KAPPA DDA Book v5.12

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Dynamic Data Analysis – v5.12.01 - © KAPPA 1988-2017 Chapte - p148/743
Integrating the material balance correction in an analytical model
The model includes a reservoir size and an initial pressure. So the initial gas in place can be
calculated as an integral part of the model. At any time step the algorithm calculates the
average pressure from the cumulative production using p/Z, and replaces the viscosity and
total compressibility in the superposition by the one coming from the average pressure. So at
any time step the simulated pressure is coherent with the material balance of the model. The
optional derivation is shown as follows:
We consider the total gas in place at initial pressure. Vres is the pore volume occupied by the
gas. Tres is the fluid temperature at reservoir conditions. G i is the initial gas in place at
standard conditions.
Real gas equation at initial pressure: res i res i nRT Z Vp 
Same amount of fluid at standard conditions:
sc i sc nRT Gp 
So we get immediately Gi:
res sc
sc res
i
i
i Tp
TV
Z
p
G  
We now consider, at time t, the same situation after a total cumulative production of Q(t). We
now want to calculate the average reservoir pressure:
Real gas equation at initial pressure: res res RT tnZ Vp )( 
Same amount of fluid at standard conditions:   sc i sc RT tn tQGp )( )( 
So we get immediately Gi:
res sc
sc res
i Tp
TV
Z
p
tQG   )(
We calculate the average pressure from:  
)( tQG
TV
Tp
Z
p
i
sc res
res sc  
Using a numerical model
The use of a numerical model is, conceptually, even simpler. As the gas equation is entered at
the level of each cell, the material balance is automatically honoured, not only globally, as
above, but at the level of each cell. Solving the problem numerically is by far the most rigorous
approach.
As the problem is nonlinear, this requires Topaze NL and the use of a nonlinear solver.