Analysis of Wave Force Data

., 1 Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021

Introduction The design of a structure for the marine environment data were obtained for 4 years during Wave Force is primarily dependent on the prediction of the forces Project I (1954 to 1958). Then in 1960, newer instm- generated by waves in most coastal areas. Morison ments were located in 100 ft of water where additional da?~ ~ew fihtai.nrl frm ‘4 mrwe veatw dmino wave et ai.~ reiated these forces to the kineinatk ‘wavepop- 1!, ““WAA.”W .“. “ . ..”.- J -- . ---= ~ ..-. erties, the water particle velocity and acceleration. The Force Project II (1960 to 1963). These projects were equation is composed of two parts — a drag term and supported by the California Research Co., Shell Oil an inertial term — that are related to the total force Co., and Humble OiI and Refining Co. Pure Oil Co. by means of the drag and inertial force coefficient, and the U. S. Navy entered the program at a later CD and Clf, respectively. Two primary limitations date. A complete description of the installation ap- associated with the Monson equation are: (1) the pears in Ref. 3. poorly defined values of these hydrodynamic force The theoretical determination of the wave particle coefficients for large waves and (2) the dependency velocity and acceleration and the fitting of the meas- on wave theory or theories to describe the kinematic ured data by means of the correlation coefficients water particle velocity and local acceleration. A sec- (CD and C.) in Morison’s equation was done inde- Qndarv, comideration.- —.—-.—— is the change of these wave force pendently by the participants. Shell has analyzed coefficients with changes of the kinematic flow field the measured wave force data differently for each and changes of the pile dimensions. The drag coe5- of the two wave force projects. The same wave cient has been shown to be a function of the Reynolds theories were used to determine the velocity and number for steady fluid flow.2 However, correlation acceleration terms from the measured wave profile of CD with the Reynolds number in an oscillating data; however, the force coefficients were analyzed by n-— 1-=-l-. - ---. --.11.. L,.,....-.l...:.,a q-ha _ _.ahl -e t--hnimux aQ deccrihecl in t&.e noW nds rxeu gfam WY UHUMUMVG. ~ 11=* ~Luu&eaA,. d~erent . ---.--.-l--”, -“ --”------_ section Wave clearly show the need for prototype field measure- Force Coefficient Analysis. With these calibrated co- ments of these wave properties in the high Reynolds efficients, Morison’s equation was used to predict the number range. individual wave forces from various large hurricane m...... =.*7fm. *,,..,- m..-+ll. nrl Wn”$l fmw.p dfifa ~ 11= ~Gb&Da, Ly ~ul “~.” ~1 “Auw a,,= .._.” .V._ __ waves. The relative agreement of the predcted and for large waves to establish the desired correlation observed forces is used as a measure of the degree resulted, in 1954, in the instrumentation of an off- of correctness of the Morison equation and as a measure of the confidence for the prediction of wave forces shore oii piatform in Xi ft of ‘Waki. lX’--”-Vv avG tulu‘-J lul&c“-” on a structure. The ctilbrated Morison equation is ●Presently with Ocean Science and Engineering, Inc. then used to determine the design forces for a struc-

1 I Wave profile and wave force data acquired in the Gulf of Mexico from several hurricanes have been analyzed to provide the drag and inertial coeflcients of Morison’s wave force equation. These coefficients are required to help determine design loads for o#shore structures.

MARCH, 1970 ~—fl~ 347 ture using the kinematic flow field determined for a as a function of time, thus providing the parameters design hurricane. By using the same technique for of wave height H and period T for a given water the prediction of the design force as for the analysis depth d. Assuming the waves are unidirectional and of the observed force, the chances for error are steady, the kinematic flow field cart then be computed greatly reduced. from the appropriate wave theories and this surface =__nrnfile.—_. Wave Force Theory This correlation technique calibrates Morison’s equation only in the range of wave heights and Morison’s Force Equation periods and wave forces that have been measured for The basic wave force theory used in design of off- certain pile diameters. Therefore, it is desirable to shore stmctures by Shell Oil Co. is the Morison equa- extend the calibration of the wave force coefficients, tion, which gives the total force on a structural CDand CM,to the wave conditions and pile size other member as the sum of the drag and the inertial forces. * than the experimental values. There have been many The Morison equation, expressed in terms of force attempts to correlate CD with such parameters as the per unit length (lb/ft) on a cylindrical pile, is Reynolds numbers, the Froude number, the Keulegan- Carpenter modulus and several other dimensionless Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021 zD’ au F~ = + CDIUIU + pcJf— — . . (1) parameters’ for an oscillating flow, but generally with 4 at poor success. Several of these relationships were in- where vestigated as well as other parameters such as wave height and period. ;~ mass density of sea water, lb ft-4 sees diameter of the cylindrical pile, ft Composite Wave Theories u= horizontal component of velocity, ft see-’ a24/at= horizontal component of local accelera- The other problem raised by the use of the Monson tion, ft see-z equation is the method of determining the kinematic CD= hydrodynamic drag coefficient flow field. Early investigators of wave force theory’ CM= hydrodynamic inertial coefficient quickly realized that no single theory available would properly describe the kinematic flow field for the many With this equation the measured wave force data can variations of wave conditions. These wave conditions be correlated with theoretically predicted wave force have been described by two parameters, the -~ave----- data, provided the kinematic flow field is known. steepness parameter (H/gT’) and the water depth Ideally this flow field would be measured simul- parameter (d/gT’). Consequently, Shell Development taneously with the wave force field for the most accu- Co. has developed numerical wave theory computer rate analysis of CD and CM.Practically, this has not programs to cover the ranges of wave steepness and been done successfully for very large waves (nor was wave characteristics in various water depths. A selec- it accomplished in the two wave force projects). How- tion graph (Fig. 1) displays the various regions of ever, the sea surface profile at a point can be measured applicability of the several wave theories used by Shell

0.03 r /+ = o C2732 I .< wiiv E t+l Hr, = \ 000

g=32 20 F1/Sec2

CHAP PE LEAR NUMERICAL

002 - / H/Hm, = O 675

. . m . z

& STOKES– 5~h c . I (JO, -: 0 W

OEPTH SOLUTIONS)

I I H, HT, = 005 1

c 002 c C)4 c 06 0 08 . d/gT~

Fig. l—Wave theory selection graph.

348 JOURNAL OF PETROLEUM TECHNOLOGY I ,

as dictated by the dimensionless parameters, H/gT2 u = wave frequency (2zT-1), see-l and d/gT2. ~s does not imply that other wave k = wave number (27rL-’), ft’ theories are not applicable to similar regions. Fig. 1 shows that two forms of the Stokes (1847) wave Vj = velocity constants in Stokes’ iifth order theories, the first order (Airy) theory and the more wave theory complex tifth order theory, were applicable to most of the ranges of these parameters. The ChappelearO Aj = acceleration constants in Stokes’ fifth numerical wave theory and the McCowan’ solitary order wave theory wave theory were used for the extreme conditions of d = dep@ ft very steep waves in deep and shallow water, respectively. With the exception of the Airy and McCmvan When x = O (at the wave crest), sin (kjx) = O, and wave theones, the computations in determining z the inertial force term is likewise O. The velocity single set of water particle velocities and accelerations (hence, drag force) is then a maximum, since cos from the theories in Fig. 1 for a given location were kj(0) = 1 at the wave crest. At x = L/2 (at the wave .... . —— -.. -L) .:- l.:tT 19) h -1 - n .~A th- fn~p ic amin nearly prohiiiitive untii they were cornpuwr pru- Uuugu], 3UI #b](&/ 6) la W&) v, -- w“ . ..- .“ -~— grammed for rapid compilation. purely drag. The velocity is a maximum in the nega- Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021 We cannot begin to delve here into these wave tive direction at the trough but not so strong as at the theories, All appear in the literature in great detail. crest since -m, ~....,. +La ~a.r.in WI..+ .f th. Aim wgy~ Au w a Lue Uw v “.+-S . “. -“ . ——., .- , theory. The Stokes fifth or&r theory is described by ; Vi cosh ki (d + q) cos kj(0) > Skjelbreia et al.,’ and the numerical wave theory ap- j=l crest pears in Chappelear.6 A more recent stream function wave theory by Dean’” is also available as it has ; VJcoshkj(d–q)coskj ~ . . (3) specific application for the analysis of the kinematic Ij=l ( )1 flow field of an irregular sea. The continuing effort Trough Of many WLIV-..w.. thenri. ...”...... ctc nn--- rd%e~~~f..—- ~~ &velop- where 7 is the eievation above mean sea ieVei. ment of new wave theory is perhaps the best indica- The inequality of the crest velocity relative to the tion of the inadequacy of the state of the art. How- trough velocity is responsible for the “wave induced ever for engineering purposes, the foregoing tech- current” characteristic of nonliiear waves (j > 1) of niques represent a good foundation for the deter- finite height, mination and analysis of wave forces. Conversely, the net force is predominantly inertial Wave Force Coefficient AnaIysis when x = L/4, since cos kjx = O for j = 1, 3, or 5. Wave Force Project I The remaining terms (for j = 2 or 4) are small and the inertial force coefficient can be evaluated assuming The analytical techniques used to determine the the drag force is zero. The sea surface profle of the hydrodynamic force coefficients from Wave Force Project I were based on the physical properties of Stokesian wave with reference to mean sea level is the kinematic flow field, The water particle velocity, &~~~ =. u, and the local water particle acceleration, Zh@, ~;qjcoskjx...... (4) become zero at specific positions in the wave profile, 7 = k,=l thus there are specfic locations where the fome is purely drag and purely inertial. Since these kinematic where q~is the coefficient to the j order, and aLlother properties are 90° out of phase, the maximum of one terms are as before. Since at x = L/4, q s O and the term occurs when the other term is zero (or extremely wave elevation approaches mean sea level. Therefore, close to zero for nonlinear waves). This can be readily we can estimate the inertial force coefficient at seen for the Stokes lifth order expression in finite x = L/4. Note that in Eqs, 2 and 4 for j = 1 (first water depth where we substituted the values of order or Airy theory) and for x = L/4, cos (2T/L) velocity and acceleration (in generalized form) into (1) (L/4) = O, and the force is purely inertial. Also h!forisoxl’sEq. 1: the wave elevation is exactly mean sea level (~ = O). Using the applicable wave theones and the above 2 F~ = CDA~ ; ~j cosh kj (d – Z) COSkjx assumptions, the wave force coefficients were deter- 2k2 [ j=, 1 mined for 556 waves for Wave Force Project I. The Drag Force inertial coefficient was determined at two locations, a quarter of wavelength (L/4) ahead of the wave crest and the same distance behind it. The drag coefficient + C#. ~ ,:1 Ajcoshkj(d – Z) Sin kjx [ 1 1 was determined for the crest position only. The total measured force over the entire length of the pile was Inertial Force compared with the total predicted force. A summary ...... (2) of the results and their interpretation is presented in a later section. where Wave Force Project II A = area, sq ft The force coefficients of Wave Force Project II data v. = volume, cu ft were analyzed using the kinematic wave theories de-

MARCH, 1970 349 H?15 FEET

1 OVER-ALL ANALYSIS I — N:460

I I II I In I

I I I I 1 1 1 1 1 I I 1 1 1 1 t 0 , , I 0.5 10 1.5 0 0.2 04 0.6 0.8 1.0 12 I.4 16 ORAG FORCE COEFFICIENT (Co) ORAG FORCE COEFFICIENT {Col Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021 Fig. 2A-Drag force coefficient histogram. Fig. 3A—Drag force coefficient histogram, higher waves.

500 —

$ 4W —

s a :X0 — x a ‘2 H-5 !0 22 FEET : 200 —

100 —

1.0 5 o ‘:L0 0.5 0 0.2 04 06 OB 10 1.2 14 ORAG FORCE COEFFICIENT (co) ORAG FORCE COEFFICIENT [CD)

Fig. 23-Drag force coefficient frequency Fig. 33--Drag force coefficient frequency distribution distribution curve. curve, higher waves.

9s.5

: z ..-, ---- ...... / / 3 lllWILal Wevca ~ 95E / 0 /’

a u! m A:, .,”,s 2+60

2 !7 / A / @ + / 0

: 40 0

+ t 20 / g w a /’ s

5 — ?(,,, ‘o 0.2 0.4 0.6 0.8 I.W !.20 I o

‘o 02 04 06 08 100 I 20 140 1 DRAG FORCE COEFFICIENT (CD)

DRAG F~R5E COEFFICIENT (CO1 .-–.?X, -, —-A ------L-k:l:*. # Fig. X&hag force coe:nclem Trcquency pIUUUUIIILY Fig. 2C-Drag force coefficient frequency probability cu we. curve, higher waves.

350 JOURNAL OF PETROLEUM TECHNOLOGY veloped for Wave Force Project I and the method of sive analysis of some 200 waves. A statistical and a least-squares to correlate the measured wave force parametncal analysis were performed on the results with the theoretical force. The wave force coefficients obtained from the least-squares analysis. were determined for the best fit values of the pre- Results of Wave Force Analysis dicted and the observed forces for a quarter of a wavelength before the wave crest passed the pile. The force Wave Force Project I correlations were made at five eleva%ons, where pres- The drag and inertial force coefficients were deter- sure dynamometers were located, from 6 ft above mined for 460 waves from 4 years of data using the mean low water down to 23 ft above the bottom. The methods described in Wave Force Data Analysis. least-squares method was felt to be an improvement Results of the drag force coefhcient analysis are dis- over the method used in the previous analysis in that played in a histogram in Fig. 2A, which shows a less subjective techniques were used to determine modal value of 0.5 for CD. The frequency distribu- actual wave positions from the recorded wave profile tion curve in Fig. 2B and the probability curve (Fig. data to evaluate the drag and the inertial coefficients. 2C) give a median of 0.5 for CD. The average value The method of least squares fits the theoretical of CDfor the 460 waves was 0.585. forces (or pressures) developed from Morison’s The separate analysis of 69 waves whose heights Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021 equation to the measured forces (or pressures) for exceeded 15 ft (up to the maximum observed 22-ft each wave analyzed using CD and CM as the vari- wave) gave a distribution with a modal value of CD ables. In this analysis, pressures were used. Desig- of 0.4 (Fig. 3A) and a median of 0.45 (Fig. 3B). The nating Pm (CD, CM) as the measured wave pressure average value was 0.495. A frequency probability and PT (CD, CM) as the theoretical pressure from the curve (Fig. 3C) of these higher waves compared with Monson equation, the least-squares method forms an all waves shows a steeper slope (less scatter of data) equation for the sum, S, of the squares of the differ- and generally smaller values of the drag coefficient. ence between the two: The results of a parameter analysis of CD vs wave height and wave period are shown in Fig. 4. A linear s = : [pm (CD, CM) – p. (C~, CM)]’ , (5) regression line has been fitted to the averaged values i=l of CDfor intervals of 1 ft and 1 second in the range where n is the total number of data points compared. of observed wave heights and wave periods. A distinct We evaluate Eq. 5 for C. and CMwhen S is a mini- trend in the wave height analysis is toward lower mum, hence when the partial derivatives of S with values of CD,with increasing wave height in the range respect to these variables are zero: of these observed heights. A similar decrease in CD n ~,,m. with kt.rc+Icku wave.. . . w rwrim-1yw..”- fnr.U. ~hae~~p-g~ of as “c-u. . “ ,.., A.Awl~eo,..e —= Oand~=O ...... (6) the observed periods. acD A parametric analysis of considerable interest in Substituting Eq. 1 into Eq. 5 after dividing through the study of hydrodynamic forces is that of the non- by D and then dfierentiating with respect to CD and dimensional Reynolds numbers (Re). Many attempts CM,we obtain: have been made to correlate Re with CDfor oscillatory motion as has been done for steady flow. Table 1 and as \3 C.pu’ A~io.-. -5 “------cllmmarize SOrn.S of fh~ Lrn.po_*nt drag fOme —=,:,ac. {- U’[P” J1 investigation.s, showing the Reynolds number range in each study. The CD curve for steady flow is also +;CMpzDd; = O . . . . (7) shown. Only three of the tests cover the area of in- )1} terest for large storm waves and one of these is for as steady fluid flow conditions. The other two tests are —= ,:,~[Pm - CCDPU’ Wlr 12 described in this report. Fig. 6 (Curw A) shrews tile .,3 individual values of C? for 460 waves. The average +; CUP.D: =0 . . . . (8) value of CD for small increments of Reynolds num- )! bers is shown in Fig. 6 (Curve B). Only the average Rearrangement of Eqs. 7 and 8 results in the final equations:

=; Pnz# ...... (9) icl C+)i:, ++ c.(+) ; (a’ — h ]i=l\dt/ =;p&4 (lo) i=l ‘ at”-----” These equations are solved simultaneously for CD and Cm, since all other parameters are known. A Fig. 4-Drag force coefficient relation to (a) wave period computer program was written to facilitate the mas- and (b) wave height.

MARCH, 1970 351 .. .

TABLE 14UMMARY OF INVESTIGATIONS OF WAVE FORCE COEFFICIENT

Yea r Agancy Ref. No. Tyw of Data Pile Size (in.) NB CD Clr ——No. 1 1954 U. of California 11 Oscillatory flow 1 <2X104 1.6 1.1 wave tank 2 1956 @!f Qf M~~~~o 9 5xl@–2xlW 0.53 1.47 waves 3 1956 National Bureau 6 Oscillatory flow 1,2,3 <2X104 1.43 1.49 of Standards wave tank 4 1956 Texas A&M U. 4 Accelerated flow 3 <2X104 1.10 1.46 lab test 5 1956 U. of California 19 Pacific Coast 7, 13,24 3X104 -9X1W 0.60 2.50 ocean waves 6 1957 Texas A&M U. 2 Gulf of Mexico 16 lxl@–7xlo5 0.50 1.50 waves 7 195s U. of California 8 &celelkad flow 2,4 2 x 105 –4X105 0.65 1.0

8 1960 U.S. Navy 16 Gulf of Mexico 12,24,36,48 0.80 1.00 waves Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021 9 1961 Humble 011 & 1 Steady.state flow 36 1x1 O–6X1O$ 0.59 Refining Co. towing tank

values bear some correlation to the steady flow curve of CD. The inertial force coefficient analysis of the 460 waves produced a total of 825 determinations of Cx, using the measured forces and theoretical accelerations ahead of and behind the wave crest. A histogram (Fig. 7A) of CMindicates a modal value of 1.2. A frequency distribution curve (Fig. 7B) and a frequency probability curve (Fig. 7C) both show a median value of C = 1.05. The histogram in Fig. 7A does not show a weIl defined peak, which indicates that there is considerable scatter of inertial force for

I I I I 111111 I I I 111111 a small dhuneter pile. The dynamometer response to ,.s ,.6

% the inertial force on the l-ft diameter pile was extremely poor to nonexistent. No parametric analyses Fig. 5-Reynolds number range of various drag coefficient analyses. were ~~ormed on CM.

cum”, . I . . .L- . . I . .

REYNoLDS NuMBER

Fig. &Drag force coefficient, Reynolds number curve with steady flow curve superimposed. Fig. 7A—lneftial force coefficient histogram.

352 JOURNAL OF PETROLEUM TECHNOLOGY ——

and 8B, respectively, for each dynamometer level. Wave Force Project II The sample mean and standard deviation of the aver- The analysis of the wave force data from the deeper age value of all waves was 0.880 and 0.650, re- locations was described previously in Wave Force spectively. The same analysis for higher waves only Data Analysis. For this project there was a differ- (H >25 ft and T ~ 10 seconds) resulted in a sample ence in the method of correlation — the least-squares mean of 0.578 and a standard deviation of 0.332. Tine method was used. And instead of obtaining the total sample mean and standard deviation of the drag co- force on the pile, as in the earlier analysis, the drag efficient at each dynamometer level for all waves and and inertial coefficients were determined at each for the higher waves are summarized in Tables 2 dynamometer location. The 200 waves anaiyzed, aii and 3. I.._.. .~=_ *n c. . . . . s ,.- Lr,,mp. =. P9F19 Idrger UMII 1u L% b.~Jc ~rulll IJU4 .I-u. ~-x- ~d L!.- paP_me~:c ana!yses of the average drag coefficient cluded 65 waves higher than 25 ft. The maximum for wave heights and for wave pe~ods are shown in wave height observed was 42.9 ft. The drag force Tables 4 and 5, respectively. The drag coefficient coefficient probabfity curves for all waves and for decreases with increasing wave heights at all levels waves higher than 2S ft are shown in Figs. 8A observed. The most rapid decrease of CDoccurs near Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021

!0, f!

,, 1! ,7 f, /— *“f”A., or ●LL “Awls

/ 231, / 35 It / / /

300 — /

200 —

I 00 —

I I I I o I o 0.5 1.0 1.5 20 2.5 30 3.5 INERTIAL FORCE COEFFICIENT (Cul

CD ,,OO.., ,””$, ,.,,, , Fig. 7B-inertial force coefficient frequency

distribution curve. Fig. M—Probability curve of CDfor all waves.

/ ,ou_L&+u#

------.-,-.,- 051 INERTIAL FWIi2E C6EFFILILN r iCM) 0.1 0.s 0, 0s ..* ,. ,, ,0 ,0 ● ,0 ,,e ** ,*, o co ILOSARIIIIUIC SCa,c , Fig. 7C-lnetial force coefficient frequency probability curve. Fig. 8B-Probability curve of CD for higher waves.

MARCH, 1970 353 TABLE 2—SAMPLE MEAN AND STANDARD DEVIATION TABLE -AVERAGE DRAG COEFFICIENTS FOR OF DRAG COEFFICIENTS FOR ALL WAVES VARYING WAVE HEIGHTS AND ELEVATION

Elevation Above Standard Elflb::i:n Wave Height Groups, ft Mud Line Sample Mean Deviation Mu~fi~ne 1::5 1::0 2::5 25..0 3;-:5 3::5

23-ft level 1.008 0.842 —— —— —. — 55-ft level 1.081 0.617 23 1.14 1.09 0.91 0.90 0.91 0.66 78-ft level 0.791 0.663 ~~ ~.~~ ~:11 I ,03 1,05 1.06 0.97

97-ft level 0.814 0.481 78 1.38 0.93 0.60 0.52 0.59 0.56 106-ft level 0.592 0.377 97 1.31 0.84 0.71 0.62 0.65 0.53 All waves, all levels 0.880 0.650 106 — 0.89 0.60 0.55 0.47 0.58

TABLE =SAMPLE MEAN AND STANDARD DEVIATION TABLE 5-AVERAGE DRAG COEFFICIENT VS OF DRAG COEFFICIENTS FOR HIGHER WAVES WAVE PERIOD AND ELEVATION

Elevation Above Standard E18&i:n Wave Period Groups, seconds Mud Line Sample Mean Deviation Mu~fi\ine 5.0...5 7.5C:0.0 10.0.; 2.5 12.5-:7.5 23-ft level 0.583 0.301 —. — 55-ft level 0.829 0.371 23 2.05 1.10 0.66 0.68 Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021 78-ft level 0.466 0.223 55 1.73 1.21 0.88 0.66 97-ft level 0.614 0.303 78 1.19 0.76 0.67 0.57 106-ft level 0.397 0.256 97 0.84 0.80 0.78 0.62 All levels 0.578 0.332 106 0.47 0.67 0.61 0.47

the surface, with the wave height increasing from 10 since 89 percent of the waves were actually between to 25 ft, and then remains fairly constant at about 10.0 and 13.0 seconds. 0.56 for the larger waves. CD decreases with eleva- The results of the statistical analysis for the inertial tion above the mud line for most wave height groups. coefficient CMare nearly similar for all waves and for The parametric analysis for wave periods also higher waves (H ~ 25 ft and T = 10 seconds). The shows a decreasing drag coefficient with increasing sample mean and standard deviation of average CM wave periods. Table 5 shows this decrease and also for all waves was 1.682 and 0.916, and for higher shows that for most wave period groups, CDdecreases waves it was 1.765 and 1.056, respectively. The re- toward the sea surface, which is simiiar to the wave mits are summarized in Tables 7 and 8. Tiiere is no height relationship. significant change in C~f at each dynamometer level A combined parametric analysis of CD for both between the values in Table 7 and those in Table 8. large wave height and long periods at various levels Probability curves for average C,, are shown in Figs. (Table 6) shows the tendency toward a lower average 9A and 9B for the two cases just discussed. Both drag coefficient (0.43) at the sea surface. The range curves show increasing values of C,,~toward the sea of periods (10 to 17 seconds) is somewhat misleading surface and also an increasing scatter of data points

9 —— 99 ———~

99 ~

98 t-

93 }

// /:/’”‘ ///// - 05, , : 30 / } W

,0 .- 10L 1 ‘t

, ! , o,. — I 0 00 !0 20 ,0 .0

c“

Fig. 9A—Probability curve of CM for all waves. Fig. 9B-Probability curve of CMfor higher waves.

354 JOURNAL OF PETROLEUM TECHNOLOGY TABLE -AVERAGE DRAG COEFFICIENT VS COMBINED temal wave force field or the total integrated loading WAVE HEIGHT AND WAVE PERIOD GROUPS on the pile as a function of time. For comparison with Wave Characteristics the theoretical forces, the total measured forces and E18#$~ H = 25.30 ft H = 30-35 ft H = 35-45 ft moments can be compiled by simple integration Mu~ti;ne T = 10-17 sec. T = 10-17 sec. T = 10.17 sec. CD CD CD methods such as Simpson’s rule. 23 0.60 0.54 0.51 A typical analysis of the Wave Force Project I 55 0.85 0.77 0.70 data showing a simultaneous comparison of the total 78 0.46 0.48 0.50 measured and theoretical forces and moments for 97 0.57 0.48 0.47 several large waves is reproduced in Figs. 10A through 106 0.46 0.40 0.43 10D. Fig. 10A represents one of the better fits of the measured force data and Fig. 10B one of the better fits of the moment data. Figs. 10C and 10D, respec- at the sea surface. The 97- and 106-ft dynamometers tively, represent a poorer fit of these two loadings. were continuously subjected to the extreme turbulence The comparisons were generally good to excellent on of the wave splash zone. most of the waves analyzed.

The parametric analysis of CMis shown in Table 9 A comparison of the internal force fields was made Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021 for wave heights and in Table 10 for wave periods. on the larger waves of Wave Force Project H. A good Tine iarger wave heights have slightly sma!!er average ~ow.pafi.~on~f the predicted and observed force fields values of CM at all elevations; however, the longer of one large wave is shown in Figs. 11A and 1IB. wave periods usually have slightly larger values of For the 10 largest waves of Hurricane Carla, the total z-- -. . ..A .-+. fin th - ~-Qt mile w~~e ~~?~rm~~~d CM. The net resuit is that tine two baiance and iio lUI@S tW ~l@klLa u,, .,,- u.. I.,.. - conclusions or trends could be drawn about these from the observed data and from the theoretical “big” waves relative to the entire sample. However, equations. The comparison of all 10 waves was good, the two parametric analyses do show a strong increase with any errors usually appearing on the safe side. in average C,~twith distance above the mud line. The The maximums of these two forces and the total force surface value at the 106-ft level nearly doubles the at the wave crest were compared. The results are bottom value at the 23-ft level. The standard deviation shown in Table 11 where the ratios are the measured nearly triples in the same respective locations. forces divided by the theoretical forces. The comparisons of the maximum forces and Comparison of Theoretical moment ratios were more favorable than for the crest And Observed Forces forces and moment ratios as there is considerably The analysis of the Wave Force Project I data resulted lower standard deviation. The agreement should be in the selection of the average value of the drag co- better at the position of maximum force, since the efficient of 0.5 and a value of the inertial coefficient rate of change of force per unit length along the wave of 1.5. The latter was slightly larger than the modal profile is much less there than at the crest. In analyzing value of 1.2. These values of CD and C.ir Used in the data, it was often difficult to locate the exact crest Eq. 1 with the proper wave theories constitute the position accurately with a step-type wave record. The necessary theoretical equations to determine the force wave crests often appeared constant for upwards of field. The required input parameters are wave height a second on some waves. This would result in a con- H, wave period T and water depth d. The results have siderable change in force calculations (see Fig. 11B) been expressed in terms of the two-dimensional in- depending on the final location chosen for the analysis.

TABLE -AVERAGE INERTIAL COEFFICIENTS TABLE 7—SAMPLE MEAN AND STANDARD VS WAVE HEIGHT GROUPS DEVIATION FOR C.,, FOR ALL WAVES Wave Height Groups, ft Ele&$m Above Mu~fi~ne 2::0 3$:5 3::5 Mud Line Sample Mean Standard Deviation 23-ft level 1.184 0.415 23 1.23 1.20 1.06 55-ft level 1.477 0.458 55 1.47 1.39 1.08 78-ft level 1.607 0.744 78 1.48 1.40 1.07 97-ft level 2.003 0.955 97 2.07 1.63 1.50 106-ft level 2.478 1.432 106 2.44 2.30 1.57 All levels 1.682 0.916

TABLE -SAMPLE MEAN AND STANDARD DEVIATION TABLE 1O-AVERAGE INERTIAL COEFFICIENTS OF INERTIAL COEFFICIENTS FOR HIGHER WAVES VS WAVE PERIOD El~b:::n Elevation Wave Period Groups Mud Line Sample Mean Standard Deviation Above Mu:ti~ne 5-1o 10-17 sec. sec. 23-ft level 1.251 0.459 554t level 1.515 0.613 23 1.20 1.15 78-ft level 1.636 0.739 55 1.40 1.55 97-ft level 1.988 1.201 78 1.42 1.85 106-ft level 2.675 1.449 97 1.80 2.32 All levels 1.763 1.056 106 2.07 2.75

MARCH, 1970 355 change appreciably for a separate analysis of high Summary and Conclusions waves only. When used with the appropriate wave theories, the As in the first parametric analysis, the drag co- analysis of the wave force data of both projects pro- efficient decreased for increasing wave heights and vided the correlation coefficients, CD and Cx, neces- increasing wave periods. Since the parametric period sary for the determination of design loads on small analysis of CDcovers the range of design wave periods diameter pile. Analysis of Wave Force Project I data (10 to 17 seconds), no additional change in CD can gave modal values of CD and CH of 0.5 and 1.2, re- be expected from this parameter. But the wave height spectively. This last value (1.2) was perhaps low be- analysis of CL)extends only to the 42.9-ft maximum cause of the poor results from the smallest (1 ft) wave height; hence, CDcould be expected to continue diameter test pile where inertial forces were small. A the established downward trend for larger wave value of CM= 1.5 was used for wave force predic- heights. Consequently, the values of CDfound in these tions in Wave Force Project I. Analysis of the highest analyses could be conservative. . (IC 4- 99 **\ 6*. r nvnrn a Valllt= nf ‘wz-vea~~~ LwAA AL)~UL~ ~ gfwe an ~. -.-gw ...... While. , ..-W the--- inertial------coefficient---_ -.._— ~.~ lK)t Chi311=& tl~~~- 0.5 and less spread of the data. CD was also shown ciably with wave size, it more than doubled in value

to decrease with both increasing wave period and from the 23-ft level to the 106-ft level (2.478), Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021 height, which is extremely important in designing off- whereas the standard deviation more than tripled, shore installations, since larger wave heights wotid 0.415 to 1.432 for the respective elevations. The mean be used. The correlation between Reynolds number value of 1.68 is in the range of classical values of CM; and CD was generally poor. Averaged values of CD however, the changes with elevation bear additional over small intervals of Reynolds numbers also gave a considerations in the design of deepwater structures. slightly better correlation with the curves developed The accuracy of the calibration of Morison’s equa- for steady fluid flow conditions. tion coupled with the theoretical prediction of the The drag and inertial coefficients determined from kinematic fluid flow properties is reflected by the fit the Wave Force Project II were somewhat higher. of the measured and predicted forces. Generally For the higher waves, CDwas 0.58 vs 0.50 of the pre- speaking, this comparison is favorable (measured vious project. Tiie average value of CMwas i .’7vs i.2 force/theoretical %= = 0.89 ~iil fi,~a~~ietf ~10- in the first project. The inertial coefficient did not ment/theoretical moment = O.85) in the analysis of

REcORDEDFORCE

‘--- THEORYFORCF.

4 \ m. 9.5 fj — RECORDED m“cl \ T.m%ec ---- 1.CORY PoncC \ / / —d.33 f! ,/ d.32 II \ -\ H. 92 f! D.4ft / 2 I! \ g / T . 6,5 SW x / // / E / / so / u. /’ / / / i ‘\ / : -2 / \ 0 / : / . \ \ -4 E \d 5 -4- ,

I I I I I I I -60 1 -6 I I ! I I I I 40 80 (60 1.?0 200 240 280 320 0 .0 00 /20 (60 200 240 2#o ,20 , 0 “1~DISTANCE FROM ADV4NCING TROUGH, f? DISTANCE ~R014 .0vANC,N6 lR.Juo.. ft Fig. 10A+omparison of recorded and theoretical Fig. 10C-Comparison of recorded and theoretical predicted force variation. predicted force variation.

I /’m’\ ‘--- ::;,7”’ I \ “.9, ft ,.a, ,ec 11 ii f II D. 2 fl

280 320 3’0 OISIANCEfem .OV.NCING TROUGH, It Fig. 10D-Comparison of recorded and Jtheoretical predicted moment variation.

356 JOURNAL OF PETROLEUM TECHNOLOGY .

TABLE 11—FORCE AND MOMENT RATlO ANALYSIS Aj = Stokes wave theory local acceleration SUMMARY,...... 10. ..- HIGHEST---- . .RECORDED...... coefficients nuuuicnrm UAKU WAVES D= pile diameter, ft d= water depth, ft

Holfit Period F~= theoretical force on a cylindrical pile, lb —— = acceleration of gravity, ft/sec2 42.9 7.7 0.22 0.10 0.84 0.63 l!?= wave height h 39.4 8.6 0.74 0,72 0.62 0.64 .= 37.8 9.6 1.16 1.02 0.98 0.90 1 order of the Stokesian wave theory (1 36.6 10.1 1.07 0.96 0.89 0.84 to 5) 35.1 9.6 1.20 1.00 0.97 0.86 k= wave number, fti 34.9 9.0 0.82 0.82 0.88 0.94 L = wave length, ft 34.5 12.6 1.19 1.34 1.06 1.21 MLw = mean low water 34.0 8.5 1.19 0.99 1.07 0.92 n = number of data points used in least- 33.4 8.6 0.70 0.56 0.85 0.75 squares method 32.9 10.8 0.61 0.66 0.82 0.79 measured wave pressure, lb/sq ft A:vey Pm= Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021 0.89 0.82 0.89 0.85 Pr = theoretical wave pressure, lb/sq ft Maximum Re = (also NR) Reynotis number - 1.20 1.34 1.07 1.21 value S = summation used in the least-squares method T = water particle velocity, ft/sec the 10 iargest waves of Hurricane Carla. The predicted u = wave period, seconds and observed internal force fields also agree reason- au — = localwater particle acceleration, ft/sec2 ably well. The fact that the predicted values of the at forces and moments were 11 and 15 percent higher Vj = Stokes wave theory velocity coefficients for these 10 waves may substantiate the earlier sus- Vo = volume of a cylindrical pile picions that CD should be smaller for the extremely u = wave frequency, see-l large wave heights. The measured maximum forces P = density of sea water, slugs/cu ft of only two of these waves exceeded the predicted ~j = Stokes wave theory elevation coefficients maximum forces, and then by only 6 and 7 percent in each of the two examples. Acknowledgments In summarizing the results of the wave force This paper represents the efforts of many individuals analysis, it is reasonable to conclude that the predic- at Shell Od Co. who have participated in the instm- tion of wave forces by the techniques outlined here mentation, collection and analysis of the wave force does give accurate and reliable design information data since the program started in 1954. It is impos- for the range of parameters encompassed. It must be sible to mention everyone concerned with this project; remembered that it is primarily a ~ystem that has ~Qw~~~~,-~ few.- . . cnntrihnttvl-- ...... ? ------cmncirlmmhle time—--- —-and been calibrated rather than one spemfic equation. CD effort. L. E. Bergman and J. E. Chappelear worked and CM will differ with varying wave theories as on the wave theories and the wave force computer dillerent kinematic fields would be predicted for the programs. L. C. S. Kobus and K. S. Chang worked same wave and depth parameters. Finally, these co- on the wave force coefficient analysis of Wave Force efficients have not been evaluated for large diameter Project I and Wave Force Project II, respectively. pile and should be treated cautiously when being used for that type of pile, References 1. Morison, S. R., OBrien, M. D., Johnson, J. W. and Schsaf, Nomenclature S. A.: ‘The Force Exerted by Surface Waves on Piles”, Trans., AIME (1950) 189, 149-154. A = surface area of a cylinder projected on a 2. ScWlchting, H.: Boundary Luyer Theory, McGraw-Hill plane normal to wave pattern, sq ft Book Co., Inc., New York (1955).

,,. )

“.,, co., vtLx!T! or ..”, .0”..,, ,6 . .,.7, . ,. 0,! T., ,,.’

w ,, ,! n / \ \ I \ / “t / \ /, / “.9. .. s.i. !,, ?,. , . 1! \ I 7,”C ,“0. c“csr, x .. .. .>2.,0’ ~J~ ~ ,!., ,.0. ,“,,:, .,,

Fig. 1 lA-Theoretical force contours and profile Fig. llB-Recorded force contours and profile for 34.O-ft wave. for 34.O-ft wave.

MARCH, 1970 357 —.

/ 1

3. Russell, T. L., Schoettle, V. and Chown, R. G.: “Ocean 13. Dean, R. G.: “F]uid Forces on Circular Cylinders”, MS Wave Force Instrumentation”, 1. Waterways and Harbors Thesis, Texas A&M U., College Station (1956). Div., Proc., ASCE ( 1966) 92, WW4, 1-16. 14. Keulegan, G. H. and Carpenter, L. H.: “Forceson CYlirs- 4. Wiegel, R. L., Beebe, K. E. and Moon, J.: “Ocean Wave ders and Plates in an Oscillating Fluid”, U. S. Natl. Forces on Circular Cylindrical Piles”, Paper 1199, J. Bureau of Standards, Report 4821 (1956). Hydrology Div., ASCE (April, 1957) 83. 15. Laird, A. D. K., Johnson, C. A. and Walker, R. W.: 5. Reid, R. O. and Bretschneider, C. L.: “Surface Waves and “Water Forces on Accelerated Cylinders”, J. Waterways Offshore Structures”, Texas A&M U. Research Founda- and Harbors Div., Proc., ASCE (1959) 85, WWf, Part I. tion Tech. Report, College Station (Oct., 1953) 36. 16. Morison, J. R., Johnson, J. W. and O’Brien, M. D.: “Ex- 6. Chappelear? J. E.: “On the Theory of the Highest Waves”, perimental Studies on Forces on Piles”, Proc., Fourth Beach ErosIon Board Tech. Mere. No. 116(1959). Conf. on Coastal Engineering, Council on Wave Research, Berkeley, Calif. (1954). 7. McCowan, J.: “On the Solitary Wave”, London, Edin- burgh and Dublin Phil. Msg. and J. Sci. (1891) 32 No. 17. Reid, R. O.: “Speciaf Report on Waves, Tides and Wave 5, 45. Forces”, Magnolia Petroleum Co., Texas A&M U. Tech. Report, Coiiege Station(1959). 8. Kinsman, B.; Wind Waves, Prentice-Hall, Inc., Englewood Cliffs, N. J. (1965) 676. 18. Skjelbreia, L., Hendrickson, J. A., Gragg, W. and Webb, L. M.: “Loading on Cylindrical Pilings Due to the Action 9. Skjelbreia, L. and Hendrickson, J. A.: “Fifth Order Grav- of Ocean Waves”, U. S. Naval Civil Eng. Lab Report ity Wave Theory”, Pmt., Seventh Cord. on Coastal Engi- (1960) Contract NBY-3 196, Natl. Eng. Sci. Co., Pasa- neering ( 1961) 1, Chap. 10. dena, Calif. Downloaded from http://onepetro.org/jpt/article-pdf/22/03/347/2228075/spe-2711-pa.pdf by guest on 23 September 2021 10. Dean, R. G.: “Stream Function Representation of Non- 19. Stokes, G. G.: “On the Theory of Oscillatory Waves”, linear Ocean Waves”, J. Geophys. Research ( 1965) 70, Trans., Cambridge Phil. Sot. ( 1847) 8 and Supplement, No. 18, 4561-4572. Sci. Papers, 1. -T 11. Blumberg, R. and Rlgg, A. M.: “Hydrodynamic Drag at Super Critical Reynolds Numbers”, paper presented at Original manuscript received in Society of Petroleum Engineers Petroleum Session, ASME Meeting, Los Angeles, June office March 17, 1969. Revised manuscript received Dec. 17, 1969. 12-14, 1961, Paper (SPE 2711) was presented at the First Annual Offshore Technology Conference held in Houston, Tex., May 18-21, 1969. 12. Bretschneider, C. L.: “Evaluation of Drag and Inertial @ Copyright 1970 American Institute of Mining, Metallurgical, and Coefficients for Maximum Range of Total Wave Force”, Petroleum Engineers, Inc. Tech. Report 55-5, Texas A&M U., College Station This paper will be printed in Transactions volume 249, which (1957). will cover 1970.

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