Uncertainty propagation in structural modeling

M.K. Elmouttie CSIRO Earth Science & Resource Engineering, Australia

G.V. Poropat CSIRO Earth Science & Resource Engineering, Australia

Abstract Knowledge of the geometrical characteristics of the discontinuities in a rock mass is vital for various geotechnical assessments of slope stability, rock mass fragmentation characteristics (predicted from in-situ block size distribution estimation) and aspects of hydrogeological properties. To model the geometry of discontinuities within the rock mass, statistical methods must be employed to infer the population’s discontinuity properties from the observed sample. Using computer simulations relying on discrete fracture network generation and polyhedral modelling, we will estimate the evolution of the uncertainties associated with typical procedures used by the practitioner to estimate discontinuity geometry from typical field data. Guidelines will be provided to help reduce the uncertainty associated with the use of such methods.

1 Introduction Modelling of natural systems requires measurement and analysis procedures capable of dealing with ‘total uncertainty’. This is defined as the combination of stochastic variability (i.e. variability inherent in random processes) as well as inherent uncertainty (i.e. lack of knowledge) and must be accounted for when undertaking all aspects of rock mass modelling. Stochastic variability is commonly addressed via assignment of statistical representations to the variables being studied. Inherent uncertainty, such as that associated with an under-represented discontinuity set, is less well defined and relies on the experience of the practitioner to determine the appropriate conceptual models to consider or studies to perform (such as sensitivity analyses) to examine less well understood processes and variables in the model.

The use of discrete representations of fracture networks to model the heterogeneity of a rock mass has greatly increased in recent years. This has been a function of both the availability of discrete fracture network (DFN) software generators as well as the widespread availability of powerful computing resources. DFNs provide a more realistic simulation of rock mass heterogeneity than the more established continuum or equivalent medium (EM) approaches in that individual fracture properties such as position, size and aperture are represented. Both deterministic and stochastic methods are used to represent the mapped sample of discontinuities and the unmapped population respectively. Further, by simultaneously simulating all relevant discontinuities in a volume of interest, the geotechnical analyses based on DFN can differ markedly from those simulating fracture intersections using combinatorial approaches based on pairs or triplets of persistent planes. The latter technique often pre-supposes structures are proximal enough for wedge or block formation to occur, thus biasing analyses towards smaller, more frequent rock-mass failures and unrealistic (e.g. convex) polyhedral representations of the blocks.

The full potential of the DFN approach to capture stochastic variability is, of course, only realised if the geometry of the DFN being generated is statistically identical (or at least similar to within some tolerance) to the actual fracture network it is being used to simulate. If this is not the case, then the DFN approach can be considered, at best, an inefficient and overly complicated alternative to the equivalent medium approach. At worst, the DFN can actually misrepresent the behaviour of the rock mass as analysed via stability, fragmentation, hydrogeological or other analyses. Importantly, the significance of the uncertainties associated with DFN generation is not widely appreciated, partly because the types of DFN utilised in the past (e.g. 2D representations or 3D representations using semi- or fully-persistent planes) failed to expose the true magnitude of the stochastic

variability in the rock mass and partly because the generation of multiple DFN realisations for statistical analysis was not computationally practical.

Critical evaluation of any procedure utilising DFN is therefore warranted. However, absence of DFN based analyses should not imply any improvement in the measure of the total uncertainty of a natural system. Indeed, the use of deterministic methods or equivalent medium methods for scenarios in which fracture properties need to be modelled discretely simply provides the practitioner with a false sense of security. It is merely the explicit nature of the requirement for fracture geometry representation in DFN based methods that highlight the inherent uncertainties in the modelling process (Jing 2003).

The DFN is usually constructed after the statistical parameters of the fracture sets observed in the field have been analysed. These data are usually acquired via core logging, exposure mapping (e.g. scan line or window mapping) or other methods. Of course, all such methods are sampling based with inherent biases that must be corrected, particularly with respect to estimates of fracture density (or frequency) and trace lengths (Mauldon 1998). Further, since most return data on fracture traces, well established inversion techniques based on certain stereological assumptions must be used to assess a most critical fracture property which is size (Lyman 2003). It is only when these processing stages are accomplished that one can base a geotechnical analysis on a series of DFN simulations using sampling methods such as Monte Carlo. However, the question of how the uncertainty inherent in each of these stages propagates through to the analysis results remains. Further, given the complexity and time required to run numerical modelling simulations, how can one utilise the potentially hundreds or thousands of DFN realisations generated?

To address these issues, we have structured this paper as follows. Section 2 will outline the methodology used, section 3 presents the results and section 4 offers discussion and gives some recommendations.

2 Methodology To answer the previous questions, one would ideally have access to a site where the structural characteristics of the rock mass were known precisely. A large rock mass, however, cannot be deconstructed so to analyse fracture positions, sizes and other properties. Further, although comparison of geotechnical analysis predictions such as stability, block size distribution (BSD) and hydrogeological analysis do provide some degree of confidence that the simulated DFN properties are representative of the rock mass, they can be insensitive to DFN properties and therefore not useful for quantifying this uncertainty propagation. We have therefore chosen to use simulated rock masses with both simplified and realistic rock mass characteristics. We have used the BSD measure extensively to characterise the DFN because it provides a global measure of the resulting rock mass fragmentation and has been used extensively in related studies (Hudson & Priest 1979, Lu & Latham 1999, Wang et al 2003, Latham et al. 2006, Kim et al. 2007, Rogers et al. 2007)

Simplified rock mass simulations

To understand the sensitivity of the metrics used in this paper to the uncertainties present in structural data, some simple but elucidating scenarios have been modelled based on a 100m rock mass cube geometry intersected by persistent joints. Various joint properties (orientation variance, spacing distribution) have been modified to examine the sensitivity of the BSD to these variables. These results are useful in interpreting the more realistic scenarios as well as framing the point made earlier regarding the dangers of using overly simplified geometric representations to understand uncertainty propagation.

Realistic rock mass simulations

Since this research is focussed on uncertainty propagation characteristics rather than absolute uncertainties in a particular DFN generation strategy, we have chosen to create the DFN for the realistic rock mass simulations using the typically adopted Baecher DFN model (Baecher et al. 1977). This model represents one of several possible choices (Dershowitz & Einstein 1988) and assumes

• Families of joints can be assigned to ‘sets’ based on common orientations

• Joints are represented as circular discs (or polygonal approximations)

• Joint centroids are spatially distributed in a random fashion (Poisson process)

• Joint sizes are distributed log-normally

• Joint spacings following negative-exponential distribution

We have used square polygonal representations of joints for computational efficiency. For brevity, we have refrained from extending this analysis to include the effects of a so-called isotropic or randomly oriented (small Fisher constant) fracture ‘set’. However, there can be no doubt that such sets do occur in nature and can form up to 30% or more of the total fractures under investigation, and work to quantify their effect on geotechnical analyses is currently the focus of another study.

Structural data

The structural data used for the simulated DFNs is based on the readily available example data set distributed with Sirovision 3D Photogrammetry System (Poropat 2001). A 3D model of two benches is available with 3 structure sets being digitally mapped (Figure 1). The region of interest consists of 130m of strike length with bench strike approximately 107º from North, 60º face angles and 34m bench heights. The simulation volume has been extended 100m into the rock mass. The joint set parameters for the structural sets are shows in Table 1. We have chosen to modify this data so as to represent two scenarios, each of which has been modelled using 100 DFN realisations.

The first (referred to as scenario ‘A’) represents a ‘poor statistics’ scenario often encountered in the field where, for a variety of reasons, the acquired structural data is sparse. For this scenario, we have used the data in Table 1 in unmodified form to drive the DFN generator.

We have also modelled a more ideal scenario ‘B’ which is representative of a desirable but not onerous mapping exercise. A general rule of thumb regarding exposure mapping is to include approximately one hundred to two hundred structures per joint set (ISRM 1978, p324). Therefore, we have increased the model dimensions to 500m strike length to allow the use of the same fracture frequency as in scenario A but satisfy the ISRM requirement. Further, one might expect that given structural homogeneity exists, the measured uncertainty associated with orientations of the structure sets might be significantly lower than those used in scenario A. We have chosen to reduce the standard deviation in orientations to 5˚. Note that no deterministic structures (and corresponding DFN conditioning) were used in this set of simulations due to the changed model geometry.

Bias correction

There is a significant amount of data regarding the need to perform bias correction on mapping of trace data, especially significant for trace lengths (e.g. Mauldon 1998). Given the simulated nature of this analysis, no attempt to formally bias correct the data was made other than a doubling in the observed number of traces to account for potential biases. However note that in general, failure to bias correct data, particularly when small numbers of structures are available, can introduce considerable errors to estimates of fracture frequency and trace-length. The latter is directly related to the sizes of the fractures and therefore greatly influences predictions of fracture connectivity, rock mass fragmentation and block size distribution and slope stability analyses. For severely censored data, under-estimates of mean trace length will have obvious consequences regarding the propagation of uncertainties into the resulting geotechnical analyses.

Figure 1 A screen-shot of the digitally mapped surface used for this analysis.

Table 1 Joint set parameters for the original data used in scenario A

Set ID Number

of traces

Dip (º) Dip

Direction (º)

Trace

Length (m)

Spacing (m)

1 28 62±28 40±18 25.5±8.7 13.2±14.0

2 20 41±24 255±53 21.8±9.3 15.8±16.5

3 44 46±14 188±30 23.2±8.6 Unspecified

DFN simulations

The structural data was imported into Siromodel, a software tool developed for the Large Open Pit Mine Slope Stability Project (CSIRO 2011). The software has facilities for the generation of DFN, running Monte Carlo simulations, detection of polyhedra resulting from intersection of structures within the DFN and performing limit equilibrium stability analyses at free surfaces. The DFN can contain polygons or triangulated surfaces representing both stochastic and deterministic structures with defined persistence and termination relationships. Details regarding the algorithms used in this polyhedral modeller are available in Elmouttie et al. (2010). For the scenario A analysis, the three structure sets were utilised in two ways. The digitally mapped traces were imported directly to generate deterministic structures. The joint set parameters derived from these data were then used to drive the stochastic structure generation. The stochastic structures were conditioned to not daylight on the free surface so as not to corrupt the statistical validity of the resulting DFN. For the scenario B analysis, the strike length of the model was increased to 500m and the stochastic variance in structure properties greatly reduced (Figure 2).

A Monte Carlo method was employed to sample the probability space associated with each of the stochastic parameters. This involved generation of 100 simulations per scenario, each simulation being based on a different DFN ‘realisation’ which was nonetheless statistically equivalent to the others.

Figure 2 More ideal conditions (scenario B) represented via this 500m strike length model. DFN for a single realisation (left) and traces (right) are shown.

For each simulation, a block size distribution was calculated for the resulting polyhedra. A limit equilibrium stability analysis at the free surface was also performed using a vector analysis method (Warburton 1981) and the classification schema defined by Goodman & Shi (1985). Through the predicted in-situ block size distribution and stability analyses, conclusions regarding the uncertainty associated with any one realisation of the DFN can be made.

3 Results

Simplified rock mass simulations

We have investigated the significance of variance in discontinuity orientation by first modelling the trivial geometry of an equi-partitioned cube. A DFN was generated consisting of three orthogonal sets of equi-spaced discontinuities using a spacing of 10 metres, yielding cubes of 1000m3 volume (Figure 3, left).

Figure 3 Equi-partitioned cube (left), single realisations of σσσσorient = 10 degree & σσσσspacing = 20% case with banded (middle) and exponential (right) spacing distributions

Gradually, the geometry of the DFN has been modified by increasing the variance in orientations of the structures. We have used a normal distribution (for ease in interpretation) to vary the discontinuity orientations with standard deviations (σorient) from 0 degrees to 10 degrees (Figure 4). The 0 degree case is represented by the vertical line corresponding to all blocks having volumes of 1000m3. As σorient is increased, the BSD curve becomes progressively more ‘S’ shaped as more and more variation in block volumes is encountered, with the 10 degree case corresponding to the left most curve. Note that these curves represent BSD measured from single realisations of the DFN.

Figure 4 Migration of block size distribution curves for single DGN realisations with orientation variations ranging from σσσσ = 0 degree case (vertical line at 1000m3) to σσσσorient = 10 degree case (left most curve). Blocks smaller than 0.1 m3 were excluded.

To understand the stochastic uncertainty, we have perform a Monte Carlo simulation based on 100 realisations of σorient = 10 degree case. The BSD curves for all simulations are shown in Figure 5. The range in 90% volumes is from 1600 to 2200 m3. The average block number generated per simulation was 1496 ± 65 blocks.

We have also introduced a variance in the spacing to study the effect of this parameter. A standard deviation to the spacing (σspacing) of 20% did not dramatically change the results with the 90% volume range increasing only slightly from 1500 to 2200 m3. The mean block number was 1522 ± 104 blocks. An example of a single realisation for this case is shown in Figure 3 (middle).

Finally, we investigated the effect of a more realistic spacing model. A negatively exponential spacing distribution with a mean spacing of 10m and σspacing = 20% was used whilst retaining σorient = 10degrees. The range in 90% volumes increased markedly from 500 to 9000 m3 (5000m3 if outlier curve rejected). A larger variance in block numbers was also observed with the mean block number being 1610 ± 704 blocks. A single realisation for this case is shown in Figure 3 (right). The BSD curves are shown in Figure 6.

Figure 5 Block size distributions for 100 realisations of the σσσσorient = 10 degree case.

Figure 6 Block size distributions for 100 realisations of the σσσσorient = 10 degree, σσσσspacing = 20% and negative-exponential spacing distribution case.

Realistic rock mass simulations

The stochastic variability of the problem is highlighted most clearly by the number of blocks or polyhedra detected in each simulation. The distribution shape associated with the number of blocks detected per simulation is lognormal as predicted by the central limit theorem for products of positive random variates (e.g. Vose 2001). The variation in block numbers detected most clearly highlights the problems associated with deterministic approaches for modelling geometrically discrete but stochastic processes. Some simulations show marked deviation from the mean block number (approx 100±100 blocks) and the modal value around 30 blocks.

The data for the idealised scenario B reveals increased numbers of structures and the reduced stochastic variance in the data has greatly reduced the uncertainty with the mean block number estimate (approx 90±25 blocks) and a model value around 80. However note that a considerable range in block number estimates remains (47 to 152).

Figure 7 shows the block size distribution curves for all 100 simulations. All blocks detected in these simulations have been used including those formed at the free surface but excluding those in contact with simulation volume boundaries (to avoid edge effects) and those with volumes less than 0.1 m3. It is clear that the curves are distributed over at least two orders of magnitude in volumes, with the 50th percentile volume ranging from around 0.2 to 2 m3. A more significant measure for slope stability considerations is the 90th percentile value showing volumes ranging from 2 to 50 m3, well over an order of magnitude.

Figure 8 shows the scenario B data excluding blocks with volumes less than 0.1m3. Now the variation in the 90th percentile value is from 2 to 20 m3, a reduced range but still covering an order of magnitude.

Figure 9 shows the predicted hazards greater than 0.1 m3 (unstable blocks) for all 100 simulations in scenario B. Only 15 events were registered highlighting the sparse nature of the statistics associated with this measure. For scenario B, we recall that the deterministic structures were not included in the analysis and therefore the stochastic structures were responsible for daylighting at the surface and generating instabilities. This, in combination with the reduced stochastic variability and increased exposure area has resulted in a greatly increased total number of events (190).

Figure 7 Block size distribution curves for the 100 simulations excluding blocks with volumes less than 0.1 m3 for scenario A. Bottom chart shows frequency of 90th percentile block volume.

Figure 8 Block size distribution curves for the 100 simulations excluding blocks with volumes less than 0.1 m3 for scenario B. Bottom chart shows frequency of 90th percentile block volume.

Figure 9 Hazard map for scenario B showing the total number of block failures detected for all simulations.

4 Discussion

Stochastic variability

It is clear that for the data analysed herein the stochastic variability has a considerable effect on DFN properties such as block size distribution. For the simplified rock mass simulations, the introduction of a realistic spacing model was the significant factor in increasing the uncertainty in BSD estimates from ‘manageable’ to significant. For the more realistic simulations, it may not be entirely surprising that scenario A showed similarly large uncertainty in BSD estimates given the small number statistics associated with this data and the limited ‘conditioning’ of the DFN to exposure mapping alone. Further constraints such as geotechnical domains, deterministic structures etc would likely reduce the magnitude of the uncertainty. Nonetheless, scenario B also

showed similar uncertainty associated with the block size distribution. These two scenarios, although representative of two very different data acquisition exercises, are indicative of the typical uncertainty propagation in analyses dependent upon discrete structure representation.

Inherent uncertainty

Due to the typically sparse nature of the data being gathered in the field, the lack of knowledge of the rock mass geometry will often be just as important to the total uncertainty as the stochastic variability of the properties being studied. This is a very different situation to other engineering domains such as manufacturing where the stochastic variability dominates. For this reason, exclusion of the realisations associated with the ‘tails’ of a particular parameter distribution (e.g. BSD) is often not plausible. Figure 10 shows a schematic outlining this point. The outlier curves are associated with two particular DFN realisations referred to as X & Y.

Figure 10 Schematic showing family of BSD curves with outliers associated with realisation X and Y.

Although from a statistical point of view, these realisations are representative of a confidence interval of potential realisations, this is not necessarily true if the lack of knowledge of the system is taken into account. To account for this, multiple models of the rock mass must be simulated, a practise often referred to as multiple model analysis or MMA (Ye et al. 2010). For example, an alternative model for the rock mass under study here could include different distribution types for joint spacings. This is a plausible variable in exposure mapping, particularly when structure numbers are small.

There are other critical variables in such an analysis that, due to insufficient data or inconclusive analyses, must often be considered ‘unknowns’. These include trace length distribution, trace length bias correction itself (when data is insufficient to perform a formal bias correction), termination relationships between joint sets, the significance of random (isotropic) joints and the presence of suspected or poorly understood large-scale structures (e.g. beds, faults or voids) in the rock mass. Some of these have been discussed in other work (Elmouttie et al. 2009). The sensitivity of the model to these unknowns must be studied and the use of parallel analyses as demonstrated above is a simple way to do this. A more thorough analysis of all sensitivities in a model would utilise a methodology similar to orthogonal array testing (e.g. Devore 2000) but given the typical lack of resources or time, the experience of the practitioner is vital in determining which scenarios to investigate.

Guidelines to quantifying the stochastic uncertainty associated with the use of such methods.

The use of digital mapping can greatly improve the statistical accuracy associated with joint set characterisation. Although not alleviating the need to inspect and map individual structures on an exposure, the ability to safely, accurately and rapidly build digital representations of large rock mass exposures should reduce the need to rely on small structure samples for analysis.

However, in the attempt to increase sample sizes for statistical robustness, one must also refrain from concatenating data associated with distinct geotechnical domains or structure geometries. In such cases where variation in structural characteristics is significant, data concatenation will simply result in stochastic variability being ‘hidden’ from the model.

As briefly mentioned earlier, if appropriate, the inclusion of random or isotropic structures to represent those not associated with a joint set is also recommended to accurately describe the rock mass. If sufficient data is available, inclusion of termination probabilities, precedence relationships, deterministic structures etc will further reduce the stochastic variability in the model outputs to those measured in the field.

Of course, the usage of DFN generation versus pair-wise or similarly isolated fracture interaction calculators to predict the formation of blocks within the rock mass is highly recommended. Although the latter method can often be implemented as a ‘spread-sheet’ solution and is useful as first-pass analysis, only simultaneous generation of structures in 3D space will properly characterise the interactivity and frequency of occurrence of blocks.

For complex, time intensive numerical simulations, usage of selected DFN realisations that ‘bracket’ the parameter distribution being studied is useful. Realisations corresponding to upper and lower tails of the parameter being investigated can be used individually in numerical analyses to more accurately model the range of possibilities. Of course, such realisations will not necessarily represent the tails of other physical parameters of interest.

Implications for hydrogeological studies

Unforeseen or unidentified uncertainty in stability analysis may have significant safety implications. For example slope failures are sometimes associated with rain events and therefore understanding the behaviour of a rock mass in relation to the injection of water is critical. Uncertainty in modeling the flow of water through the rock mass and the consequent effects on stability makes predicting the likelihood of failure and the possible timing of a failure prone to very large errors. Calibration of hydrogeological models based on insufficient data can lead to errors in excess of 100% in predictions of transport times (Moore 2005) and relatively small numbers of stochastically generated structures can also produce large uncertainty in percolation parameters (Ji et al. 2011). Since the presence of fractures in a rock mass leads to a two-phase fluid flow, structure uncertainty in modeling the rock mass discontinuities may have a major effect on the ability to predict the influence of groundwater on the behaviour of the rock mass.

5 Conclusions We have presented an analysis of the propagation of total uncertainty associated with structural modelling dependent upon DFN generation. The inclusion of DFN generation from a geotechnical analysis does not imply more uncertainty in the modelling process than an equivalent medium approach, but merely exposes and quantifies the sources of uncertainty. Realisations of DFN cannot be used in isolation since each realisation is merely a sampling of the total probability space of possible realities. Rather, a Monte Carlo or similar sampling approach should be utilised to build up confidence intervals in the parameters or physical processes being studied. It is currently not practical to apply full Monte Carlo analyses to complex numerical models which individually may take hours, days or even weeks to run. Therefore the use of simplified modelling techniques which still ‘interrogate’ the full DFN realisations is recommended. Polyhedral modelling offers one possible approach for the quantitative assessment of DFN realisations. Properties such as in-situ block size distribution and block stability can be used to determine the realisations which ‘bracket’ the range of possibilities covering the upper and lower tails of the parameter distribution of interest. These individual DFN can then be singled out for more detailed numerical analyses. The use of multiple model analysis techniques has also been recommended as an effective way to quantify inherent uncertainty.

6 Acknowledgements The authors would like to acknowledge the support of the Australian Coal Association Research Program in the development of the prototype versions of the algorithms used in elements of the structural modelling. The sponsors of the Large Open Pit Mine Slope Stability Project, which is managed by the CSIRO, are acknowledged for their support in developing applications to utilise the algorithm for the analysis of slope stability phenomena.

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