# 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

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

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

Erlend Øian – Dept. of Math., Univ. of Bergen                     prev   back next
Parallelization                                                            12 of 14

Parallelization

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

• Oil migration

• 4 subdomains, 5 CPUs

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

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

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

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