Reservoir Modeling and Simulation

Dr. Siroos Azizmohammadi

Summer Course 2016

Department of Petroleum Engineering

Chair of Reservoir Engineering

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

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”

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

Constraint equations

•

Mole constraints

•

Saturation constraint

Other equations

•

Source/Sink

•

Adsorption equations

Balance (governing) equations

•

Mass or mole balance (continuity)

•

Momentum balance (Darcy’s Law)

•

Energy balance

Rock and fluid equations

•

Rock properties

•

Fluid properties (PVT)

Initial and boundary conditions

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,

and computer scientists.

Mathematiacl formulation

Non-linear PDEs

Discretization

System of non-linear algebraic equations

Linearization

System of linear algebraic equations

Numeical methods

•

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

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, ….

Pre-Processing

Processing

Numerical errors

•

Round-off error

•

Truncation error

•

Numerical dispersion

Non uniqueness of solution

•

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

Averaging problems

•

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

Discretization methods

Analytical

Numerical

Distance

Sat

ur

at

ion

Averaging Method 𝑘𝑘1 100 100 100 𝑘𝑘2 200 0 100 Arithmetic 𝑘𝑘avg=� � 𝑖𝑖=1 𝑛𝑛 ℎ𝑖𝑖𝑘𝑘𝑖𝑖 � 𝑖𝑖=1 𝑛𝑛 ℎ𝑖𝑖 150 50 100 Geometric ln 𝑘𝑘avg=� � 𝑖𝑖=1 𝑛𝑛 ℎ𝑖𝑖ln 𝑘𝑘𝑖𝑖 � 𝑖𝑖=1 𝑛𝑛 ℎ𝑖𝑖 141 0 100 Harmonic 𝑘𝑘avg= � � 𝑖𝑖=1 𝑛𝑛 𝐿𝐿𝑖𝑖 � 𝑖𝑖=1 𝑛𝑛 ⁄ 𝐿𝐿𝑖𝑖 𝑘𝑘𝑖𝑖 133 0 100

Fluid types

•

Isothermal compressibility defined as:

𝑐𝑐 = −

_𝑉𝑉

1

𝜕𝜕𝑉𝑉

_{𝜕𝜕𝜕𝜕}

𝑇𝑇

=

1

_𝜌𝜌

𝜕𝜕𝜌𝜌

_{𝜕𝜕𝜕𝜕}

𝑇𝑇

Flow regimes

•

Steady state flow

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

_𝑟𝑟

= 0

•

Pseudo (quasi or semi) steady state flow

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

_𝑟𝑟

= costant

•

Unsteady state (transient) flow

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

_𝑟𝑟

= 𝑓𝑓 𝑟𝑟, 𝜕𝜕

Flow geometries

Pressure Vol um e Pressure Vol um e Pressure Vol um e Incompressible fluids (𝑐𝑐 = 0) 𝜕𝜕𝑉𝑉 𝜕𝜕𝜕𝜕 = 𝜕𝜕𝜌𝜌 𝜕𝜕𝜕𝜕 = 0

Slightly compressible fluids (𝑐𝑐 = constant and small) 𝑉𝑉 = 𝑉𝑉ref 1 + 𝑐𝑐 𝜕𝜕ref− 𝜕𝜕 Compressible fluids (𝑐𝑐 ≠ constant) 𝑐𝑐g=1_{𝜕𝜕 −}1_𝑍𝑍 𝜕𝜕𝑍𝑍_{𝜕𝜕𝜕𝜕} 𝑇𝑇 Linear flow Plan View Side View Wellbore

Flow Lines

Radial flow

Spherical flow

Conservation of mass (continuity equation)

𝜕𝜕

𝜕𝜕𝜕𝜕 𝜌𝜌𝜌𝜌 + 𝛻𝛻 � 𝜌𝜌𝐯𝐯 = 𝑞𝑞

𝑠𝑠

Conservation of momentum (microscopic approach)

𝜌𝜌

𝜕𝜕𝐮𝐮

_{𝜕𝜕𝜕𝜕 + 𝐮𝐮 � 𝛻𝛻𝐮𝐮}

inertial force

+ �

𝛻𝛻𝜕𝜕

pressure force

+ �

𝛻𝛻 � 𝝉𝝉

viscous force

+ �

𝜌𝜌𝐠𝐠

gravity force

= 0

Darcy’s law (macroscopic momentum balance)

𝜌𝜌𝐮𝐮 = 𝐯𝐯 = −

𝐤𝐤

_{𝜇𝜇 𝛻𝛻𝜕𝜕 + 𝜌𝜌g𝛻𝛻ℎ = −}

𝐤𝐤

_{𝜇𝜇 𝛻𝛻𝛷𝛷}

Assumptions of Darcy’s law

•

steady state flow,

•

incompressible fluid,

•

constant viscosity,

•

laminar creeping flow,

Combination of continuity equation and Darcy’s law results in pressure equation

𝜕𝜕 𝜌𝜌𝜌𝜌

𝜕𝜕𝜕𝜕 − 𝛻𝛻. 𝜌𝜌

k

𝜇𝜇 𝛻𝛻𝛷𝛷 = 𝑞𝑞

𝑠𝑠

𝜕𝜕 𝜌𝜌𝜌𝜌

𝜕𝜕𝜕𝜕 =

𝑑𝑑 𝜌𝜌𝜌𝜌

𝑑𝑑𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕 = 𝜌𝜌

𝑑𝑑𝜌𝜌

𝑑𝑑𝜕𝜕 + 𝜌𝜌

𝑑𝑑𝜌𝜌

𝑑𝑑𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕 = 𝜌𝜌𝜌𝜌

1

𝜌𝜌

𝑑𝑑𝜌𝜌

𝑑𝑑𝜕𝜕 +

1

𝜌𝜌

𝑑𝑑𝜌𝜌

𝑑𝑑𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕 = 𝜌𝜌𝜌𝜌 𝑐𝑐

𝑓𝑓

+ 𝑐𝑐

𝑟𝑟

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕 = 𝜌𝜌𝜌𝜌𝑐𝑐

𝑡𝑡

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜌𝜌𝜌𝜌𝑐𝑐

_𝑡𝑡

𝜕𝜕𝜕𝜕

_{𝜕𝜕𝜕𝜕 − 𝛻𝛻. 𝜌𝜌}

_{𝜇𝜇 𝛻𝛻𝜕𝜕 + 𝜌𝜌g𝛻𝛻ℎ = 𝑞𝑞}

k

_𝑠𝑠

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

𝜕𝜕 𝑥𝑥, 𝜕𝜕 − 𝜕𝜕

₀

𝜕𝜕

_𝑖𝑖

− 𝜕𝜕

₀

=

𝑥𝑥

𝐿𝐿 +

2

𝜋𝜋 �

_𝑛𝑛=1 ∞

1

𝑛𝑛 sin 𝜆𝜆

𝑛𝑛

𝑥𝑥 exp −𝜆𝜆

𝑛𝑛2

𝛼𝛼𝜕𝜕

𝜕𝜕 𝑥𝑥, 𝜕𝜕 = 𝜕𝜕

𝑖𝑖

− 𝜕𝜕

0 𝑥𝑥_𝐿𝐿

+ 𝜕𝜕

0

steady state solution

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.

𝜕𝜕

2

_𝜕𝜕

𝜕𝜕𝑥𝑥

2

=

1

𝛼𝛼

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜆𝜆

𝑛𝑛

=

𝑛𝑛𝜋𝜋

_{𝐿𝐿 , 𝑛𝑛 = 1, 2, 3, …}

characteristic values

𝜕𝜕 𝜕𝜕 = 0 = 𝜕𝜕

𝑖𝑖

𝜕𝜕 𝑥𝑥 = 0 = 𝜕𝜕

0

𝜕𝜕 𝑥𝑥 = 𝐿𝐿 = 𝜕𝜕

𝑖𝑖

𝑥𝑥

𝜕𝜕

Steady state

Transient

𝜕𝜕

𝑖𝑖

𝜕𝜕

0

𝛼𝛼 = 0.1

m

2

⁄

_s

𝛼𝛼 =

_{𝜌𝜌𝜇𝜇𝑐𝑐}

𝑘𝑘

𝑡𝑡

hydraulic diffusivity

𝑥𝑥

𝑦𝑦

𝑧𝑧

Saturation equation (phase continuity)

𝜕𝜕

𝜕𝜕𝜕𝜕 𝑆𝑆

𝛼𝛼

𝜌𝜌

𝛼𝛼

𝜌𝜌 + 𝛻𝛻. 𝜌𝜌

𝛼𝛼

𝐯𝐯

𝛼𝛼

= 𝑞𝑞

𝑠𝑠𝛼𝛼

Extended Darcy’s law for phase 𝛼𝛼

𝐯𝐯

_𝛼𝛼

= −

𝐤𝐤

_𝜇𝜇

𝛼𝛼

𝛼𝛼

𝛻𝛻𝛷𝛷

𝛼𝛼

= −

𝐤𝐤𝑘𝑘

_{𝑟𝑟𝛼𝛼}

𝜇𝜇

_𝛼𝛼

𝛻𝛻𝜕𝜕

𝛼𝛼

+ 𝜌𝜌

𝛼𝛼

g𝛻𝛻ℎ

Relative permeability

0 ≤ 𝑘𝑘

_{𝑟𝑟𝛼𝛼}

= 𝑓𝑓 𝑆𝑆

_𝑤𝑤

≤1

Fluid properties

𝜌𝜌

𝛼𝛼

= 𝑓𝑓 𝜕𝜕 ,

𝜇𝜇

𝛼𝛼

= 𝑓𝑓 𝜕𝜕 ,

…

Capillary pressure constraint

𝜕𝜕

_𝑐𝑐

= 𝜕𝜕

_𝑛𝑛

− 𝜕𝜕

_𝑤𝑤

= 𝑓𝑓 𝑆𝑆

_𝑤𝑤

Saturation constraint

�

𝛼𝛼=1 𝑛𝑛

𝑆𝑆

𝛼𝛼

= 1

Concentration constraint

�

𝛼𝛼=1 𝑛𝑛

𝑆𝑆

_𝛼𝛼

𝐶𝐶

_{𝑖𝑖𝛼𝛼}

= 𝐶𝐶

_𝑖𝑖

Transport equation

𝜕𝜕

𝜕𝜕𝜕𝜕 𝑆𝑆

𝛼𝛼

𝜌𝜌

𝛼𝛼

𝜌𝜌 − 𝛻𝛻. 𝜌𝜌

𝛼𝛼

𝐤𝐤𝑘𝑘

_{𝑟𝑟𝛼𝛼}

𝜇𝜇

𝛼𝛼

𝛻𝛻𝜕𝜕

𝛼𝛼

+ 𝜌𝜌

𝛼𝛼

g𝛻𝛻ℎ = 𝑞𝑞

𝑠𝑠𝛼𝛼

𝜕𝜕 𝜕𝜕𝜕𝜕 𝑆𝑆𝑤𝑤𝜌𝜌𝑤𝑤𝜌𝜌 − 𝛻𝛻. 𝜌𝜌𝑤𝑤 𝐤𝐤𝑘𝑘𝑟𝑟𝑤𝑤 𝜇𝜇𝑤𝑤 𝛻𝛻𝜕𝜕𝑤𝑤+ 𝜌𝜌𝑤𝑤𝑔𝑔𝛻𝛻ℎ = 𝑞𝑞𝑠𝑠𝑤𝑤 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜌𝜌𝑆𝑆𝑤𝑤 𝐵𝐵𝑤𝑤 − 𝛻𝛻. 𝐤𝐤 𝜇𝜇𝑤𝑤 𝑘𝑘𝑟𝑟𝑤𝑤 𝐵𝐵𝑤𝑤 𝛻𝛻𝜕𝜕𝑤𝑤+ 𝜌𝜌𝑤𝑤𝑔𝑔𝛻𝛻ℎ = 𝑞𝑞𝑠𝑠𝑤𝑤 𝜕𝜕 𝜕𝜕𝜕𝜕 𝑆𝑆𝑜𝑜𝜌𝜌𝑜𝑜𝜌𝜌 − 𝛻𝛻. 𝜌𝜌𝑜𝑜 𝐤𝐤𝑘𝑘𝑟𝑟𝑜𝑜 𝜇𝜇𝑜𝑜 𝛻𝛻𝜕𝜕𝑜𝑜+ 𝜌𝜌𝑜𝑜𝑔𝑔𝛻𝛻ℎ = 𝑞𝑞𝑠𝑠𝑜𝑜 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜌𝜌𝑆𝑆𝑜𝑜 𝐵𝐵𝑜𝑜 − 𝛻𝛻. 𝐤𝐤 𝜇𝜇𝑜𝑜 𝑘𝑘𝑟𝑟𝑜𝑜 𝐵𝐵𝑜𝑜 𝛻𝛻𝜕𝜕𝑜𝑜+ 𝜌𝜌𝑜𝑜𝑔𝑔𝛻𝛻ℎ = 𝑞𝑞𝑠𝑠𝑜𝑜 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜌𝜌𝑆𝑆𝑤𝑤 𝐵𝐵𝑤𝑤 = 𝜌𝜌 𝐵𝐵𝑤𝑤 𝜕𝜕𝑆𝑆𝑤𝑤 𝜕𝜕𝜕𝜕 + 𝜌𝜌𝑆𝑆𝑤𝑤 𝜕𝜕 𝜕𝜕𝜕𝜕𝑜𝑜 1 𝐵𝐵𝑤𝑤 + 𝑆𝑆𝑤𝑤 𝐵𝐵𝑤𝑤 𝜕𝜕𝜌𝜌 𝜕𝜕𝜕𝜕𝑜𝑜 𝜕𝜕𝜕𝜕𝑜𝑜 𝜕𝜕𝜕𝜕 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜌𝜌𝑆𝑆𝑜𝑜 𝐵𝐵𝑜𝑜 = 𝜌𝜌 𝐵𝐵𝑜𝑜 𝜕𝜕𝑆𝑆𝑜𝑜 𝜕𝜕𝜕𝜕 + 𝜌𝜌𝑆𝑆𝑜𝑜 𝜕𝜕 𝜕𝜕𝜕𝜕𝑜𝑜 1 𝐵𝐵𝑜𝑜 + 𝑆𝑆𝑜𝑜 𝐵𝐵𝑜𝑜 𝜕𝜕𝜌𝜌 𝜕𝜕𝜕𝜕𝑜𝑜 𝜕𝜕𝜕𝜕𝑜𝑜 𝜕𝜕𝜕𝜕 𝑆𝑆𝑤𝑤+ 𝑆𝑆𝑜𝑜= 1 ∴ 𝜕𝜕𝑆𝑆_{𝜕𝜕𝜕𝜕}𝑤𝑤+𝜕𝜕𝑆𝑆_{𝜕𝜕𝜕𝜕}𝑜𝑜= 0 𝐵𝐵𝑜𝑜 𝜌𝜌 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜌𝜌𝑆𝑆𝑜𝑜 𝐵𝐵𝑜𝑜 + 𝐵𝐵𝑤𝑤 𝜌𝜌 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜌𝜌𝑆𝑆𝑤𝑤 𝐵𝐵𝑤𝑤 = 𝐵𝐵𝑜𝑜𝑆𝑆𝑜𝑜 𝜕𝜕 𝜕𝜕𝜕𝜕𝑜𝑜 1 𝐵𝐵𝑜𝑜 + 𝐵𝐵𝑤𝑤𝑆𝑆𝑤𝑤 𝜕𝜕 𝜕𝜕𝜕𝜕𝑜𝑜 1 𝐵𝐵𝑤𝑤 + 1 𝜌𝜌 𝜕𝜕𝜌𝜌 𝜕𝜕𝜕𝜕𝑜𝑜 compressibility terms 𝛼𝛼 𝑆𝑆o,𝑝𝑝o 𝜕𝜕𝜕𝜕𝑜𝑜 𝜕𝜕𝜕𝜕 𝜕𝜕𝑐𝑐= 𝜕𝜕𝑜𝑜− 𝜕𝜕𝑤𝑤 ∴ 𝛻𝛻𝜕𝜕𝑤𝑤 = 𝛻𝛻𝜕𝜕𝑜𝑜− 𝛻𝛻𝜕𝜕𝑐𝑐 𝜌𝜌𝛼𝛼 𝑆𝑆𝑜𝑜, 𝜕𝜕𝑜𝑜 𝜕𝜕𝜕𝜕_{𝜕𝜕𝜕𝜕}𝑜𝑜 = 𝐵𝐵𝑜𝑜𝑞𝑞𝑠𝑠𝑜𝑜+ 𝐵𝐵𝑤𝑤𝑞𝑞𝑠𝑠𝑤𝑤 source/sink + 𝐵𝐵𝑜𝑜𝛻𝛻 � _𝜇𝜇𝐤𝐤 𝑜𝑜 𝑘𝑘𝑟𝑟𝑜𝑜 𝐵𝐵𝑜𝑜 𝛻𝛻𝜕𝜕𝑜𝑜 oil flow + 𝐵𝐵𝑤𝑤𝛻𝛻 � _𝜇𝜇𝐤𝐤 𝑤𝑤 𝑘𝑘𝑟𝑟𝑤𝑤 𝐵𝐵𝑤𝑤 𝛻𝛻𝜕𝜕𝑜𝑜 water flow + 𝐵𝐵𝑜𝑜𝛻𝛻 � _𝜇𝜇𝐤𝐤 𝑜𝑜 𝑘𝑘𝑟𝑟𝑜𝑜 𝐵𝐵𝑜𝑜 𝜌𝜌𝑜𝑜𝑔𝑔𝛻𝛻ℎ + 𝐵𝐵𝑤𝑤𝛻𝛻 � 𝐤𝐤 𝜇𝜇𝑤𝑤 𝑘𝑘𝑟𝑟𝑤𝑤 𝐵𝐵𝑤𝑤 𝜌𝜌𝑤𝑤𝑔𝑔𝛻𝛻ℎ gravity terms + 𝐵𝐵𝑤𝑤𝛻𝛻 � _𝜇𝜇𝐤𝐤 𝑤𝑤 𝑘𝑘𝑟𝑟𝑤𝑤 𝐵𝐵𝑤𝑤 𝛻𝛻𝜕𝜕𝑐𝑐 capillary term Mobility 𝜆𝜆𝑜𝑜 𝑆𝑆𝑜𝑜 =𝐤𝐤𝑘𝑘_𝜇𝜇𝑟𝑟𝑜𝑜 𝑜𝑜 and 𝜆𝜆𝑤𝑤 𝑆𝑆𝑜𝑜 = 𝐤𝐤𝑘𝑘𝑟𝑟𝑤𝑤 𝜇𝜇𝑤𝑤 Pressure equation 𝜌𝜌𝛼𝛼 𝑆𝑆𝑜𝑜, 𝜕𝜕𝑜𝑜 𝜕𝜕𝜕𝜕_{𝜕𝜕𝜕𝜕}𝑜𝑜 = 𝐵𝐵𝑜𝑜𝑞𝑞𝑠𝑠𝑜𝑜+ 𝐵𝐵𝑤𝑤𝑞𝑞𝑠𝑠𝑤𝑤 + 𝐵𝐵𝑜𝑜𝛻𝛻 � 𝜆𝜆_𝐵𝐵𝑜𝑜 𝑜𝑜𝛻𝛻𝜕𝜕𝑜𝑜 + 𝐵𝐵𝑤𝑤𝛻𝛻 � 𝜆𝜆𝑤𝑤 𝐵𝐵𝑤𝑤𝛻𝛻𝜕𝜕𝑜𝑜 + 𝐵𝐵𝑜𝑜𝛻𝛻 � 𝜆𝜆𝑜𝑜 𝐵𝐵𝑜𝑜𝜌𝜌𝑜𝑜𝑔𝑔𝛻𝛻ℎ + 𝐵𝐵𝑤𝑤𝛻𝛻 � 𝜆𝜆𝑤𝑤 𝐵𝐵𝑤𝑤𝜌𝜌𝑤𝑤𝑔𝑔𝛻𝛻ℎ + 𝐵𝐵𝑤𝑤𝛻𝛻 � 𝜆𝜆𝑤𝑤 𝐵𝐵𝑤𝑤𝛻𝛻𝜕𝜕𝑐𝑐 Saturation equation 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜌𝜌𝑆𝑆𝑜𝑜 𝐵𝐵𝑜𝑜 − 𝛻𝛻. 𝜆𝜆𝑜𝑜 𝐵𝐵𝑜𝑜 𝛻𝛻𝜕𝜕𝑜𝑜+ 𝜌𝜌𝑜𝑜𝑔𝑔𝛻𝛻ℎ = 𝑞𝑞𝑠𝑠𝑜𝑜

Initial condition

•

specifies the initial state of the primary variables of the system. For the simple case (1-D), a constant initial pressure.

𝜕𝜕 𝑥𝑥, 𝜕𝜕 = 0 = 𝜕𝜕

𝑖𝑖

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).

𝜕𝜕 𝑥𝑥 = 0, 𝜕𝜕 = 𝜕𝜕

0

Neumann (second type) boundary condition

•

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).

−

𝑘𝑘𝑘𝑘

_𝜇𝜇

𝜕𝜕𝜕𝜕

_{𝜕𝜕𝑥𝑥}

𝑥𝑥=0

= 𝑞𝑞

0

Robin (third type) boundary condition

Time level 𝑛𝑛, initialize 𝑆𝑆𝑜𝑜𝑖𝑖𝑛𝑛 , 𝜕𝜕𝑖𝑖𝑛𝑛

Solve pressure equation for current time step 𝛻𝛻 � 𝜆𝜆𝑡𝑡 𝑆𝑆𝑜𝑜𝑖𝑖𝑛𝑛 𝛻𝛻𝜕𝜕 = 0

to obtain 𝜕𝜕𝑖𝑖𝑛𝑛+1

Use 𝜕𝜕𝑖𝑖𝑛𝑛+1and 𝜆𝜆𝑜𝑜 𝑆𝑆𝑜𝑜𝑖𝑖𝑛𝑛 to solve saturation equation

𝜌𝜌𝜕𝜕𝑆𝑆_{𝜕𝜕𝜕𝜕 − 𝛻𝛻. 𝜆𝜆}𝑜𝑜 𝑜𝑜 𝑆𝑆𝑜𝑜𝑖𝑖𝑛𝑛 𝛻𝛻𝜕𝜕 = 0 to obtain 𝑆𝑆𝑜𝑜𝑖𝑖𝑛𝑛+1

Are these 𝜕𝜕𝑖𝑖𝑛𝑛+1and 𝑆𝑆𝑜𝑜𝑖𝑖𝑛𝑛+1satisfactory? (converged?) Final time reached? Calculate mobilities 𝜆𝜆𝑡𝑡 𝑆𝑆𝑜𝑜𝑖𝑖𝑛𝑛 = 𝜆𝜆𝑜𝑜 𝑆𝑆𝑜𝑜𝑖𝑖𝑛𝑛 + 𝜆𝜆𝑤𝑤 𝑆𝑆𝑜𝑜𝑖𝑖𝑛𝑛

Keep 𝜕𝜕𝑖𝑖𝑛𝑛+1and 𝑆𝑆𝑜𝑜𝑖𝑖𝑛𝑛+1and set to “current” values. Take the next time step

YES Set the latest 𝜕𝜕𝑖𝑖𝑛𝑛+1and 𝑆𝑆𝑜𝑜𝑖𝑖𝑛𝑛+1to

"current" values and ITERATE through calculation again

NO

NO

START

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

Pressure equation

𝜕𝜕

𝜕𝜕𝑥𝑥 𝜆𝜆

𝑡𝑡

𝜕𝜕𝜕𝜕

𝑜𝑜

𝜕𝜕𝑥𝑥 = 0

where 𝜆𝜆

𝑡𝑡

𝑆𝑆

𝑜𝑜

= 𝜆𝜆

𝑜𝑜

+ 𝜆𝜆

𝑤𝑤

is total mobility.

Saturation equation

𝜌𝜌

𝜕𝜕𝑆𝑆

𝑜𝑜

𝜕𝜕𝜕𝜕 −

𝜕𝜕

𝜕𝜕𝑥𝑥 𝜆𝜆

𝑜𝑜

𝜕𝜕𝜕𝜕

𝑜𝑜

𝜕𝜕𝑥𝑥 = 0

𝜌𝜌

𝜕𝜕𝑆𝑆

_{𝜕𝜕𝜕𝜕 +}

𝑜𝑜

𝜕𝜕𝑣𝑣

_{𝜕𝜕𝑥𝑥 = 0}

𝑜𝑜

and 𝜌𝜌

𝜕𝜕𝑆𝑆

_{𝜕𝜕𝜕𝜕 +}

𝑤𝑤

𝜕𝜕𝑣𝑣

_{𝜕𝜕𝑥𝑥 = 0}

𝑤𝑤

Buckley-Leverett solution

𝑣𝑣

𝑆𝑆𝑤𝑤𝑤𝑤

=

𝑑𝑑𝑥𝑥

𝑑𝑑𝜕𝜕

_𝑆𝑆 𝑤𝑤𝑤𝑤

=

_{𝑘𝑘𝜌𝜌}

𝑞𝑞

𝑡𝑡

_{𝜕𝜕𝑆𝑆}

𝜕𝜕𝑓𝑓

𝑤𝑤 𝑤𝑤 𝑆𝑆_{𝑤𝑤𝑤𝑤}

𝑥𝑥

𝑤𝑤𝑓𝑓

=

_{𝑘𝑘𝜌𝜌}

𝑊𝑊

𝑖𝑖

_{𝑑𝑑𝑆𝑆}

𝑑𝑑𝑓𝑓

𝑤𝑤 𝑤𝑤 𝑆𝑆_{𝑤𝑤𝑤𝑤}

1

0.8

0.6

0.4

0.2

0

5

4

3

2

1

0

0.2

0.4

0.6

0.8

1

𝑑𝑑𝑓𝑓

_𝑤𝑤

𝑑𝑑𝑆𝑆

𝑤𝑤

𝑆𝑆

_w

𝑓𝑓

_𝑤𝑤

𝑆𝑆

_𝑤𝑤

1 − 𝑆𝑆

𝑜𝑜𝑟𝑟

𝑆𝑆

𝑤𝑤𝑖𝑖

𝑥𝑥

_{𝑤𝑤𝑓𝑓}

⁄

𝑥𝑥 𝐿𝐿

1

0.8

0.6

0.4

0.2

0

Saturation profile

Shock front

Welge (1952)

𝑊𝑊

_𝑖𝑖

= 𝑘𝑘𝜌𝜌𝑥𝑥

_{𝑤𝑤𝑓𝑓}

𝑆𝑆

_𝑤𝑤

−𝑆𝑆

_{𝑤𝑤𝑖𝑖}

Buckley-Leverett solution:

𝑥𝑥

𝑤𝑤𝑓𝑓

=

_{𝑘𝑘𝜌𝜌}

𝑊𝑊

𝑖𝑖

_{𝑑𝑑𝑆𝑆}

𝑑𝑑𝑓𝑓

𝑤𝑤 𝑤𝑤 𝑆𝑆𝑤𝑤𝑤𝑤

1

𝑤𝑤

̅𝑆𝑆

= 𝑆𝑆

𝑤𝑤𝑖𝑖

+ �

1

_{𝑑𝑑𝑆𝑆}

𝑑𝑑𝑓𝑓

𝑤𝑤 𝑤𝑤 𝑆𝑆𝑤𝑤𝑤𝑤

2 𝑆𝑆

𝑤𝑤

=

�

�

0 𝑥𝑥𝑤𝑤𝑤𝑤

𝑆𝑆

𝑤𝑤

𝑑𝑑𝑥𝑥 �

0 𝑥𝑥𝑤𝑤𝑤𝑤

𝑑𝑑𝑥𝑥 = 𝑆𝑆

𝑤𝑤𝑓𝑓

+

1 − 𝑓𝑓

_{𝑑𝑑𝑓𝑓}

𝑤𝑤𝑓𝑓 𝑤𝑤

𝑑𝑑𝑆𝑆

_{𝑤𝑤 𝑆𝑆𝑤𝑤𝑤𝑤}

𝑑𝑑𝑓𝑓

_𝑤𝑤

𝑑𝑑𝑆𝑆

_{𝑤𝑤 𝑆𝑆} 𝑤𝑤𝑤𝑤

=

1 − 𝑓𝑓

𝑤𝑤𝑓𝑓

𝑆𝑆

_𝑤𝑤

− 𝑆𝑆

_{𝑤𝑤𝑓𝑓}

=

1

𝑆𝑆

_𝑤𝑤

− 𝑆𝑆

_{𝑤𝑤𝑖𝑖} 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 𝑆𝑆𝑤𝑤 1 − 𝑆𝑆𝑜𝑜𝑟𝑟 𝑆𝑆𝑤𝑤𝑖𝑖 𝑥𝑥𝑤𝑤𝑓𝑓 𝑆𝑆𝑤𝑤 𝑥𝑥 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 𝑓𝑓𝑤𝑤 𝑆𝑆𝑤𝑤 𝑆𝑆𝑤𝑤𝑓𝑓 𝑆𝑆𝑤𝑤𝑖𝑖 𝑆𝑆𝑤𝑤 𝑓𝑓𝑤𝑤𝑓𝑓 𝑓𝑓𝑤𝑤= 1 1 + 𝑘𝑘𝑟𝑟𝑜𝑜 𝑘𝑘𝑟𝑟𝑤𝑤 𝜇𝜇𝑤𝑤 𝜇𝜇𝑜𝑜

𝑆𝑆

𝑤𝑤𝑓𝑓

= Water saturation at flood front before water breakthrough

𝑓𝑓

_{𝑤𝑤𝑓𝑓}

= Producing water cut at flood front before water breakthrough

𝑆𝑆

𝑤𝑤

= Average water saturation behind the front at water breakthrough

Front location

𝑥𝑥

_{𝑆𝑆𝑤𝑤𝑤𝑤}

=

_{𝑘𝑘𝜌𝜌}

𝑊𝑊

𝑖𝑖

_{𝑑𝑑𝑆𝑆}

𝑑𝑑𝑓𝑓

𝑤𝑤 𝑤𝑤 𝑆𝑆𝑤𝑤𝑤𝑤

Front velocity

𝑣𝑣

𝑆𝑆_{𝑤𝑤𝑤𝑤}

=

𝑑𝑑𝑥𝑥

_{𝑑𝑑𝜕𝜕}

𝑆𝑆𝑤𝑤𝑤𝑤

=

_{𝑘𝑘𝜌𝜌}

𝑞𝑞

𝑡𝑡

_{𝜕𝜕𝑆𝑆}

𝜕𝜕𝑓𝑓

𝑤𝑤 𝑤𝑤 𝑡𝑡 𝑆𝑆𝑤𝑤𝑤𝑤

Injected pore volume

𝑄𝑄

_{𝑤𝑤𝑖𝑖}

=

_{𝐿𝐿𝑘𝑘𝜌𝜌 = �}

𝑊𝑊

𝑖𝑖

1

_{𝑑𝑑𝑆𝑆}

𝑑𝑑𝑓𝑓

𝑤𝑤

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.

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).

Taylor series expansion

𝜕𝜕 𝑥𝑥 + ∆𝑥𝑥, 𝜕𝜕 = 𝜕𝜕 𝑥𝑥, 𝜕𝜕 +

∆𝑥𝑥

_1!

𝜕𝜕𝜕𝜕 𝑥𝑥, 𝜕𝜕

_{𝜕𝜕𝑥𝑥 +}

∆𝑥𝑥

_2!

2

𝜕𝜕

2

_{𝜕𝜕𝑥𝑥}

𝜕𝜕 𝑥𝑥, 𝜕𝜕

₂

+

∆𝑥𝑥

_3!

3

𝜕𝜕

3

𝜕𝜕 𝑥𝑥, 𝜕𝜕

_{𝜕𝜕𝑥𝑥}

₃

+ ⋯

𝜕𝜕 𝑥𝑥 − ∆𝑥𝑥, 𝜕𝜕 = 𝜕𝜕 𝑥𝑥, 𝜕𝜕 +

−∆𝑥𝑥

_1!

𝜕𝜕𝜕𝜕 𝑥𝑥, 𝜕𝜕

_{𝜕𝜕𝑥𝑥 +}

−∆𝑥𝑥

_2!

2

𝜕𝜕

2

𝜕𝜕 𝑥𝑥, 𝜕𝜕

_{𝜕𝜕𝑥𝑥}

₂

+

−∆𝑥𝑥

_3!

3

𝜕𝜕

3

𝜕𝜕 𝑥𝑥, 𝜕𝜕

_{𝜕𝜕𝑥𝑥}

₃

+ ⋯

Scheme

Spatial

Time

Forward difference

𝜕𝜕𝜕𝜕

_{𝜕𝜕𝑥𝑥}

𝑖𝑖 𝑛𝑛

=

𝜕𝜕

𝑖𝑖+1𝑛𝑛

_∆𝑥𝑥

− 𝜕𝜕

𝑖𝑖𝑛𝑛

+ 𝑂𝑂 ∆𝑥𝑥

𝜕𝜕𝜕𝜕

_{𝜕𝜕𝜕𝜕}

𝑖𝑖 𝑛𝑛

=

𝜕𝜕

𝑖𝑖𝑛𝑛+1

_∆𝜕𝜕

− 𝜕𝜕

𝑖𝑖𝑛𝑛

+ 𝑂𝑂 ∆𝜕𝜕

Backward difference

𝜕𝜕𝜕𝜕

_{𝜕𝜕𝑥𝑥}

𝑖𝑖 𝑛𝑛

=

𝜕𝜕

𝑖𝑖𝑛𝑛

− 𝜕𝜕

_∆𝑥𝑥

𝑖𝑖−1𝑛𝑛

+ 𝑂𝑂 ∆𝑥𝑥

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

_𝑖𝑖 𝑛𝑛+1

=

𝜕𝜕

𝑖𝑖𝑛𝑛+1

_∆𝜕𝜕

− 𝜕𝜕

𝑖𝑖𝑛𝑛

+ 𝑂𝑂 ∆𝜕𝜕

Central difference

𝜕𝜕𝜕𝜕

_{𝜕𝜕𝑥𝑥}

𝑖𝑖 𝑛𝑛

=

𝜕𝜕

𝑖𝑖+1𝑛𝑛

_2∆𝑥𝑥

− 𝜕𝜕

𝑖𝑖−1𝑛𝑛

+ 𝑂𝑂 ∆𝑥𝑥

2

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

_𝑖𝑖 𝑛𝑛

=

𝜕𝜕

𝑖𝑖𝑛𝑛+1

_2∆𝜕𝜕

− 𝜕𝜕

𝑖𝑖𝑛𝑛−1

+ 𝑂𝑂 ∆𝜕𝜕

2

Second order derivative

𝜕𝜕

2

𝜕𝜕

𝜕𝜕𝑥𝑥

2

𝑖𝑖 𝑛𝑛

Scheme

Spatial

Time

Explicit

𝜕𝜕

2

𝜕𝜕

𝜕𝜕𝑥𝑥

2 𝑖𝑖 𝑛𝑛

=

𝜕𝜕

𝑖𝑖+1𝑛𝑛

− 2𝜕𝜕

_∆𝑥𝑥

𝑖𝑖𝑛𝑛₂

+ 𝜕𝜕

𝑖𝑖−1𝑛𝑛

𝜕𝜕𝜕𝜕

_{𝜕𝜕𝜕𝜕}

𝑖𝑖 𝑛𝑛

=

𝜕𝜕

𝑖𝑖𝑛𝑛+1

− 𝜕𝜕

𝑖𝑖𝑛𝑛

∆𝜕𝜕

Implicit

𝜕𝜕

2

𝜕𝜕

𝜕𝜕𝑥𝑥

2 𝑖𝑖 𝑛𝑛+1

=

𝜕𝜕

𝑖𝑖+1𝑛𝑛+1

− 2𝜕𝜕

_∆𝑥𝑥

𝑖𝑖𝑛𝑛+1₂

+ 𝜕𝜕

𝑖𝑖−1𝑛𝑛+1

𝜕𝜕𝜕𝜕

_{𝜕𝜕𝜕𝜕}

𝑖𝑖 𝑛𝑛+1

=

𝜕𝜕

𝑖𝑖𝑛𝑛+1

_∆𝜕𝜕

− 𝜕𝜕

𝑖𝑖𝑛𝑛

Crank-Nicholson

𝜕𝜕

2

_𝜕𝜕

𝜕𝜕𝑥𝑥

2 𝑖𝑖 𝑛𝑛+12

=

₂

1

𝜕𝜕

𝑖𝑖+1𝑛𝑛

− 2𝜕𝜕

_∆𝑥𝑥

𝑖𝑖𝑛𝑛₂

+ 𝜕𝜕

𝑖𝑖−1𝑛𝑛

+

𝜕𝜕

𝑖𝑖+1𝑛𝑛+1

− 2𝜕𝜕

_∆𝑥𝑥

𝑖𝑖𝑛𝑛+1₂

+ 𝜕𝜕

𝑖𝑖−1𝑛𝑛+1

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

_𝑖𝑖 𝑛𝑛+12

=

𝜕𝜕

𝑖𝑖𝑛𝑛+1

_∆𝜕𝜕

− 𝜕𝜕

𝑖𝑖𝑛𝑛

𝑛𝑛

𝑛𝑛 + 1

𝑖𝑖 − 1

𝑖𝑖

𝑖𝑖 + 1

Explicit Scheme

𝑛𝑛

𝑛𝑛 + 1

𝑖𝑖 − 1

𝑖𝑖

𝑖𝑖 + 1

Implicit Scheme

𝜕𝜕

2

_𝜕𝜕

𝜕𝜕𝑥𝑥

2

=

1

𝛼𝛼

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

Explicit: 𝜕𝜕

_𝑖𝑖𝑛𝑛+1

_{= 𝜕𝜕}

𝑖𝑖𝑛𝑛

+

𝛼𝛼∆𝑡𝑡_∆𝑥𝑥2

𝜕𝜕

𝑖𝑖+1𝑛𝑛

− 2𝜕𝜕

𝑖𝑖𝑛𝑛

+ 𝜕𝜕

𝑖𝑖−1𝑛𝑛

Implicit:

𝜕𝜕

_𝑖𝑖−1𝑛𝑛+1

− 2 +

∆𝑥𝑥_{𝛼𝛼∆𝑡𝑡}2

𝜕𝜕

_𝑖𝑖𝑛𝑛+1

+ 𝜕𝜕

_𝑖𝑖+1𝑛𝑛+1

= −

∆𝑥𝑥_{𝛼𝛼∆𝑡𝑡}2

𝜕𝜕

_𝑖𝑖𝑛𝑛

𝑎𝑎

𝑖𝑖−1

𝜕𝜕

𝑖𝑖−1𝑛𝑛+1

+ 𝑎𝑎

𝑖𝑖

𝜕𝜕

𝑖𝑖𝑛𝑛+1

+ 𝑎𝑎

𝑖𝑖+1

𝜕𝜕

𝑖𝑖+1𝑛𝑛+1

= 𝑏𝑏

𝑖𝑖

𝑘𝑘 𝜕𝜕 = 𝑏𝑏

𝑖𝑖 = 2

3

𝑚𝑚 − 2

𝑚𝑚 − 1

𝑎𝑎

2

𝑎𝑎

2

𝑎𝑎

3

𝑎𝑎

3

𝑎𝑎

₄

⋱

⋱

𝑎𝑎

𝑚𝑚−3

⋱

𝑎𝑎

𝑚𝑚−2

𝑎𝑎

𝑚𝑚−2

𝑎𝑎

𝑚𝑚−1

𝑎𝑎

𝑚𝑚−1

𝜕𝜕

2

𝜕𝜕

3

⋮

𝜕𝜕

𝑚𝑚−2

𝜕𝜕

𝑚𝑚−1

=

𝑏𝑏

2

− 𝑎𝑎

1

𝜕𝜕

1𝑛𝑛+1

𝑏𝑏

3

⋮

𝑏𝑏

𝑚𝑚−2

𝑏𝑏

𝑚𝑚−1

− 𝑎𝑎

𝑚𝑚

𝜕𝜕

m𝑛𝑛+1

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

Space index: 𝑖𝑖 = 2, 3, … , 𝑚𝑚 − 1

Time index : 𝑛𝑛 = 1, 2, 3 …

∆𝑥𝑥: space interval

∆𝜕𝜕: time interval

𝛼𝛼: hydraulic diffusivity

For all times:

𝜕𝜕

1

= 𝜕𝜕

0

(left boundary)

𝜕𝜕

𝑚𝑚

= 𝜕𝜕

i

(right boundary)

space tim e

𝑛𝑛 = 0

1

2

3

𝑖𝑖 = 1

2

𝑖𝑖

𝑚𝑚 − 1 𝑚𝑚

left

boundary

boundary

right

in

itia

l

condi

tion

1 2 3 4 5 6 7 8 1 × × 2 × × × 3 × × × 4 × × × 5 × × × 6 × × × 7 × × × 8 × ×

𝜕𝜕 = 0.2 0.4 0.6 1.0 2.0

Explicit

𝜕𝜕 = 0.2 0.4 0.6 1.0 2.0

Implicit

𝑥𝑥

𝜕𝜕

Steady state

Transient

𝜕𝜕

i

𝜕𝜕

0

Analytical solution

𝜕𝜕

2

𝜕𝜕

𝜕𝜕𝑥𝑥

2

=

1

𝛼𝛼

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕 𝜕𝜕 = 0 = 𝜕𝜕

i

𝜕𝜕 𝑥𝑥 = 0 = 𝜕𝜕

0

𝜕𝜕 𝑥𝑥 = 𝐿𝐿 = 𝜕𝜕

_i

𝑚𝑚 = 5

∆𝑥𝑥 = 0.2

∆𝜕𝜕 = 0.2

𝛼𝛼 = 0.1

m

2

⁄

_s

𝑥𝑥

𝑦𝑦

𝑧𝑧

𝜕𝜕

𝜕𝜕𝑥𝑥 𝑘𝑘

𝑥𝑥

𝜕𝜕𝜕𝜕

𝜕𝜕𝑥𝑥 = 𝛽𝛽

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝛽𝛽 = 𝜇𝜇𝜌𝜌𝑐𝑐

𝑡𝑡

𝑘𝑘

_𝑥𝑥

𝜕𝜕𝜕𝜕

_{𝜕𝜕𝑥𝑥}

𝑖𝑖+ ⁄1 2 𝑛𝑛+1

− 𝑘𝑘

_𝑥𝑥

𝜕𝜕𝜕𝜕

_{𝜕𝜕𝑥𝑥}

𝑖𝑖− ⁄1 2 𝑛𝑛+1

∆𝑥𝑥

= 𝛽𝛽

𝜕𝜕

_𝑖𝑖𝑛𝑛+1

− 𝜕𝜕

_𝑖𝑖𝑛𝑛

∆𝜕𝜕

𝑘𝑘

_{𝑥𝑥 𝑖𝑖+ ⁄1 2}

𝜕𝜕

𝑖𝑖+1𝑛𝑛+1

_∆𝑥𝑥

− 𝜕𝜕

₂ 𝑖𝑖𝑛𝑛+1

− 𝑘𝑘

_{𝑥𝑥 𝑖𝑖− ⁄1 2}

𝜕𝜕

𝑖𝑖𝑛𝑛+1

_∆𝑥𝑥

− 𝜕𝜕

₂ 𝑖𝑖−1𝑛𝑛+1

= 𝛽𝛽

𝜕𝜕

𝑖𝑖𝑛𝑛+1

_∆𝜕𝜕

− 𝜕𝜕

𝑖𝑖𝑛𝑛

−

𝑘𝑘

𝑥𝑥 𝑖𝑖− ⁄

_∆𝑥𝑥

₂1 2

𝜕𝜕

_𝑖𝑖−1𝑛𝑛+1

+

_∆𝑥𝑥

1

₂

𝑘𝑘

_{𝑥𝑥 𝑖𝑖− ⁄1 2}

+ 𝑘𝑘

_{𝑥𝑥 𝑖𝑖+ ⁄1 2}

−

_{∆𝜕𝜕 𝜕𝜕}

𝛽𝛽

_𝑖𝑖𝑛𝑛+1

−

𝑘𝑘

𝑥𝑥 𝑖𝑖+ ⁄

_∆𝑥𝑥

₂1 2

𝜕𝜕

_𝑖𝑖+1𝑛𝑛+1

=

_{∆𝜕𝜕 𝜕𝜕}

𝛽𝛽

_𝑖𝑖𝑛𝑛

System of equations can be shown in simplified form as:

𝑎𝑎

_𝑖𝑖−1

𝜕𝜕

_𝑖𝑖−1𝑛𝑛+1

+ 𝑎𝑎

_𝑖𝑖

𝜕𝜕

_𝑖𝑖𝑛𝑛+1

+ 𝑎𝑎

_𝑖𝑖+1

𝜕𝜕

_𝑖𝑖+1𝑛𝑛+1

= 𝑏𝑏

_𝑖𝑖

Permeability is evaluated at the boundary faces by harmonic averaging.

𝑖𝑖

𝑖𝑖 + 1

𝑖𝑖 − 1

𝑖𝑖 − ⁄

1 2

𝑖𝑖 + ⁄

1 2

𝑘𝑘

_{𝑥𝑥 𝑖𝑖− ⁄1 2}

=

2 𝑘𝑘

_𝑘𝑘

𝑥𝑥 𝑖𝑖−1

𝑘𝑘

𝑥𝑥 𝑖𝑖 𝑥𝑥 𝑖𝑖−1

+ 𝑘𝑘

𝑥𝑥 𝑖𝑖

𝑘𝑘

_{𝑥𝑥 𝑖𝑖+ ⁄1 2}

=

2 𝑘𝑘

_𝑘𝑘

𝑥𝑥 𝑖𝑖

𝑘𝑘

𝑥𝑥 𝑖𝑖+1 𝑥𝑥 𝑖𝑖

+ 𝑘𝑘

𝑥𝑥 𝑖𝑖+1

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 (∆𝜕𝜕) 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 [ 𝑘𝑘

_𝑟𝑟 𝑛𝑛

_{and 𝜕𝜕}

𝑐𝑐 𝑛𝑛

]. A solution for

instability is to take smaller time steps or go to a fully implicit model.

Numerical dispersion

is caused by saturation discontinuity within a cell. The solutions for numerical

dispersion are: (1) smaller grid block size (∆𝑥𝑥), (2) modify 𝑘𝑘

_𝑟𝑟

calculations (using upstream

weighting instead of average weighting between two grid blocks), (3) select proper time step (∆𝜕𝜕).

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).