reservoir simulation
TPG4160 Reservoir Simulation 2012
Lecture note 1
page 1 of 11
INTRODUCTION TO RESERVOIR SIMULATION
Analytical and numerical solutions of simple one-dimensional, one-phase flow equations
As an introduction to reservoir simulation, we will review the simplest one-dimensional flow equations for horizontal flow of one fluid, and look at analytical and numerical solutions of pressure as function of position and time. These equations are derived using the continuity equation, Darcy's equation, and compressibility definitions for rock and fluid, assuming constant permeability and viscosity. They are the simplest equations we can have, which involve transient fluid flow inside the reservoir.
Linear flow
Consider a simple horizontal slab of porous material, where initially the pressure everywhere is P , and then at
0
time zero, the left side pressure (at x = 0 ) is raised to PL while the right side pressure (at x = L ) is kept at
PR = P . The system is shown on the next figure:
0
x q Partial differential equation (PDE)
The linear, one dimensional, horizontal, one phase, partial differential flow equation for a liquid, assuming constant permeability, viscosity and compressibility is:
∂2 P φµc ∂P
)
2 =(
∂x
k ∂t
Transient vs. steady state flow
The equation above includes time dependency through the right hand side term. Thus, it can describe transient, or time dependent flow. If the flow reaches a state where it is no longer time dependent, we denote the flow as steady state. The equation then simplifies to:
d2 P
=0
dx 2
Transient and steady state pressure distributions are illustrated graphically in the figure below for a system where initial and right hand pressures are equal. As can be observed, for some period of time, depending on the properties of the system, the pressure will increase in all parts of the system (transient solution), for then to approach a final distribution (steady state), described by a straight line
Lecture note 1
page 1 of 11
INTRODUCTION TO RESERVOIR SIMULATION
Analytical and numerical solutions of simple one-dimensional, one-phase flow equations
As an introduction to reservoir simulation, we will review the simplest one-dimensional flow equations for horizontal flow of one fluid, and look at analytical and numerical solutions of pressure as function of position and time. These equations are derived using the continuity equation, Darcy's equation, and compressibility definitions for rock and fluid, assuming constant permeability and viscosity. They are the simplest equations we can have, which involve transient fluid flow inside the reservoir.
Linear flow
Consider a simple horizontal slab of porous material, where initially the pressure everywhere is P , and then at
0
time zero, the left side pressure (at x = 0 ) is raised to PL while the right side pressure (at x = L ) is kept at
PR = P . The system is shown on the next figure:
0
x q Partial differential equation (PDE)
The linear, one dimensional, horizontal, one phase, partial differential flow equation for a liquid, assuming constant permeability, viscosity and compressibility is:
∂2 P φµc ∂P
)
2 =(
∂x
k ∂t
Transient vs. steady state flow
The equation above includes time dependency through the right hand side term. Thus, it can describe transient, or time dependent flow. If the flow reaches a state where it is no longer time dependent, we denote the flow as steady state. The equation then simplifies to:
d2 P
=0
dx 2
Transient and steady state pressure distributions are illustrated graphically in the figure below for a system where initial and right hand pressures are equal. As can be observed, for some period of time, depending on the properties of the system, the pressure will increase in all parts of the system (transient solution), for then to approach a final distribution (steady state), described by a straight line