Numerical Reservoir Simulation

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Numerical Reservoir Simulation
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Topic Overview
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• An introduction to standard numerical solution
  techniques for reservoir flow equations.

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Introduction
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Gridding
Stability analyses
Differential equations for mass flow
Reservoir equations
Numerical Modell
Reservoir Performance
Difference Approximation
Discretization Error
For more information click on the subject you
want to learn more about.
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Discretization Techniques
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• General partial differential equations for
  reservoir fluid flow must be discretized before
  they can be treated computationally.
• The most common techniques are
• - finite differences
• - finite elements
• We will in in this module learn about the finite
  difference technique.

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Finite Differences
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• Finite difference approximations are used in most
  commercial reservoir simulation software to solve
  fluid flow equations numerically.
• Main steps in a discretization procedure
• replace differential operators by algebraic
  ciexpressions
• compute approximate solution at given points and
  iiispecified times

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Differential Equations for Mass Flow
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Mass conservation equations for Black Oil models
Where Ql are sink/source term
Discretization Techniques
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Reservoir Equations
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Discrete equations for Black Oil models for block
i,j,k
For more information click on the equation you
want to learn more about.
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Water Equation
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The water equation consists of three parts a
flow term, a well term and an accumulation term.
Flow term well term accumulation term
For more information click on the term of the
water equation you want to learn more about.
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Flow Term for Water
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The flow term for water consists of three terms,
one for each coordinate direction.
For more information click on the term of the
equation you want to learn more about.
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Flow Term for Water in x- direction
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• The x-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox.
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Flow Term for Water in y- direction
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• The y-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox.
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Flow Term for Water in z- direction
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• The z-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox.
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Well Term for Water
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Specification are different for production and
injection wells.
water
Click here to see how the production term for
water is given.
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Up
Well Equations for Black Oil Model
Pwell pressure in the well
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Well Equations for Black Oil Model
Pwell pressure in the well
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Well Equations for Black Oil Model
Pwell pressure in the well
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Accumulation Term for Water
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The change of mass of water in block i,j,k
during time ?t between step n and n1 is given by
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Evaluation on Block Boundaries
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Oil Equation
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The oil equation consists of three parts a flow
term, a well term and an accumulation term.
Flow term well term accumulation term
For more information click on the term of the oil
equation you want to learn more about.
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Flow Term for Oil
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The flow term for oil consists of three terms,
one for each coordinate direction.
For more information click on the term of the
equation you want to learn more about.
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Flow Term for Oil in x- direction
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• The x-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox.
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Flow Term for Oil in y- direction
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• The y-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox.
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Flow Term for Oil in z- direction
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• The z-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox.
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Well Term for Oil
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Specification are different for production and
injection wells.
oil
Click here to see how the production term for oil
is given.
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Accumulation Term for Oil
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The change of mass of water in block i,j,k
during time ?t between step n and n1 is given by
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Gas Equation
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The gas equation consists of a flow term for gas
and dissolved gas, a well term and an
accumulation term for gas and dissolved gas.
Flow terms well term accumulation terms
For more information click on the term of the
equation you want to learn more about.
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Flow Term for Gas
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The flow term for gas consists of three terms,
one for each coordinate direction.
For more information click on the term of the
equation you want to learn more about.
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Flow Term for Gas in x- direction
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• The x-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox.
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Flow Term for Gas in y- direction
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• The y-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox. (not active yet)
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Flow Term for Gas in z- direction
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• The x-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox.
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Flow Term for Dissolved Gas
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The flow term for dissolved gas consists of
three terms, one for each coordinate direction.
For more information click on the term of the
equation you want to learn more about.
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Flow Term for Dissolved Gas in x- direction
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• The x-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox.
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Flow Term for Dissolved Gas in y- direction
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• The y-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox.
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Flow Term for Dissolved Gas in z- direction
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• The z-part consists of two terms one to compute
  flow to neighbour block in the positive direction
  and one for flow in the negative direction.

For information on block boundaries, click on the
textbox.
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Well Term for Gas
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Specification are different for production and
injection wells.
gas
Click here to see how the production term for gas
is given.
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Accumulation Term for Gas and Dissolved Gas
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The change of mass of water in block i,j,k
during time ?t between step n and n1 is given by
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Definition of Symbols
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• l o,w,g
• s x,y,z
• p i,j,k
• ql,i,j,k
• Ql,i,j,k
• ?
• Sl
• Bl
• k
• k
• ?l
• Vi,j,k
• ?t
• ?t
• ?Rs
• Rs
• ?sTls

• ?s?ls
• WIp
• pi
• pwell

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Difference Approximations
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Taylor series can be used to derive a difference
formula for single and double derivates.
Taylor series of f(x?x) and f(x-?x) are given
by
With these expansion we can deduce - first
order approximation of f - second order
approximation of f - second order
approximation of f
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First Order Approximation of f
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From the expansion of f(x?x) we get an
expression for f(x)
From the expansion of f(x-?x) we get an
expression for f(x)
This difference formula is used for discretizing
time derivative in the mass equations
Click on the box to see how the approximation
changes when the step size is halved.
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Difference Formula
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A first order approximation of ut at the point
n1 is given by
The time axis is divided into points at distance
?t
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First Order Approximation of f
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From the serie f(x?x)
From the serie f(x-?x)
The step size reduction produces more accurate
approximations.
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Second Order Approximation of f
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Adding expansion of f(x?x) and f(x-?x) results
in the approximations
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Click on the box to see how the approximation
changes when the time step is halved.
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Second Order Approximation of f
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The sum of f(x) of the series f(x?x) and
f(x-?x)
Step size reduction will produce more accurate
approximations.
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Second Order Approximation of f
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The sum of the Taylor series f(x?x) and f(x-?x)
is used to deduced a second order approximation
of f
This approximation is frequently used and the
numerator is written
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Difference Approximation
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Uxx can be approximated at each point i by the
formula
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Discretization Error
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The order of a difference approximation can by
analysed using Taylor expansions.
The discretization error approaches zero faster
for a high order approximation then for a low
order approximation.
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Gridding
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A faulted reservoir
Well locations
An imposed grid
Initial fluid distribution
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A Faulted Reservoir
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Well Locations
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An Imposed Grid
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Main criteria for grid selection

• The ability to identify saturations and
  pressures ii at specific locations (existing and
  planned well i iiiilocations).
• The ability to produce a solution with the i
  iiiirequired accuracy (numerical dispersion and
  iiiigrid orientation effects).
• The ability to represent geometry, geology and
  iiiphysical properties of the reservoir (external
  iiiboundaries, faults, permeability distribution
  iiiincluding vertical layering).

• Keep the number of grid blocks small in order to
  iiimeet requirements of limited money and time
  iiiavailable for the study.

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Initial Fluid Distribution
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Stability Analyses
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Stable
Unstable
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Stable
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Animation of the stable solution
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Unstable
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Reservoir Performance
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