Efficient and Accurate simulation of nonlinearly coupled multiphase flow in porous media 25

74 Pages·2015·3.67 MB·English

Efficient and Accurate simulation of nonlinearly coupled multiphase flow
in porous media

25 August 2015 Meeneesh Kardale

Title : Efficient and accurate simulation of nonlinearly coupled
multiphase flow in porous media
Author(s) : Meeneesh Kardale
Date : 25 August 2015
Professor(s) : Dr. Hadi Hajibeygi
Supervisor(s) : Matteo Cusini
Postal Address : Section for Petroleum Engineering
Department of Geoscience and Engineering
Delft University of Technology
P.O. Box 5028
The Netherlands
Telephone : [phone] (secretary)
Telefax : (31) 15 2781189
Copyright (cid:13)c 2015 Section of Petroleum Engineering
All rights reserved. No parts of this publication may be reproduced, Stored in a
retrieval system, or transmitted, In any form or by any means, electronic, Mechanical,
photocopying, recording, or otherwise, Without the prior written permission of the Sec-
tion of Petroleum Engineering

Delft University of Technology
Faculty of Civil Engineering and Geosciences
Department of Geoscience & Engineering
Efficient and Accurate simulation of
nonlinearly coupled multiphase flow in
porous media
Thesis submitted to the
Delft University of Technology
in partial fulfilment of the requirements
for the degree
by
Meeneesh Kardale
[email protected]
Delft, The Netherlands
25 August 2015
Copyright (cid:13)c Meeneesh Kardale. All rights reserved.
An electronic version of this dissertation is available at
http://repository.tudelft.nl/
2
Author
Meeneesh Kardale MSc. Student
[email protected] Track: Petroleum Engineering & Geosciences
Specialization: Petroleum Engineering
Department of Geoscience & Engineering
Delft University of Technology
Title
Efficient and Accurate simulation of nonlinearly coupled multiphase flow
in porous media
Committee Members
Dr. Hadi Hajibeygi Assistant Professor
[email protected] Department of Geoscience & Engineering
Delft University of Technology
Prof.dr.ir. J.D. Jansen Professor and Department Head
[email protected] Department of Geoscience & Engineering
Delft University of Technology
Dr.ir. M.I. Gerritsma Associate Professor
[email protected] Faculty of Aerospace Engineering
Delft University of Technology
Matteo Cusini PhD. Candidate
[email protected] Department of Geoscience & Engineering
Delft University of Technology
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Contents
List of Figures iii
Abstract v
Acknowledgement vii
1 Introduction 1
1.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Peaceman Well Model . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.4 Interface properties: upwind and harmonic-average . . . . . . . . 9
1.2.5 Capillary Treatment . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Simulation Strategies for Coupled P-S Systems . . . . . . . . . . . . . . . 11
1.3.1 IMPES Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Sequential Implicit Formulation (SQ-Imp) . . . . . . . . . . . . . 11
1.3.3 Fully Implicit Formulation (FIM) . . . . . . . . . . . . . . . . . . 13
2 Stability Analysis 15
2.1 Non-Linear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Upwind Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Viscous forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Viscous and buoyancy forces . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Viscos forces and Capillarity forces . . . . . . . . . . . . . . . . . 24
3 Flux Correction Methods 27
3.1 Non-Linear Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 Trust Region Method . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.2 Locating Trust Regions . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.3 Trust Region Chopping . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Results 31
4.1 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.1 Viscous Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.2 Viscos and Buoyancy Forces . . . . . . . . . . . . . . . . . . . . . 34
5 Enhanced Oil Recovery 43
5.1 Foam in Permeable Media . . . . . . . . . . . . . . . . . . . . . . . . . . 43
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5.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3.1 Flux correction strategy for foam flow . . . . . . . . . . . . . . . . 46
6 Conclusion 49
Bibliography III
A FIM formulation VI
A.1 Derivation of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . VI
A.1.1 Flow Residual w.r.t Pw . . . . . . . . . . . . . . . . . . . . . . . . VI
A.1.2 Flow Residual w.r.t Sw . . . . . . . . . . . . . . . . . . . . . . . . VII
A.1.3 Transport Residual w.r.t Pw . . . . . . . . . . . . . . . . . . . . . VIII
A.1.4 Transport Residual w.r.t Sw . . . . . . . . . . . . . . . . . . . . . VIII
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