Abstract
The modeling of subsurface geometry and properties is a key element to understand Earth processes and manage natural hazards and resources. In this paper, we suggest this field should evolve beyond pure data fitting approaches by integrating geological concepts to constrain interpretations or test their consistency. This process necessarily calls for adding the time dimension to 3D modeling, both at the geological and human time scales. Also, instead of striving for one single best model, it is appropriate to generate several possible subsurface models in order to convey a quantitative sense of uncertainty. Depending on the modeling objective (e.g., quantification of natural resources, production forecast), this population of models can be ranked. Inverse theory then provides a framework to validate (or rather invalidate) models which are not compatible with certain types of observations. We review recent methods to better achieve both stochastic and time-varying geomodeling and advocate that the application of inversion should rely not only on random field models, but also on geological concepts and parameters.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aanonsen SI, Nœvdal G, Oliver DS, Reynolds AC, Vallès B (2009) The ensemble Kalman filter in reservoir engineering—a review. SPE J 14(3):393–412
Abrahamsen P (1992) Bayesian kriging for seismic depth conversion of a multi-layer reservoir. In: Soares A (ed) Geostatistics Troia’92, vol 1. Kluwer, Dordrecht, pp 385–398
Apel M (2006) From 3D geomodelling systems towards 3D geoscience information systems: Data model, query functionality, and data management. Comp Geosci 32(2):222–229
Aug C, Chiles JP, Courrioux G, Lajaunie C (2004) 3D geological modeling and uncertainty: the potential field method. In: Leuangthong O, Deutsch CV (eds) Geostatistics Banff, proc seventh international geostatistics congress. Kluwer, Dordrecht, pp 145–154
Bardossy G, Fodor J (2001) Traditional and new ways to handle uncertainty in geology. Nat Resour Res 10(3):179–187
Bennis C, Sassi W, Faure JL, Hage Chehade F (1996) One more step in gocad stratigraphic grid generation: taking into account faults and pinchouts. In: Proc European 3D reservoir modelling conference (SPE 35526)
Bond C, Gibbs A, Shipton Z, Jones S, Stracher G (2007) What do you think this is? Conceptual uncertainty in geoscience interpretation. GSA Today 17(11):4–10
Braun J (1993) Three-dimensional numerical modeling of compressional orogenies: Thrust geometry and oblique convergence. Geology 21(2):153
Brin S (1995) Near neighbor search in large metric spaces. In: Proc international conference on very large data bases. Morgan Kaufmann, San Francisco, pp 574–584
Caers J, Hoffman T (2006) The probability perturbation method: a new look at Bayesian inverse modeling. Math Geol 38(1):81–100
Calcagno P, Chilès JP, Courrioux G, Guillen A (2008) Geological modelling from field data and geological knowledge Part I. Modelling method coupling 3D potential-field interpolation and geological rules. Phys Earth Planet Inter 171(1–4):147–157
Caumon G, Grosse O, Mallet JL (2004a) High resolution geostatistics on coarse unstructured flow grids. In: Leuangthong O, Deutsch CV (eds) Geostatistics Banff, proc seventh international geostatistics congress. Kluwer, Dordrecht, pp 703–712
Caumon G, Strebelle S, Caers JK, Journel AG (2004b) Assessment of global uncertainty for early appraisal of hydrocarbon fields. In: Proc SPE annual technical conference and exhibition (SPE 89943)
Caumon G, Sword CH, Mallet JL (2004c) Building and editing a Sealed Geological Model. Math Geol 36(4):405–424
Caumon G, Tertois AL, Zhang L (2007) Elements for stochastic structural perturbation of stratigraphic models. In: Proc petroleum geostatistics A02, EAGE
Caumon G, Collon-Drouaillet P, le Carlier de Veslud C, Sausse J, Viseur S (2009) Teacher’s aide: 3D modeling of geological structures. Math Geosci 41(9):927–945
Chambers KT, DeBaun DR, Durlofsky LJ, Taggart IJ, Bernath A, Shen AY, Legarre HA, Goggin DJ (1999) Geologic modeling, upscaling and simulation of faulted reservoirs using complex, faulted stratigraphic grids. In: Proc SPE reservoir simulation symposium (SPE 51889)
Charles T, Guéméné J, Corre B, Vincent G, Dubrule O (2001) Experience with the quantification of subsurface uncertainties. In: Proc SPE annual technical conference and exhibition (SPE 68703)
Cherpeau N, Caumon G, Lévy B (2009) Stochastic simulations of structural models. In: Proc IAMG’09, Stanford, California
Chilès JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Series in Probability and Statistics. Wiley, New York
Chilès JP, Aug C, Guillen A, Lees T (2004) Modelling the geometry of geological units and its uncertainty in 3D from structural data: the potential-field method. In: Proc orebody modelling and strategic mine planning, pp 313–320
Cowan EJ, Beatson RK, Ross HJ, Fright WR, McLennan TJ, Evans TR, Carr JC, Lane RG, Bright DV, Gillman AJ, Oshust PA, Titley M (2003) Practical implicit geological modeling. In: Dominy S (ed) Proc 5th international mining conference, Australian Inst Mining and Metallurgy, pp 89–99
Cross TA, Lessenger MA (1999) Construction and application of a stratigraphic inverse model. In: Harbaugh J, Watney W, Rankey E, Slingerland R, Goldstein R, Franseen E (eds) Numerical experiments in stratigraphy: recent advances in stratigraphic and sedimentologic computer simulations, vol 62. SEPM Special Publication, pp 69–83
Dahlstrom C (1969) Balanced cross section. Can J Earth Sci 6:743–757
de Kemp EA (1999) Visualization of complex geological structures using 3-D Bézier construction tools. Comput Geosci 25(5):581–597
de Kemp EA, Sprague KB (2003) Interpretive tools for 3D structural geological modeling parti: Bézier-based curves, ribbons and grip frames. GeoInformatica 7(1):55–71
Deutsch CV (2002) Geostatistical reservoir modeling. Oxford University Press, New York
Dunbar J, Cook R (2003) Palinspastic reconstruction of structure maps: an automated finite element approach with heterogeneous strain. J Struct Geol 26:1021–1036
Durand-Riard P, Caumon G, Muron P (2010) Balanced restoration of geological volumes with relaxed meshing constraints. Comp Geosci 36(4):441–452
Dynamic Graphics (2009) EarthVision software http://www.dgi.com
Elliott D (1983) The construction of balanced cross-sections. J Struct Geol 5(2):101
Euler N, Sword CH, Dulac JC (1998) A new tool to seal a 3D earth model: a cut with constraints. In: Proc 68th SEG annual meeting, pp 710–713
Evensen G (2007) Data assimilation: the ensemble Kalman filter. Springer, Berlin
Fisher TR, Wales RQ (1992) Rational splines and multidimensional geologic modeling. In: Computer Graphics in Geology, vol 41. Springer, Berlin, pp 17–28
Frank T, Tertois AL, Mallet JL (2007) 3D-reconstruction of complex geological interfaces from irregularly distributed and noisy point data. Comp Geosci 33:932–943
Fremming N (2002) 3D geological model construction using a 3D grid. In: Proc ECMOR VIII European Conference on mathematics of oil recovery
Frøyland LA, Lakså A, Strom K, Pajchel J (1993) A 3D cellular, smooth boundary representation modelling system for geological structures. In: EAGE 55th meeting and technical exhibition
Gemcom (2009) Gems and Surpack software http://www.gemcomsoftware.com
Gjøystdal H, Reinhardsen JE, Åstebøl K (1985) Computer representation of complex three-dimensional geological structures using a new solid modeling technique. Geophys Prospect 33(8):1195–1211
Goff JA (2000) Simulation of stratigraphic architecture from statistical and geometrical characterizations. Math Geol 32(7):765–786
Goovaerts P (1997) Geostatistics for natural resources evaluation. Applied Geostatistics. Oxford University Press, New York
Gratier J, Guillier B, Delorme A (1991) Restoration and balance of a folded and faulted surface by best-fitting of finite elements: principles and applications. J Struct Geol 13(1):111–1115
Gringarten E, Arpat G, Haouesse M, Dutranois A, Deny L, Jayr S, Tertois A, Mallet J, Bernal A, Nghiem L (2008) New grids for robust reservoir modeling. In: SPE annual technical conference and exhibition (SPE 116649)
Guillen A, Calcagno P, Courrioux G, Joly A, Ledru P (2008) Geological modelling from field data and geological knowledge Part II: modelling validation using gravity and magnetic data inversion. Phys Earth Planet Inter 171(1-4):158–169
Guiziou JL, Mallet JL, Madriaga R (1996) 3-d seismic reflection tomography on top of the GOCAD depth modeler. Geophysics 61(5):1499–1510
Guzofski CA, Mueller JP, Shaw JH, Muron P, Medwedeff DA, Bilotti F, Rivero C (2009) Insights into the mechanisms of fault-related folding provided by volumetric structural restorations using spatially varying mechanical constraints. AAPG Bull 93(4):479–502
Harbaugh J, Watney W, Rankey E, Slingerland R, Goldstein R, Franseen E (eds) (1999) Numerical experiments in stratigraphy: recent advances. In: Stratigraphic and sedimentologic computer simulations, vol 62. SEPM Spec Pub
Hoaglund J, Pollard D (2005) Dip and anisotropy effects on flow using a vertically skewed model grid. Ground Water 41(6):841–846
Hoffmann CM (1989) Geometric and solid modeling: an introduction. Morgan Kaufmann, San Francisco
Hoffman KS, Neave JW, Klein RT (2003) Streamlining the workflow from structure model to reservoir grid. In: SPE annual conference and technical exhibition (SPE 66384)
Holden L, Mostad P, Nielsen B, Gjerde J, Townsend C, Ottesen S (2003) Stochastic structural modeling. Math Geol 35(8):899–914
Hollund K, Mostad PF, Nielsen BF, Holden L, Gjerde J, Contursi MG, McCann AJ, Townsend C, Sverdrup E (2002) Havana: a fault modeling tool. In: Hydrocarbon seal quantification, Norwegian petroleum society conference. NPF Spec Pub, vol 11. Elsevier, Amsterdam
Houlding S (1994) 3D geoscience modeling, computer techniques for geological characterization. Springer, Berlin
Hu L, Blanc G, Noetinger B (2001) Gradual deformation and iterative calibration of sequential stochastic simulations. Math Geol 33(4):475–489
Intrepid Geophysics (2009) 3D Geomodeller software http://www.intrepid-geophysics.com
Jayr S, Gringarten E, Tertois AL, Mallet J, Dulac JC (2009) The need for a correct geological modelling support: the advent of the uvt-transform. First Break 26(10):73–79
Jessell M (2001) Three-dimensional geological modelling of potential-field data. Comput Geosci 27(4):455–465
Jessell M, Valenta R (1996) Structural geophysics: integrated structural and geophysical modeling. In: de Paor DG (ed) Structural geology and personal computers. Elsevier, Amsterdam, pp 303–323
JOA (2009) JOA Jewel Suite software http://www.jewelsuite.com
Johnson C, Jones T (1988) Putting geology into reservoir simulations: a three-dimensional modeling approach. In: SPE annual conference and technical exhibition (SPE 18321)
Journel AG (1993) Resampling from stochastic simulations. Environ Ecol Stat 1:63–83
Journel AG, Huijbregts C (1978) Mining geostatistics. Academic Press, New York. Blackburn Press, NJ (reprint, 2004)
Kaven J, Mazzeo R, Pollard D (2009) Constraining surface interpolations using elastic plate bending solutions with applications to geologic folding. Math Geosci 41(1):1–14
Kedzierski P, Caumon G, Mallet JL, Royer JJ, Durand-Riard P (2008) 3D marine sedimentary reservoir stochastic simulation accounting for high resolution sequence stratigraphy and sedimentological rules. In: Ortiz J, Emery X (eds) Proc eighth geostatistical geostatistics congress, vol 2. Gecamin, Santiago, pp 657–666
Lasseter T, Jackson S (2004) Improving integrated interpretation accuracy and efficiency using a single consistent reservoir model from seismic to simulation. Lead Edge 23(11):1118–1121
Lecour M, Cognot R, Duvinage I, Thore P, Dulac JC (2001) Modeling of stochastic faults and fault networks in a structural uncertainty study. Petrol Geosci 7:S31–S42
Ledez D (2002) Euclidean distance mapping: geological applications. In: Terra Nostra (Proc IAMG), vol 04. Berlin, pp 25–30
Lelièvre P, Oldenburg D, Williams N (2008) Constraining geophysical inversions with geologic information. In: SEG technical program expanded abstracts, vol 27(1), pp 1223–1227
Lemon AM, Jones NL (2003) Building solid models from boreholes and user-defined cross-sections. Comput Geosci 29:547–555
Maerten F, Pollard DD, Karpuz R (2000) How to constrain 3-D fault continuity and linkage using reflection seismic data: a geomechanical approach. AAPG Bull 84(9):1311–1324
Maerten L, Maerten F (2006) Chronologic modeling of faulted and fractured reservoirs using geomechanically based restoration: technique and industry applications. AAPG Bull 90(8):1201
Maharaja A, Journel AG, Caumon G, Strebelle S (2008) Assessment of net-to-gross uncertainty at reservoir appraisal stage: application to a turbidite reservoir offshore west Africa. In: Ortiz J, Emery X (eds) Proc eighth geostatistics congress, vol 2. Gecamin, Santiago, pp 707–716
Mallet JL (1988) Three dimensional graphic display of disconnected bodies. Math Geol 122:977–990
Mallet JL (2002) Geomodeling. Applied geostatistics. Oxford University Press, New York
Mallet JL (2004) Space-time mathematical framework for sedimentary geology. Math Geol 36(1):1–32
Mann CJ (1993) Uncertainty in geology. In: Davis JC, Herzfeld UC (eds) Computers in geology—25 years of progress number 5. IAMG studies in mathematical geology, vol 20. Oxford University Press, New York, pp 241–254
Manzocchi T, Carter J, Skorstad A, Fjellvoll B, Stephen K, Howell JA, Matthews JD, Walsh JJ, Nepveu M, Bos C et al (2008) Sensitivity of the impact of geological uncertainty on production from faulted and unfaulted shallow-marine oil reservoirs: objectives and methods. Petrol Geosci 14(1):3–15
Massonnat G (1999) Breaking of a paradigm: geology can provide 3D complex probability fields for stochastic facies modeling. In: Annual technical conference and exhibition (SPE 56652)
Matheron G (1970) La théorie des variables régionalisées et ses applications: Fascicule 5, Centre de Géostatistique, Ecole des Mines de Paris, Fontainebleau
Mello UT, Henderson ME (1997) Techniques for including large deformations associated with salt and fault motion in basin modeling. Mar Petrol Geol 14(5):551–564
Midland Valley (2009) 4DMove software http://www.mve.com
Moretti I (2008) Working in complex areas: New restoration workflow based on quality control. 3D and 3D restorations. Mar Petrol Geol 25(3):205–218
Moyen R, Mallet JL, Frank T, Leflon B, Royer JJ (2004) 3D-parameterization of the 3D geological space—the GeoChron model. In: Proc European conference on the mathematics of oil recovery (ECMOR IX)
Muron P (2005) Méthodes numériques 3D de restauration des structures géologiques faillées. PhD dissertation, INPL, Nancy, France
Mustapha H, Mustapha K (2007) A new approach to simulating flow in discrete fracture networks with an optimized mesh. SIAM J Sci Comput 29:1439–1459
Norris R, Massonnat G, Alabert F (1993) Early quantification of uncertainty in the estimation of oil-in-place in a turbidite reservoir. In: SPE annual technical conference and exhibition (SPE 26490)
Owen S (1998) A survey of unstructured mesh generation technology. In: Proc 7th international meshing round table, pp 239–267
Paradigm Geophysical (2009) Gocad and SKUA Suite software http://www.pdgm.com
Piegl LA, Tiller W (1997) The NURBS book, 2nd edn. Monographs in visual communications. Springer, Berlin
Rémy N, Boucher A, Wu J (2008) Applied geostatistics with SGeMS—a user’s guide. Cambridge University Press, New York
Rouby D, Xiao H, Suppe J (2000) 3-D restoration of complexly folded and faulted surfaces using multiple unfolding mechanisms. AAPG Bull 84(6):805–829
Roxar (2009) Irap rms software http://www.roxar.com
Sambridge M (1999a) Geophysical inversion with a neighbourhood algorithm I: searching a parameter space. Geophys J Int 138:479–494
Sambridge M (1999b) Geophysical inversion with a neighbourhood algorithm II: appraising the ensemble. Geophys J Int 138(3):727–746
Scheidt C, Caers JK (2009) Representing spatial uncertainty using distances and kernels. Math Geosci 41(4):397–419
Schlumberger (2009) Petrel software http://www.slb.com/content/services/software
Sprague KB, de Kemp EA (2005) Interpretive tools for 3-D structural geological modelling Part II: surface design from sparse spatial data. GeoInformatica 9(1):5–32
Suzuki S, Caers J (2008) A distance-based prior model parameterization for constraining solutions of spatial inverse problems. Math Geosci 40(4):445–469
Suzuki S, Caumon G, Caers J (2008) Dynamic data integration for structural modeling: model screening approach using a distance-based model parameterization. Comput Geosci 12(1):105–119
Swanson D (1988) A new geological volume computer modeling system for reservoir desciption. In: SPE annual conference and technical exhibition (SPE 17579)
Sword CH (1991) Building flexible interactive, geologic models. In: Proc 61st SEG annual international meeting, pp 1465–1467
Tarantola A (1987) Inverse Problem Theory. Elsevier, New York
Tarantola A (2006) Popper, Bayes and the inverse problem. Nature Phys 2(8):492–494
Tertois AL, Mallet JL (2007) Editing faults within tetrahedral volumes in real time. In: Jolley S, Barr D, Walsh J, Knipe R (eds) Structurally complex reservoirs spec pub, vol 292. Geological Society, London, pp 89–101
Thibault M, Gratier JP, Léger M, Morvan JM (1996) An inverse method for determining three dimensional fault with thread criterion: strike slip and thrust faults. J Struct Geol 18:1127–1138
Thibert B, Gratier JP, Morvan JM (2005) A direct method for modeling and unfolding developable surfaces and its application to the Ventura Basin (California). J Struct Geol 27(2):303–316
Thomas AL (1993) Poly3D: a three-dimensional, polygonal element, displacement discontinuity boundary element computer program with applications to fractures, faults, and cavities in the earth’s crust. Master’s dissertation, Stanford University
Thore P, Shtuka A, Lecour M, Ait-Ettajer T, Cognot R (2002) Structural uncertainties: determination, management and applications. Geophysics 67(3):840–852
Tournois J, Wormser C, Alliez P, Desbrun M (2009) Interleaving Delaunay refinement and optimization for practical isotropic tetrahedron mesh generation. ACM Trans Graph 28(3):75
Turner AK (ed) (1992) Three-dimensional modeling with geoscientific information systems. NATO-ASI, Math and phys sciences, vol 354. Kluwer Academic, Dordrecht
Vail PR (1987) Seismic stratigraphy interpretation using sequence stratigraphy, part 1: seismic stratigraphy interpretation procedure. In: Bally AW (ed) Atlas of seismic stratigraphy, vol 27(1). AAPG Studies in Geology, pp 1–10
Wheeler HF (1958) Time-stratigraphy. Bull AAPG 42(5):1047–1063
Wu XH, Parashkevov RR (2009) Effect of grid deviation on flow solutions. SPE J 14(1):67–77
Zanchi A, de Donatis M, Gibbs A, Mallet JL (2009) Imaging geology in 3D. Comput Geosci 35(1)
Zaparo (2009) Leapfrog software http://www.leapfrog3d.com
Zhang H, Thurber C (2005) Adaptive mesh seismic tomography based on tetrahedral and Voronoi diagrams: application to Parkfield, California. J Geophys Res 110:B04303
Zhong D, Li M, Song L, Wang G (2006) Enhanced NURBS modeling and visualization for large 3D geoengineering applications: an example from the Jinping first-level hydropower engineering project, China. Comput Geosci 32(9):1270–1282
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Caumon, G. Towards Stochastic Time-Varying Geological Modeling. Math Geosci 42, 555–569 (2010). https://doi.org/10.1007/s11004-010-9280-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11004-010-9280-y