Selçuk J. Appl. Math. Selçuk Journal of Vol. 5. No.1. pp. 31-37, 2004 Applied Mathematics
Optimization of the Allowed Drilling Domain in the Exploitation of Marine Floating Platforms
H. Mamedov1 and Ch. Hajiyev2
1Azerbaijan Technical University, Faculty of Automatics and Calculation Techniques,
H.Javid str.,25, Baku, Azerbaijan;
2Istanbul Technical University, Faculty of Aeronautics and Astronautics, Maslak,
34469 Istanbul, Turkey; email: cingiz@itu.edu.tr
Received: May 5, 2004
Summary. An algorithm for optimizing the allowed drilling domain with allowance for the errors of measuring instruments has been developed to allow more reliable control of the positioning of Marine Floating Platforms. Minimum risk is used as the optimization criterion and the optimal limits of the feasible drilling area are found for di¤erent positioning systems.
Key words: Marine ‡oating platform, reliable control, optimization
1. Introduction
The growing demand for oil and gas has made it necessary to create special technical means of ensuring e¤ective exploratory deepwater drilling and oil and gas recovery. Besides drilling rigs mounted on legs right on the sea bottom there are hundreds of ‡oating semisubmersible drilling rigs, drilling ships, semi-submersible platforms with excess buoyancy on preloaded legs, multipurpose semisubmersible self-propelled units, and operating systems of the tanker and barge type. All of them can be called marine ‡oating platforms (MFPs).
When an MFP is in operation at sea, its horizontal displacement relative to the wellhead must be limited to prevent the drilling equipment and the marine riser from being broken as well as to ensure the safety of the operating personnel. During drilling operations, the allowable horizontal displacements of the center of gravity of the MFP relative to the wellhead should not exceed 4% of the sea depth (Gadzhiev, 1996).
The position of the MFP is monitored by the results of measurements of the coordinates of the center of gravity of the MFP relative to the wellhead. The MFP coordinates can be measured on the basis of the various principles by which positioning systems operate (Gadzhiev, 1996): hydroacoustic, incli-nometric, dynamometric, etc. Since the measurements contain errors, incorrect conclusions may be made about the position of the MFP in the allowed area and, therefore, incorrect decisions may be made about the safety of the drilling operations done and this may lead to an irreparable accident situation. In view of this, it becomes necessary to study various methods of increasing the relia-bility with which the location of the MFP is monitored, minimizing the losses due to incorrect decisions.
Methods of improving the reliability of MFP position monitoring by means of equipment are described in (Aliev and Gadzhiev 1988; Abdullayev and Gadzhiev, 1990; Krinetskii, and Gadzhiev,1990). As a result of studies done in mentioned works, requirements for the accuracy of primary information sensors were deter-mined from the condition for ensuring the required reliability of MFP position monitoring. Those results make it possible to determine what the accuracy of direct measurements in MFP position monitoring should be to ensure a given level of errors of the …rst and second kind. Such accuracy, however, may be unattainable in practice. In order to increase the reliability of decisions to continue or stop drilling operations, it is necessary to develop a monitoring al-gorithm that takes into account the real MFP operating conditions and the error of the measuring instruments.
2. Optimization of the Allowed Drilling Domain
One possible method of increasing the reliability of the monitoring is to in-troduce monitoring tolerances, di¤erent from the operating tolerances, and to optimize them according to a chosen criterion. It is useful to choose the average risk for the optimization criterion because it is the most general characteristic of monitoring reliability and takes into account both the errors in the functioning of the monitoring system and the in‡uence of those errors on the e¢ ciency of the system being monitored.
Taking this into account, we use the following conditions to solve the problem of optimizing the allowed drilling region:
the allowed drilling region is given as a circle of radius R, with the wellhead at its center;
the coordinates x; y of the MFP location and the errors of measurement have a normal distribution (Hajiyev, 1991)
f1(x) = 1=( p 2 x) expf (x mx)2=(2 2x)g; f1(y) = 1=( p 2 y) expf (y my)2=(2 2y)g; f ( ) = 1=(p2 ) expf 2=(2 2)g;
where mx; my; x and y are the mathematical expectations and standard deviations of the measured coordinates, respectively, and is the standard deviation of the errors of measurement;
as the optimality criterion we have taken the minimum average risk for the case when the losses with the correct solution are zero, i.e.,
(1) C = L1Pf:a+ L2Pm:d
where Pf:aand Pm:d, respectively, are the probability of a false alarm and miss detection, L1 and L2 are the losses that correspond to errors of the …rst and second kind
L1= 1T1
L2= 2T2+ 1T3+ S
where T1 is the time when drilling was not done because an error of the …rst kind appeared; T2 and T3 are the times when drilling was done and not done, respectively, up to the appearance of an error of the second kind; S is the cost of using drilling equipment until an error of the second kind appears; and 1 and 2are the hourly costs when drilling operations are and are not conducted, respectively).
Optimal limits for the allowable region for drilling operations must be de-termined from the condition of minimum average risk C.
Since the monitored MFP displacements are written as r =px2+ y2 and the coordinates x; y are subject to the normal distribution law with cyclic disper-sion ( x= y = ), the parameter r has a Rayleigh distribution with probability density
(2) f1(r) =
r
2exp( r2=2 2), r > 0
0 ; r < 0
The accuracy of r can be estimated by using the cyclic standard error. In the given case, it has the form
r =
p 2:
The equation of measurements of the monitored displacement r is written as
z = r + r
Taking into account many factors that a¤ect the displacement r and basing ourselves on the central theorem of probability theory, we assume that the error
r is asymptotically normal with zero mathematical expectation and variance 2 r: (3) f2( r) = 1= p 2 rexpf 2r=(2 2 r)g:
The probabilities Pf:a and Pm:d can be calculated with allowance for the deviation " of the monitoring tolerance for the monitored displacement r relative to the operating tolerance (see Figure 1) by using the relations
Figure1. Optimization of the allowable drilling region
Pf:a = R Z 0 f1(r) 1 Z R r " f2( r)d rdr; (4) Pm:d = 1 Z R f1(r) R rZ " 0 f2( r)d rdr
The function C= C("), which satis…es Eq. (1), is continuous in the range of de…nition of the vector r and is di¤erentiable over all the points of that range. The variation of the monitoring tolerance e, for which the function C reaches its minimum, can be calculated from the condition
(5) dC
d" = 0 We write Eq. (5) with allowance for (1),
d
d"(L1Pf:a+ L2Pm:d) = 0
and express the errors Pf:a and Pm:d on the basis of (4). After di¤erentiating C, we have (6) L1 R Z 0 f1(r)f2(R r ")dr L2 1 Z R f1(r)f2(R r ")dr = 0
We substitute the values of the densities (2) and (3) into (6) and, making the necessary transformations, we obtain
L1 p 2 2 r R Z 0 r exp r 2( 2+ 2 r) 2r(R 2 " 2)+R2 2+"2 2 2R" 2 2 2 2 r dr = 0 (7) p L2 2 2 r 1 Z R r exp r 2 ( 2+ 2r) 2r(R 2 " 2)+R2 2+"2 2 2R" 2 2 2 2 r dr = 0
We complement the exponent in (7) to a complete square and change vari-ables: a = R 2 " 2 2+ 2 r ; b = v u u t 2 2 2r 2+ 2 r
In that case, we have
L1 p 2 2 r exp ( (R ")2 2( 2+ 2 r) )ZR 0 r exp ( r a b 2) dr L2 p 2 2 r exp ( (R ")2 2( 2+ 2 r) )Z1 R r exp ( r a b 2) dr
On integrating the respective expressions, we …nally have
(L1+ L2) " a R a b b 2p2 exp ( R a b 2)# + (8) L1 2 4b exp a 2 b 2 2p2 a a b 3 5 L2a 2p2 = 0
Bearing in mind that errors of the second kind are the most dangerous during the operation of an MFP, we …nd the optimal limits of the allowable drilling region for L1= 0. In that case, we rewrite Eq. (8) as
(9) a R a b b 2p2 exp ( R a b 2) a 2p2 = 0 3. Computational Results
For an operating tolerance R= 4 m, using Eq. (9) we found the following optimal limits of the allowable drilling region that minimize the average risk:
when a hydroacoustic system is used ( 0:3) " 0:46 and, therefore, the optimal value of the monitoring tolerance is Rm 3,54;
when an inclimometric system is used ( 0:51) " 0:72 and, therefore, the optimal value of the monitoring tolerance is Rm 3,28;
when a dynamometric system is used ( 0:85) " 1:35 and, therefore, the optimal value of the monitoring tolerance is Rm 2,65.
As it passes through the point e, the derivative of the function C changes sign (from minus to plus) and, therefore, the extremum of the average risk at the point " corresponds to the minimum.
4. Conclusion
In drilling operations the minimum average risk can be attained by shrinking the monitored allowable drilling region relative to the given size of the operating region.
When the error of determination of the MFP coordinates increases, the optimal monitoring tolerance decreases. When the error of determination of the MFP coordinate becomes very large relative to the given operating tolerance, the probability of the correct decision being made as a result of the monitoring drops to its minimum.
References
1. Gadzhiev (Hajiyev), Ch. M. (1996): The Information Provision of O¤shore Plat-form Supervision and Control, Elm, Baku (in Russian with abstract in English). 2. Aliev, R. M., Gadzhiev (Hajiyev), Ch. M. (1988): Validation of the Requirements for the Accuracy of Results of Direct Measurements to Ensure a Given Reliability of the Monitoring of the Location of O¤shore Oil Rigs by the Dilatometric Method, Deposited at VINITI as No: 8052–V88, November 14, 1988, All-Union Institute of Scienti…c and Technical Information, Academy of Sciences of the USSR, (in Russian).
3. Abdullayev, A. A., Gadzhiyev (Hajiyev), Ch. M. (1990): Design of Inclinometrical System of SFDR Position Determination from Preset Supervision Certainty, Azerbai-janian Oil Economy Journ., No: 4, 42-45 (in Russian).
4. Krinetski, Ye. I., Gadzhiyev (Hajiyev), Ch. M. (1990): Justi…cation of Require-ments to Primary Informatics Sensors Accuracy for Provision of Predetermined Cer-tainty of SFDR Position Supervision by Hydroacoustic Methods, USSR High Educa-tion Proceedings “Oil and Gaz”, No: 9, 76-80 (in Russian).
5. Hajiyev, Ch. M. (1991): Adaptive Digitization of Floating Drilling Rigs Coordi-nates, In Digest of 12 Triennial World Congress of International Measurement Con-federation (IMECO), Sept. 5-10, Beijing, China, Vol. 3, 164-165.
6. Mamedov, H., Hajiyev, Ch. (2004): Optimization of the Allowed Drilling Domain in the Explotation of Marine Floating Platforms, Proc. of the 9th Biennial ASCE Aerospace Division International Conference on Engineering, Construction, and Op-erations in Challenging Environments, March 7-10, 2004, Houston, Texas, USA, pp. 696-702.
