Mathematical Geosciences

, Volume 46, Issue 5, pp 597–623 | Cite as

Simultaneous Estimation of Geologic and Reservoir State Variables Within an Ensemble-Based Multiple-Point Statistic Framework

  • Liangping Li
  • Sanjay Srinivasan
  • Haiyan Zhou
  • J. Jaime Gómez-Hernández
Special Issue


Assessment of uncertainty due to inadequate data and imperfect geological knowledge is an essential aspect of the subsurface model building process. In this work, a novel methodology for characterizing complex geological structures is presented that integrates dynamic data. The procedure results in the assessment of uncertainty associated with the predictions of flow and transport. The methodology is an extension of a previously developed pattern search-based inverse method that models the spatial variation in flow parameters by searching for patterns in an ensemble of reservoir models. More specifically, the pattern-searching algorithm is extended in two directions: (1) state values (such as piezometric head) and parameters (such as conductivities) are simultaneously and sequentially estimated, which implies that real-time assimilation of dynamic data is possible as in ensemble filtering approaches; and (2) both the estimated parameter and state variables are considered when pattern searching is implemented. The new scheme results in two main advantages—better characterization of parameters, especially for delineating small scale features, and an ensemble of head states that can be used to update the parameter field using the dynamic data at the next instant, without running expensive flow simulations. An efficient algorithm for pattern search is developed, which works with a flexible search radius and can be optimized for the estimation of either large- or small-scale structures. Synthetic examples are employed to demonstrate the effectiveness and robustness of the proposed approach.


Multiple-point statistics Pattern Inverse method Ensemble-based method History matching 



The first three authors gratefully acknowledge the financial support by US Department of Energy through project DE-FE0004962. The fourth author acknowledges the financial support by Spanish Ministry of Science and Innovation through project CGL2011-23295. The authors also wish to thank the guest editors, Philippe Renard and Grégoire Mariethoz, as well as three anonymous reviewers for their comments, which substantially helped improving the final version of the manuscript.


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Copyright information

© International Association for Mathematical Geosciences 2013

Authors and Affiliations

  • Liangping Li
    • 1
  • Sanjay Srinivasan
    • 1
  • Haiyan Zhou
    • 1
  • J. Jaime Gómez-Hernández
    • 2
  1. 1.Center for Petroleum and Geosystems Engineering ResearchThe University of Texas at AustinAustinUSA
  2. 2.Research Institute of Water and Environmental EngineeringUniversitat Politècnica de ValènciaValenciaSpain

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