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Some problems of computational geophysics Yu.M. Laevsky, B.G. Mikhaylenko, G.V. Reshetova Institute of Computational Mathematics and Mathematical Geophysics SB RAS V.A. Tcheverda Institute of Petroleum Geology and Geophysics SB RAS Moscow 2013 (simulation of oil exploration and production) 1

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Outline: 1. Preliminaries and motivation 2. Oil exploration: seismic waves propagation in multiscale media 3. Oil production: filtration of two-phase fluid in inhomogeneous media 4. Parallel implementation 5. Outlook 2

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3 1. Preliminaries and motivation Fracture corridors

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4 1. Preliminaries and motivation Fracture corridors

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Samples from cavernous/fractured reservoirs 1. Preliminaries and motivation Subvertical fractures (main streamlines) Caverns along the fractures (reservoir capacitive properties) Impermeable rock matrix 5

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Fracture corridors 1. Preliminaries and motivation 6

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FC fracture corridors BFC bed controlled fracture MBF multibed fractures HPF highly persistent fractures 7 Fractures variety of carbonate collectors (J.-P.Petit, L.Bazalgette Fracture corridors: What they are?) 1. Preliminaries and motivation

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8 Scattered waves are one of the main indicator in seismic exploration of fractured structure of oil reservoir Scattered waves 1/2 1/4 1/8 One needs to take into account macro- and microheterogeneities! Solution: usage a coarse mesh for smooth background, and a fine mesh for the microscale description

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1. Preliminaries and motivation Fractured/porous media two-porous homogenization Fractures Porous blocks 9

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Injection well Production well OilOil WaterWater Oil production 1. Preliminaries and motivation 10

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2.1. Mathematical model 2.2. Numerical algorithm 2.3. Seismic waves propagation 2. Oil exploration: seismic waves propagation in multiscale media 11

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2.1. Mathematical model 12 Fluid (oil): stress tensor Skeleton (carbonate): velocity

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2.2. Numerical algorithm 13 Main requirements: The algorithm must take into account macro- and microheterogeneities to describe the scattered waves The algorithm must take into account macro- and microheterogeneities to describe the scattered waves Algorithmic artificial reflections must be small in comparison with the scattered waves Algorithmic artificial reflections must be small in comparison with the scattered waves The algorithm must have feasibility of parallel implementation The algorithm must have feasibility of parallel implementation

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2.2. Numerical algorithm 14 space time Simultaneous time-space refinement DisplacementStress

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2.3. Seismic waves propagation 15 Microscale (scattered waves) within realistic environment

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2.3. Seismic waves propagation 16 V p in XZ plane at Y=1100m V p in YZ plane at X=1100m

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2.3. Seismic waves propagation 17 V p in XY plane at Z=1650m

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2.3. Seismic waves propagation 18 Azimuthal distribution of scattered energy

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3. Oil production: filtration of two-phase fluid in inhomogeneous media 3.1. Mathematical models 3.2. Numerical algorithms 3.3. 2D examples 3.4. 3D examples 3.5. Fractured/porous media examples 19

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3.1. Mathematical models 20 2-velocity 2-pase model filtration of incompressible fluid (Masket-Leverett model): conservation law (separately in fractures and porous blocks) Darcy law Darcy law capillary pressure; partial pressure; mass exchange;

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3.2. Numerical algorithms 21 Spatial approximation: MFEM

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3.2. Numerical algorithms 22 . Integration in time: IMPES-like algorithm 2 nd order of accuracy predictor-corrector with only one calculation of r.h.s. in time step

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5-point location 3.3. 2D examples 23

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7-point location 3.3. 2D examples 24

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9-point location 3.3. 2D examples 25

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9-point location (5+4)-point location 3.3. 2D examples Control of wells: oil recovery optimization 26

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3.4. 3D examples 27 Water saturation near production wells at different porosity

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28 Fractures with small porosity Fractures with increased permeability 3.5. Fractured/porous media examples

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4. Parallel implementation 4.1. Parallelization for the problem of seismic waves propagation 4.2. Parallelization for the problem of two-phase filtration 29

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4.1. Parallelization for the problem of seismic waves propagation Domain Decomposition Domain Decomposition (separately for the coarse and fine meshes) 30

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4.1. Parallelization for the problem of seismic waves propagation Dimensional Domain Decomposition 3D 2D 1D Model volume 31

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4.1. Parallelization for the problem of seismic waves propagation Theoretical acceleration via DD 1D 2D 3D 32

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4.2. Parallelization for the problem of two-phase filtration Distribution of memory 33

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4.2. Parallelization for the problem of two-phase filtration 2D 3D 34

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5. Outlook 35 Implementation of the approach for elastic media with attenuation and anisotropy Joint simulation of oil exploration and production with taking into account movement of oil-water interface Further development of the software and access to petaflops massive computing with the assessment of the performance of exaflops computer systems At the moment, the grant for 32 million cores-hours in HRLS is received from the Partnership for Advanced Computing in Europe HRLS: Hermit Cray XE6, University of Stuttgart, No. 26 in Top 500 November 2012

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Acknowledgments Russian Foundation for Basic Research: 12-05-00943 13-01-00019 13-05-12051 36 Partnership for Advanced Computing in Europe

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! Thank you for attention! Q & A 37