Peter P. Valkó, associate professor
Harold Vance Department of Petroleum Engineering, Texas A&M University
web (personal): http://pumpjack.tamu.edu/~valko/
web (department): http://pumpjack.tamu.edu/faculty/valko/
email: mailto:p-valko@tamu.edu
office: 501K
mail: 3116 TAMU,
phone: (
fax: (
Reservoir simulation has been a territory of traditional “number crunching”. But even this traditional territory is gradually penetrated by computer algebra systems (CAS). Computer-aided mathematics has reached a level where it can support effectively many of the computations in petroleum engineering research. This course is a first attempt to make extensive use of the CAS system Mathematica in providing foundations for modeling flow in porous media.
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1-hour Examinations (35 % each, 2) |
70 % |
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Research Paper* |
30 % |
*SPE
quality and format, with references etc.
Reservoir
Simulation:
Selected SPE papers:
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Mathematica:
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A1. Single phase
flow in homogenous and isotropic porous media
Reservoir-well
systems: flow equations Driving
mechanisms, under-saturated, gas cap, water
drive Outer
boundary conditions Inner
boundary conditions / well models Single and multiple dimension
models/ cross-sectional model, areal
model Dimensionless variables Fluid/Rock properties |
B1. Introduction
to MATHEMATICA 4.1
Download all material in a zip file: z.zip B1-1-1. First notebook (conversion and
cells) B1-1-2
Numbers, symbolics, plots B1-1-3 Palettes B1-1-4 Keystrokes B1-2-1 Variables and expressions B1-2-2 Lists, tables, matrices B1-3-1 Plots and help B1-3-2 Animation B1-4-1 More 2D visualization B1-4-2 Algebraic solve B1-5-1 More visualization in 3D B1-6-1 Calculus (Limit, derivative and
integral) B1-7-1 Algebra revisited B1-8-1 Unifying idea B1-8-2 Programming B1-8-3 Techniques (Tricks) B1-9-1 Add-ons B1-9-2 Data handling B1-9-3 Formatting notebooks |
A2. Simplified
Reservoir Modeling – Well Centered Single Phase Models (Analytical and
semi-analytical methods)
Flow
regimes and time invariant characteristics, PI (HW\HW3_PVT_RelPerm_Matbal.nb) Solutions
in Laplace space Numerical
inversion of Laplace transform Transient
flow regimes Wellbore
storage and skin Solving
for the time invariant component: PI Boundary
element method for finite conductivity fracture |
B2. Solving differential equations with Mathematica: analytical,
semi-analytical methods. Time invariant PI
B2-1_1 2nd order DE and Special functions B2-2-1 Sturm-Liouville problem and
eigenvalue-eigenfunction expansion for the diffusivity (heat) equation B2-2-2 Non-homogenuous problem B2-3-1 B2-3-2 B2-3-3 Numerical inversion
of the Download:
NumericalLaplaceInversion.m B2-4-1 Transient Diffusivity equation: 1D lin
semi-infinite. Constant rate. B2-4-2 Transient Diffusivity equation: 1D lin
semi-infinite. Constant pressure. B2-4-3
Transient Diffusivity equation: 1D radial semi-infinite - line source
approximation. Constant rate. B2-4-4
Transient Diffusivity equation: 1D radial semi-infinite – finite well. Constant rate. B2-4-5
Transient Diffusivity equation: 1D radial semi-infinite- finite well.
Constant pressure. B2-5-1 Transient Diffusivity equation: 1D linear,
finite drainage area – Left: Constant Pressure; Right: Constant Pressure B2-5-2 Transient Diffusivity equation: 1D radial
finite drainage area - finite well. Constant rate. B2-5-3 Transient
Diffusivity equation: 1D radial finite drainage area - finite well. Constant
pressure. B2-6-1 PI Radial Analytic Formulae B2-6-2 PI From Boundary
element method |
A3. Reservoir
Modeling – Finite difference/ finite volume
Classic
discretization-based numerical methods: Finite difference (1D) and Finite volume element (1D) Handling boundary conditions and
wells 2D steady-state well representation A3-3
(no slides) Matrix
formalism and linear algebra solution methods Solution schemes |
B3. Finite
difference/ finite volume
B3-1-1 Reservoir simulation: finite
difference (point centered, block
centered); Transient: (explicit vs. implicit) B3-2-1 Reservoir simulation:
well representation: The Peaceman problem: isotropic B3-2-2 Reservoir
simulation: well representation: The Peaceman problem: anisotropic B3-3-1 3D Transient B3-3-2 Solver Techniques |
A4. Reservoir
Modeling – Other Approaches
Introduction
to Streamline simulation A4-2 A4-3 Higher
Order / Weighted residuals -Finite element (see handout) |
B4. Streamline
and Finite elements
B4-1-1 Streamline: Introduction to potential field B4-1-2 Streamline: Five spot pattern
(injection-production system) B4-2-1 Higher order
schemes in 1D B4-2-2 Weighted residual/finite
element, introduction |
A5. Multiphase,
multicomponent, structured reservoir modeling
Multi-phase
flow Primary and secondary variables
in Black-Oil Displacement problem , Numerical dispersion, grid
effects Primary and secondary variables
in compositional simulation A5-2 Various scales / upscaling Layers, flow units, flow compartments Handling gravity and capillary
pressure effect, vertical equilibrium Heterogeneous and anisotropic
media / fractured reservoirs Horizontal, deviated, multi-lateral
and fracture treated wells Reduction of dimensions and
scaling by using "pseudo-functions" and “well functions” Coupling
Reservoir Simulation with Geomechanical Modeling |
B5.
B5-1_OW_1D_IMPES.nb Oil displacement by water flooding, 1D, IMPES B5-3_PowerLawPermeabilityDerivation.nb B5-5_How_To_Get_Streamline_Coordinates?.nb |
Class
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Date
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Topics and
Reading Material for Class
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1 |
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A1-1 |
B1-1, B1-2, B1-3, B1-4 |
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2 |
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A1-2 (Reading
Material: M-D 133-139) |
B1-5, B1-6, B1-7 |
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3 |
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A1-3 |
B1-8, B1-9 |
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4 |
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A2-1 |
B2-1, B2-2 |
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5 |
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A2-2 |
B2-3, B2-4 |
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6 |
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A2-3 |
B2-5, B2-6 |
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7 |
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A3-1 Last hour: Exam 1 (Class
1-6) |
B3-1 |
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8 |
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A3-2 (Reading
Material: P1 and P2) |
B3-2 |
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9 |
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A3-3 (Reading
Material: M-D 141-153) |
B3-3-1 |
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10 |
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A3-4 |
B3-3-2 |
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11 |
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A4-1 |
B4-1-1 and B4-1-2 |
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12 |
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Research Paper Project Assignment Last hour: Exam 2 (Class
7-11) |
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13 |
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A5-1 |
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No Class |
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14 |
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A5-2 |
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15 |
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Projects Due |
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