Modelling the Positive Feedback Mechanism of a Karst Aquifer Using Surface Reconstruction

Modelling the Positive Feedback Mechanism of a Karst Aquifer Using Surface Reconstruction

Kyffin K. Bradshaw Tane S. Ray

Department of Computer Science, Mathematics & Physics, University of West Indies Cave Hill Campus, Barbados

Available online: 
| Citation



Karst conduit network modelling is particularly difficult because the location of the conduits within an aquifer is often unknown. To address this, many mathematical models of karst aquifers use stochastically simulated conduit networks to try to extract certain geometrical and hydraulic connectivity properties that may prevail within the aquifer. Such idealized representations of a karst aquifer do not adequately represent the positive feedback mechanism that exists between the distribution of hydraulic head and the growth of the solution conduits that determine the geometry and the interconnectedness of the resulting conduit network. In this paper, Poisson surface reconstruction is presented as a simple method for constructing a realistic model of a karst aquifer by simulating the positive feedback mechanism between dissolution and flow. Direct application of the Poisson technique to a tropical karst limestone aquifer of the island of Barbados highlights how the complete conduit geometry and the feedback mechanism of a real aquifer system may be interpolated. The result suggests that applying surface reconstruction to a good calibrated point cloud sampling taken from an aquifer itself is an efficient methodology for generating realistic karstic networks. Additionally, Poisson surface reconstruction replicates an aquifer without directly solving complex hydrogeological and speleological equations and oversimplifying the hydrogeological and geological complexities of a karst environment in the way that hypothesized conduit network models do. As a result, it is believed that this conceptual model provides a utility for characterizing a karst aquifer in terms of the well-established theoretical foundations of the surface reconstruction problem even when the input data is sparse.


conceptual model, conduit network, feedback mechanism, Poisson surface reconstruction


[1] Collon-Drouaillet, P., Henrion, V. & Pellerin, J., An algorithm for 3D simulation ofbranchwork karst networks using Horton parameters and A-star: application to asynthetic case. Geol Soc Lond., Spec. Pub, 370(1), pp. 295–306, 2012.

[2] Jaquet, O., Siegel, P., Klubertanz, G. & Benabderrhamane, H., Stochastic discretemodel of karstic networks. Adv Water Resour, 27(7), pp. 751–760, 2004.

[3] Ronayne, M., Influence of conduit network geometry on solute transport in karst aquiferswith a permeable matrix. Adv Water Resour, 56, pp. 27–34, 2013.

[4] Vuilleumier, C., Borghi, A., Renard, P., Ottowitz, D., Schiller A, Supper R. & Cornaton,F., A method for the stochastic modeling of karstic systems accounting for geophysicaldata: an example of application in the region of Tulum, Yucatan Peninsula (Mexico).Hydrogeol. J, 21, pp. 529–44, 2013.

[5] Borghi, A., Philippe, R. & Cornaton, F., Can one identify karst conduit networks geometryand properties from hydraulic and tracer test data? Adv Water Resour, 90, pp.99–115, 2016.

[6] Borghi, A., Philippe, R. & Sandra, J., A pseudo-genetic stochastic model to generatekarstic networks. J Hydrol, 414, pp. 516–529, 2012.

[7] De Rooij, R. & Graham, W., Generation of complex karstic conduit networks witha hydrochemical model. Water Resour. Res., 53, pp. 6993–7011, 2017.

[8] Western, A.W., Blöschl, G., & Grayson, R.B., Toward capturing hydrologically significantconnectivity in spatial patterns. Water Resour. Res. 37, pp. 83–97, 2001.

[9] Filipponi, M., Jeannin, P.Y. & Tacher, L., Understanding cave genesis along favourablebedding planes: the role of the primary rock permeability, Z. Geomorphology, 54, pp.91–114, 2010.

[10] Bonacci, O., Pipan, T., & Culver, D.C., A framework for karst ecohydrology. EnvironmentalGeology, 56, pp. 891–900, 2009.

[11] Pardo-Igúzquiza E., Dowd, P.A., Chaoshui, X. & Durán-Valsero, J.J., Stochastic simulationof karst conduit networks. Adv. Water Resour, 35, pp. 141–150, 2012.

[12] Pinault, J.L., Bakalowicz, M., Plagnes, V. & Aquilina, L., Inverse modelling of thehydrological and the hydrochemical behaviour of hydrosystems: characterization ofkarst system functioning. Water Resour. Res, 37, pp. 2191–2204, 2001.

[13] Shaban, A. & Darich, T., The role of sinkholes in groundwater recharge in the highmountains of Lebanon. J. Environ. Hydrol, pp. 19, 1–11, 2011.

[14] Goldscheider, N., Meiman, J., Pronk, M. & Smart, C., Tracer tests in karst hydrogeologyand speleology. International Journal of Speleology, 37(1), pp. 27–40, 2008.

[15] Kaufmann, G., A model comparison of karst aquifer evolution for different matrix-flowformulations. J. Hydrol., 283(1–4), pp. 281–289, 2003.

[16] Kaufmann, G., Modelling karst geomorphology on different time scales. Geomorphology,106(1–2), pp. 62–77, 2009.

[17] Kaufmann, G., Romanov, D. & Hiller, T., Modeling three-dimensional karst aquiferevolution using different matrix-flow contributions. J. Hydrol., 388(3–4), pp. 241–250,2010.

[18] Kazhdan, M., Bolitho, M. & Hoppe, H., Poisson surface reconstruction. Proceedings of4th Eurographics Symposium on Geometry Processing, pp. 61–70, 2006.

[19] Aronov, B., Brönnimann, H., Chang, A.Y. & Chiang, Y.J., Cost-driven octree constructionschemes: an experimental study, Computational Geometry, 31, pp. 127–148, 2005.

[20] Kambesis, P.N. & Machel, H.G., Caves and Karst of Barbados. In Lace, M.J. &Mylroie, J.E., eds., Coastal Karst Landforms, Dordrecht: Springer, Coastal ResearchLibrary 5, pp. 227–244, 2013.

[21] Stafford, K.W., Mylroie, J.E., Taborosi, D., Jenson, J.W. & Mylroie, J.R., Karst developmenton Tinian, Commonwealth of the Northern Mariana Islands: Controls on dissolutionin relation to the carbonate island karst model. Journal of Cave and KarstStudies, 67(1), pp. 14–27, 2005.

[22] Klein, A. & Godunov, A., Introductory Computational Physics, Cambridge UniversityPress, 2006.

[23] Cazalas, F. & Giesen, J., Delaunay Triangulation-based Surface Reconstruction. InEffective Computational Geometry for Curves and Surfaces, Springer, 2006.

[24] Lace, M.J. & Mylorie, J.E., Coastal Cave and Karst Resource Management. In CoastalKarst Landforms, Dordrecht: Springer, 2013.

[25] Pardo-Iguzquiza, E., Durán-Valsero, J.J. & Rodríguez-Galiano, V., Morphometric analysisof three-dimensional networks of karst conduits. Geomorphology, 132(1), 17–28,2011.

[26] Kambesis, P.N., Larson, E.B. & Mylroie, J.E., Morphometric analysis of cave patternsusing fractal indices. In Feinberg, J., Gao, Y., & Alexander, E.C., Jr., eds., Caves andKarst Across Time: Geological Society of America Special Paper 516, pp. 67–86, 2015a.

[27] Palmer, A.N., Distinction between epigenic and hypogenic maze caves. Geomorphology,134, pp. 9–22, 2011.

[28] Jameson, R.A., Identification and analysis of early flow paths in branchwork caves inwest virginia, usa. Geological Society of America Special Papers, 404, pp. 23–30, 2006

[29] Farrell, D.A., Sandberg, S.K., Mayers, B.L., Sutherland, A., Barnes, H., Nurse, J. &Moseley, L., Characterizing and Modeling of Seawater Intrusion in an Aquifer alongthe West Coast of Barbados. EEGS 17th Symposium on the Application of Geophysicsto Engineering and Environmental Problems, pp. 1216–12795, 2004.

[30] Covington, M.D., Wicks, C.M., & Saar, M.O., A dimension-less number describingthe effects of recharge and geometry on discharge from simple karstic aquifers.WaterResour. Res, 45, pp. 1216–12795, 2009.

[31] Taboroši, D., Jenson, J.W., & Mylroie, J.E., Zones of enhanced dissolution and associatedcave morphology in an uplifted carbonate island karst aquifer, northern Guam,Mariana Islands. Speleogenesis Evol. Karst Aquifers, 1(4), pp. 1–16, 2003.

[32] Collon, P., Bernasconi, D., Vuilleumier, C., & Renard, P., Statistical metrics for thecharacterization of karst network geometry and topology. Geomorphology, 283, pp.122–142, 2017.

[33] Cashman, A.C., Water Policy Development and Governance in the Caribbean: Anoverview of regional progress. Water Policy, 14(1), pp. 14–30, 2012.

[34] Doerfliger, N., Jeannin, P.Y. & Zwahlen, F., Water Vulnerability Assessment in KarstEnvironments: A new method of defining protection areas using a multi-attributeapproach and GIS tools (EPIK method). Environmental Geology, 39(2), pp. 165–176,1999.

[35] Foster, S. & Skinner, A.C., Groundwater protection: The science and practice of landsurface zoning. IAHS Publications-Series of Proceedings and Reports-Intern AssocHydrological Sciences, 225, pp. 471–482, 1995.

[36] Onody, R.N. & Zara, R.A., Cluster, backbone, and elastic backbone structures of themultiple invasion percolation. Physical Review E, 56(3), pp. 1063–6510, 1997.

[37] Berkowitz, B. & Balberg, I., Percolation theory and its application to groundwaterhydrology. Water Resources Research, 29(4), pp. 775–794, 1993.

[38] Jacobsen, L.J. & Zinn-Justin, P., Monochromatic path crossing exponents and graphconnectivity in 2D percolation. Phys. Rev. E. 66(5), pp. 1–3, 2002a.

[39] Jacobsen, L.J. & Zinn-Justin, P., A transfer matrix for the backbone exponent of twodimensionalpercolation. J. Phys. A: Math Gen., 35, pp. 2131–2144, 2002b.

[40] Ali-Bray., N., Moore, J.E., Senthil, T., and Vishwanath, A., Ordering near the percolationthreshold in models of two-dimensional bosons with quenched dilution. PhysicalReview B, 73(6), pp. 1–10, 2006.

[41] Song, C., Halvin, S. & Makse, H., Origins of fractality in growth complex networks.Nature Physics, 2(4), pp. 275–281, 2006.

[42] Mohammadi, Z. & Raeisi, E., Hydrogeological uncertainties in delineation of leakageat karst dam sites, the Zagros Region, Iran. Journal of Cave and Karst Studies, 69(3),pp. 305–317, 2007.

[43] Jones, I.C. & Banner, J.L., Estimating recharge thresholds in tropical Karst Island Aquifers:Barbados, Puerto Rico and Guam. Journal of Hydrology, 278(1), pp. 131–143,2003.

[44] Antcheva, I., et al., ROOT: A C++ framework for petabyte data storage, statistical analysisand visualization. Comput. Phys. Commun., 182(6), pp. 1384–1385, 2011.