correlation association, the cross plot of RHA vs. RFL pairs will be located in figure's upper
f (x ) = 1 − abx (b −1) e−ax
b
(5)
an inverse Gaussian function,
f (x ) = 1 − ae−(
x−b 2 c )
(6)
a power-law function (Eq. (2)),and an exponential function (Eq. (4)).respectively. Since the exponential function has the best R-square goodness of fit (> 0.96) with 95% confidence bounds, we use its fit result as the fitted trend for the correlation model. The fitted curves as shown in Fig. 12(b) can be treated as a template to estimate the correlation radius between in-panel fault traces and high-stress areas for the study panel.
Fig. 9. Cross plot of RHA vs. RFL pairs for anti-correlation model. The larger the anti-correlation radius, the further are the points from the diagonal.
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Fig. 10. Fitted RHA-RFL cross plots with Weibull function. In (a), the anti-correlation radius of scattered points and fitted curve is 30 m; in (b), the labels on the fitted curves indicate the anti-correlation radii.
Fig. 11. A realization of in-panel fault traces and initial high-stress seed areas for the correlation model. Black lines are fault traces, and gray areas are high-stress areas. The correlation radius used in this model is 20 m.
artifacts. Those artifacts may come from the inaccurate picking of first breaks, limited seismic bandwidth, and the azimuthal limitations of observation. Because of the existence of those artifacts, the estimate of correlation radius may be different from the previous estimate. Consequently, we build a noisy correlation model to calibrate the influence of velocity artifacts. The modeling procedures of the noisy correlation model are similar to the procedures of correlation model. At first, we randomly generate fault traces with a uniform distribution function as we did for all scenarios. As in the correlation model, we generate high-stress areas distributed near the faults traces within the correlation radius. But in addition, to simulate noise artifacts a small fraction of ‘false’ high-stress areas are uniformly sprinkled throughout the fault zone irrespective of their distance from the fault traces. In the following simulations we assume this fraction to be 30% of the high-stress areas. As an example, Fig. 13 shows one realization. We carry out 100 simulations for each set of correlation radii (10 m, 20 m, 30 m, 40 m and 50 m) and fit every set of symbols for a fixed correlation radius with an exponential function (Eq. (4)) as we did in Section 3.4. The RHA-RFL cross plots and the fitted curves are shown in Fig. 14. The R-squares, goodness of fit, are > 0.96 with 95% confidence bounds. In order to analyze the influence of random artifacts on the estimate of correlation radius, we overlay the observed scatter points of the fault zone on the trend curves estimated from ‘noisy’ simulations. The solid points are along and around the trend curve of 20 m, indicating the correlation radius is estimated to be 20 m if tomographic velocities contain 30% random artifacts. In comparison with Fig. 12(b), the solid points and the 20 m trend curve have closer association. In sum, the actual correlation radius will be smaller than the estimate of correlation model from high-stress areas inferred from tomographic velocities containing random artifacts. The larger the percentage of random artifacts, greater is the overestimate because the random artifacts make the observed curve move closer to the diagonal. In the limit (when the tomographically interpreted high stress areas are completely unreliable
3.5. In situ estimates of correlation radius Through the simulations of Sections 3.2–3.4, we achieved three basic observations about the cross plot of RHA vs. RFL pairs. (1) If the cross-plotted points are below the diagonal, fault traces and high-stress areas will have anti-correlation spatial association. (2) If the crossplotted points are along the diagonal with a small deviation, fault traces and high-stress areas will have no spatial correlation association. (3) If the cross-plotted points are above the diagonal, fault traces and highstress areas will have positive spatial correlation association. If other factors remain unchanged, a further distance from the diagonal will imply a smaller correlation radius. Since we already have the templates to estimate the correlation radius or anti-correlation radius between in-panel fault traces and highstress areas for the study panel, we first count and cross plot the RHA and RFL of the fault zone in163L02C panel using the measured faults and estimated high-stress areas and overlay them over the template as shown in Fig. 12(b). Because the black points are far above the diagonal, this implies that the fault traces and high-stress areas in the fault zone have positive correlation association. This outcome is in accordance with the geological analysis of Chen et al. (2015). To estimate the correlation radius quantitatively, we see the observed black points being scattered between the 20 m curve and the 30 m curve. This implies that the fault traces and high-stress areas in the studied fault zone have a correlation radius between 20–30 m. 4. The influence of model inputs 4.1. Velocity artifacts The data we used to estimate the correlation association between high-stress areas and faults are the measured in-panel faults and subsurface tomographic seismic velocities as shown in Fig. 5. As an indirect measurement, the tomographic velocities may contain some 57
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Fig. 12. Cross plot of RHA vs. RFL pairs for correlation model (a) and fitted curves for RHA-RFL cross plots with an exponential function (b). In (b), every labeled curve is a fitted result of a set of single-colour symbols. The labels on the curves are the corresponding correlation radii. The solid points in (b) are observed RHA-RFL cross plot of the fault zone in Fig. 5.
4.2. Interpreted faults
and are just randomly distributed) the actual spatial correlation is lost and it appears (incorrectly) to be uncorrelated. Thus the estimate with velocity artifacts (provided it is not too large) provides an upper bound on the correlation radius. This maximum estimate can be a useful parameter to design the plan for the mine and to prevent occurrences of stress related rock failures and gas outbursts.
In comparison with the underground measured in-panel faults, interpreted faults from 3D seismic and/or underground seismic data are often more easily accessible for most practical applications. In such cases, the interpreted faults cannot be 100% correct. In this section, we analyze the influence of uncertainties in interpreted faults on the estimate of correlation radius between in-panel faults and high-stress areas.
Fig. 13. A realization of the noisy correlation model with a correlation radius of 20 m.
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Fig. 14. Cross plot of RHA vs. RFL pairs for noisy correlation model (a) and fitted curves for RHA-RFL cross plots with an exponential function (b). In (b), every labeled curve is a fitted result of a set of single-colour symbols. The labels on the curves are the corresponding correlation radii. The solid points in (b) are observed RHA-RFL cross plot of the fault zone in Fig. 5.
Fig. 15. A realization of in-panel fault traces and initial high-stress seed areas for the correlation model with − 90° ~ 90° fault strike distribution. Black lines are fault traces, and gray areas are high-stress areas. The correlation radius used in this model is 30 m.
30 m, 40 m and 50 m). As an example, Fig. 15 shows one realization. Unlike the previously simulated fault traces, the simulated fault traces in Fig. 15 have a much wider range of strikes as expected. The cross plot of RHA vs. RFL pairs for this new model are shown in Fig. 16. In comparison with the correlation model, the scatter points of this new model have a much larger deviation range. Fitting the scatter points of each set of correlation radii (10 m, 20 m, 30 m, 40 m and 50 m) with an exponential function (Eq. (4)) as we did in Section 3.4, the fitted R-
First of all, we analyze the influence of deviation in interpreted fault strike on the estimate. The modeling procedures are similar to the procedures of the correlation model. The only difference is the deviation range of simulated fault strikes. In this experiment, we expand the limit of the fault strikes to a radical range of − 90° ~ 90° rather than the previous range of − 30° ~ 30°, implying a large uncertainty in the fault interpretation. As we did in Section 3.4, we carry out 100 simulations for each set of correlation radii (10 m, 20 m, 59
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Fig. 16. Cross plot of RHA vs. RFL pairs for correlation model with different range of fault strikes.
smaller influence on the accuracy of the estimate compared to the uncertainties in fault strikes. In sum, both inaccurate fault strikes and lengths will affect the accuracy of the estimated correlation radius. In comparison with inaccurate fault lengths, inaccurate fault strikes may have a larger influence. If one hopes to estimate the correlation radius between inpanel faults and high-stress areas, it is crucial to provide good fault strike distribution.
squares of this new model are < 0.86 with 95% confidence bounds. This result is much worse than the correlation model, indicating inaccurate inputs of fault strikes have strong influence on the accuracy of the estimate. Fortunately, few true panel have this wide (− 90° to + 90°) of an uncertainty in fault strike distribution. Then, we analyze the influence of fault length deviation on the estimate. The modeling procedures are similar to the procedures of correlation model. The only difference is the deviation range of simulated fault lengths. In this experiment, we limit the fault lengths to a range of 100–200 m rather than the previous range of 30–200 m because small faults with short lengths are not interpretable from seismic data. As we did in Section 3.4, we carry out 100 simulations for each set of correlation radii (10 m, 20 m, 30 m, 40 m and 50 m). As an example, Fig. 17 shows one realization. Unlike the previous simulated fault traces, the simulated fault traces in Fig. 17 have a longer averaged fault trace. The cross plot of RHA vs. RFL pairs for this new model is shown in Fig. 18. In comparison with the correlation model, the scatter points of this new model have a little bit larger deviation range. By fitting the scatter points of each set of correlation radii (10 m, 20 m, 30 m, 40 m and 50 m) with an exponential function (Eq. (4)) as we did in Section 3.4, the fitted R-squares of this new model are about 0.94 with 95% confidence bounds. This result is only a little bit worse than the correlation model, indicating inaccurate inputs of fault lengths have
5. Conclusions In this paper, we presented a workflow to quantitatively estimate the correlation association between in-panel faults and high-stress areas in an underground coalmine panel using true measured in-panel faults, tomographic seismic velocities, Monte Carlo simulations and point process statistics. Through modeling, simulation and comparison, we can conclude: (1) The proposed workflow of this paper can produce a deterministic estimate of correlation radius rather than a qualitative description of correlation association between in-panel faults and high-stress areas. This estimate could be a practicable parameter for coalmine operators to optimize mining methods and to minimize stress
Fig. 17. A realization of in-panel fault traces and initial high-stress seed areas for the correlation model with 100–200 m fault lengths. Black lines are fault traces, and gray areas are highstress areas. The correlation radius used in this model is 30 m.
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Fig. 18. Cross plot of RHA vs. RFL pairs for correlation model with different range of fault lengths.
consortia sponsors.
related rock failures and gas outbursts. (2) In the fault zone of the 163L02C panel, Monte Carlo simulations show that high-stress areas have positive correlation with in-panel faults. This result is in accordance with the geological analysis of this panel (Chen et al., 2015). In this specific example, the estimated correlation radius is 20–30 m. If ones hope to estimate a template for their own application, ones should follow the workflow and generate their own template calibrated to the geology of the specific application area. (3) Velocity artifacts affect the estimate of correlation radius between in-panel faults and high-stress areas. If those artifacts are uniformly random, the correlation radius estimate using a template constructed with no noise will be an overestimation with increasing percentage of artifacts. Therefore, the estimate of correlation model is an upper bound estimate of correlation radius between in-panel faults and high-stress areas. (4) In comparison with the variation of fault lengths, too large a variation of fault strikes will decrease the reliability of fitted correlation radius, although this situation does not often appear in true underground panels. (5) The preconditions of this proposed workflow are a relative flat panel, small fault throws and small fault-throw deviations, and small lithologic variability. If these preconditions are not satisfied, the applicability of this proposed workflow is unknown. Conceptually the workflow can be easily expanded from a 2-D flat panel to a full 3-D volume. But incorporating large fault throws, complex faulting regimes and very heterogeneous lithologies are much more challenging. A further research is a compulsive option for more complex faulting regimes.
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Acknowledgement Financial support for this work, provided by the National Natural Science Foundation of China (No. 41374140 and No. 41430317), Natural Science Foundation of Jiangsu Province (BK20130175) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), are gratefully acknowledged. We also acknowledge Stanford Center for Reservoir Forecasting (SCRF) and Stanford Rock Physics and Borehole Geophysics (SRB) 61
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