Table Of ContentSPRINGER BRIEFS IN EARTH SCIENCES
Jan Dirk Jansen
A Systems
Description of
Flow Through
Porous Media
SpringerBriefs in Earth Sciences
For furthervolumes:
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Jan Dirk Jansen
A Systems Description
of Flow Through
Porous Media
123
Jan DirkJansen
Department of Geoscience andEngineering
Delft Universityof Technology
Delft
The Netherlands
ISSN 2191-5369 ISSN 2191-5377 (electronic)
ISBN 978-3-319-00259-0 ISBN 978-3-319-00260-6 (eBook)
DOI 10.1007/978-3-319-00260-6
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013935769
(cid:2)TheAuthor(s)2013
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Preface
This text forms part of material taught during an elective course in advanced
reservoir simulation at Delft University of Technology over the past 10 years.
Thecourseisaimedatstudentswithanappliedmathematicsorphysicsbackground
interested in petroleum engineering applications, and petroleum engineering
studentsseekingtodeepentheirunderstandingofthesystemsandcontrolaspectsof
reservoirmanagement.Partofthematerialhasalsobeenpresentedatvariousshort
courses for industrial and academic researchers interested in background knowl-
edgeneededtoperformresearchintheareaofclosed-loopreservoirmanagement,
also known as smart fields, related to, e.g., model-based production optimization,
data assimilation (or history matching), model reduction or upscaling techniques.
All these topics have connections to system-theoretical concepts such as observ-
ability,controllability,oridentifiability.Inordertofacilitatetheaccessibilityofthe
systems-and-controlliterature,thecoursestartswithareviewoftheequationsfor
flowthroughporousmediainasystems-and-controlnotation.Althoughthetheory
islimitedtotherelativelysimplesituationofhorizontaltwo-phase(oil–water)flow,
it covers several typical aspects of porous media flow. Over the year, various
editions of the course notes ‘‘Systems theory for reservoir management’’, which
contain much more material than presented in this text, have found their way to
interested parties outside the university. The introductory part, i.e., the Systems
Description of Flow Through Porous Media, forms the topic of this brief mono-
graph.Chapterorsectionheadingsmarkedwithastarsuperscriptindicatematerial
thatcanbeskippedwithoutlosingtrackofthegeneraltrainofthoughts.Ihopethe
contents of this text will be of value to students and researchers interested in the
applicationofsystemsandcontrolconceptstooilandgasreservoirsimulationand
other application areas of subsurface flow simulation such as CO storage,
2
geothermal energy, or groundwater remediation. I like to acknowledge the many
studentsandcolleagues,withinandoutsideTUDelft,whosecriticalcommentsand
constructive remarks have helped to improve the text over the years. If you have
any further comments, please letme knowat [email protected].
Delft, February 2013 Jan Dirk Jansen
v
Contents
1 Porous-Media Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Single-Phase Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.2 Finite-Difference Discretization. . . . . . . . . . . . . . . . . . 7
1.3.3 Example 1: Single-Phase Flow in a Simple
Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4 Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.5 Mass-Conservative Formulation*. . . . . . . . . . . . . . . . . 12
1.3.6 Well Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.2 Nature of the Equations . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.3 Relative Permeabilities. . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.4 Example 2: Two-Phase Flow in a Simple Reservoir. . . . 21
1.4.5 Buckley-Leverett Equation* . . . . . . . . . . . . . . . . . . . . 21
1.4.6 Linear Approximation* . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.7 Formation Volume Factors* . . . . . . . . . . . . . . . . . . . . 26
1.4.8 Finite-Difference Discretization. . . . . . . . . . . . . . . . . . 28
1.4.9 Example 3: Inverted Five-Spot . . . . . . . . . . . . . . . . . . 30
1.4.10 Sources of Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 31
1.4.11 Incompressible Flow . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.4.12 Fluid Velocities*. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 System Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1 System Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1.1 Partial-Differential Equations . . . . . . . . . . . . . . . . . . . 39
2.1.2 Ordinary-Differential Equations. . . . . . . . . . . . . . . . . . 40
2.1.3 State-Space Representation. . . . . . . . . . . . . . . . . . . . . 41
2.1.4 Linearized Equations . . . . . . . . . . . . . . . . . . . . . . . . . 44
vii
viii Contents
2.2 Single-Phase Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.1 System Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.2 Example 1 Continued—Location Matrix. . . . . . . . . . . . 50
2.2.3 Prescribed Pressures and Flow Rates . . . . . . . . . . . . . . 51
2.2.4 Well Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2.5 Example 1 Continued—Well Model . . . . . . . . . . . . . . 55
2.2.6 Elimination of Prescribed Pressures* . . . . . . . . . . . . . . 56
2.2.7 System Energy* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.3 Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.3.1 System Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.3.2 Well Operating Constraints. . . . . . . . . . . . . . . . . . . . . 68
2.3.3 Computational Aspects. . . . . . . . . . . . . . . . . . . . . . . . 69
2.3.4 Lift Tables* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.3.5 Control Valves* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.3.6 Streamlines*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.3.7 System Energy* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3 System Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.1 Free Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.1.1 Homogeneous Equation . . . . . . . . . . . . . . . . . . . . . . . 79
3.1.2 Diagonalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.1.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.1.4 Singular System Matrix . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.5 Example 1 Continued: Free Response . . . . . . . . . . . . . 84
3.2 Forced Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2.1 Nonhomogeneous Equation. . . . . . . . . . . . . . . . . . . . . 87
3.2.2 Diagonalization and Modal Analysis . . . . . . . . . . . . . . 88
3.2.3 Singular System Matrix . . . . . . . . . . . . . . . . . . . . . . . 89
3.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.3.1 Explicit Euler Discretization. . . . . . . . . . . . . . . . . . . . 90
3.3.2 Implicit Euler Discretization. . . . . . . . . . . . . . . . . . . . 92
3.3.3 Picard and Newton–Raphson Iteration . . . . . . . . . . . . . 93
3.3.4 Numerical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.3.5 IMPES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3.6 Computational Aspects. . . . . . . . . . . . . . . . . . . . . . . . 99
3.3.7 Control Aspects*. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.3.8 Stream Line Simulation* . . . . . . . . . . . . . . . . . . . . . . 101
3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.4.1 Example 1 Continued: Stability. . . . . . . . . . . . . . . . . . 103
3.4.2 Example 2 Continued: Mobility Effects . . . . . . . . . . . . 105
3.4.3 Example 3 Continued: Well Constraints. . . . . . . . . . . . 107
Contents ix
3.4.4 Example 3 Continued: Time-Stepping Statistics . . . . . . 109
3.4.5 Example 3 Continued: System Energy* . . . . . . . . . . . . 111
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Symbols
Notes:
• Severalsymbolsoccurmorethanoncebecausetheyhaveadifferentmeaningin
different parts of the text.
• The dimensions of vectors and matrices have only been indicated when all
elements have the same dimensions.
Symbol Description Dimensions SIunits
a Coefficient – –
A Area L2 m2
A Systemmatrix – –
A Jacobianmatrix – –
b Coefficient – –
B Oilformationvolumefactor – –
B Inputmatrix – –
c Compressibility t2m-1L-1 1/Pa
C Integrationconstant L-3m kg/m3
C Outputmatrix – –
d Depth L m
D Diffusionconstant L2t-1 m2/s
D Direct-throughputmatrix – –
E Energy L2mt-2 J
E Accumulationmatrix – –
e Nonlinearfunction – –
e Nonlinearvector-valuedfunction – –
f Nonlinearfunction – –
f Fractionalflow – –
f Nonlinearvector-valuedfunction – –
F Fractional-flowmatrix – –
g Accelerationofgravity Lt-2 m/s2
g Velocitygradient t-1 1/s
g Nonlinearvector-valuedfunction – –
h Reservoirheight L m
(continued)
xi
xii Symbols
(continued)
h Nonlinearvector-valuedoutputfunction – –
i Counter – –
i Unitvector – –
j Counter – –
j Nonlinearvector-valuedoutputfunction – –
I Identitymatrix – –
J Wellindex,productivityindex L2m-1 m3/
(Pas)
J Wellindexmatrix L2m-1 m3/
(Pas)
k Permeability L2 m2
k Counter – –
k Discretetime – –
K Totalnumberoftimesteps – –
K~ Permeabilitytensor L2 m2
L Length m m
L Spatialdifferentialoperator – –
L Locationmatrix(selectionmatrix) – –
m Numberofelementsininputvectoru – –
m Eigenvector – –
M Matrixofeigenvectors – –
n Numberofelementsinstatevectorx – –
n Coreyexponent – –
n Unitvectornormaltoboundary – –
p Numberofelementsinoutputvectory – –
p Pressure L-1mt-2 Pa
p Pressurevector L-1mt-2 Pa
p Vectorofpressuredifferences – –
P Power L2mt-3 W
q Flowrate(sourceterm) L3t-1 m3/s
~q Flowrateoveragrid-blockboundary L3t-1 m3/s
q00 Flowrateperunitarea(sourceterm) Lt-1 m/s
q000 Flowrateperunitvolume(sourceterm) t-1 1/s
q Vectorofflowrates(sourceterms) L3t-1 m3/s
Q Scaledflowrate L-1mt-3 Pa/s
r Radius L m
r Residual – –
r Residualvector – –
s Coordinatealongacurve – –
s Saturationvector – –
S Saturation – –
S Matrixtocomputev~fromp L2M-1t m/
(Pas)
t Time t s
T Temperature T K
(continued)
