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Mathematical and Numerical Modelling

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					Mathematical and Numerical Modelling

               Erlend Øian

        Department of Mathematics




            December 4, 2002
Outline                                                             1 of 14



                                                Outline

• Fault Zone Modelling and Upscaling

• Mathematical Model

• Numerical Approach: Domain Decomposition & LGR

• Numerical Experiment

• Parallelization

• Areas of Interest




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Introduction                                                                  2 of 14



                                                     Introduction
• Characteristics of flow in frac-
  tured fault zones

• 3D description of complex geo-
  metries

• Domain decomposition based
  local grid refinement

• Implicit formulation of mass
  transport equations

• Numerical                flow           simulator
  Athene


Erlend Øian – Dept. of Math., Univ. of Bergen                       prev   back next
Fault Zone Modelling                                                                               3 of 14



                                                Fault Zone Modelling

• Detailed discretization
                                                              • Conceptual fault model




Erlend Øian – Dept. of Math., Univ. of Bergen                                            prev   back next
Upscaling                                                                                      4 of 14



                                                        Upscaling

• Treating parameters individually

• Multiscale modelling (Peder Langlo, Jørg Aarnes)

                                   ˜
                                  ∂S        ¯ ˜ ˜
                                      + · [f (S)v ] = 0
                                  ∂t
                                  ↓ from fine to coarse model
                                  ∂S               ¯             ¯     1
                                     +          · [f (S)v] −   · f (S) Dm ·   S =0
                                  ∂t                                  |v| ˜


• Fractured damage zone


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Mathematical Model                                                                                                5 of 14



                                                Mathematical Model


                     ∂
                                 (ρu)dV −           (k T ) · dS = −          hρu · dS +         qdV                (1)
                     ∂t     CV                    CS ˜                  CS                 CV



                                c       n
                     ∂R ∂pw       ∂R ∂Nν    R   ∂R ∂W
                            +            =−   −       ,                           (R = Vp − Vf )                   (2)
                     ∂pw ∂t   ν=1
                                  ∂Nν ∂t    ∆t ∂W ∂t



                     ∂Nν
                         +                      a Nν v   · dS =        Qν dV,    ν = 1, 2, . . . , nc              (3)
                      ∂t           CS                             CV




Erlend Øian – Dept. of Math., Univ. of Bergen                                                           prev   back next
Numerical Approach                                                             6 of 14



                                                Numerical Approach

• Domain Decomposition
    – Enables large problems to be
      solved on parallel computers
    – Different solution techniques
      can be used in different do-
      mains

• Local Grid Refinement
    – Resolve geological features
      and regions with important
      phenomena (e.g. large gradi-
      ents)


Erlend Øian – Dept. of Math., Univ. of Bergen                        prev   back next
Numerical Approach                                                             7 of 14



                                                Numerical Approach



• Underlying coarse grid




• Final composite grid




Erlend Øian – Dept. of Math., Univ. of Bergen                        prev   back next
Numerical Experiment                                                             8 of 14



                                                Numerical Experiment

 3D fault zone
• 15 domains

• 2 sandstone layers

• 1 fault core, 4 “lenses”

• oil and gas flux into
  lower sand layer

• water flux out of upper
  sand layer


Erlend Øian – Dept. of Math., Univ. of Bergen                          prev   back next
Numerical Experiment                                                             9 of 14



                                                Numerical Experiment

 3D fault zone




Erlend Øian – Dept. of Math., Univ. of Bergen                          prev   back next
Numerical Experiment                                                            10 of 14



                                                Numerical Experiment

 3D fault zone




Erlend Øian – Dept. of Math., Univ. of Bergen                          prev   back next
Parallelization                                                            11 of 14



                                                Parallelization




• Parallelization on the subdomain level

• MPI for communication



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Parallelization                                                            12 of 14



                                                Parallelization


• Running on several platforms
     – HP
     – SGI Origin 2000
     – Linux clusters
     – IBM Regatta

• Oil migration

• 4 subdomains, 5 CPUs


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Areas of Interest                                                                   13 of 14



                                                Areas of Interest

  Fault zone modelling
     • Building more complex models, Roxar provides Irap RMS

  Multiscale models
     • Upscaling of damage zone, treating full tensors with MPFA methods

  Solving the convection/diffusion equations
     • ELLAM-like methods




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Areas of Interest                                                            14 of 14



                                                Areas of Interest

  Parallelization
     • Large cases, scale-up, load balancing
     • Parallel in time

  Inverse modelling
     • Co-operation with Rogalandsforskning

  Communication
     • http://www.uib.no/geol/SUP




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posted:11/2/2011
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