Reservoir simulation course

FUNDAMENTALS OF RESERVOIR
SIMULATION

Dr.
 Mai
 Cao
 Lan,
 
GEOPET,
 HCMUT,
 Vietnam
Jan,
 2014


ABOUT THE COURSE
COURSE OBJECTIVE
COURSE OUTLINE
REFERENCES

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum

 Engineering,
 HCMUT,
  Vietnam

2


Course
 Objective
•

To review the background of petroleum reservoir
simulation with an intensive focus on what and how
things are done in reservoir simulations

•

To provide guidelines for hands-­‐‑on practices with
Microsoft Excel

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &

 Petroleum
 Engineering,
 HCMUT,
  Vietnam

3


COURSE OUTLINE
INTRODUCTION
FLOW EQUATIONS FOR PETROLEUM RESERVOIRS
FINITE DIFFERENCE METHOD &
 NUMERICAL SOLUTION FOR
FLOW EQUATIONS
SINGLE-­‐‑PHASE FLOW SIMULATION
MULTIPHASE FLOW SIMULATION


References
§

T.
 Eterkin et
 al.,
 2001.
  Basic
 Applied
 Reservoir
 Simulation,
 

SPE,
 Texas

§

J.H.
 Abou-­‐‑Kassem et
 al.,
 2005.
  Petroleum
 Reservoir
 
Simulation
 – A
 Basic
 Approach,
 Gulf
 Publishing
 Company,
 
Houston,
 Texas.

§

C.Mattax &
 R.
 Dalton,
 1990.
  Reservoir

 Simulation,
 SPE,
 
Texas.
 

9/11/16

Mai Cao Lân – Faculty of Geology & Petroleum Engineering - HCMUT

5


INTRODUCTION
NUMERICAL SIMULATION – AN OVERVIEW
COMPONENTS OF A
 RESERVOIR SIMULATOR
RESERVOIR SIMULATION BASICS

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum

 Engineering,
 HCMUT,
  Vietnam

6


Numerical
 Simulation
 – An
 Overview

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

7

Mathematical
 Formulation

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

8


Numerical
 Methods
 for
 PDEs

9/11/16

Dr.
 Mai

 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

9


Numerical
 Methods
 for
  Linear
 Equations

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &

 Petroleum
 Engineering,
 HCMUT,
  Vietnam

10


Components
 of
 a
 Reservoir
 Simulator
Computer
 Code

Physical
 Model

Reservoir
 
Simulator

Mathematical
 Model

9/11/16

Numerical
 Model

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

11


What
 is
 Reservoir
 Simulation?
•

A powerful tool for evaluating reservoir performance
with the purpose of establishing a sound field
development plan

•

A helpful tool for investigating problems associated with

the petroleum recovery process and searching for
appropriate solutions to the problems

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

12


Reservoir
 Simulation
 Basics
• The
  reservoir
 is
 divided
 into
 a

 number
  of
 cells
• Basic
 data
 is
 provided
 for
 each
 cell
• Wells
 are
 positioned
  within
 the
 cells
• The
  required
 well
 production
 rates
 are
 specified
 as
 a
 
function
 of
 time
• The

  equations
 are
 solved
 to
 give
 the
 pressure
  and
 
saturations
 for
 each
 block
 as
 well
 as
 the
 production
 of
 
each
 phase
 from
 each
 well.
9/11/16

Dr.
 Mai
 Cao

 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

13


Simulating
 Flow
 in
 Reservoirs
• Flow
 from
 one
 grid
 block
 to
 the
 next
• Flow
 from
 a
 grid
 block

 to
 the
 well
 completion
• Flow
 within
 the
 wells
 (and
 surface
 networks)
Flow
 =
 Transmissibility
 *
 Mobility
 *
 Potential
 Difference
Geometry
 &
 
Properties

9/11/16

Fluid
 
Properties

Well
 
Production

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

14


SINGLE-­‐‑PHASE FLOW
EQUATIONS
ESSENTIAL PHYSICS
CONTINUITY EQUATION
MOMENTUM EQUATION
CONSTITUTIVE EQUATION
GENERAL 3D
 SINGLE-­‐‑PHASE FLOW EQUATION
BOUNDARY &
 INITIAL CONDITIONS

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

15


Essential
 Physics
The basic differential equations are derived from the
following essential laws:
§

Mass
 conservation
 law

§

Momentum
 conservation
 law

§

Material
 behavior
 principles

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

16


Conservation
 of

 Mass
Mass
 conservation
 may
 be
 formulated
 across
 a
 control
 element
 with
 one
 fluid
 
of
 density
 ρ, flowing
 through
 it
 at
 a
 velocity
 u:

u
ρ
Δx

⎧Mass into the ⎫ ⎧Mass out of the ⎫ ⎧Rate of change of mass⎫
⎨

⎬ − ⎨
⎬ = ⎨
⎬
element
at
x
element
at
x
+
Dx
inside
the
element
⎩
⎭ ⎩
⎭ ⎩
⎭

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum

 Engineering,
 HCMUT,
  Vietnam

17


Continuity
  Equation
Based
 on
 the
 mass
 conservation
 law,
 the
 continuity
 equation
 can
 be
 
expressed
 as
 follow:

∂
∂
− ( Aρ u ) = A (φρ )
∂x
∂t

For
 constant
 cross
 section
 area,
 one
 has:

∂
∂
− ( ρ u ) = (φρ )
∂x
∂t
9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

18

Conservation
 of
 Momentum
Conservation of momentum for fluid flow in porous materials
is governed by the semi-­‐‑empirical Darcy's equation, which for
one dimensional, horizontal flow is:

k ∂P
u=−
µ ∂x

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

19

Equation
  Governing
  Material
 Behaviors
 
§ The behaviors of rock and fluid during the production
phase of a reservoir are governed by the constitutive
equations or also known as the equations of state.
§ In general, these equations express the relationships
between rock & fluid properties with respect to the
reservoir pressure.

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

20

Constitutive
  Equation
  of
 Rock
The behavior of reservoir rock corresponding to the
pressure declines can be expressed by the definition of the
formation compaction

⎛ 1 ⎞ ⎛ ∂φ ⎞
c f = ⎜ ⎟ ⎜ ⎟
⎝ φ ⎠ ⎝ ∂P ⎠T
For
 isothermal
 processes,
 the
 constitutive
 equation
 of
 rock
 
becomes

dφ
= φc f
dP
9/11/16

Dr.
 Mai
 Cao

 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

21


Constitutive
  Equation
  of
 Fluids
The behavior of reservoir fluids corresponding to the
pressure declines can be expressed by the definition of fluid
compressibility (for liquid)

1 ⎛ ∂V ⎞
cl = − ⎜
⎟ , l = o, w, g
V ⎝ ∂P ⎠T
For natural gas, the well-­‐‑known equation of state is used:

PV = nZRT
9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

22


Single-­‐‑Phase
  Fluid
  System
Normally, in single-­‐‑phase reservoir simulation, we would
deal with one of the following fluids:
Fluid
 System

One
 Phase
 Gas

9/11/16

One
 Phase
 Water

One
 Phase
 Oil

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

23


Single-­‐‑Phase
  Gas
The gas must be single phase in the reservoir, which means
that crossing of the dew point line is not permitted in order
to avoid condensate fall-­‐‑out in the pores. Gas behavior is
governed by:

ρ gs constant
ρg =
=
Bg
Bg
9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

24


Single-­‐‑Phase
  Water
One phase water, which strictly speaking means that the
reservoir pressure is higher than the saturation pressure of
the water in case gas is dissolved in it, has a density
described by:

ρws constant
ρw =
=
Bw
Bw

9/11/16

Dr.
 Mai
 Cao
 Lan,
 F aculty
 of
 G eology
 &
 Petroleum
 Engineering,
 HCMUT,
  Vietnam

25