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?