Reservoir Modeling and Simulation

Dr. Siroos Azizmohammadi

Summer Course 2016 Department of Petroleum Engineering Chair of Reservoir Engineering September 16, 2016

1. Introduction 2. Reservoir modeling, simulation, history, workflow, and challenges 3. Fluid flow characteristics 4. Flow equations (single-phase flow, multiphase flow) 5. Two-phase flow system and Buckley-Leverett equation 6. Discretization methods 7. Finite difference method - explicit vs. Implicit scheme 8. Accuracy of solution

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

2

What is mathematical modeling?

• Mathematical modeling is the use of mathematical language to describe the behavior of a system. In other words it is mathematical description of the physical processes.

The main goal of mathematical modeling is to model

• Transport phenomena (Fluid flow, Heat transfer, Mass transfer) Mathematical model is a:

• Set of partial (ordinary) differential equations • Initial and boundary conditions • Constrains Mathematical models are derived from two general approaches

• Lumped formulation (material balance or tank model) • Distributed formulation (differential or integral form)

Arpaci, V. S., 1966, “Conduction Heat Transfer” Dietrich, P.,Helmig, R., Sauter, M., Hotzl, H., Kongeter, J., and Teutsch, G., 2005,“Flow and Transport in Fractured Porous Media”

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

3

A Reservoir is a: (1) hydrocarbon bearing zone, (2) three dimensional (3D) domain, (3) heterogeneous and anisotropic rock, (4) saturated with fluids of different composition

Balance (governing) equations • Mass or mole balance (continuity) • Momentum balance (Darcy’s Law) • Energy balance Rock and fluid equations • Rock properties • Fluid properties (PVT)

Constraint equations • Mole constraints • Saturation constraint Other equations • Source/Sink • Adsorption equations

Initial and boundary conditions • Dirichlet and Neumann boundary conditions

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

4

Reservoir simulator

• A computer program that solves governing equations for mass,

momentum (fluid flow) and heat in porous media with appropriate initial and boundary conditions and constraints, “numerically”.

Major goals of reservoir simulation • • • • • • •

Prediction of future performance of the reservoir Optimizing the recovery under various operating conditions Development plan Sensitivity and risk analysis Reservoir management Better understanding about the reservoir heterogeneity Flow units, …

Reservoir simulation

• combination of skills: physicists, mathematicians, reservoir engineers,

Mathematiacl formulation

Non-linear PDEs

Discretization

System of non-linear algebraic equations

Linearization

System of linear algebraic equations

Numeical methods

and computer scientists.

Computer algorithms and codes

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

5

•

No other solutions available (complex physics)

•

Accurate geology and petrophysics

•

Cheaper and more available than other methods

•

It is always possible to simulate the reservoir

•

Increase profitability through improved reservoir management

•

Assess economic and technical risks

•

Optimize well locations, type and spacing, conditions, …

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

6

Traditional Reservoir Engineering (1930 - 1960)

• Representation of reservoir by single block (Tank models) • One dimensional, analytical solutions for linear two-phase and radial single-phase flow

Early Reservoir Simulation (1960 - 1970)

• • • •

First generation of digital computers Simulation in research labs, high costs Limited by speed and storage Poor reliability and confidence in technology

Modern Reservoir Simulation (1970 - 1985)

• • • • • •

Decreasing hardware costs Increasing confidence in technology 3D models, large numbers of grid cells Availability of supercomputers Applications available to reservoir engineers in operating companies Multi-component fluid descriptions

Reservoir Simulation (1985 - today)

• • • • • • •

Graphical User Interfaces (GUI) Personal Computers (PC) Parallelization Multi-purpose simulation models Internet applications Reservoir simulation has become a tool for “reservoir management” Integration, integration, integration, ….

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

7

Pre-Processing

Processing

Post-Processing

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

8

Numerical errors Round-off error

Analytical Numerical

Saturation

• • •

Truncation error Numerical dispersion

Non uniqueness of solution

•

History matching is an “inverse” modeling approach (no unique

Distance

solution!)

•

Unknowns are the input parameter. We attempt to find the best set of input data to reproduce past performance.

•

Many different sets of input data may reproduce the same performance even “non-physical” values!

•

Dependent on good engineers judgment and experience

Grid orientation effects

•

Orientation of the grid may have considerable influence on the results.

Arithmetic

Averaging problems

•

What is the best way to calculate the flux between 2 blocks with different permeabilities?

Discretization methods Dr. Siroos Azizmohammadi

100

100

100

𝑘𝑘2

200

0

100

150

50

100

ln 𝑘𝑘avg = � ℎ𝑖𝑖 ln 𝑘𝑘𝑖𝑖 �� ℎ𝑖𝑖

141

0

100

𝑘𝑘avg = � 𝐿𝐿𝑖𝑖 �� 𝐿𝐿𝑖𝑖 ⁄𝑘𝑘𝑖𝑖

133

0

100

𝑘𝑘1

Averaging Method

Geometric

Harmonic

𝑛𝑛

𝑛𝑛

𝑖𝑖=1

𝑖𝑖=1

𝑘𝑘avg = � ℎ𝑖𝑖 𝑘𝑘𝑖𝑖 �� ℎ𝑖𝑖

𝑛𝑛

𝑖𝑖=1

𝑖𝑖=1

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

𝑛𝑛

𝑛𝑛

𝑛𝑛

𝑖𝑖=1

𝑖𝑖=1

September 16, 2016

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𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

𝑟𝑟

Slightly compressible fluids (𝑐𝑐 = constant and small) 𝑉𝑉 = 𝑉𝑉ref 1 + 𝑐𝑐 𝑝𝑝ref − 𝑝𝑝

Pressure

=0

Volume

Incompressible fluids (𝑐𝑐 = 0) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 = =0 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

Volume

Flow regimes • Steady state flow

Volume

Fluid types • Isothermal compressibility defined as: 1 𝜕𝜕𝜕𝜕 1 𝜕𝜕𝜕𝜕 𝑐𝑐 = − = 𝑉𝑉 𝜕𝜕𝜕𝜕 𝑇𝑇 𝜌𝜌 𝜕𝜕𝜕𝜕 𝑇𝑇

Compressible fluids (𝑐𝑐 ≠ constant) 1 1 𝜕𝜕𝜕𝜕 𝑐𝑐g = − 𝑝𝑝 𝑍𝑍 𝜕𝜕𝜕𝜕 𝑇𝑇

Pressure

Pressure

• Pseudo (quasi or semi) steady state flow 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

𝑟𝑟

= costant

Radial flow

• Unsteady state (transient) flow 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

𝑟𝑟

Spherical flow

= 𝑓𝑓 𝑟𝑟, 𝑡𝑡

Flow geometries

Linear flow Plan View Side View

Hemispherical flow

Wellbore Flow Lines

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

10

Conservation of mass (continuity equation) 𝜕𝜕 𝜌𝜌𝜌𝜌 + 𝛻𝛻 � 𝜌𝜌𝐯𝐯 = 𝑞𝑞𝑠𝑠 𝜕𝜕𝜕𝜕

Conservation of momentum (microscopic approach) 𝜌𝜌

𝜕𝜕𝐮𝐮 � � 𝝉𝝉 + 𝜌𝜌 + 𝐮𝐮 � 𝛻𝛻𝐮𝐮 + 𝛻𝛻 �𝑝𝑝 + 𝛻𝛻 �𝐠𝐠 = 0 𝜕𝜕𝜕𝜕 pressure viscous gravity inertial force

force

force

Darcy’s law (macroscopic momentum balance) 𝜙𝜙𝐮𝐮 = 𝐯𝐯 = −

Assumptions of Darcy’s law • • • • •

force

𝐤𝐤 𝐤𝐤 𝛻𝛻𝑝𝑝 + 𝜌𝜌g𝛻𝛻ℎ = − 𝛻𝛻𝛷𝛷 𝜇𝜇 𝜇𝜇

steady state flow, incompressible fluid, constant viscosity, laminar creeping flow, the pore volume is saturated with 100% of the fluid, (single phase flow, 𝐤𝐤 is absolute permeability)

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

11

Combination of continuity equation and Darcy’s law results in pressure equation 𝜕𝜕 𝜌𝜌𝜌𝜌 k − 𝛻𝛻. 𝜌𝜌 𝛻𝛻𝛻𝛻 = 𝑞𝑞𝑠𝑠 𝜕𝜕𝜕𝜕 𝜇𝜇 𝜕𝜕 𝜌𝜌𝜌𝜌 𝑑𝑑 𝜌𝜌𝜌𝜌 𝜕𝜕𝜕𝜕 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕 1 𝑑𝑑𝑑𝑑 1 𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 = = 𝜙𝜙 + 𝜌𝜌 = 𝜌𝜌𝜌𝜌 + = 𝜌𝜌𝜌𝜌 𝑐𝑐𝑓𝑓 + 𝑐𝑐𝑟𝑟 = 𝜌𝜌𝜌𝜌𝑐𝑐𝑡𝑡 𝜕𝜕𝜕𝜕 𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕 𝜌𝜌 𝑑𝑑𝑑𝑑 𝜙𝜙 𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜌𝜌𝜌𝜌𝑐𝑐𝑡𝑡

𝜕𝜕𝜕𝜕 k − 𝛻𝛻. 𝜌𝜌 𝛻𝛻𝛻𝛻 + 𝜌𝜌g𝛻𝛻ℎ 𝜕𝜕𝜕𝜕 𝜇𝜇

= 𝑞𝑞𝑠𝑠

which is called pressure diffusion (diffusivity) equation for transient flow.

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

12

One-Dimensional: • Laminar creeping flow (Darcy’s law) • Linear flow (Cartesian) • Single-phase flow • Homogeneous rock (constant permeability) • Constant viscosity • No gravity effects (horizontal flow) • Without sources/sinks ∞

𝑝𝑝 𝑥𝑥, 𝑡𝑡 − 𝑝𝑝0 𝑥𝑥 2 1 = + � sin 𝜆𝜆𝑛𝑛 𝑥𝑥 exp −𝜆𝜆𝑛𝑛 2 𝛼𝛼𝑡𝑡 𝐿𝐿 𝜋𝜋 𝑛𝑛 𝑝𝑝𝑖𝑖 − 𝑝𝑝0 𝑝𝑝 𝑥𝑥, 𝑡𝑡 = 𝑝𝑝𝑖𝑖 − 𝑝𝑝0

𝑥𝑥 𝐿𝐿

𝑛𝑛=1

+ 𝑝𝑝0

steady state solution

𝜕𝜕 2 𝑝𝑝 1 𝜕𝜕𝜕𝜕 = 𝜕𝜕𝑥𝑥 2 𝛼𝛼 𝜕𝜕𝜕𝜕 𝑝𝑝 𝑡𝑡 = 0 = 𝑝𝑝𝑖𝑖

𝑦𝑦

𝑝𝑝 𝑥𝑥 = 0 = 𝑝𝑝0 𝑝𝑝 𝑥𝑥 = 𝐿𝐿 = 𝑝𝑝𝑖𝑖

𝑛𝑛𝜋𝜋 𝜆𝜆𝑛𝑛 = , 𝑛𝑛 = 1, 2, 3, … 𝐿𝐿 characteristic values

𝑧𝑧

𝑥𝑥

𝑝𝑝𝑖𝑖

𝑝𝑝

𝑝𝑝0

𝛼𝛼 = 0.1 m2 ⁄s Transient Steady state

𝑥𝑥

𝛼𝛼 =

𝑘𝑘 𝜙𝜙𝜙𝜙𝑐𝑐𝑡𝑡

hydraulic diffusivity

Analytical methods: only for simplified cases (simple geometries, constant properties, simple initial and boundary conditions, …) but not for realistic models (complex geometries and complex systems of equations, non-linear effects and coupling between physical and chemical effects) Numerical methods have been developed to address these issues.

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

13

Saturation equation (phase continuity) Extended Darcy’s law for phase 𝛼𝛼 Relative permeability Fluid properties Capillary pressure constraint Saturation constraint Concentration constraint Transport equation

Dr. Siroos Azizmohammadi

𝜕𝜕 𝑆𝑆 𝜌𝜌 𝜙𝜙 + 𝛻𝛻. 𝜌𝜌𝛼𝛼 𝐯𝐯𝛼𝛼 = 𝑞𝑞𝑠𝑠𝑠𝑠 𝜕𝜕𝜕𝜕 𝛼𝛼 𝛼𝛼 𝐯𝐯𝛼𝛼 = −

𝐤𝐤 𝛼𝛼 𝐤𝐤𝑘𝑘𝑟𝑟𝑟𝑟 𝛻𝛻𝛷𝛷𝛼𝛼 = − 𝛻𝛻𝑝𝑝𝛼𝛼 + 𝜌𝜌𝛼𝛼 g𝛻𝛻ℎ 𝜇𝜇𝛼𝛼 𝜇𝜇𝛼𝛼

0 ≤ 𝑘𝑘𝑟𝑟𝑟𝑟 = 𝑓𝑓 𝑆𝑆𝑤𝑤 ≤1 𝜌𝜌𝛼𝛼 = 𝑓𝑓 𝑝𝑝 ,

𝜇𝜇𝛼𝛼 = 𝑓𝑓 𝑝𝑝 ,

𝑝𝑝𝑐𝑐 = 𝑝𝑝𝑛𝑛 − 𝑝𝑝𝑤𝑤 = 𝑓𝑓 𝑆𝑆𝑤𝑤

…

𝑛𝑛

� 𝑆𝑆𝛼𝛼 = 1

𝛼𝛼=1 𝑛𝑛

� 𝑆𝑆𝛼𝛼 𝐶𝐶𝑖𝑖𝑖𝑖 = 𝐶𝐶𝑖𝑖

𝛼𝛼=1

𝜕𝜕 𝐤𝐤𝑘𝑘𝑟𝑟𝑟𝑟 𝑆𝑆𝛼𝛼 𝜌𝜌𝛼𝛼 𝜙𝜙 − 𝛻𝛻. 𝜌𝜌𝛼𝛼 𝛻𝛻𝑝𝑝𝛼𝛼 + 𝜌𝜌𝛼𝛼 g𝛻𝛻ℎ 𝜕𝜕𝜕𝜕 𝜇𝜇𝛼𝛼 Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

= 𝑞𝑞𝑠𝑠𝑠𝑠

September 16, 2016

14

𝜕𝜕 𝐤𝐤𝑘𝑘𝑟𝑟𝑟𝑟 𝑆𝑆 𝜌𝜌 𝜙𝜙 − 𝛻𝛻. 𝜌𝜌𝑤𝑤 𝛻𝛻𝑝𝑝𝑤𝑤 + 𝜌𝜌𝑤𝑤 𝑔𝑔𝑔𝑔𝑔 𝜇𝜇𝑤𝑤 𝜕𝜕𝜕𝜕 𝑤𝑤 𝑤𝑤 𝜕𝜕 𝐤𝐤𝑘𝑘𝑟𝑟𝑟𝑟 𝑆𝑆 𝜌𝜌 𝜙𝜙 − 𝛻𝛻. 𝜌𝜌𝑜𝑜 𝛻𝛻𝑝𝑝𝑜𝑜 + 𝜌𝜌𝑜𝑜 𝑔𝑔𝑔𝑔𝑔 𝜇𝜇𝑜𝑜 𝜕𝜕𝜕𝜕 𝑜𝑜 𝑜𝑜

𝑆𝑆𝑤𝑤 + 𝑆𝑆𝑜𝑜 = 1

𝑝𝑝𝑐𝑐 = 𝑝𝑝𝑜𝑜 − 𝑝𝑝𝑤𝑤 𝜙𝜙𝛼𝛼 𝑆𝑆𝑜𝑜 , 𝑝𝑝𝑜𝑜 Mobility

𝜕𝜕 𝜙𝜙𝑆𝑆𝑤𝑤 𝜙𝜙 𝜕𝜕𝑆𝑆𝑤𝑤 = 𝜕𝜕𝜕𝜕 𝐵𝐵𝑤𝑤 𝐵𝐵𝑤𝑤 𝜕𝜕𝜕𝜕 ∴

∴

𝜕𝜕𝑆𝑆𝑤𝑤 𝜕𝜕𝑆𝑆𝑜𝑜 + =0 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

+ 𝜙𝜙𝑆𝑆𝑤𝑤

𝜕𝜕 1 𝑆𝑆𝑤𝑤 𝜕𝜕𝜕𝜕 + 𝐵𝐵𝑤𝑤 𝜕𝜕𝑝𝑝𝑜𝑜 𝜕𝜕𝑝𝑝𝑜𝑜 𝐵𝐵𝑤𝑤

𝜕𝜕 𝜙𝜙𝑆𝑆𝑤𝑤 𝐤𝐤 𝑘𝑘𝑟𝑟𝑟𝑟 − 𝛻𝛻. 𝛻𝛻𝑝𝑝𝑤𝑤 + 𝜌𝜌𝑤𝑤 𝑔𝑔𝑔𝑔𝑔 𝜕𝜕𝜕𝜕 𝐵𝐵𝑤𝑤 𝜇𝜇𝑤𝑤 𝐵𝐵𝑤𝑤

= 𝑞𝑞𝑠𝑠𝑠𝑠

𝜕𝜕 𝜙𝜙𝑆𝑆𝑜𝑜 𝐤𝐤 𝑘𝑘𝑟𝑟𝑟𝑟 − 𝛻𝛻. 𝛻𝛻𝑝𝑝𝑜𝑜 + 𝜌𝜌𝑜𝑜 𝑔𝑔𝑔𝑔𝑔 𝜕𝜕𝜕𝜕 𝐵𝐵𝑜𝑜 𝜇𝜇𝑜𝑜 𝐵𝐵𝑜𝑜

= 𝑞𝑞𝑠𝑠𝑠𝑠

𝜕𝜕𝑝𝑝𝑜𝑜 𝜕𝜕𝜕𝜕

𝜕𝜕 𝜙𝜙𝑆𝑆𝑜𝑜 𝜙𝜙 𝜕𝜕𝑆𝑆𝑜𝑜 = 𝜕𝜕𝜕𝜕 𝐵𝐵𝑜𝑜 𝐵𝐵𝑜𝑜 𝜕𝜕𝜕𝜕

+ 𝜙𝜙𝑆𝑆𝑜𝑜

= 𝑞𝑞𝑠𝑠𝑠𝑠

𝜕𝜕 1 𝑆𝑆𝑜𝑜 𝜕𝜕𝜕𝜕 + 𝐵𝐵𝑜𝑜 𝜕𝜕𝑝𝑝𝑜𝑜 𝜕𝜕𝑝𝑝𝑜𝑜 𝐵𝐵𝑜𝑜

𝐵𝐵𝑜𝑜 𝜕𝜕 𝜙𝜙𝑆𝑆𝑜𝑜 𝐵𝐵𝑤𝑤 𝜕𝜕 𝜙𝜙𝑆𝑆𝑤𝑤 𝜕𝜕 1 𝜕𝜕 1 1 𝜕𝜕𝜕𝜕 + = 𝐵𝐵𝑜𝑜 𝑆𝑆𝑜𝑜 + 𝐵𝐵𝑤𝑤 𝑆𝑆𝑤𝑤 + 𝜙𝜙 𝜕𝜕𝜕𝜕 𝐵𝐵𝑜𝑜 𝜙𝜙 𝜕𝜕𝜕𝜕 𝐵𝐵𝑤𝑤 𝜕𝜕𝑝𝑝𝑜𝑜 𝐵𝐵𝑜𝑜 𝜕𝜕𝑝𝑝𝑜𝑜 𝐵𝐵𝑤𝑤 𝜙𝜙 𝜕𝜕𝑝𝑝𝑜𝑜 compressibility terms 𝛼𝛼 𝑆𝑆o ,𝑝𝑝o

𝛻𝛻𝑝𝑝𝑤𝑤 = 𝛻𝛻𝑝𝑝𝑜𝑜 − 𝛻𝛻𝑝𝑝𝑐𝑐

= 𝑞𝑞𝑠𝑠𝑠𝑠

𝜕𝜕𝑝𝑝𝑜𝑜 𝜕𝜕𝜕𝜕

𝜕𝜕𝑝𝑝𝑜𝑜 𝜕𝜕𝜕𝜕

𝜕𝜕𝑝𝑝𝑜𝑜 𝐤𝐤 𝑘𝑘𝑟𝑟𝑟𝑟 𝐤𝐤 𝑘𝑘𝑟𝑟𝑟𝑟 𝐤𝐤 𝑘𝑘𝑟𝑟𝑟𝑟 𝐤𝐤 𝑘𝑘𝑟𝑟𝑟𝑟 𝐤𝐤 𝑘𝑘𝑟𝑟𝑟𝑟 = 𝐵𝐵𝑜𝑜 𝑞𝑞𝑠𝑠𝑠𝑠 + 𝐵𝐵𝑤𝑤 𝑞𝑞𝑠𝑠𝑠𝑠 + 𝐵𝐵𝑜𝑜 𝛻𝛻 � 𝛻𝛻𝑝𝑝𝑜𝑜 + 𝐵𝐵𝑤𝑤 𝛻𝛻 � 𝛻𝛻𝑝𝑝𝑜𝑜 + 𝐵𝐵𝑜𝑜 𝛻𝛻 � 𝜌𝜌 𝑔𝑔𝑔𝑔𝑔 + 𝐵𝐵𝑤𝑤 𝛻𝛻 � 𝜌𝜌 𝑔𝑔𝑔𝑔𝑔 + 𝐵𝐵𝑤𝑤 𝛻𝛻 � 𝛻𝛻𝑝𝑝𝑐𝑐 𝜕𝜕𝜕𝜕 𝜇𝜇𝑜𝑜 𝐵𝐵𝑜𝑜 𝜇𝜇𝑤𝑤 𝐵𝐵𝑤𝑤 𝜇𝜇𝑜𝑜 𝐵𝐵𝑜𝑜 𝑜𝑜 𝜇𝜇𝑤𝑤 𝐵𝐵𝑤𝑤 𝑤𝑤 𝜇𝜇𝑤𝑤 𝐵𝐵𝑤𝑤 source/sink

Saturation equation

gravity terms

water flow

𝜆𝜆𝑜𝑜 𝑆𝑆𝑜𝑜 =

Pressure equation 𝜙𝜙𝛼𝛼 𝑆𝑆𝑜𝑜 , 𝑝𝑝𝑜𝑜

oil flow

𝐤𝐤𝑘𝑘𝑟𝑟𝑟𝑟 𝜇𝜇𝑜𝑜

and

𝜆𝜆𝑤𝑤 𝑆𝑆𝑜𝑜 =

capillary term

𝐤𝐤𝑘𝑘𝑟𝑟𝑟𝑟 𝜇𝜇𝑤𝑤

𝜕𝜕𝑝𝑝𝑜𝑜 𝜆𝜆𝑜𝑜 𝜆𝜆𝑤𝑤 𝜆𝜆𝑜𝑜 𝜆𝜆𝑤𝑤 𝜆𝜆𝑤𝑤 = 𝐵𝐵𝑜𝑜 𝑞𝑞𝑠𝑠𝑠𝑠 + 𝐵𝐵𝑤𝑤 𝑞𝑞𝑠𝑠𝑠𝑠 + 𝐵𝐵𝑜𝑜 𝛻𝛻 � 𝛻𝛻𝑝𝑝 + 𝐵𝐵𝑤𝑤 𝛻𝛻 � 𝛻𝛻𝑝𝑝 + 𝐵𝐵𝑜𝑜 𝛻𝛻 � 𝜌𝜌 𝑔𝑔𝑔𝑔𝑔 + 𝐵𝐵𝑤𝑤 𝛻𝛻 � 𝜌𝜌 𝑔𝑔𝑔𝑔𝑔 + 𝐵𝐵𝑤𝑤 𝛻𝛻 � 𝛻𝛻𝑝𝑝 𝜕𝜕𝜕𝜕 𝐵𝐵𝑜𝑜 𝑜𝑜 𝐵𝐵𝑤𝑤 𝑜𝑜 𝐵𝐵𝑜𝑜 𝑜𝑜 𝐵𝐵𝑤𝑤 𝑤𝑤 𝐵𝐵𝑤𝑤 𝑐𝑐

They are coupled equations.

Dr. Siroos Azizmohammadi

𝜕𝜕 𝜙𝜙𝑆𝑆𝑜𝑜 𝜆𝜆𝑜𝑜 − 𝛻𝛻. 𝛻𝛻𝑝𝑝𝑜𝑜 + 𝜌𝜌𝑜𝑜 𝑔𝑔𝑔𝑔𝑔 𝐵𝐵𝑜𝑜 𝜕𝜕𝜕𝜕 𝐵𝐵𝑜𝑜

= 𝑞𝑞𝑠𝑠𝑠𝑠

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

15

Initial condition

• specifies the initial state of the primary variables of the system. For the simple case (1-D), a constant initial pressure. 𝑝𝑝 𝑥𝑥, 𝑡𝑡 = 0 = 𝑝𝑝𝑖𝑖

Boundary conditions

• Basically there are two types of BCs in reservoir engineering. Pressure conditions (Dirichlet conditions) and rate conditions (Neumann conditions). Dirichlet (first type) boundary condition

• specifies the value of the solution variable at the boundary of the domain (pressures at the end faces of the system). Neumann (second type) boundary condition

𝑝𝑝 𝑥𝑥 = 0, 𝑡𝑡 = 𝑝𝑝0

• specifies the gradient of the solution variable at the domain boundary. This gradient is always specified in the direction normal to the boundary (flow rates at the end faces of the system).

Robin (third type) boundary condition

−

𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝜇𝜇 𝜕𝜕𝜕𝜕

𝑥𝑥=0

= 𝑞𝑞0

• specifies a linear combination of the solution variable and its gradient at the domain boundary.

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

16

START

𝑛𝑛 Time level 𝑛𝑛, initialize 𝑆𝑆𝑜𝑜𝑜𝑜 , 𝑝𝑝𝑖𝑖𝑛𝑛

Calculate mobilities 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝜆𝜆𝑡𝑡 𝑆𝑆𝑜𝑜𝑜𝑜 = 𝜆𝜆𝑜𝑜 𝑆𝑆𝑜𝑜𝑜𝑜 + 𝜆𝜆𝑤𝑤 𝑆𝑆𝑜𝑜𝑜𝑜 𝑛𝑛+1 Set the latest 𝑝𝑝𝑖𝑖𝑛𝑛+1 and 𝑆𝑆𝑜𝑜𝑜𝑜 to

Solve pressure equation for current time step

through calculation again

to obtain 𝑝𝑝𝑖𝑖𝑛𝑛+1

"current" values and ITERATE

set to “current” values.

Take the next time step

𝑛𝑛 𝛻𝛻 � 𝜆𝜆𝑡𝑡 𝑆𝑆𝑜𝑜𝑜𝑜 𝛻𝛻𝛻𝛻 = 0

NO

𝑛𝑛 Use 𝑝𝑝𝑖𝑖𝑛𝑛+1 and 𝜆𝜆𝑜𝑜 𝑆𝑆𝑜𝑜𝑜𝑜 to solve saturation equation

Final time

𝜙𝜙

NO

𝑛𝑛+1 Keep 𝑝𝑝𝑖𝑖𝑛𝑛+1 and 𝑆𝑆𝑜𝑜𝑜𝑜 and

𝜕𝜕𝑆𝑆𝑜𝑜 𝑛𝑛 − 𝛻𝛻. 𝜆𝜆𝑜𝑜 𝑆𝑆𝑜𝑜𝑜𝑜 𝛻𝛻𝛻𝛻 = 0 𝜕𝜕𝜕𝜕

reached?

END

𝑛𝑛+1 to obtain 𝑆𝑆𝑜𝑜𝑜𝑜

YES Are these 𝑝𝑝𝑖𝑖𝑛𝑛+1 and 𝑛𝑛+1 𝑆𝑆𝑜𝑜𝑜𝑜 satisfactory?

(converged?)

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

17

One-Dimensional:

• • • • • • •

Two-phase flow Laminar creeping flow (Darcy’s law) Constant viscosity (oil and water) Incompressible rock and fluids (𝐵𝐵o = 𝐵𝐵w = 1) No gravity effects (horizontal flow) No capillary effects (displacement occurs at a high injection rate) Without sources/sinks

𝑓𝑓𝑤𝑤

Saturation equation 𝜕𝜕𝑆𝑆𝑜𝑜 𝜕𝜕 𝜕𝜕𝑝𝑝𝑜𝑜 𝜙𝜙 − 𝜆𝜆𝑜𝑜 =0 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝑆𝑆𝑜𝑜 𝜕𝜕𝑣𝑣𝑜𝑜 + =0 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

Buckley-Leverett solution 𝑣𝑣𝑆𝑆𝑤𝑤𝑤𝑤 𝑥𝑥𝑤𝑤𝑤𝑤

𝑆𝑆𝑤𝑤𝑤𝑤

𝑊𝑊𝑖𝑖 𝑑𝑑𝑓𝑓𝑤𝑤 = 𝐴𝐴𝐴𝐴 𝑑𝑑𝑆𝑆𝑤𝑤

0.8

4

0.6

3

𝑑𝑑𝑓𝑓𝑤𝑤 𝑑𝑑𝑆𝑆 2 𝑤𝑤

0.4

1

0

and

𝜙𝜙

𝑞𝑞𝑡𝑡 𝜕𝜕𝑓𝑓𝑤𝑤 = 𝐴𝐴𝐴𝐴 𝜕𝜕𝑆𝑆𝑤𝑤 𝑆𝑆𝑤𝑤𝑤𝑤

𝜕𝜕𝑆𝑆𝑤𝑤 𝜕𝜕𝑣𝑣𝑤𝑤 + =0 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑆𝑆𝑤𝑤𝑤𝑤

𝑊𝑊𝑖𝑖 = 𝑡𝑡𝑞𝑞𝑡𝑡 is the cumulative water injected at time t

Dr. Siroos Azizmohammadi

0.2

1

where 𝜆𝜆𝑡𝑡 𝑆𝑆𝑜𝑜 = 𝜆𝜆𝑜𝑜 + 𝜆𝜆𝑤𝑤 is total mobility.

𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑

5

0.2

Pressure equation 𝜕𝜕 𝜕𝜕𝑝𝑝𝑜𝑜 =0 𝜆𝜆𝑡𝑡 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

𝜙𝜙

1

0.8

𝑆𝑆𝑤𝑤

1 − 𝑆𝑆𝑜𝑜𝑜𝑜

0.4

𝑆𝑆w

0.8

0.6

1

0

Saturation profile

0.6

Shock front

0.4 0.2 0

𝑆𝑆𝑤𝑤𝑖𝑖

0.2

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

0.4

𝑥𝑥 ⁄𝐿𝐿

0.6

𝑥𝑥𝑤𝑤𝑓𝑓

0.8

September 16, 2016

1 18

Welge (1952)

1

𝑊𝑊𝑖𝑖 = 𝐴𝐴𝐴𝐴𝑥𝑥𝑤𝑤𝑤𝑤 𝑆𝑆𝑤𝑤 −𝑆𝑆𝑤𝑤𝑤𝑤

0.8

Buckley-Leverett solution:

0.6

𝑥𝑥𝑤𝑤𝑤𝑤 1 2

𝑊𝑊𝑖𝑖 𝑑𝑑𝑓𝑓𝑤𝑤 = 𝐴𝐴𝐴𝐴 𝑑𝑑𝑆𝑆𝑤𝑤

𝑆𝑆𝑤𝑤

𝑆𝑆𝑤𝑤𝑤𝑤

0.2

𝑑𝑑𝑓𝑓𝑤𝑤 ̅ = 𝑆𝑆𝑤𝑤𝑤𝑤 + 1� 𝑆𝑆𝑤𝑤 𝑑𝑑𝑆𝑆𝑤𝑤 𝑆𝑆𝑤𝑤 = �

𝑥𝑥𝑤𝑤𝑤𝑤

0

𝑑𝑑𝑓𝑓𝑤𝑤 𝑑𝑑𝑆𝑆𝑤𝑤

𝑆𝑆𝑤𝑤𝑤𝑤

=

𝑆𝑆𝑤𝑤 𝑑𝑑𝑑𝑑��

1 − 𝑓𝑓𝑤𝑤𝑤𝑤

𝑆𝑆𝑤𝑤 − 𝑆𝑆𝑤𝑤𝑤𝑤

𝑥𝑥𝑤𝑤𝑤𝑤

0

=

𝑆𝑆𝑤𝑤𝑤𝑤

0

𝑑𝑑𝑑𝑑 = 𝑆𝑆𝑤𝑤𝑤𝑤 + 1

𝑆𝑆𝑤𝑤 − 𝑆𝑆𝑤𝑤𝑤𝑤

Injected pore volume 𝑊𝑊𝑖𝑖 𝑑𝑑𝑓𝑓𝑤𝑤 𝑄𝑄𝑤𝑤𝑤𝑤 = = 1� 𝐿𝐿𝐴𝐴𝐴𝐴 𝑑𝑑𝑆𝑆𝑤𝑤 𝑆𝑆

𝑤𝑤𝑤𝑤

Dr. Siroos Azizmohammadi

0.4

𝑆𝑆𝑤𝑤

1 0.8

1 − 𝑆𝑆𝑜𝑜𝑜𝑜 𝑆𝑆𝑤𝑤

0.6

𝑓𝑓𝑤𝑤𝑓𝑓

𝑓𝑓𝑤𝑤 =

𝑓𝑓𝑤𝑤

𝑆𝑆𝑤𝑤𝑖𝑖

1 𝑘𝑘 𝜇𝜇 1 + 𝑟𝑟𝑟𝑟 𝜇𝜇𝑤𝑤 𝑘𝑘𝑟𝑟𝑟𝑟 𝑜𝑜

0.4

0.2

1 − 𝑓𝑓𝑤𝑤𝑤𝑤 𝑑𝑑𝑓𝑓𝑤𝑤 𝑑𝑑𝑆𝑆𝑤𝑤 𝑆𝑆

𝑤𝑤𝑤𝑤

0.4

𝑥𝑥

0.6

𝑥𝑥𝑤𝑤𝑤𝑤

0.8

1

0.2 𝑆𝑆𝑤𝑤𝑖𝑖

0

0.2

0.4

𝑆𝑆𝑤𝑤

𝑆𝑆𝑤𝑤𝑓𝑓

0.6

0.8

1

𝑆𝑆𝑤𝑤𝑤𝑤 = Water saturation at flood front before water breakthrough

𝑓𝑓𝑤𝑤𝑤𝑤 = Producing water cut at flood front before water breakthrough

𝑆𝑆𝑤𝑤 = Average water saturation behind the front at water breakthrough Front location 𝑊𝑊𝑖𝑖 𝑑𝑑𝑓𝑓𝑤𝑤 𝑥𝑥𝑆𝑆𝑤𝑤𝑤𝑤 = 𝐴𝐴𝐴𝐴 𝑑𝑑𝑆𝑆𝑤𝑤 𝑆𝑆

𝑤𝑤𝑤𝑤

𝑣𝑣𝑆𝑆𝑤𝑤𝑤𝑤

Front velocity 𝑑𝑑𝑑𝑑 𝑞𝑞𝑡𝑡 𝜕𝜕𝑓𝑓𝑤𝑤 = = 𝑑𝑑𝑑𝑑 𝑆𝑆 𝐴𝐴𝐴𝐴 𝜕𝜕𝑆𝑆𝑤𝑤

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

𝑤𝑤𝑤𝑤

𝑡𝑡 𝑆𝑆 𝑤𝑤𝑤𝑤

September 16, 2016

19

Equations: all equations (partial differential equations, boundary conditions, constraints, …) which are obtained from mathematical modeling of fluid flow in reservoir must be discretized before they can be treated numerically. The most common discretization techniques for equations are:

• Finite Difference (FD) method • Finite Volume (FV) method • Finite Element (FE) method

Geometry: also reservoir geometry (solution domain) must be discretized. Reservoir discretization means that the reservoir is described by a set of grid blocks whose properties, dimensions, boundaries, and locations in the reservoir are well defined.

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

20

Finite difference approximations are used in most commercial reservoir simulators to solve fluid flow equations numerically.

Finite difference scheme is a way of approximating derivatives which are involved in flow equation.

Each finite difference method has the following steps: Discretization of domain: construction of a grid with points on which we are interested in solving the equation(s). Difference equation for each point: replacing the continuous derivatives of equation with their finite difference approximations. System of equations: Rearrangement of the discretized equation, so that all known quantities (i.e. pressure at time 𝑛𝑛) are on the right hand side and the unknown quantities on the left-hand side (properties at time

𝑛𝑛 + 1).

Solution method: solving the system of equations by direct or iterative methods.

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

21

Taylor series expansion ∆𝑥𝑥 𝜕𝜕𝜕𝜕 𝑥𝑥, 𝑡𝑡 ∆𝑥𝑥 2 𝜕𝜕 2 𝑝𝑝 𝑥𝑥, 𝑡𝑡 ∆𝑥𝑥 3 𝜕𝜕 3 𝑝𝑝 𝑥𝑥, 𝑡𝑡 + 𝑝𝑝 𝑥𝑥 + ∆𝑥𝑥, 𝑡𝑡 = 𝑝𝑝 𝑥𝑥, 𝑡𝑡 + + +⋯ 1! 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 2 𝜕𝜕𝜕𝜕 3 2! 3!

−∆𝑥𝑥 𝜕𝜕𝜕𝜕 𝑥𝑥, 𝑡𝑡 −∆𝑥𝑥 + 𝑝𝑝 𝑥𝑥 − ∆𝑥𝑥, 𝑡𝑡 = 𝑝𝑝 𝑥𝑥, 𝑡𝑡 + 1! 𝜕𝜕𝜕𝜕 2! Scheme Forward difference

Backward difference

Central difference

Second order derivative

Dr. Siroos Azizmohammadi

2

𝜕𝜕 2 𝑝𝑝 𝑥𝑥, 𝑡𝑡 −∆𝑥𝑥 + 𝜕𝜕𝜕𝜕 2 3!

Spatial 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

𝜕𝜕 2 𝑝𝑝 𝜕𝜕𝜕𝜕 2

𝑛𝑛

𝑛𝑛 𝑝𝑝𝑖𝑖+1 − 𝑝𝑝𝑖𝑖𝑛𝑛 = + 𝑂𝑂 ∆𝑥𝑥 ∆𝑥𝑥

𝑖𝑖

𝑛𝑛

𝑛𝑛 𝑝𝑝𝑖𝑖𝑛𝑛 − 𝑝𝑝𝑖𝑖−1 = + 𝑂𝑂 ∆𝑥𝑥 ∆𝑥𝑥

𝑖𝑖

𝑛𝑛 𝑖𝑖

𝑛𝑛 𝑛𝑛 𝑝𝑝𝑖𝑖+1 − 𝑝𝑝𝑖𝑖−1 = + 𝑂𝑂 ∆𝑥𝑥 2 2∆𝑥𝑥

𝑛𝑛 𝑖𝑖

𝑛𝑛 𝑛𝑛 𝑝𝑝𝑖𝑖+1 − 2𝑝𝑝𝑖𝑖𝑛𝑛 + 𝑝𝑝𝑖𝑖−1 = + 𝑂𝑂 ∆𝑥𝑥 2 ∆𝑥𝑥 2

3

𝜕𝜕 3 𝑝𝑝 𝑥𝑥, 𝑡𝑡 +⋯ 𝜕𝜕𝜕𝜕 3

Time 𝜕𝜕𝜕𝜕 𝜕𝜕𝑡𝑡 𝜕𝜕𝜕𝜕 𝜕𝜕𝑡𝑡 𝜕𝜕𝜕𝜕 𝜕𝜕𝑡𝑡

𝑛𝑛 𝑖𝑖

=

𝑛𝑛+1 𝑖𝑖

𝑛𝑛 𝑖𝑖

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

𝑝𝑝𝑖𝑖𝑛𝑛+1 − 𝑝𝑝𝑖𝑖𝑛𝑛 + 𝑂𝑂 ∆𝑡𝑡 ∆𝑡𝑡

𝑝𝑝𝑖𝑖𝑛𝑛+1 − 𝑝𝑝𝑖𝑖𝑛𝑛 = + 𝑂𝑂 ∆𝑡𝑡 ∆𝑡𝑡

𝑝𝑝𝑖𝑖𝑛𝑛+1 − 𝑝𝑝𝑖𝑖𝑛𝑛−1 = + 𝑂𝑂 ∆𝑡𝑡 2 2∆𝑡𝑡

September 16, 2016

22

Explicit Scheme 𝑛𝑛 + 1

𝑛𝑛 + 1

𝑛𝑛

𝑖𝑖 − 1

Scheme Explicit Implicit

Crank-Nicholson

Dr. Siroos Azizmohammadi

Implicit Scheme

𝑖𝑖

𝑖𝑖 + 1

𝑛𝑛

𝑖𝑖 − 1

Spatial 𝜕𝜕 2 𝑝𝑝 𝜕𝜕𝜕𝜕 2 𝜕𝜕 2 𝑝𝑝 𝜕𝜕𝜕𝜕 2 𝜕𝜕 2 𝑝𝑝 𝜕𝜕𝜕𝜕 2

𝑛𝑛 𝑖𝑖 𝑖𝑖

1 𝑛𝑛+ 2 𝑖𝑖

𝑖𝑖 + 1

Time

𝑛𝑛 𝑛𝑛 𝑝𝑝𝑖𝑖+1 − 2𝑝𝑝𝑖𝑖𝑛𝑛 + 𝑝𝑝𝑖𝑖−1 = ∆𝑥𝑥 2

𝑛𝑛+1

𝑖𝑖

𝑛𝑛+1 𝑛𝑛+1 𝑝𝑝𝑖𝑖+1 − 2𝑝𝑝𝑖𝑖𝑛𝑛+1 + 𝑝𝑝𝑖𝑖−1 = ∆𝑥𝑥 2

=

𝑛𝑛 𝑛𝑛 𝑛𝑛+1 𝑛𝑛+1 1 𝑝𝑝𝑖𝑖+1 − 2𝑝𝑝𝑖𝑖𝑛𝑛 + 𝑝𝑝𝑖𝑖−1 𝑝𝑝𝑖𝑖+1 − 2𝑝𝑝𝑖𝑖𝑛𝑛+1 + 𝑝𝑝𝑖𝑖−1 + 2 ∆𝑥𝑥 2 ∆𝑥𝑥 2

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

𝜕𝜕𝜕𝜕 𝜕𝜕𝑡𝑡 𝜕𝜕𝜕𝜕 𝜕𝜕𝑡𝑡 𝜕𝜕𝜕𝜕 𝜕𝜕𝑡𝑡

𝑛𝑛 𝑖𝑖

=

𝑛𝑛+1 𝑖𝑖

1 𝑛𝑛+ 2 𝑖𝑖

𝑝𝑝𝑖𝑖𝑛𝑛+1 − 𝑝𝑝𝑖𝑖𝑛𝑛 ∆𝑡𝑡

𝑝𝑝𝑖𝑖𝑛𝑛+1 − 𝑝𝑝𝑖𝑖𝑛𝑛 = ∆𝑡𝑡 𝑝𝑝𝑖𝑖𝑛𝑛+1 − 𝑝𝑝𝑖𝑖𝑛𝑛 = ∆𝑡𝑡

September 16, 2016

23

𝑛𝑛+1 𝑝𝑝𝑖𝑖−1

=

𝑝𝑝𝑖𝑖𝑛𝑛

+

− 2+

𝛼𝛼∆𝑡𝑡 ∆𝑥𝑥 2

∆𝑥𝑥 2 𝛼𝛼∆𝑡𝑡

𝑛𝑛 𝑝𝑝𝑖𝑖+1

𝑝𝑝𝑖𝑖𝑛𝑛+1

−

+

2𝑝𝑝𝑖𝑖𝑛𝑛

+

𝑛𝑛+1 𝑝𝑝𝑖𝑖+1

3

𝑛𝑛 𝑝𝑝𝑖𝑖−1

=

∆𝑥𝑥 2 𝑛𝑛 − 𝑝𝑝 𝛼𝛼∆𝑡𝑡 𝑖𝑖

𝑛𝑛+1 𝑛𝑛+1 𝑎𝑎𝑖𝑖−1 𝑝𝑝𝑖𝑖−1 + 𝑎𝑎𝑖𝑖 𝑝𝑝𝑖𝑖𝑛𝑛+1 + 𝑎𝑎𝑖𝑖+1 𝑝𝑝𝑖𝑖+1 = 𝑏𝑏𝑖𝑖

𝑖𝑖 = 2 3

𝐴𝐴 𝑝𝑝 = 𝑏𝑏

𝑎𝑎2 𝑎𝑎3 𝑎𝑎2 𝑎𝑎3 𝑎𝑎4

𝑚𝑚 − 2 𝑚𝑚 − 1

⋱

𝑝𝑝2 𝑝𝑝3

⋱

⋱

𝑎𝑎𝑚𝑚−3 𝑎𝑎𝑚𝑚−2 𝑎𝑎𝑚𝑚−1 𝑎𝑎𝑚𝑚−2 𝑎𝑎𝑚𝑚−1

⋮

𝑝𝑝𝑚𝑚−2 𝑝𝑝𝑚𝑚−1

=

2

1

𝑛𝑛 = 0 𝑏𝑏2 − 𝑎𝑎1 𝑝𝑝1𝑛𝑛+1 𝑏𝑏3 ⋮

𝑏𝑏𝑚𝑚−2 𝑛𝑛+1 𝑏𝑏𝑚𝑚−1 − 𝑎𝑎𝑚𝑚 𝑝𝑝m

Space index: 𝑖𝑖 = 2, 3, … , 𝑚𝑚 − 1

Time index : 𝑛𝑛 = 1, 2, 3 …

For all times:

𝑝𝑝𝑚𝑚 = 𝑝𝑝i (right boundary)

Dr. Siroos Azizmohammadi

∆𝑡𝑡: time interval

𝑝𝑝1 = 𝑝𝑝0 (left boundary)

2

𝑖𝑖 = 1

Tridiagonal matrix (solution method: direct Thomas algorithm or iterative methods).

∆𝑥𝑥: space interval

right boundary

𝛼𝛼: hydraulic diffusivity

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

𝑖𝑖

space

2

1

×

×

2

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

4 5 6 7 8

4

5

6

𝑚𝑚

1

3

3

𝑚𝑚 − 1 7

September 16, 2016

initial condition

Implicit:

𝑝𝑝𝑖𝑖𝑛𝑛+1

left boundary

time

Explicit:

𝜕𝜕 2 𝑝𝑝 1 𝜕𝜕𝜕𝜕 = 𝜕𝜕𝑥𝑥 2 𝛼𝛼 𝜕𝜕𝜕𝜕

8

24

𝑝𝑝i

𝜕𝜕 2 𝑝𝑝 1 𝜕𝜕𝜕𝜕 = 𝜕𝜕𝑥𝑥 2 𝛼𝛼 𝜕𝜕𝜕𝜕 𝑝𝑝 𝑡𝑡 = 0 = 𝑝𝑝i

𝑝𝑝 𝑥𝑥 = 0 = 𝑝𝑝0 𝑝𝑝 𝑥𝑥 = 𝐿𝐿 = 𝑝𝑝i

𝑚𝑚 = 5

∆𝑥𝑥 = 0.2 ∆𝑡𝑡 = 0.2

Dr. Siroos Azizmohammadi

Analytical solution

Transient

𝑦𝑦

𝑧𝑧

𝑥𝑥

𝑝𝑝

Steady state

𝛼𝛼 = 0.1 m2 ⁄s 𝑝𝑝0

𝑡𝑡 = 0.2 0.4 0.6 1.0

2.0

Explicit

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

𝑥𝑥 𝑡𝑡 = 0.2 0.4 0.6 1.0

2.0

Implicit

September 16, 2016

25

𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 = 𝛽𝛽 𝑘𝑘𝑥𝑥 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑘𝑘𝑥𝑥 𝜕𝜕𝜕𝜕

𝑘𝑘𝑥𝑥 −

𝑛𝑛+1

𝜕𝜕𝜕𝜕 − 𝑘𝑘𝑥𝑥 𝜕𝜕𝜕𝜕 𝑖𝑖+1⁄2

𝑖𝑖+1⁄2

∆𝑥𝑥

𝑛𝑛+1

𝑖𝑖−1⁄2

𝑝𝑝𝑖𝑖𝑛𝑛+1 − 𝑝𝑝𝑖𝑖𝑛𝑛 = 𝛽𝛽 ∆𝑡𝑡

𝑛𝑛+1 𝑝𝑝𝑖𝑖+1 − 𝑝𝑝𝑖𝑖𝑛𝑛+1 − 𝑘𝑘𝑥𝑥 ∆𝑥𝑥 2

𝑘𝑘𝑥𝑥 𝑖𝑖−1⁄2 𝑛𝑛+1 1 𝑝𝑝 + 𝑖𝑖−1 ∆𝑥𝑥 2 ∆𝑥𝑥 2

𝑘𝑘𝑥𝑥

𝑖𝑖−1⁄2

𝑛𝑛+1 𝑝𝑝𝑖𝑖𝑛𝑛+1 − 𝑝𝑝𝑖𝑖−1 𝑝𝑝𝑖𝑖𝑛𝑛+1 − 𝑝𝑝𝑖𝑖𝑛𝑛 = 𝛽𝛽 ∆𝑥𝑥 2 ∆𝑡𝑡

𝑖𝑖−1⁄2 + 𝑘𝑘𝑥𝑥

𝑖𝑖+1⁄2 −

𝑖𝑖 − 1

𝑖𝑖

𝑖𝑖 + 1

2 𝑘𝑘𝑥𝑥 𝑖𝑖−1 𝑘𝑘𝑥𝑥 𝑖𝑖 𝑘𝑘𝑥𝑥 𝑖𝑖−1 + 𝑘𝑘𝑥𝑥 𝑖𝑖

𝑘𝑘𝑥𝑥 𝑖𝑖+1⁄2 𝑛𝑛+1 𝛽𝛽 𝑛𝑛+1 𝛽𝛽 𝑛𝑛 𝑝𝑝𝑖𝑖 − 𝑝𝑝 𝑝𝑝 = 𝑖𝑖+1 ∆𝑡𝑡 ∆𝑡𝑡 𝑖𝑖 ∆𝑥𝑥 2

System of equations can be shown in simplified form as: 𝑛𝑛+1 𝑛𝑛+1 + 𝑎𝑎𝑖𝑖 𝑝𝑝𝑖𝑖𝑛𝑛+1 + 𝑎𝑎𝑖𝑖+1 𝑝𝑝𝑖𝑖+1 = 𝑏𝑏𝑖𝑖 𝑎𝑎𝑖𝑖−1 𝑝𝑝𝑖𝑖−1

Permeability is evaluated at the boundary faces by harmonic averaging. Dr. Siroos Azizmohammadi

𝑖𝑖 + 1⁄2

𝑖𝑖 − 1⁄2

𝛽𝛽 = 𝜇𝜇𝜇𝜇𝑐𝑐𝑡𝑡

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

𝑘𝑘𝑥𝑥

𝑖𝑖−1⁄2

=

𝑘𝑘𝑥𝑥

𝑖𝑖+1⁄2

=

2 𝑘𝑘𝑥𝑥 𝑖𝑖 𝑘𝑘𝑥𝑥 𝑖𝑖+1 𝑘𝑘𝑥𝑥 𝑖𝑖 + 𝑘𝑘𝑥𝑥 𝑖𝑖+1

September 16, 2016

26

In any type of computer simulation work, it is important to determine the accuracy of the solution generated. Some type of errors in the solution are: Round-off error can occur when using single precision accuracy when double precision is required or by mixing single and double precision variables. Truncation error is caused by truncating the Taylor series. A solution to truncation error is to vary time step (∆𝑡𝑡) and grid block size (∆𝑥𝑥) by trial and error until the solution converges. Non-linear error occurs when using a linear approximation to find a value at the 𝑛𝑛 + 1 time level of a non-linear function such as formation volume factors. Instability error is caused by explicit saturation dependent variables [ 𝑘𝑘𝑟𝑟 instability is to take smaller time steps or go to a fully implicit model.

𝑛𝑛

and 𝑝𝑝𝑐𝑐

𝑛𝑛

]. A solution for

Numerical dispersion is caused by saturation discontinuity within a cell. The solutions for numerical dispersion are: (1) smaller grid block size (∆𝑥𝑥), (2) modify 𝑘𝑘𝑟𝑟 calculations (using upstream weighting instead of average weighting between two grid blocks), (3) select proper time step (∆𝑡𝑡). Grid orientation can change the final answer. Grid orientation is generally important in calculating saturation distributions in a water flood. Typically, a diagonal grid system will result in better recoveries (more optimistic).

Dr. Siroos Azizmohammadi

Summer Course 2016 | Department of Petroleum Engineering Reservoir Engineering Module: Reservoir Modeling and Simulation

September 16, 2016

27