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Henry Neeman, 2 Dimitrios V. Papavassiliou 1 1 School of Chemical Engineering and Materials Science 2 School of Computer Science The University of Oklahoma OSCER Symposium on High Performance Computing Norman,Oklahoma, September 12-13 Use of HPC to Predict Porous Media Properties and Flow in Hydrocarbon Reservoirs

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P P S S Deliverability of Gas Condensate Reservoirs Reservoir Simulation Fundamentals of Rock Physics and Fluid Flow Integrated Surface Facilities Data Input for Wellbore Corrosion Models LNG LPG Plant Field Integrated Reservoir Simulation well l l Condensate wellhead r r r r S S iw The University of Oklahoma

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Method LBM (Pores) Pore Network Model (Representative Volume) Multiscale Simulation (Reservoir Scale) Hierarchical Multiscale Simulator Technology (HiMuST) Data NMR digital images Pore size distribution Porosity Permeability tensor - Allocate resources -Decide on methods Results Develop an integrated, multiscale Problem Solving Environment consisting of software components that simulate, using a hierarchy of scales, the 3D flow through anisotropic porous media, both for Darcy and non-Darcy flow. The University of Oklahoma

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Hierarchical Multiscale Simulator Technology (HiMuST) The University of Oklahoma

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Level 1: Pore scale simulation Advantages of the Lattice Boltzmann Method It is based on first principles. It is substantially parallelizable. It can handle irregular boundaries. It is equivalent to 2 nd order accurate Navier-Stokes solver (Chen et al., 1992) Discretization The porous medium is divided into a 3D network of nodes. Each node inside the solid material is set as a “wall” node and is unavailable for flow. The University of Oklahoma

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Autoparallelization Speedup The University of Oklahoma SGI Origin 2000 (@NCSA), MIPSpro compiler. 41 3 : 863s on 1 proc 105 3 : 30358s on 1 proc

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Current Parallelization Effort Done: Code has been translated to standard Fortran90 code. Optimization of computational efficiency of the program. Reordering of arrays (e.g. A(k,j,i) instead of A(i,j,k)) to improve data locality (23% increase in execution speed) Change to array notation and nested loops (e.g. avoid A=B, where A,B are arrays) to improve data locality (330% increase in execution speed !! ) Simulation of single phase flow through unconsolidated media. To Do: Simulate two-phase flow through realistic porous material (X- ray tomography images). Use Message Passing Interface to improve parallel performance The University of Oklahoma

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Level 2: Pore Network Representation The University of Oklahoma Pores and throats Array geometry Collection of Pipes (e.g., Haring & Greenkorn) (e.g., Dias & Payatakes; Mohanty et al)

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Pore Network Modeling The Code Network geometry module Two versions: 2D with two geometries (Fortran 90) 2D and 3D with fully random configuration (C++) Capability to use experimentally observed pore size distributions Common output format for network geometry Flow module Network geometry read interface Single phase, incompressible, maximum C=3 Momentum, mass and energy balances Non-linear system of equations (order of 3N) Newton’s method with direct solvers Solving for velocity and pressure drop Goal: obtain permeability and Forchheimer coefficient The University of Oklahoma

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Original Flow Matrix Structure The University of Oklahoma Matrix Structure (Newton’s Method) Problem: system of equations is very hard to solve! Failed: LAPACK PETSc QMRPack Succeeded: IMSL (sometimes) Example Pipe Network 3 5 6 1 2 4

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Improved Flow Matrix Structure The University of Oklahoma Improved Flow Matrix Structure Improved Matrix Structure The new matrix structure is smaller, but that’s not what makes it interesting.... Original Matrix Structure

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The University of Oklahoma Permuted Flow Matrix Structure After Permuting Rows The permuted matrix is lower triangular! Properties: 1 – 4 nonzero entries per row # of rows < 3 * N pr Therefore: can solve in O(N pr ) via substitution! Improved Matrix Structure

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Level 3: Macroscopic scale Anisotropic Non-Darcy Integrated Reservoir Simulator ANDIRS Done: Code has been written in standard Fortran90. Implicit methods. 2D simulation with anisotropic permeability. Validation of 2D results To Do: Further improve the numerical scheme - Use iterative solvers Incorporate well equations Incorporate anisotropic, non-Darcy flow cases The University of Oklahoma

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ANDIRS – Implicit methods, Anisitropy Kxx=120mD, Kyy=80mD Kxy=Kyx=0 Kxx=100mD, Kyy=100mD Kyx=Kxy=20mD Single – spot pattern, 20 days, Lx=Ly=100m, =25%

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Summary Scope: Develop multiscale, hierarchical simulator. Each software component (scale) is stand-alone. Prediction of porous media properties with a computational approach based on measurable properties, such as pore size distribution and porosity. The University of Oklahoma Acknowledgements: National Computational Science Alliance CTS990021N (NCSA SGI Origin2000) National Science Foundation, NSF-CMS-00084554 American Chemical Society, ACS-PRF35103-G9

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