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RESERVOIR SIMULATION
Hussein Y. Ali
Basrah University for Oil and Gas
Explicit Formulation
LECTURE
SEVEN
Stability Analysis
Truncation Error
Consistency and Convergence
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PART 3
EXPLICIT FORMULATION
INTRODUCTION
• The forward-difference approximation to the flow equation results in an explicit
calculation procedure for the new-time-level pressures (designated n+ 1 in the finitedifference equations).
• Solving the forward-difference equation (Eq.9) for the unknown quantity ,
expression:
yields the
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INTRODUCTION
• All terms on the right side of the previous equation are known because all pressures
appearing on this side are at the known (old) time level,n.
• In this equation, the pressures at the new-time level can be obtained explicitly by use of
these known pressures.
EXAMPLE 1
• For the 1D, block-centered grid shown in Fig. below, determine the pressure distribution
during the first year of production. The initial reservoir pressure is 6,000 psia. The rock
and fluid properties for this problem are Δx = 1,000 ft, Δ y= 1,000 ft, Δz=75 ft, B = 1
RB/STB, c = 3.5 x 10^(-6) psi^(-1) , kx= 15 md, ø = 0 .18, µ =10 cp, and
= 1 RB/STB. Use
timestep sizes of Δt= 10, 15, and 30 days. Assume B acts as a constant within the
pressure range of interest.
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EXAMPLE 2
• This example is identical to example 1, except that the no flow boundary on the left side
of the reservoir is replaced by a constant-pressure boundary. Use the same rock and
fluid properties as in example 1. Although this is a block-centered grid system, assume
that the pressure in Grid block 1 is equal to the boundary pressure. se a timestep of 15
days.
STABILITY ANALYSIS
• Stability is a property that describes the capacity of a small error to propagate and
grow with subsequent calculations.
• Explicit method is only stable (errors do not propagate over time) if:
Where
• There are several procedures for analyzing the stability of a given finite difference
approximation such as Fourier Series and Matrix methods.
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CONSISTENCY AND CONVERGENCE
• Consistency: The discretization of a differential equation (e.g. by finite difference
approximation) should become identical to the original PDE as the mesh size and the
size of the time interval approach zero( Tr. Error should vanish).
• Convergence: a numerical solution must convergent to the solution of the PDE when Δx
and Δt are decreased in size.
• Generally: consistency + stability = convergence
TRUNCATION ERROR
• Its the error that introduced when a continuous PDE replaced with a finite-difference
approximation through Taylor series expansion
• These errors may be introduced at a certain stage in the computations which grow
uncontrollably to dominate the solution.
• The problems we need to address are the magnitude of the truncation error and how
can we increase the accuracy of an approximation by decreasing the magnitude of this
error.
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TRUNCATION ERROR
• the local truncation error or local discretization error is the deviation of the PDE from
its corresponding finite-difference approximation at a given point in space and at a
given instant in the time.
• Where
EXAMPLE.3
• Determine the local truncation error of the explicit finite-difference approximation to
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THANK YOU
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