Research on appropriate borehole density for establishing reliable geological model based on quantitative uncertainty analysis

  • Qian Sun
  • Jingli ShaoEmail author
  • Yulong Wang
  • Tao Ma
Original Paper


Geological structure is an important factor to explore the underground geological conditions for hydrogeological purpose. Borehole density has great influence on the accuracy and application of geological model. In this paper, Transition Probability Geostatistical Software (T-PROGS) has been used to simulate the four facies distribution of West Liao River Plain. And a quantitative uncertainty model of entropy method is introduced. For getting a reliable geological model with as few as the boreholes, two parts have been given. One is the vertical lithologic variability analysis, and the other is the model correct rate and uncertainty analysis. In geological modeling, the borehole data is too sparse to characterize the lateral heterogeneity, so the actual profiles are added. At last, many equal probability realizations of the geological model using 350 boreholes are built. Depending on the model calibration, uncertainty analysis and simulated profile comparison, the geological models are reliable. Thus, for the simple and single stratigraphy study area without complex fault structures and graben structures of several thousands to tens of thousands of square kilometer scale, establishing a reliable geological structure model requires one borehole at least within an average area of 120.81 km2. It is of great significance for decision maker to save manpower and material resources. And we present a workflow to build a 3D Markov chain using boreholes and actual profiles and develop a reliable geological model.


T-PROGS Uncertainty Borehole density Markov chain Conditional simulation 



The authors would like to express their thanks to Junhuan Xue at Inner Mongolia Autonomous Region, the Fourth Hydrogeologic and Engineering Geological Prospecting Institute, and Di Yu at Beijing Qingliu Technology Co., Ltd. for the assistance in the field survey. We are grateful to Yali Cui and Qiulan Zhang at China University of Geosciences (Beijing) for the comments on the manuscript.

Funding information

This research was funded by National Key R&D Program of China (grand number 2017YFC0406106).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Barchielli A, Gregoratti M, Toigo A (2018) Measurement uncertainty relations for discrete observables: relative entropy formulation. Commun Math Phys 357(3):1253–1304. CrossRefGoogle Scholar
  2. Carle SF, Fogg GE (1996) Transition probability-based indicator geostatistics. Math Geol 28(4):453–476. CrossRefGoogle Scholar
  3. Carle SF, Fogg GE (1997) Modeling spatial variability with one and multidimensional continuous-lag Markov chains. Math Geol 29(7):891–918. CrossRefGoogle Scholar
  4. Chen G, Zhu J, Qiang M, Gong W (2018) Three-dimensional site characterization with borehole data—a case study of Suzhou area. Eng Geol 234:65–82. CrossRefGoogle Scholar
  5. De Cesare L, Myers DE, Posa D (2002) FORTRAN programs for space-time modeling. Comput Geosci 28(2):205–212. CrossRefGoogle Scholar
  6. de Marsily G, Delay F, Gonçalvès J, Renard P, Teles V, Violette S (2005) Dealing with spatial heterogeneity. Hydrogeol J 13(1):161–183. CrossRefGoogle Scholar
  7. dell’Arciprete D, Bersezio R, Felletti F, Giudici M, Comunian A, Renard P (2011) Comparison of three geostatistical methods for hydrofacies simulation: a test on alluvial sediments. Hydrogeol J 20(2):299–311. CrossRefGoogle Scholar
  8. Deutsch CV, Journel AJ (1992) Geostatistical software library and user’s guide. Oxford University Press, OxfordGoogle Scholar
  9. Elfeki AMM, Dekking FM (2005) Modelling subsurface heterogeneity by coupled Markov chains: directional dependency, Walther’s law and entropy. Geotech Geol Eng 23(6):721–756. CrossRefGoogle Scholar
  10. Engdahl NB, Vogler ET, Weissmann GS (2010) Evaluation of aquifer heterogeneity effects on river flow loss using a transition probability framework. Water Resour Res 46(1):W01506. CrossRefGoogle Scholar
  11. Fleckenstein JH, Niswonger RG, Fogg GE (2006) River-aquifer interactions, geologic heterogeneity, and low-flow management. Ground Water 44(6):837–852. CrossRefGoogle Scholar
  12. Goovaerts P (1997) Geostatistics for natural resources evaluation. Applied Geostatistics. Oxford University Press, New YorkGoogle Scholar
  13. Hansen AL, Gunderman D, He X, Refsgaard JC (2014) Uncertainty assessment of spatially distributed nitrate reduction potential in groundwater using multiple geological realizations. J Hydrol 519:225–237. CrossRefGoogle Scholar
  14. Harp DR, Vesselinov VV (2010) Stochastic inverse method for estimation of geostatistical representation of hydrogeologic stratigraphy using borehole logs and pressure observations. Stoch Env Res Risk A 24(7):1023–1042. CrossRefGoogle Scholar
  15. Harp DR, Vesselinov VV (2012) Analysis of hydrogeological structure uncertainty by estimation of hydrogeological acceptance probability of geostatistical models. Adv Water Resour 36:64–74. CrossRefGoogle Scholar
  16. He F, Wu J (2003) Markov chain-based multi-indicator geostatistical model. Hydrogeol Eng Geol (05):28–32Google Scholar
  17. He Y, Hu K, Li B, Chen D, Suter HC, Huang Y (2009) Comparison of sequential Indicator simulation and transition probability Indicator simulation used to model clay content in microscale surface soil. Soil Sci 174(7):395–402. CrossRefGoogle Scholar
  18. He X, Koch J, Sonnenborg TO, Jorgensen F, Schamper C, Refsgaard JC (2014) Transition probability-based stochastic geological modeling using airborne geophysical data and borehole data. Water Resour Res 50(4):3147–3169. CrossRefGoogle Scholar
  19. He X, Højberg AL, Jørgensen F, Refsgaard JC (2015) Assessing hydrological model predictive uncertainty using stochastically generated geological models. Hydrol Process 29(19):4293–4311. CrossRefGoogle Scholar
  20. Hölzl J (2016) Markov chains and Markov decision processes in Isabelle/HOL. J Autom Reason 59(3):1–43Google Scholar
  21. Høyer A-S, Vignoli G, Hansen TM, Vu LT, Keefer DA, Jørgensen F (2017) Multiple-point statistical simulation for hydrogeological models: 3-D training image development and conditioning strategies. Hydrol Earth Syst Sci 21(12):6069–6089. CrossRefGoogle Scholar
  22. Jin P, Shao J, Li C, Cui Y, Zhang L (2009) Application of T-PROGS to a 3-D numerical simulation of groundwater flow. Hydrogeol Eng Geol (04):21–26Google Scholar
  23. Karamouz M, Nokhandan AK, Kerachian R, Maksimovic C (2009) Design of on-line river water quality monitoring systems using the entropy theory: a case study. Environ Monit Assess 155(1–4):63–81. CrossRefGoogle Scholar
  24. Koch J, He X, Jensen KH, Refsgaard JC (2014) Challenges in conditioning a stochastic geological model of a heterogeneous glacial aquifer to a comprehensive soft data set. Hydrol Earth Syst Sci 18(8):2907–2923. CrossRefGoogle Scholar
  25. Lee SY, Carle SF, Fogg GE (2007) Geologic heterogeneity and a comparison of two geostatistical models: sequential Gaussian and transition probability-based geostatistical simulation. Adv Water Resour 30(9):1914–1932. CrossRefGoogle Scholar
  26. Lelliott MR, Cave MR, Wealthall GR (2009) A structured approach to the measurement of uncertainty in 3D geological models. Q J Eng Geol Hydrogeol 42(1):95–105. CrossRefGoogle Scholar
  27. Lerche I, Noeth S (2009) Value change in oil and gas production: I. additional information at fixed cost but variable resolution probability. Energy Explor Exploit 20(1):1–16CrossRefGoogle Scholar
  28. Ma L (2013) Research on multiscale model of heterogeneous porous medium and its application in groundwater modeling. PhD, Hefei University of TechnologyGoogle Scholar
  29. Ortiz JDO, Felgueiras CA, Camargo ECG, Rennó CD, Ortiz MJ (2017) Spatial modeling of soil lime requirements with uncertainty assessment using geostatistical sequential Indicator simulation. Open J Soil Sci 07(7):133–148CrossRefGoogle Scholar
  30. Refsgaard JC, Auken E, Bamberg CA, Christensen BS, Clausen T, Dalgaard E, Efferso F, Ernstsen V, Gertz F, Hansen AL, He X, Jacobsen BH, Jensen KH, Jorgensen F, Jorgensen LF, Koch J, Nilsson B, Petersen C, De Schepper G, Schamper C, Sorensen KI, Therrien R, Thirup C, Viezzoli A (2014) Nitrate reduction in geologically heterogeneous catchments—a framework for assessing the scale of predictive capability of hydrological models. Sci Total Environ 468-469:1278–1288. CrossRefGoogle Scholar
  31. Skeel RD, Fang YH (2017) Comparing Markov chain samplers for molecular simulation. Entropy 19(10):561. ARTN 561. CrossRefGoogle Scholar
  32. Sun WX, Zhao YC, Huang B, Shi XZ, Darilek JL, Yang JS, Wang ZG, Zhang BE (2012) Effect of sampling density on regional soil organic carbon estimation for cultivated soils. J Plant Nutr Soil Sci 175(5):671–680. CrossRefGoogle Scholar
  33. Swan A (1996) Stochastic modeling and geostatistics. Principles, methods, and case studies. Geol Mag 133(2)Google Scholar
  34. Tan VYF, Hayashi M (2018) Analysis of remaining uncertainties and exponents under various conditional Rényi entropies. IEEE Trans Inf Theory 64(5):3734–3755. CrossRefGoogle Scholar
  35. Weissmann GS, Carle SF, Fogg GE (1999) Three dimensional hydrofacies modeling based on soil surveys and transition probability geostatistics. Water Resour Res 35(6):1761–1770. CrossRefGoogle Scholar
  36. Wellmann JF, Horowitz FG, Schill E, Regenauer-Lieb K (2010) Towards incorporating uncertainty of structural data in 3D geological inversion. Tectonophysics 490(3–4):141–151. CrossRefGoogle Scholar
  37. Yamamoto JK, Landim PMB, Kikuda AT, Leite CBB, Lopez SD (2015) Post-processing of sequential indicator simulation realizations for modeling geologic bodies. Comput Geosci 19(1):257–266CrossRefGoogle Scholar
  38. Yang L, Hou W, Cui C, Cui J (2016) GOSIM: a multi-scale iterative multiple-point statistics algorithm with global optimization. Comput Geosci 89:57–70. CrossRefGoogle Scholar
  39. Ye M, Khaleel R (2008) A Markov chain model for characterizing medium heterogeneity and sediment layering structure. Water Resour Res 44(9):W09427. CrossRefGoogle Scholar
  40. Yin L, Deng Y (2018) Toward uncertainty of weighted networks: an entropy-based model. Phys A Stat Mech Appl 508:176–186. CrossRefGoogle Scholar

Copyright information

© Saudi Society for Geosciences 2019

Authors and Affiliations

  1. 1.School of Water Resources and EnvironmentChina University of GeosciencesBeijingChina
  2. 2.Inner Mongolia Autonomous Region, the Fourth Hydrogeologic and Engineering Geological Prospecting InstituteTongliaoChina
  3. 3.Center for Hydrogeology and Environmental Geology SurveyBaodingChina

Personalised recommendations