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Applied math in the oil industry
April 21, 2015
Jeremy Brandman
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• Career path and industrial experience
• Overview of the oil industry
• Case studies:
• Simulating flow in an oil reservoir
• Calibrating a geologic model
Outline
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Yale, B.S. Math 1998-2002
Chicagoland Jewish High
School, Instructor
Math 2002-2003
UCLA, Ph.D. Applied Math 2003-2008
Courant Institute, NYU,
Postdoc
Applied Math 2008-2011
ExxonMobil, Researcher Corporate Strategic
Research
2011-Present
Career Path
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• Wanted to work on real-world problems.
• Background in PDE and numerical analysis.
• Looked into opportunities in many areas, including:
• Medical imaging
• National labs
• Data analytics (e.g. Google, Facebook)
• Pharmaceuticals
Why industry?
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My experience:
• Good fit with my skill set.
• Exciting to use mathematics to solve real-world problems.
• Enjoy working in teams with engineers and scientists.
Changes from university experience:
• Mathematics is only valuable insofar as it benefits the company.
• Provide value through careful reasoning and a solid understanding of the
fundamentals.
• Interdisciplinary teamwork is exciting and challenging.
• Learning the basics of other disciplines (e.g. engineering, geology) is vital.
• Clear communication is extremely important.
• High performance computing, for certain problems, is essential.
Working in industry
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Overview of the oil industry
Exploration Production Transportation
RefiningConsumer products
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Case studies
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Reservoir simulation
Oil
Rate
Cu
mu
lativ
e O
il
Predictions
Geologic model Reservoir simulation
Other applications:
• Determining injection type (gas, water)
• Optimizing well placement
• Determining the appropriate facilities
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Flow in porous media
http://www.britannica.com
Rock cores
http://eprints.maths.ox.ac.uk
Gas/oil reservoir Oil trapped in rock pores
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Rock properties:
• Rock porosity 𝜙 =fraction of void pore space
• Rock permeability 𝑘 = how permeable rock is to fluid flow
Fluid properties:
• Water saturation sa = fraction of void space occupied by water
• Fluid pressure 𝑝
• Fluid velocity 𝑣
Key fluid and rock quantities
Model inputs
Model
outputs
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Example: injecting water into an oil reservoir
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• Darcy’s law relates flow rate to rock permeability and pressure drop:
• Incompressibility:
• Darcy’s law and incompressibility leads to Laplace’s equation:
• How to determine 𝑝 and 𝑣? Different strategies: mixed methods, multi-
point flux approximations, mimetic finite differences,…
One-phase flow: PDE model
𝑣 = −𝑘
𝜇𝛻𝑝
𝑘: permeability tensor
𝛻 ⋅ 𝑣 = 0
−𝛻 ⋅𝑘
𝜇𝛻𝑝 = 0
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• Conservation of mass:
• All pore volume occupied:
• Darcy’s law:
Two-phase flow: PDE model
𝜕
𝜕𝑡(𝜙𝑠𝑎𝜌𝑎) + 𝛻 ⋅ 𝜌𝑎𝑣𝑎 = 0
𝜕
𝜕𝑡(𝜙𝑠𝑜𝜌𝑜) + 𝛻 ⋅ 𝜌𝑜𝑣𝑜 = 0
𝑠𝑎 + 𝑠𝑜 = 1
𝑣𝑎 = −𝑘 ⋅ 𝑘𝑟,𝑎(𝑠𝑎)
𝜇𝑎𝛻𝑝
𝑣𝑜 = −𝑘 ⋅ 𝑘𝑟,𝑜(𝑠𝑎)
𝜇𝑜𝛻𝑝
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• Constant density saturation equation:
•𝑑
𝑑𝑡𝑠𝑎 + 𝑠𝑜 = 1 pressure-velocity system:
Two-phase flow: simplified system
𝜕
𝜕𝑡(𝜙𝑠𝑎) + 𝛻 ⋅
𝑘𝑟,𝑎 𝑠𝑎𝜇𝑎
𝑘𝑟,𝑎 𝑠𝑎𝜇𝑎
+𝑘𝑟,𝑜 𝑠𝑎
𝜇𝑜
𝑣𝑇 = 0
𝛻 ⋅ 𝑣𝑇 = 0
𝑣𝑇 = −𝑘𝑘𝑟,𝑎(𝑠𝑎)
𝜇𝑎+𝑘𝑟,𝑜(𝑠𝑎)
𝜇𝑜𝛻𝑝
Hyperbolic: non-convex,
possibly degenerate
Elliptic system
2-way
coupling!
Rewrite system in terms of 𝑠𝑎, 𝑝, 𝑣𝑇 ≔ 𝑣𝑎 + 𝑣0
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Reservoir simulation: challenges
Complex physical
recovery processesComplex geology Large ill-conditioned
matrices
• Effective preconditioners
• Scalable linear solversEffective meshingRobust and accurate
numerical methods
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Example: increase solution accuracy
Evolution of water
front (in red) within
oil reservoir
Mesh used for computations
(colors indicate different processors)
Rese
rvo
ir M
od
el
permeability
production
injection
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Calibrating a geologic model
?Seismic
Geologic Analogs
Well
Logs
Core
SamplesProduction
Data
Geologic model
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Example: Wu et al. 2012
Large-scale trend“True” reservoir model
Injection profile
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Differences in production curves
Entire field Platform B Platform D
Observations:
• Large-scale heterogeneities are sufficient to capture overall production
curve.
• Significant differences in production exist at individual wells. Additional
data is needed to resolve finer-scale details.
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Questions?