Statistic Analysis of Karst Geometry

Cecile Vuilleumier and Pauline Collon. ( 2012 )
in: GeoEnv 2012, 9th Conference on Geostatistics for Environmental Applications,Valencia, Spain, September 19–21, pages 325--326

Abstract

Karstic systems have a complex geometry which results from the succession of several genetic phases. Simulating these highly heterogeneous and anisotropic systems is an active research topic which recently leads to several efficient static methods. However, assessment of these methods calls for a quantification of the similarity between numerical and actual cave morphologies. According to the widely-used classification of Palmer, six specific cave patterns can be discriminated and are mainly linked to the type of recharge and the initial aquifer permeability field. These factors determine the type of connectivity of the network (dendritic or anastomotic) and the distribution of passage orientations (rectilinear/angular or curvilinear). In this paper, we present several parameters that aim at numerically describing these cave pattern morphologies and apply them to several cave systems. The 6 2D karst networks of the Palmers classification have been skeletonised with ImageJ and vectorized as curves in Gocad. 22 3D karst networks mapped by speleologists are available as a sequence of topographic stations and have been imported as curves in Gocad. The network connectivity analysis is done regarding the systems as graphs. By default, a segment, or graph, represents a line-of-sight between two stations, corresponding to graph nodes. A karst branch represents the portion of the curve connecting an extremity or a junction node with another junction or extremity node. To correct data imprecision, a pre- processing tool allows to merge or link nodes and edges that are closer to each other than a user-defined tolerance distance. A depth-first-search algorithm explores the graph in order to identify its connected components. The degree of the graph nodes (i.e. the number of neighbours) is computed. With the number of external nodes (extremities), internal nodes (junctions) and cycles, the Howard parameters alpha, beta and gamma are computed and provide a degree of connectivity. Passage orientation distributions are tackled by the automatic computation of azimuths, plunges, map segment lengths and real segment lengths. This can be addressed regardless the measurement step by densifying the network. These regularized orientation data are plotted using a stereographic projection to obtain Wulff or Schmidt nets and Rose diagrams. Families of poles can be identified and extracted and circular statistics can be calculated. The tortuosity of the karst branches (the ratio between Euclidean and curvilinear distance between two nodes) is also computed to give an insight on the network morphology. To complete these parameters and explore the relation between the karst occurrence and the local stratigraphy it is often proposed to compute a Z-histogram representing the repartition of karst occurrence relative to their elevation and to compare it to the local stratigraphy. When the geological context is characterised by folded layers or high dip values, elevation is not representative. Therefore, we propose using W- histograms, where the W value represents an elevation relative to geological horizons. On Palmers 2D karsts, the connectivity degree reflects clearly if a karst follows a branchwork or maze pattern whereas orientations are good indicators of the rectilinear or curvilinear character of the network. The application on real 3D networks reveals more subtle differences reflecting the complexity of the networks which cannot always be easily classified. Moreover, preferential directions may be induced by a high hydraulic gradient and not only by fracturation. Tortuosity is also hard to interpret in terms of classification. The numerical pre-processing of the data also appears to have a potentially high impact on the results. Stability of these parameters at different scales of observation (locally or globally) is still studied. Nevertheless these preliminary results on real 3d karst networks are encouraging.

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    BibTeX Reference

    @INPROCEEDINGS{Vuilleumier2012c,
        author = { Vuilleumier, Cecile and Collon, Pauline },
         title = { Statistic Analysis of Karst Geometry },
     booktitle = { GeoEnv 2012, 9th Conference on Geostatistics for Environmental Applications,Valencia, Spain, September 19–21 },
          year = { 2012 },
         pages = { 325--326 },
      abstract = { Karstic systems have a complex geometry which results from the succession of several genetic phases. Simulating these highly heterogeneous and anisotropic systems is an active research topic which recently leads to several efficient static methods. However, assessment of these methods calls for a quantification of the similarity between numerical and actual cave morphologies. According to the widely-used classification of Palmer, six specific cave patterns can be discriminated and are mainly linked to the type of recharge and the initial aquifer permeability field. These factors determine the type of connectivity of the network (dendritic or anastomotic) and the distribution of passage orientations (rectilinear/angular or curvilinear). In this paper, we present several parameters that aim at numerically describing these cave pattern morphologies and apply them to several cave systems.
    
    The 6 2D karst networks of the Palmers classification have been skeletonised with ImageJ and vectorized as curves in Gocad. 22 3D karst networks mapped by speleologists are available as a sequence of topographic stations and have been imported as curves in Gocad. The network connectivity analysis is done regarding the systems as graphs. By default, a segment, or graph, represents a line-of-sight between two stations, corresponding to graph nodes. A karst branch represents the portion of the curve connecting an extremity or a junction node with another junction or extremity node. To correct data imprecision, a pre- processing tool allows to merge or link nodes and edges that are closer to each other than a user-defined tolerance distance.
    
    A depth-first-search algorithm explores the graph in order to identify its connected components. The degree of the graph nodes (i.e. the number of
    neighbours) is computed. With the number of external nodes (extremities), internal nodes (junctions) and cycles, the Howard parameters alpha, beta and gamma are computed and provide a degree of connectivity. Passage orientation distributions are tackled by the automatic computation of azimuths, plunges, map segment lengths and real segment lengths. This can be addressed regardless the measurement step by densifying the network. These regularized orientation data are plotted using a stereographic projection to obtain Wulff or Schmidt nets and Rose diagrams. Families of poles can be identified and extracted and circular statistics can be calculated. The tortuosity of the karst branches (the ratio between Euclidean and curvilinear distance between two nodes) is also computed to give an insight on the network morphology. To complete these parameters and explore the relation between the karst occurrence and the local stratigraphy it is often proposed to compute a Z-histogram representing the repartition of karst occurrence relative to their elevation and to compare it to the local stratigraphy. When the geological context is characterised by folded layers or high dip values, elevation is not representative. Therefore, we propose using W- histograms, where the W value represents an elevation relative to geological horizons.
    
    On Palmers 2D karsts, the connectivity degree reflects clearly if a karst follows a branchwork or maze pattern whereas orientations are good indicators of the rectilinear or curvilinear character of the network. The application on real 3D networks reveals more subtle differences reflecting the complexity of the networks which cannot always be easily classified. Moreover, preferential directions may be induced by a high hydraulic gradient and not only by fracturation. Tortuosity is also hard to interpret in terms of classification. The numerical pre-processing of the data also appears to have a potentially high impact on the results. Stability of these parameters at different scales of observation (locally or globally) is still studied. Nevertheless these preliminary results on real 3d karst networks are encouraging. }
    }