On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs Page: 117 of 188
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and roundoff errors. The truncation error arises because of omitting the second
and higher order terms in the Taylor series expansion shown below.
f(x + h) = f (x) + hf'(x) + -h2f"(x) + -h3f'(x) + ... (8.28)
The truncation error, as a result, is of the order of hf".
The roundoff error arises primarily due to choosing h with a high effective value
which refers to the difference between x + h and x as represented in the machine.
This value is of the order of Emx where em is the machine's floating-point precision.
The fractional error in the order of h can be expressed as an order of '. A large
fractional error translates into a large error in the derivative.
A significantly more accurate way to evaluate the derivative is using a central
difference approximation to estimate it as follows,
, f(x + h) - f(x - h) (8.29)
In this case, the roundoff error is about the same as before but the truncation
error is now of the order of h2f"'. However, this too depends on how wisely the
value of h is chosen.
An improved method to compute the derivative was suggested by Ridders .
In this method, the derivatives computed using finite-difference calculations are
extrapolated, using Neville's algorithm for fitting a polynomial, with progressively
smaller values of h so that eventually h -- 0. A subroutine, dfridr, in C has been
reported in  and will be used for gradient computation in the optimization
algorithm. As a side note, a forward difference scheme is used to compute the
gradient at the lower control bound and a backward difference scheme is used at
the upper control bound.
It is quite obvious that the gradient computation requires the function to
be evaluated several times. During the entire optimization algorithm, the cost
function is evaluated hundreds of times.
8.3.5 Optimization Algorithm
Water injection rate was chosen as the control variable for the optimization prob-
lem. The simulations were performed with a total fluid rate constraint. The
total volume of water injected equalled the total volume of fluids produced. The
simulation period was divided into multiple stages and the algorithm was used
to optimize the water injection rates in all those periods. The number of stages
determined the number of control variables. The gradient of the cost function
with respect to the control variable was computed using the numerical scheme
described in Section 8.3.4. The set of control variables obtained after each op-
timization iteration is referred to as the control policy. The initial set of values
assigned to the water injection rates is known as the nominal control policy and
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Deo, Milind D. On-line Optimization-Based Simulators for Fractured and Non-fractured Reservoirs, report, August 31, 2005; Utah. (https://digital.library.unt.edu/ark:/67531/metadc877059/m1/117/: accessed June 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.