Streamline Simulation

283 Pages·2011·36.858 MB·English

by  Marco R. Thiele (Ed.), Margot Gerritsen (Ed.), Martin Blunt (Ed.)

Society of Petroleum Engineers
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Streamline Simulation
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S Marco R. Thiele
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Margot Gerritsen

Stanford University

Martin Blunt

Imperial College London

Society of Petroleum Engineers
© Copyright 2011 Society of Petroleum Engineers

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The purpose of this collection of papers is to introduce the subject of streamline simulation to
engineers interested in gaining a basic overview of the technology and its applications. It is
intended not to offer a comprehensive look at the topic but, rather, to provide the reader with
enough knowledge to allow for more advanced study or work in the field.

Foreword
The purpose of this collection of papers on streamline simulation (SLS) is to introduce the subject
to geoscientists and engineers interested in gaining a quick overview of this technology and its
applications. In selecting these papers we favored breadth over depth, with the focus being on
the applicability of SLS, as opposed to the numerical details. It is not a collection of best papers,
but rather a best overview of the subject through a limited number of publications. For the
specialist interested in the numerical aspects of SLS, outstanding technical papers and books are
available on that specific topic.
Streamlines have been in the petroleum literature since the mid-1930s and were popular until
the early 1970s (Morel-Seytoux 1965). Modern SLS has come into prominence since the early
1990s, and has been continuously improved since then by various research groups and
practitioners looking for workable solutions to challenging modeling problems. The usefulness of
SLS is clearly documented by the papers in this collection (Thiele et al. 2010a). Our hope is that
readers will be inspired to apply some of the lessons learned and to help widen the usability of
the methodology.
The most immediate application of SLS is in helping reservoir engineers proactively manage
mature floods (Batycky et al. 2008; Thiele and Batycky 2006; Grinestaff and Caffrey 2000).
Streamlines are particularly useful for identifying preferential flow paths and injection/production
fluid allocations, and for estimating sweep patterns. It is noteworthy that
in SLS, individual well patterns are time varying and dynamic, driven by geological connectivity
and well rates. For pro-active management of mature fields, quantifying well patterns in this
way represents a notable step toward the more common ad hoc, static and geometric definitions
of patterns.
The strong connection between geology and streamlines has allowed important progress in
the area of history matching—particularly as it pertains to modification of underlying geological
properties in a sensible and consistent way (Caers et al. 2002; Wu and Datta-Gupta 2002). The
use of SLS in history matching remains an active area of research, because proper model
calibration using historical production data requires a feedback loop to upstream geomodeling
algorithms that are themselves in constant evolution.
Proper tracing of 3D streamlines, mapping to and from the underlying static Cartesian grid
and including fluid/rock compressibility data, are at the center of modern SLS. To be sure, there
are many additional details that make SLS work as well as it does, but the selected papers give a
good overview of some of the key cornerstones (Bratvedt et al. 1996; Batycky et al. 1997;
Matringe and Gerritsen 2004; Mallison et al. 2006; Cheng et al. 2006; Jimenez and Datta-Gupta
2010).
Enhanced-oil-recovery methods center on the principle of improving the local sweep
efficiency along individual streamlines. Indeed, modern SLS originated from the need to model
miscible displacements through highly heterogeneous reservoirs (Jessen and Orr 2002; Seto et
al. 2007). Even today, these problems remain extremely challenging for traditional numerical
methods and, in many cases, timely field-scale modeling is not even possible. The extension of
SLS for thermal displacements is an example of trying to exploit the numerical efficiency of SLS
(Zhu et al. 2009). There is a dire need to make management decisions using credible field-scale
models that run in reasonable time. SLS has also been extended to solute transport (Crane and
Blunt 1999), carbon dioxide storage (Qi et al. 2009), and polymer flooding (Thiele et al. 2010b).
Fractured reservoirs are also difficult to model and SLS represents an opening in this
challenging area. Considering the enormous uncertainties that exist in defining a fracture network
and its associated properties that affect oil production, the ability to assess the impact of flow
paths that have the potential to short circuit a flood is extremely useful (Di Donato et al. 2003).
Using a single reservoir model—itself a crude approximation of the real field—is unlikely to
lead to good reservoir-management decisions. Using ensembles of models that embrace the
concept of uncertainty is a more useful approach. Running many models efficiently and using
different proxies judiciously is in line with modern reservoir-engineering thinking. SLS is an
excellent proxy for many workflows centered on quantifying uncertainty (Gilman et al. 2002;
Scheidt and Caers 2009).
Field-scale modeling is a recognized strength of SLS (Lolomari et al. 2000; McKishnie et al.
2005), and use of SLS for real-time reservoir management will continue to evolve. But there are
also many outstanding technical developments yet to be made to improve usability, applicability,
and accuracy of the approach, and we envision the next ten years bringing further refinement to
the technology.

About the Editors

Marco R. Thiele is cofounder of Streamsim Technologies and a consulting professor at Stanford
University. He holds BS and MS degrees from the University of Texas at Austin and a PhD
degree from Stanford University, all in petroleum engineering. Thiele is a recipient of the 1996
SPE Cedric K. Ferguson Medal and has served on SPE’s Books Development and Editorial
Review committees.

Margot Gerritsen is an associate professor in the Department of Energy Resources Engineering
at Stanford University. She specializes in the development of numerical algorithms for fluid flow
applications, particularly in enhanced oil recovery processes. Gerritsen holds BS and MS degrees
from Delft University of Technology and a PhD degree from Stanford University in scientific
computing and computational mathematics.

Martin Blunt is a professor of petroleum engineering and head of the Department of Earth
Science and Engineering at Imperial College London. Previously, he was an associate professor
in the Department of Petroleum Engineering at Stanford University and worked at the BP
Research Centre in Sunbury-on-Thames, U.K. Blunt holds MA and PhD degrees in physics from
Cambridge University. He is a recipient of the 1996 Cedric K. Ferguson Medal, served as
Associate Executive Editor of SPE Journal from 1996 to 1998, and was a 2001 Distinguished
Lecturer.
Contents

Historical and Background
Thiele, M.R., Batycky, R.P., and Fenwick, D.H. 2010a. Streamline Simulation for Modern
Reservoir-Engineering Workflows. J Pet Technol 62 (1): 64–70. SPE-118608-MS. DOI:
10.2118/118608-MS.

Morel-Seytoux, H.J. 1965. Analytical-Numerical Method in Waterflooding Predictions. SPE J. 5
(3): 247–258. SPE-985-PA. DOI: 10.2118/985-PA.

Application—Waterflood Management
Batycky, R.P., Thiele, M.R., Baker, R.O., and Chugh, S.H. 2008. Revisiting Reservoir Flood-
Surveillance Methods Using Streamlines. SPE Res Eval & Eng 11 (2): 387–394. SPE-95402-
PA. DOI: 10.2118/95402-PA.

Thiele, M.R. and Batycky, R.P. 2006. Using Streamline-Derived Injection Efficiencies for
Improved Waterflood Management. SPE Res Eval & Eng 9 (2): 187–196. SPE-84080-PA.
DOI: 10.2118/84080-PA.

Grinestaff, G.H. and Caffrey, D.J. 2000. Waterflood Management: A Case Study of the Northwest
Fault Block Area of Prudhoe Bay, Alaska, Using Streamline Simulation and Traditional
Waterflood Analysis. Paper SPE 63152 presented at the SPE Annual Technical Conference
and Exhibition, Dallas, 1–4 October. DOI: 10.2118/63152-MS.

Methodology
Jimenez, E.A. and Datta-Gupta, A. 2010. Full-Field Streamline Tracing in Complex Faulted
Systems With Nonneighbor Connections. SPE J. 15 (1): 7–17. SPE-113425-PA. DOI:
10.2118/113425-PA.

Mallison, B.T., Gerritsen, M.G., and Matringe, S.F. 2006. Improved Mappings for Streamline-
Based Simulation. SPE J. 11 (3): 294–302. SPE-89352-PA. DOI: 10.2118/89352-PA.

Cheng, H., Osako, I., Datta-Gupta, A., and King, M.J. 2006. A Rigorous Compressible Streamline
Formulation for Two- and Three-Phase Black-Oil Simulation. SPE J. 11 (4): 407–417. SPE-
96866-PA. DOI: 10.2118/96866-PA.

Matringe, S.F. and Gerritsen, M.G. 2004. On Accurate Tracing of Streamlines. Paper SPE 89920
presented at the SPE Annual Technical Conference and Exhibition, Houston, 26–29
September. DOI: 10.2118/89920-MS.

Batycky, R.P., Blunt, M.J., and Thiele, M.R. 1997. A 3D Field-Scale Streamline-Based Reservoir
Simulator. SPE Res Eng 12 (4): 246–254. SPE-36726-PA. DOI: 10.2118/36726-PA.

Bratvedt, F., Gimse, T., and Tegnander, C. 1996. Streamline Computations for Porous Media
Flow Including Gravity. Transport in Porous Media 25 (1): 63–78. DOI: 10.1007/BF00141262.

Application—History Matching
Caers, J., Krishnan, S., Wang, Y., and Kovscek, A.R. 2002. A Geostatistical Approach to
Streamline-Based History Matching. SPE J. 7 (3): 250–266. SPE-73144-PA. DOI:
10.2118/73144-PA.

Wu, Z. and Datta-Gupta, A. 2002. Rapid History Matching Using a Generalized Travel-Time
Inversion Method. SPE J. 7 (2): 113–122. SPE-78359-PA. DOI: 10.2118/78359-PA.
Application—Enhanced Oil Recovery
Thiele, M.R., Batycky, R.P., Pöllitzer, S., and Clemens, T. 2010b. Polymer-Flood Modeling Using
Streamlines. SPE Res Eval & Eng 13 (2): 313–322. SPE-115545-PA. DOI: 10.2118/115545-
PA.

Zhu, Z., Gerritsen, M.G., and Thiele, M.R. 2009. Thermal Streamline Simulation for Hot Water
Flooding. Paper SPE 119200 presented at the SPE Reservoir Simulation Symposium, The
Woodlands, Texas, USA, 2–4 February. DOI: 10.2118/119200-MS.

Seto, C.J., Jessen, K., and Orr, F.M. 2007. Using Analytical Solutions in Compositional
Streamline Simulation of a Field-Scale CO -Injection Project in a Condensate Reservoir. SPE
2
Res Eval & Eng 10 (4): 393–405. SPE-79690-PA. DOI: 10.2118/79690-PA.

Jessen, K. and Orr, F.M. 2002. Compositional Streamline Simulation. Paper 77379 presented at
the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, USA, 29
September–2 October. DOI: 10.2118/77379.

Application—Fractured Reservoirs
Di Donato, G., Huang, W., and Blunt, M. 2003. Streamline-Based Dual Porosity Simulation of
Fractured Reservoirs. Paper SPE 84036 presented at the SPE Annual Technical Conference
and Exhibition, Denver, 5–8 October. DOI: 10.2118/84036-MS.

Application—Field Studies
McKishnie, R.A., Malik, S., Chugh, S., Lavoie, R.G., and Griffith, P.J. 2005. Streamline
Technology for the Evaluation of Full-Field Compositional Processes: Midale, A Case Study.
SPE Res Eval & Eng 8 (5): 404–417. SPE-89363-PA. DOI: 10.2118/89363-PA.

Lolomari, T., Bratvedt, K., Crane, M., Milliken, W.J., and Tyrie, J.J. 2000. The Use of Streamline
Simulation in Reservoir Management: Methodology and Case Studies. Paper 63157
presented at the SPE Annual Technical Conference and Exhibition, Dallas, 1–4 October.
DOI: 10.2118/63157-MS.

Application—Uncertainty/Optimization
Scheidt, C. and Caers, J. 2009. Uncertainty Quantification in Reservoir Performance Using
Distances and Kernel Methods—Application to a West Africa Deepwater Turbidite Reservoir.
SPE J. 14 (4): 680–692. SPE-118740-PA. DOI: 10.2118/118740-PA.

Gilman, J.R., Meng, H.-Z., Uland, M.J., Dzurman, P.J., and Cosic, S. 2002. Statistical Ranking of
Stochastic Geomodels Using Streamline Simulation: A Field Application. Paper SPE 77374
presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas,
USA, 29 September–2 October. DOI: 10.2118/77374-MS.

Application—Other
Crane, M.J. and Blunt, M.J. 1999. Streamline-based simulation of solute transport. Water
Resources Research 35 (10): 3061–3078. DOI: 10.1029/1999WR900145.

Application—Carbon Sequestration
Qi, R., LaForce, T.C., and Blunt, M.J. 2009. Design of carbon dioxide storage in aquifers. Intl J.
Greenhouse Gas Control 3 (2): 195–205.
DISTINGUISHED AUTHOR SERIES
Streamline Simulation for Modern
Reservoir-Engineering Workflows
M.R. Thiele, Streamsim Technologies and Stanford University, and R.P. Batycky and D.H. Fenwick, Streamsim Technologies
Abstract good representation of streamlines under the assumption of
In this article, we present a high-level description of streamline- steady state.
based flow simulation and focus on four areas in which the Modern SLS used in the oil and gas industry has its roots in
technology has proved valuable: reservoir-flow surveillance, the analytical and semianalytical streamline and streamtube
flow simulation, history matching, and flood management. methods that date back to the work of Muskat and Wyckoff
We highlight the advantages and disadvantages of streamline (1934). Since then, important early contributions were made
simulation (SLS) throughout the article and conclude with a by several authors (see Datta-Gupta and King 2007 for a ref-
look at possible SLS evolution. SLS re-emerged in the early erence list). Streamlines also have a long history in the areas
1990s to alleviate some computational problems faced by of fluid mechanics and groundwater flow, and petroleum
finite-difference (FD) simulation when confronted with high- literature has drawn heavily from those sources.
resolution geological models characterized by heterogeneous In contrast to the early semianalytical streamtube work of
spatial distributions of static properties. Since then, develop- the 1970s and 1980s, modern SLS generally is understood
ment and application of SLS has advanced the technology sig- to be associated with work published after 1990 and is char-
nificantly, such that SLS complements conventional-modeling acterized by six important ideas: tracing 3D streamlines by
approaches in many reservoir-engineering (RE) workflows. use of the concept of “time of flight” (TOF) rather than arc
length, expressing the mass-conservation equations in terms
What Is SLS? of TOF, periodic updating of the streamlines in time, solving
A streamline is defined as a line that is everywhere tangent to the transport problems numerically along the streamlines
the local velocity field at a given instant in time. The smoke rather than analytically, accounting for gravity effects, and
lines generated in a wind tunnel and shown in advertise- extension to compressible flow. All of these improvements
ments to demonstrate aerodynamic qualities of cars are a originated from the need to relax the limiting assumptions
inherent in the early semianalytical streamtube methods and
adapt the method to more-realistic and -complex reservoir
Marco Thiele, SPE, is cofounder of Streamsim Technologies and scenarios (Batycky et al. 1997).
a Consulting Professor at Stanford University. He earned BS and The distinguishing feature of streamline-based flow simu-
MS degrees from the University of Texas at Austin and a PhD lation is that fluids are transported over a timestep (t to t+Δt)
degree from Stanford University, all in petroleum engineering. along streamlines rather than from cell-to-cell as in conven-
Thiele is a recipient of the 1996 SPE Cedrick K. Ferguson Medal, tional FD methods. Because streamlines represent an image
serves on the SPE Books Committee, and is an Associate Editor of the instantaneous velocity field, anything assumed to
for SPEREE. move with the total velocity field will follow the streamlines
until the velocity field is updated to account for its changing
Rod Batycky, SPE, is cofounder of Streamsim Technologies. He behavior in time. The geometry of the streamlines and the
earned a BS degree in chemical engineering from the University of velocity at which components travel along each individual
Calgary and MS and PhD degrees in petroleum engineering from streamline result directly from the spatial distribution of the
Stanford University. Batycky is a recipient of the 1996 SPE Cedrick static petrophysical properties (e.g., permeability, porosity,
K. Ferguson Medal. Previously, he worked as a reservoir engineer and relative permeability regions) and the volumes produced/
with Shell Canada. Batycky is a Technical Editor for SPEREE and injected at the wells. The ability of streamlines to visualize
JCPT. He is a registered Professional Engineer in Alberta. flow is unmistakable, even to the untrained eye.
To trace the streamlines at a particular time t, the total
Darryl H Fenwick, SPE, is project leader of the Streamsim/ velocity field must be known at that instant. This idea high-
Stanford History Matching Joint Industry Project (JIP) that
fast tracks advanced history-matching technology developed at
Copyright 2010 Society of Petroleum Engineers
Stanford University to members of the JIP. He earned BS, MS, and This is paper SPE 118608. Distinguished Author Series articles are general, descriptive
PhD degrees, all from Stanford University in petroleum engineer- representations that summarize the state of the art in an area of technology by describing recent
developments for readers who are not specialists in the topics discussed. Written by individuals
ing. Fenwick is a Technical Editor for SPEREE. recognized as experts in the area, these articles provide key references to more definitive work
and present specific details only to illustrate the technology. Purpose: to inform the general
readership of recent advances in various areas of petroleum engineering.
64 JPT • JANUARY 2010
Fig. 1—The dual-grid approach in SLS: (left) The static Eulerian grid is used to calculate the velocity field by use
of the pressure field and Darcy’s law; (right) the dynamic Lagrangian grid (streamlines) is used to transport
components from an upstream to a downstream location over a timestep, Δt.
lights the first key characteristic of SLS: It is a dual-grid until the desired final simulation time is reached. How many
approach. A traditional Eulerian (static) time-invariant grid times the streamlines are updated (i.e., the number of Δt’s)
is used to calculate the total velocity field, while a Lagrangian is user defined, although there are guidelines and numeri-
(dynamic) time-variant grid is used to transport components cal constraints to ensure a sufficiently accurate solution.
from an upstream to a downstream location along stream- Large changes in well rates, or new wells coming on line,
lines. Fig. 1 shows the 3D pressure field (left), calculated for example, typically force an update of the streamlines
assuming spatially varying static petrophysical properties (Fig. 2).
and well conditions. Once the pressure field is known, the One way to frame SLS is the concept of overall sweep effi-
spatial velocity field is constructed by use of Darcy’s law and ciency, E, as a product of the volumetric sweep, E , times the
V
the streamlines (right) traced in 3D. By displaying water displacement efficiency, E ,
D
(blue) and oil (red) saturations along each streamline, fast-
vs. slow-fluid paths are clearly recognizable. E=E ×E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
V D
Reservoir-flow simulation involves constructing a spatial
and temporal distribution of pressure(s) and fluid compo- The volumetric sweep efficiency of an injection well is the
sitions, given a static petrophysical description, an initial reservoir volume contacted by the streamlines associated
state of the reservoir, and the temporal injection/production with that well, while the displacement efficiency is given by
of fluid volumes. SLS accomplishes this by first solving the the transport equation(s) solved along individual stream-
pressure on the static Eulerian grid with a conventional lines. This points to one of the most appealing characteristics
FD approach, then constructing the total velocity field of SLS: breaking up the original 3D domain into a series of
from the newly obtained spatial pressure distribution, the 1D independent objects along which the relevant transport
static petrophysical description, and Darcy’s law and finally equations are solved, then reassembling the 3D solution by
tracing the streamlines that form the Lagrangian grid. The mapping back from the streamlines to the original static grid.
streamlines are assumed to remain fixed for a period Δt, and This breakup and reassembly for each timestep to generate
components are transported along this grid from t to t+Δt. the new distributions of compositions and pressures can be
At this point, the new state of the system (pressures and numerically efficient and at the same time can provide useful
compositions) at t+Δt is known and the process is repeated insight into the displacement process.
Fig. 2—Streamline change as new wells are brought on line. Here, the streamlines are colored by terminating
producers, and the streamline geometries change as new wells are added over time.
JPT • JANUARY 2010 65
DISTINGUISHED AUTHOR SERIES
Advantages and Drawbacks by FD methods, underscoring the complementary nature of
Much has been written on the benefits of SLS, such as speed SLS and FD approaches. Problems that fall in between are
and the ability to process high-resolution grids efficiently. more difficult to decide on. Initial depletion of a reservoir that
However, understanding the limitations of SLS is equally creates a gas cap, followed by repressurization by water injec-
important for proper application of the technology. Some of tion, is a classic hybrid case. With experience, such problems
the advantages and disadvantages of SLS are discussed next. can be solved sequentially, by use of FD for the expansion and
The major advantage of SLS compared with other simula- repressurization phases and then by use of SLS.
tion approaches is the information provided by the stream- The biggest drawbacks of SLS may come from its two core
lines themselves. There are two particularly useful sources architectural features: the dual grid and the assumption that
of data. First, streamlines can outline the drainage and streamlines are independent of each other. The dual grid
irrigation volumes associated with producers and injec- requires repeated mapping of the solution variables—pres-
tors, respectively. It is possible to know which gridblocks sure and overall compositions—between the static Eulerian
are associated with which well—injector or producer—at and the dynamic Lagrangian grid, which leads to a method
any particular time. These regions can be used in well- that is inherently not mass conservative. Additionally, the
level assisted-history-matching workflows to decide how to independence between streamlines does not favor capturing
modify static-grid properties to improve the match between physics that is transverse to the main direction of flow, such
simulated and historically observed volumes. Another use as might be the case with gravity (driven by density gradi-
can be as a metric to establish the effectiveness of scaleup ents), transverse-capillary-pressure (driven by saturation
methodologies. The second data source comes from sum- gradients), diffusion (driven by concentration gradients),
ming the volumetric flow rates associated with all the compressibility (in all directions), and transverse-thermal
streamlines connecting an injector/producer pair. Doing so (temperature gradients) effects. These difficulties can be
enables determining the well-rate allocation factors (WAFs) alleviated with an operator-splitting approach, which solves
(i.e., the percentage of flow from one well to each offset the convective part along the streamlines and the diffusive
well with which it communicates). Thus, streamlines offer a part on the Eulerian grid. Nevertheless, it remains a sequen-
simple solution to the challenging problem of trying to asso- tial approach to capturing nonlinearities in the flow, which
ciate produced and injected volumes. Well-allocation data might not always be appropriate. In fact, modern SLS can be
are critical to workflows that are based on pattern analysis considered a sequential multigrid method to solve nonlinear
and are critical to manage floods effectively. partial-differential equations, with the special feature that
Computational speed and memory efficiency are well- one grid is dynamic and streamline-based.
known strengths of SLS. Because the transport problem— The authors’ experience with SLS over the last 15 years
usually the more-difficult and computationally involved has been that all real-world reservoir problems exhibit char-
step in reservoir simulation—is solved efficiently along 1D acteristics that do not align perfectly with the assumptions
streamlines and the streamlines are updated at user-defined in SLS. Whether SLS can still be of benefit depends strongly
intervals (timesteps), SLS can be significantly faster than FD on the questions being asked from the model, the assump-
methods. The most immediate application of the efficiency tions engineers are willing to make, and ultimately the time
of SLS is in the simulation on fine grids with a high level available for a reservoir study. Without exception, however,
of geological detail. Because streamlines lend themselves to it has been found that SLS provides useful insights into the
easy parallelization (Batycky et al. 2009), it is also possible to dynamic behavior of a reservoir and can be inserted success-
pursue such workflows on standard multicore hardware. For fully into most traditional engineering workflows.
example, a 1.5-million-active-cell waterflood model of the
Forties field (UK), with 200+ wells and 40+ years of his- SLS-Based Workflows
tory, ran in less than 2.5 hours on a 2-CPU quad-core system SLS has found its way into many areas of RE. Four spe-
compared to approximately 6 hours for a single-core run. cific examples illustrate how SLS can aid RE workflows.
However, the improved computational speed and memory Workflows are stepwise application processes used to solve
efficiency apply to problems that are particularly tailored to RE problems. The examples are not meant to be exhaustive,
SLS: slightly compressible systems in which the principal and many other applications of SLS exist that the reader is
flow physics is displacement of resident oil by an injected encouraged to explore in published literature.
fluid—usually water, miscible gas, or both—in the presence
of strongly correlated geological features. These problems are Flood Surveillance. Flood surveillance is the pairwise associ-
referred to as being convective-dominated (i.e., principally ation of produced- and injected-well volumes from observed
governed by pressure gradients rather than absolute pres- production/injection data and usually does not involve flow
sure). These cases are traditionally difficult to model with FD simulation. Flood surveillance relies on WAFs—the percent-
methods, and the use of SLS can be an effective complemen- age of total flow at a producer that can be attributed to an off-
tary approach for a broader RE analysis. set injector. Traditionally, WAFs for a producer are estimated
If SLS is best suited for problems dominated by convection, by use of a fixed geometric pattern derived from the injectors
then problems dominated by diffusive-flow physics, such as nearest to it and the angle open to flow. For large multiwell
gas expansion and capillary pressure, are more challenging for floods, geometric-pattern definitions and WAF calculations
SLS. The reason is that diffusive problems do not have a well- are time consuming and critically exposed to the strengths
defined flow direction—the exact opposite of a streamline. and weaknesses of the engineer undertaking the study.
However, such problems are treated effectively and efficiently Except for the most regular of five- or nine-spot patterns, it
66 JPT • JANUARY 2010
B
R
d,
e
c
u
d
o
r
P
Oil
e
v
ati
ul
m
u
C
Cumulative Water Injected, RB
Fig. 3—(left) Streamlines and flux-pattern map used for determining WAFs for the pattern centered on Injector
P9-7, and (right) a traditional conformance graph for all the injectors in the field calculated from the WAFs
for all injectors over all times. RB=res bbl.
is unlikely that any two engineers will determine the same can be represented with a flux-pattern map*, a convenient
patterns for a field, much less the same WAFs. This is one abstraction to show the volumetric flux between well pairs
reason that pattern-level surveillance is rarely practiced. and its relative strength. The streamline-derived WAFs then
Because streamlines connect sources and sinks, the bundle can be used to generate a conventional conformance graph:
of streamlines connecting an injector and a producer neces- offset oil production vs. volume injected for each injector
sarily quantifies the volumetric flux between the two. This (pattern). Fig. 3 shows a close-up of the streamline-derived
is a significant improvement over the guess-work associated WAFs for Injector P9-7 and the conformance graph deter-
with geometric WAFs. Also, because streamlines change mined from the WAFs for all the injectors (patterns) in the
over time with well-rate changes, so will the WAFs. Most field. As an example, the offset oil rate associated with the
importantly, SLS changes the concept of a pattern from being injection at Injector P9-7 is prorated according to the WAFs
a fixed predefined geometric object to being a dynamic for that time period as follows:
injector-centered element. A pattern becomes an injector
and its connected producers at a particular moment in time. QP9-7=0.19×QP9-6+0.27×QP9-11+0.96×QP9-8
o o o o
The pattern is dynamic because the connections change over
time and there will be no influx or efflux from such a pattern +0.85×QH43+0.48×QH37+0.49×QH33. . . . . . . . . . (2)
o o o
(Batycky et al. 2008).
To determine the streamlines, it is necessary to calculate The oil volume produced by the pattern, Q , over that time
o
the velocity field. To do so, requires as a minimum a static period is the oil rate times the time period for which that pat-
Eulerian grid with associated petrophysical properties, well tern is assumed to hold. Summing all volumes for all pattern
locations, and historical injected/produced volumes. The configurations over time for Injector P9-7 gives the confor-
static model can range from a simple one-layer homogeneous mance curve for that pattern (injector). The elegance of a
“pancake” to a 3D faulted grid with spatially varying proper- streamline-based reservoir-surveillance model is the ease with
ties. Because WAFs reflect the connectivity of the reservoir, which the reservoir engineer can perform a pattern analysis.
assumptions about faults or other spatial properties will
necessarily affect the calculation of the WAFs. In the authors’ Flow Simulation. A flow-simulation model differs from a
experience, major flow units and gross geologic properties reservoir-surveillance model in that there is a fluid-trans-
are important and should be included, but small-scale prop- port step along each streamline, and, thus, it can be used
erties, such as interwell permeabilities, have a much smaller for modeling past and future performance. Typically, SLS
effect on well-pair WAFs. This is because the WAFs are, models are introduced when the equivalent FD versions are
largely, a function of well locations and voidage rates, which, computationally too costly, as might be the case in work-
implicitly, account for the geology and the connectivity of flows involving optimizing or screening full-field models
the reservoir; wells would not be producing/injecting at the with detailed geological descriptions, hundreds of wells,
given rates if the reservoir would not allow it. and many years of history. Given the large uncertainties
A streamline-based reservoir-surveillance model is not a inherent in the model parameters—particularly geological
flow-simulation model. Streamlines are used only to calculate parameters—many would argue that estimating the range
well-pair connectivity. There is no transport step involved of uncertainty by use of SLS is more valuable than a single,
along the individual streamlines. This restriction makes it costly full-physics FD simulation.
computationally light, but it also precludes the model being
used for forecasting. Once the WAFs are calculated, they *FPMap is protected by US Patent 6,519,531.
JPT • JANUARY 2010 67