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Re: GEOSTATS: Multiple scale structure and variogram

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  • David Garner
    ... Not necessarily. Although most practical problems have trends. In geology, overprinting of lithologic trends or multiple channel directions does this. I m
    Message 1 of 7 , Dec 9, 1997
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      >
      > 1) Is there necessarily the presence of a trend in multiscale behavior?
      Not necessarily. Although most practical problems have trends. In
      geology, overprinting of lithologic trends or multiple channel
      directions does this. I'm not a theoretician on this point and I am
      thinking of maps in cartesian space.
      >
      > 2) I am right in trying to obtain a multiple sill variogram, or is there
      > other variogram structures that could indicate multiscale behavior?
      An isotropic model could be designed from 2 structures and give what
      appear to be 2 sills. For example, a cardinal sine(sinusoidal) structure plus a
      linear(power=1) appears to give many "sills".
      >
      > 3) Does the presence of anisotropy always signals a trend? In relation
      > to that, I have been wondering, if a field is anisotropic, does it
      > mean it is nonstationary, are anisotropy and stationarity related?
      In practice anisotropy indicates a trend. Non-stationarity is a
      domain where the mean variogram rises faster than the square of
      distance between pairs of points. Anisotropy does not necessarily
      indicate non-stationarity. This can be a complicated picture in geology.
      For a depth structure map, a channel edge (0.5 kilometer) may
      represent a locally non-stationary problem and should be handled as
      such. However the map of the local area at a few kilometers may
      exhibit no non-stationarity. The channel has a prefered orientation
      and is anisotropic in behavior. The scale of the problem is
      important to the solution in this case.

      Regards,
      David


      David Garner TerraMod Consulting, Inc.

      phone: (403) 294-5136 17th Flr, 715 - 5th Ave, SW
      fax: (403) 264-4375 Calgary, AB T2P 2X6
      e-mail: garner@... Canada
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    • Pierre Goovaerts
      Basically, there are two ways to generate realizations with nested semivariogram models indicating the presence of different scales of spatial variation.
      Message 2 of 7 , Dec 10, 1997
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        Basically, there are two ways to generate realizations
        with nested semivariogram models indicating the presence of
        different scales of spatial variation.
        Consider, for example, a semivariogram model composed of two
        spherical models with a short-range (say 100m) and a large range
        (say 1000m):
        1) You could generate two realizations using the short-range and
        long-range semivariogram models, then create the final realization
        as the sum of the two fields. It works well for non-conditional
        simulation (e.g, see Goovaerts, 1992 for a one-dimensional example),
        but it is inappropriate if you want to honor observations at data
        locations.
        2) A more flexible and straightforward approach consists of
        generating directly a realization with the nested semivariogram
        model using, for example, sequential simulation algorithms,
        see Gslib book (Deutsch and Journel, 1997) or my book
        (Goovaerts, 1997).
        In the second approach, a practical obstacle to reproduction of long-range
        structures is the screening of distant data by too many data closer
        to the location being simulated. The multiple-grid concept
        (Gomez-Hernandez, 1991; Tran, 1994) allows one to reproduce long-range
        correlation structures without having to consider large search
        neighborhoods with too many conditioning data.

        consider large search neighborhoods with too many conditioning data.
        For example, a two-step simulation of a square grid 500~$\times$~500
        could proceed as follows:
        1) The attribute values are first simulated on a coarse grid (e.g.,
        25~$\times$~25) using a large search neighborhood so as to reproduce
        long-range correlation structures. Because the grid is coarse, each
        neighborhood contains few data, which reduces the screening effect.
        2) Once the coarse grid has been completed, the simulation continues on
        the finer grid 500~$\times$~500 using a smaller search neighborhood so
        as to reproduce short-range correlation structures. The previously
        simulated
        values on the coarse grid are used as data for the simulation on the
        fine grid.
        The multiple-grid concept is implemented in the new version of Gslib.

        References:
        -----------
        Deutsch CV, Journel AG (1997) GSLIB: Geostatistical
        Software Library and User's Guide: second edition. Oxford University
        Press, New-York, 369 pp.

        Gomez-Hernandez J (1991) A Stochastic Approach to the Simulation
        of Block Conductivity Fields Conditioned upon Data Measured at a
        Smaller Scale. Doctoral dissertation, Stanford University, Stanford, CA.

        Goovaerts P (1992) Factorial kriging analysis: A useful tool for
        exploring the structure of multivariate spatial soil information.
        J. Soil Sci. 43: 597--619.

        Goovaerts P (1997) Geostatistics for Natural Resources Evaluation.
        Oxford Univ. Press, New-York, 512 pp.

        Tran T (1994) Improving variogram reproduction on dense simulation grids.
        Computers & Geosciences, 20(7):1161--1168.


        <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

        ________ ________
        | \ / | Pierre Goovaerts
        |_ \ / _| Assistant professor
        __|________\/________|__ Dept of Civil & Environmental Engineering
        | | The University of Michigan
        | M I C H I G A N | EWRE Building, Room 117
        |________________________| Ann Arbor, Michigan, 48109-2125, U.S.A
        _| |_\ /_| |_
        | |\ /| | E-mail: goovaert@...
        |________| \/ |________| Phone: (313) 936-0141
        Fax: (313) 763-2275

        <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>


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      • Timothy D. Scheibe
        ... One way to do this conditionally, that is also attractive from a geological / soft-data perspective, is to use a two-stage simulation approach where the
        Message 3 of 7 , Dec 19, 1997
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          In his response to Sophie Dufresne, P. Goovaerts wrote:

          >1) You could generate two realizations using the short-range and
          > long-range semivariogram models, then create the final realization
          > as the sum of the two fields. It works well for non-conditional
          > simulation (e.g, see Goovaerts, 1992 for a one-dimensional example),
          > but it is inappropriate if you want to honor observations at data
          > locations.

          One way to do this conditionally, that is also attractive
          from a geological / soft-data perspective, is to use a two-stage
          simulation approach where the long-range model represents the
          correlation of some discrete geological entities (e.g., facies),
          and the small-scale model(s) represent local correlation of the
          property of interest (e.g., hydraulic conductivity) within facies
          bodies. You simulate the spatial distribution of facies
          first using a categorical indicator stochastic simulation method.
          Realizations can be conditioned to observations of facies type.
          Then, for each facies type you simulate a complete
          field of the variable of interest, with short-range correlation
          structure, using either continuous indicator or gaussian
          simulation method. These realizations can be conditioned to
          observations of the simulated variable (e.g., hydraulic conductivity).
          The final result is obtained by mapping each local realization into
          those spatial locations where the corresponding facies type was
          simulated, and honors observations of both facies type and
          hydraulic conductivity.

          Possible advantages of this approach are that 1) it
          allows for a distinct separation of scales rather than a smooth
          transition between scales of correlation such as you would get with
          a single multi-scale variogram model, 2) it allows direct
          conditioning to both facies type and hdyraulic conductivity,
          and 3) it incorporates qualitative geological information
          (facies classifications).

          Possible disadvantages are 1) you must simulate several fields to
          combine into one final realization - computationally demanding,
          and 2) you must have enough data to support estimation of local
          variograms within each facies type. Overall, however, this
          is a simple, intuitive, and consistent approach to simulating
          correlated fields with multiple discrete scales of structure,
          such as one expects to see in geological systems where there
          are discrete scales of structure (e.g., sedimentary structures,
          various scales and types of bedding). We are using this approach
          in a stochastic groundwater modeling study, and I noticed
          other investigators using similar approaches at the recent
          AGU conference.


          =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
          Tim Scheibe -- Senior Research Scientist -- Hydrology
          Pacific Northwest National Laboratory
          P.O. Box 999 MSIN K9-36 Richland WA 99352
          Phone: (509)372-6065 FAX: (509)372-6089 E-mail: td_scheibe@...
          WWW: http://terrassa.pnl.gov:2080/
          =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

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        • David Garner
          Tim, Your suggested approach sounds great. The physical basis is intriguing. For building geological models for reservoir simulation for the oil industry we
          Message 4 of 7 , Dec 21, 1997
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            Tim,
            Your suggested approach sounds great. The physical basis is
            intriguing. For building geological models for reservoir
            simulation for the oil industry we currently use the Heresim 3D
            software from Beicip-Franlab,Paris and Geomath, Houston. The idea that
            facies can be correlated over long range, i.e. over several well
            spacings is the first idea. Then within each facies category, the
            petrophysics is attributed to each facies.

            For Facies, the simualtion technique is the truncated gaussian using
            lithofacies proportions to constrain the volume of facies and to
            provide the cut-offs for the truncation at each stratigraphic level.
            Vertical proportion curves are a fixed probability vector along each
            level of the bed layering. Variograms provide a geometry for the
            realizations (Gaussian widths). Non-stationarity is handled through
            multiple vertical proportion curves.

            Each facies has a petrophysical model describing porosity and 3
            directions of permeability. A MonteCarlo simulation is most commonly
            used here. Core and logs provide most of the data. There is a
            volume variance/sample support arguement as to why we might not want
            to honor the exact permeability value from a core plug for a cell
            that is 25mx25mx0.25m in size.

            The approach above is focused on the goal of reservoir simulation. The
            physical basis is in close agreement with what you propose. I
            strongly agree with the facies first approach.

            Regards,
            David


            David Garner TerraMod Consulting, Inc.

            phone: (403) 294-5136 17th Flr, 715 - 5th Ave, SW
            fax: (403) 264-4375 Calgary, AB T2P 2X6
            e-mail: garner@... Canada
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          • Pierre Goovaerts
            I just want to mention that indeed, in the context of reservoir description, a two-step approach involving a prior modeling of the spatial distribution of
            Message 5 of 7 , Dec 22, 1997
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              I just want to mention that indeed, in the context of reservoir
              description, a two-step approach involving a prior modeling of the
              spatial distribution of facies is a good idea.
              Examples of the implementation of this algorithm can be
              found in these two papers:

              1) Goovaerts, P (1996) Stochastic simulation of categorical variables
              using a classification algorithm and simulated annealing,
              Mathematical Geology, 28: 909-921.

              2) Goovaerts, P & A.G. Journel (1996) Accounting for local probabilities
              in stochastic modeling of facies data, SPE Journal, 1:21-29.



              <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

              ________ ________
              | \ / | Pierre Goovaerts
              |_ \ / _| Assistant professor
              __|________\/________|__ Dept of Civil & Environmental Engineering
              | | The University of Michigan
              | M I C H I G A N | EWRE Building, Room 117
              |________________________| Ann Arbor, Michigan, 48109-2125, U.S.A
              _| |_\ /_| |_
              | |\ /| | E-mail: goovaert@...
              |________| \/ |________| Phone: (313) 936-0141
              Fax: (313) 763-2275

              <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>


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            • Gary Weissmann
              Hi all, This approach of modeling facies first makes great sense. We currently are using a transition probability/Markov approach to modeling facies in
              Message 6 of 7 , Dec 22, 1997
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                Hi all,

                This approach of modeling facies first makes great sense. We currently are
                using a transition probability/Markov approach to modeling facies in
                aquifer systems. This has been very successful in alluvial aquifer
                systems, where correlation lengths are typically less than well spacings
                and geological intuition must enter the modeling. Additionally, the Markov
                chain models allow for asymmetrical distributions of facies (e.g., as found
                in fining-upward tendencies of fluvial sediments), something that the
                standard gaussian/variogram approach can not do. This methodology has been
                published in Mathematical Geology by Steve Carle and Graham Fogg [1996, v.
                28(4), p. 453-476; 1997, v.29(7) p. 8910918, and v. 29(2), p. 231-244].
                Additionally, a SEPM volume on geostatistics will include a paper by Carle,
                et al on use of this technique. I believe that Steve Carle is planning to
                release the fortran code (a modification of GSLIB code) in the near future.


                Gary


                At 6:02 AM 12/21/97, David Garner wrote:
                >Tim,
                >Your suggested approach sounds great. The physical basis is
                >intriguing. For building geological models for reservoir
                >simulation for the oil industry we currently use the Heresim 3D
                >software from Beicip-Franlab,Paris and Geomath, Houston. The idea that
                >facies can be correlated over long range, i.e. over several well
                >spacings is the first idea. Then within each facies category, the
                >petrophysics is attributed to each facies.
                >
                >For Facies, the simualtion technique is the truncated gaussian using
                >lithofacies proportions to constrain the volume of facies and to
                >provide the cut-offs for the truncation at each stratigraphic level.
                >Vertical proportion curves are a fixed probability vector along each
                >level of the bed layering. Variograms provide a geometry for the
                >realizations (Gaussian widths). Non-stationarity is handled through
                >multiple vertical proportion curves.
                >
                >Each facies has a petrophysical model describing porosity and 3
                >directions of permeability. A MonteCarlo simulation is most commonly
                >used here. Core and logs provide most of the data. There is a
                >volume variance/sample support arguement as to why we might not want
                >to honor the exact permeability value from a core plug for a cell
                >that is 25mx25mx0.25m in size.
                >
                >The approach above is focused on the goal of reservoir simulation. The
                >physical basis is in close agreement with what you propose. I
                >strongly agree with the facies first approach.
                >
                >Regards,
                >David
                >
                >
                >David Garner TerraMod Consulting, Inc.
                >
                >phone: (403) 294-5136 17th Flr, 715 - 5th Ave, SW
                >fax: (403) 264-4375 Calgary, AB T2P 2X6
                >e-mail: garner@... Canada
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                >*To post a message to the list, send it to ai-geostats@....
                >*As a general service to list users, please remember to post a summary
                >of any useful responses to your questions.
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                Gary Weissmann

                LAWR - Hydrologic Sciences
                University of California
                One Shields Avenue
                Davis, CA 95616-8627

                gweissmann@...

                (530) 752-1372
                (530) 752-5262 FAX


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