In petroleum engineering, the oil production profiles of a reservoir can be simulated by using a finite gridded model. This profile is affected by the number and choice of wells which in turn is a result of various production limits and constraints including, for example, the economic minimum well spacing, the number of drilling rigs available and the time required to drill and complete a well. After a well is available it may be shut in because of excessive water or gas productions. In order to optimize the field performance a penalty function algorithm was developed for scheduling wells. For an example with some 343 wells and 15 different constraints, the scheduling routine vectorized for the CYBER 205 averaged 560 times faster performance than the scalar version

Absar, I.

English

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VECTORIZATION OF A PENALTY FUNCTION 
ALGORITHM FOR WELL SCHEDULING 
ILYAS ABSAR 
SOHIO PETROLEUM COMPANY 
SAN FRANCISCO, CALIFORNIA 
https://ntrs.nasa.gov/search.jsp?R=19840012165 2020-03-22T08:37:54+00:00Z
Vectorization of a Penalty Function Algorithm for tie11 Scheduling 
Ilyas Absar 
SOHIO Petroleum Co. 
San Francisco,California 
Abstract: 
In petroleum engineering, the oil production profile of a reservoir can be 
simulated by using a finite grided model. This profile is affected by the 
number and choice of wells which in turn is a result of various production 
limits and constraints including, for example, the economic minimum well 
spacing, the number of drilling rigs available and the time required to drill 
and complete a well. After a well is available it may be shut-in because of 
excessive water or gas productions. In order to optimize the field 
performance a penalty function algorithm was developed for scheduling wells. 
For an example with some 343 wells and 15 different constraints, the 
scheduling routine vectorized for the Cyber 205 averaged 560 times faster 
performance than the scalar version. 
351 
__--.-.-- - ._ . _. _- 
Introduction: 
ldlathematical nodelling of the fluid production from a naturally occurring 
reservoir involves considering the reservoir as a network of interconnected 
blocks. To each grid block is associated a ijeologic description through 
properties, e.g., thickness, porosity, permeability, etc. Each grid block is 
considered to be in material balance with its surroundings, i.e., the amount 
of fluid in the block at time t +At is equal to the amount of fluid in that 
1 At minus fluid outflux block at time t plus fluid influx in the time interva 
in the time intervalAt. 
In Figure lA, the reservoir is shown by a curved boundary. Overlaid 
areally is a rectangular grid. The sizes of the blocks can be chosen to 
represent the geological features of the reservoir as accurately as possible. 
Figure 18 shows a two dimensional cross-section of a reservoir and the yrid 
used for its simulation. Notice that the reservoir contains water, oil and 
sas in various regions, and only some blocks are in communication with the 
wells by means of perforations in the well bore. To simulate the production 
profile, the material ba7ance of the grid blocks in which wells are perforated 
must also take into account the fluid production or injection. In this manner 
one obtains pressures and saturations for each of the yrid-blocks. For 
details on mathematical modelling of oil reservoirs please refer to a standard 
text, for example, references 1 and 2. 
Once a reservoir simulator is formulated, it can be used in many ways, 
e.g.: 
1. Assist in makiny economic decisions for field operation, e.g., the 
investments to date at Prudhoe say exceed $9 billion. 
2. Desiyn of production strategy. The effect of changes in the number, 
location, spacing, or timiny of wells can be studied. 
3. Prediction of reservoir performance. 
4. iilatchiny of the production history. 
362 
When an oil field is developed, of course the most important objective is 
to maximize oil recovery. However, this objective is tempered by limitations, 
economic and physical, e.g., costs and capacities of various installations and 
devices. 
The dashed curve in Figure 2 represents oil production when all wells 
flow at their maximum capacity. The area under this curve represents 
cumulative oil production. The ratio of cumulative oil production to in-place 
oil represented as a fraction or percentage is called the Oil Recovery 
Factor. If facilities were constructed for this production profile, they 
would have to be constructed to handle oil production at the maximum rate, 
q max. Economic considerations give us a target oil rate, qy, less than qmax, 
at which oil production can be sustained for a period of time. The solid 
curve in Figure 1 represents this strategy. Note that sometimes this can be 
achieved without appreciable sacrifice in cumulative oil production. 
Well Scheduling Problem: 
Once qt is established, the problem of optimal scheduling, i.e., selecting for 
operation a given number of wells (say n) can be represented mathematically as 
follows: 
Maximize, n 
r. qi4qT 
i=l 
The waximun production rates of oil, gas and water are, however, limited to 
the capacity of the reservoir facilities. Thus, the field oil production is 
subject to constraints of the form: 
c Xiqi 6 L 
i 
363 
Suppose (k-l 1 we 11s have been already chosen. 
For choosing the k th well subject to a constraint of the form: 
where, 
qi is the oil production rate from well i, 
% is the target oil production rate for the field, 
xi is either 1 or the gas-oil ratio or the water-oil ratio for well i, 
xioi is then the oil or gas , or water production/injection rate. 
and L is the oil or gas or water production of injection constraint. 
Some examples of these limits are: 
1. Fieldwide gas handling capacity, 
2. Water injection limit, 
3. Oil production limit at a station due to pipeline size, 
4. Gas-lift capacity available. 
In order to select wells for production, each well can be assigned a 
priority. In the penalty function approach priority assignment, is made with 
a function which becomes large as a particular constraint approaches violation, 
a simple penalty function is: 
k-l 
p(k) = (zxiqj + Xkqk)/ L 
i=l 
The penalty function p(k) has a value for each of the available wells, 
and arranyes the set of available wells in order according to this particular 
constraint. 
364 
When there are several (say m) constraints, penalty functions pi(k), 
pz(k) ---pm(k) can be obtained similarly. 
Since each constraint is individually fatal for well scheduling purposes, 
the violation of one constraint is as bad as any other. 
Hence, an overall penalty function can be of the form: 
p(k) = max P j 
j=l . . . m 
Results and Discussion: 
The implementation of this scheme involves calculating for each available 
well, m different pi (k) and then obtaining an overall penalty, p(k) as the 
maximum of these m values. Thereafter the well with the lowest value of p (k) 
is selected. This procedure is repeated selecting one well at a time until 
the target rate q is achieved without violating any of the constraints. If 
the target rate cannot be achieved without violating one or more constraints, 
we are on the decline portion of the production curve. 
This scheme was programmed into a three dimensional, three phase (oil, 
gas, water) simulator. The simulator originally used a simple prioritization 
scheme based on gas-oil ratios. When a scalar version of the penalty function 
algorithm was introduced, the simulator ran appreciably slower. It was 
therefore decided to vectorize the penalty function algorithm. 
TO calculate the penalty function in a case with n wells and m 
constraints declare an array p (n, ml). Usually n is much greater than m. 
For each of the m constraints vectorize the penalty calculation, e.g., 
for constraint i, store the values of pi(k) in the elements of p (n, m), 
starting with p (1, i) and ending at p (n, i). 
Next, using a WHERE comparison statement pick out the largest of the m 
values for each well. We now have the priority p (k) for each well. Use the 
il8SnINI call to pick out the minimum value. If this value exceeds l., no well 
can be chosen without violating a constraint. 
365 
TABLE 1, 
No. of wells 
Case 1 
115 
Case 2 
343 
No. of constraints 9 15 
Average Well 
Selection Time (sees) 
Scalar: 
Vector: 
Scalar: Vector 
Ratio 
112 
.14 1.6 
.001245 .00287 
560 
A summary of results for two cases is presented in Table 1. For a 
reservoir with 119 wells and nine constraints, the vector algorithm was on the 
average 112 times faster than the scalar version. For a larger example, Case 
2 in Table 1, 343 wells with 15 constraints, the vector algorithm achieved 
even more spectacular results, an average acceleration factor of 560. 
The details of Case 1 are represented graphically in Figure 3. In the 
scalar algorithm, the time required for selection of wells increases 
monotonically for each subsequent selection. The selection of the first well 
required only .005 sets while the selection of the 65th well required .226 
sets. However, in the vector algorithm, each well selection required .001244 
sets, except for the first, which required .00155 sets. 
Similarly, for Case 2, the vector algorithm took .00287 sets for each 
bdell selection, except for the first well, for which it took .00447 sets. The 
scalar algorithm had a monotonic increase from .0185 sets for the first well, 
to 2.647 sets for the 220th. we77. This means that the selection of the 220th 
well was some 920 times faster in the vector algorithm as compared to the 
scalar version. 
366 
- 
Conclusions: 
Clearly as the number of wells and the number of constraints increase, 
the advantage of the vectorized version over the scalar version becomes 
yreater. 
The reservoir simulator with the vectorized well selection scheme, 
including the more complicated penalty function scheme, ran faster than the 
original version with the simpler scalar well selection scheme. 
In short, judicious use of vectorization can make feasible highly 
desirable enhancements to larye simulators. 
References: 
1. D. W. Peacemain, Fundamentals of Numerical Reservoir Simulation, Elsevier 
Scientific Publishing Company, R.Y., 1977 
2. K. Aziz and A. Settari, Petroleum Reservoir Simulation, Applied Science 
Publishers Ltd., London, 1979 
367 
FIGURE 1 A. 
RECTANGULAR GRID TO REPRESENT 
A RESERVOIR. EACH BLOCK MAY 
HAVE DIFFERENT THICKNESS AND 
POROSITY. 
GA 
--WI I ml 
WATER 
\ 
FIGURE 1 B. WATER 
CROSS-SECTION OR A GRID WITH 
DIFFERENT TYPES OF WELLS. 
368 
OIL 
P RODUCTION A 
RATE 
9 max - --------- --- / \ I\ 
/ -\ 
// \ 
1’ 
\ 
\ 
9, --- .-! 
/ 
I 
\ 
\ 
\ 
\ 
\ 
\ 
\ 
TIME - 
FIGURE 2. 
PRODUCTION PROFILE FOR AN OIL FIELD. 
369 
SELECTION 
TIME 
(SEC.) 
.22 
.20 
18 
.18 
.14 
.12 
.lO 
.08 
.08 
.04 
.02 
c 
. 
. 
. 
F 
. 
. 
. . 
. 
VECTOR 
I I I I I I I b 
6 10 20 30 40 50 80 70 80 90 100 
SEQUENCE NUMBER 
FIGURE 3. 
aU.S.COVERNME~PPRINTINGOFFICE:19~4-739-Ol(Y 59 REGION NO. 4 
370 


Vectorization of a penalty function algorithm for well scheduling

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