International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 48 Structural Response of a Standalone FPSO by Swell Wave in Offshore Nigeria Abam Tamunopekere Joshua*, Akaawase Bernard Teryima Abstract--- Deep water exploration has significantly increased the use of FPSOs. The reason been that FPSO provides an economic and flexible approach to exploration of oil and gas. However, these moored offshore structures are subjected to wave forces especially the swell effects in offshore Nigeria. Using mathematical and graphical computational tool, computations have been made on how the structure will respond on swell experienced in offshore Nigeria. The conditions considered in this paper include swell actions caused by the following swell waves, beam and head swell which is directly responsible for the surge, sway and heave motions. Index Terms--- swell wave, structural response, FPSO, surge, sway, heave, standalone Introduction When waves come in contact with solid matter, energy is transferred to such objects, making such bodies to produce a response which can either be one or multi direction. In respect to FPSO responds to wave it results in the combination of surge, sway, heave, pitch, yaw and roll. Which is termed the six (6) degree of freedom []. The response prediction of ocean structures in a seaway is practised in design and installation of offshore structures through the application of the linear superposition principle in stochastic processes which was first introduced in the industry. The present interest in swell wave impact analysis comes from the active deep water development which is presently taking place in offshore West Africa. In 4 Olagnon et al described the West African offshore environment which ought to be generally mild because of less wind impact as persistent and can reach fairly high amplitudes with very low periods. [] Which results from it receiving swell wave from the storms of Southern Ocean. The present interest in swell wave impact analysis comes from the active deep water development which is presently taking place in offshore West Africa. In 4 Olagnon et al described the West African offshore environment which ought to be generally mild because of less wind impact as persistent and can reach fairly high amplitudes with very low periods. [] Which results from it receiving swell wave from the storms of Southern Ocean. The FPSO model used in this paper possesses an overall length of 8.4m, Beam = 53.6m, deadweight of 553T, gross tonnage of 83T. Methodology The mean wave (drift) force coefficients are generated from the wave excitation in the hydrodynamics analysis using the theories stated in this work. The theory was modified such that it considered Newman (997) [3] [6] and Chakarbarti (987) [4] [7] which suggested the best wave model for design and operation of ship-shaped structures in West Africa region. However, the proposed coefficient for current by Jensen (4) [5] [8] is not applied to my calculations since this work considers only swell wave. The work also established a relationship between wavelength and response amplitude operator. Note, the scope of this paper is limited to the hull response as a result analysis on the mooring system have been neglected.. Frequency of Encounter. 8
International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 49 When a vessel moored oscillate at a particular frequency ω and a wave direction μ, the frequency at which it encounters the waves (ω e ) becomes important even though the significant wave height may be smaller than that of a fully developed sea, since the modal frequency is higher the heave motions tend to have higher natural frequencies. The relationship between the frequency of encounter and the wave frequency becomes: ω e = ω - kvcosμ Note that μ = for following waves.. Forces and moment responsible for FPSO response on swell The steady drift forces and moments for an FPSO subjected to arbitrary waves neglecting current coefficient is expressed as: the x-, y- and z-axes and three (3) rotations about them as given in figure. Surge = x = X a cos(ω e +Ɛ x ξ) Sway = y = Y a cos(ω e +Ɛ y ξ) Heave = α = α a cos(ω e +Ɛ α ξ) Note that each of the ξ values is at a different phase angle. Where; Ɛ = Phase gap. ω e = Encountered frequency X a = Surge Amplitude F i = ρg. ξ L L sin (Ѳ + β)ni. dl So the steady Surge and Sway drift forces and heave drift moment for a ship shaped structure can be expressed as: Surge: F = ρgξ sin (Ѳ + β). sinѳ. dl Sway: F = ρgξ Heave: F3 = ρgξ sin (Ѳ + β). (XcosѲ YsinѲ). dl 4.3 The linear response Whenever a force is induced into a body, the body tends to move or remained at rest depending on the magnitude of the exciting force. The structural response of an FPSO on swell wave is simply how far the structure displaced, when acted upon by a certain magnitude of force. This response is always in the same direction of the induced wave. [6] [3] The response of the FPSO is determined in this paper by first defining its initial points defined on the hull of the FPSO, in this case the centre of gravity. Such that the responses becomes the deviation from the centre of gravity. The FPSO motions in the steadily translating O(x,y,z) systems are defined by the three (3) translations of the vessel s centre of gravity(cg) in the direction of Figure : Defination of FPSO motions in six degree of freedom. () ω e = k(c-vcosμ) 9 μ 8 for head wave. μ 9 for following wave. And μ is 9 for beam waves sin (Ѳ + β). cosѳ. dl C = ω 3 k = L T Where V is the vessel speed C is the wave speed μ is the angle of attack T e is the encounter period T e = λ C+Vcos(μ π) = λ C Vcosμ 5 ωe = π = π(c Vcosμ) = K(C Vcosμ) T e λ 6 But note that; KC = ω ωe = ω ω ωv Vcosμ = ω( cosμ) g g 7 Where the angle of attack (μ) ranges between and 8 The structure will respond in head waves than the following waves this has been base on the 8
International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 4 assumption that the vessels considered here, is anchored at the aft (turret). The following response will occur on the six degree of freedom as presented in equation 8 to 6 having neglected roll, pitch and yaw in this analysis due to negligible. Surge displacement = X = Xacos(ωet + Ɛx) Surge velocity = V = ωe XaSin(ωet + Ɛx) = V = ωe XaCos(ωet + Ɛx + π ) 9 Surge acceleration a = ωe XaCos(ωet + Ɛx) a = ωe XaCos(ωet + Ɛx + π) 8 Figure : Harmonic wave and surge signal.4 Hydrodynamic Theory The theory which forms the basis of computations of the mean and low frequency second order drift forces (mean and low frequency) on floating structures. This theory is developed based on the assumption that the fluid surrounding the body is in-viscid, irrotational, homogeneous and incompressible. Sway displacement = Y = Yacos(ωet + Ɛy) Sway velocity = V = ωe YaSin(ωet + Ɛx) V = ωe YaCos(ωet + Ɛx + π ) Sway acceleration a = ωe YaCos(ωet + Ɛx) a = ωe YaCos(ωet + Ɛx + π) 3 Heave Displacement Z = Zacos(ωet + Ɛz) 4 Heave velocity V = ωe ZaSin(ωet + Ɛx) V = ωe ZaCos(ωet + Ɛx + π ) 5 Heave acceleration (a) = ωe ZaCos(ωet + Ɛx) a = ωe ZaCos(ωet + Ɛx + π) 6 Response = response amplitude Cos (ω e t + phase gap) λ T e = and λ= ω C+vcos(μ π) ωz t K = and ω z = c Zaωz a Response amplitude (Z a ) = Z st μ Z st = Fa zkμ and phase gap = C μ The fluid motion may be described by a velocity potential n = εi i i=i.5 Coordinate System Where εi is a small parameter (perturbation) and i is the i th order velocity potential such the φ denotes second order velocity potential. The three co-ordinate system of axes is use as presented in figure The first is a right-handed system of G-X -X -X 3 body axes with as origin the center of gravity G and with positive G- X 3 axis vertically upwards in the mean position of the oscillating vessel. The surface of the hull is uniquely defined relative to this system of axis. A point on the surface has as position the vector x. the orientation of a surface element in this system of axes is defined by the outward pointing normal vector n The second system of co-ordinate axes is a fixed O- X -X -X 3 system with axes parallel to the G-X -X - X 3 system of axes with the body in the mean position and origin O in the mean free surface. Wave amplitude (Z) = H μ = ( μ ) + 4h λ The third system of co-ordinate axes is a G-X -X - X 3 system of axes with origin in the center of gravity G of the body and axes which are at all times parallel to the axes of the fixed O-X -X -X 3 system. 8
International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 4 Considering a fixed coordinate system, the pressure at a point on the hull of the FPSO can be determined by writing down the Bernoulli s equation as: p = po ρgz ρ φ t Where: Po = atmospheric pressure Z = vertical distance of the point below the mean water surface C (t) = a function independent of the coordinates T = time ρ = mass density of the fluid The quadratic term in equation above can be extended as 7 pressure constant term ρ A steady pressure. ρ φ = ρ v + V +V3 8 Considering an idealized sea state consisting of two wave components of circular frequency ω and ω. An approximation for the x-component of the velocity can be written formally as + A represent 3 Result Analysis The first task in a global response analysis is to identify the static position of the structure that is, an act of establishing a reference point. Thus any deviation from this static position is defined as the response of the structure. The lateral motion include sway, surge and heave motion. These responses have been calculated and tabulated in table -6. Some of the parameters used in the lateral response calculations are encounter wave period, encounter frequency, wavelength, phase angle and amplitude. Since the system is linear, the resulting motion in waves can be seen as a superposition of the motion of the body in still water and the forces on the restrained body in waves. Thus, two important assumptions are made here for the loads on the right hand side of the figure 3. The so-called hydro mechanical forces and moments are induced by the harmonic oscillations of the rigid body, moving in the undisturbed surface of the fluid. V = A cos (ω t + ϵ ) + A cos (ω t + ϵ ) 9 Extending the first velocity terms of equation 3 for two wave components with different wave amplitude A and A and of circular frequencies ω and ω propagating in idealized sea state lead to: ρv = ρ A A + A + A cos(ωit + Ei) + cos(ωit + E) + AA cos (ω + ω)t + E E + AA cos (ω + ω)t + E E This equation shows that second order effects are generally those effects which are their linear with the wave amplitude or proportional to the square of the wave amplitude. It can be analyzed such that the Figure 3: Heaving Circular Cylinder. () These frequency characteristics are known, for instance via model experiments or computations. In many cases the FPSO motions have mainly been a linear behaviour. This means that, at each frequency, the ratios between the motion amplitudes and the wave amplitudes and also the phase shifts yields the motions. Doubling the input (wave) amplitude results in a doubled output amplitude making the relationship directly reciprocal, while the phase shifts 8
International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 4 between output and input does not change. [7] [4] As a consequence of the linear theory, the resulting motions in irregular waves can be obtained by adding together results from regular waves of different amplitudes, frequencies and possibly propagation Table : Surge motion response of an FPSO on following waves directions. With known wave energy spectra and the calculated frequency characteristics of the responses of the ship shaped structure, the response spectra and the statistics of these responses can be established. [8] [5] Wave period (s) Wave frequency (rad/s) ω Wavelenght Surge response 8.775.665.5598.5 8.5.794.534 5.784.9 9.688889.474568 9.83.7 9.3.666667.444444 38.6. 9.8.63653.45 53.939.4.6.3844 6.497.4.6.58496.345 8.566.43.563636.37686 93.9.68.5.5393.966.933.7.8.5544.767 3.37.64.56667.66944 3.7596.7.4.5.5 46.4.79.7.48889.3838 58.4668.93 3.47693.7456 7.8.8 Surge motion response of an FPSO on following waves.93 Surge response.9.8.7.6.5.4.3..9..7.4.4.43.68.7.7.64.79.8..5 5 5 3 Wave lenght of induced wave 8
International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 43 Figure 4: Surge motion response of an FPSO on following waves. Table : Surge motion response of an FPSO at 9 (Beam waves) T (s) f ω L Surge response 8.775.665.5598. 8.5.794.534 5.784.38 9.688889.474568 9.83.34 9.3.666667.444444 38.6.4 9.8.63653.45 53.939.8.6.3844 6.497.84.6.58496.345 8.566.86.563636.37686 93.9.36.5.5393.966.933.4.8.5544.767 3.37.8.56667.66944 3.7596.4.4.5.5 46.4.58.7.48889.3838 58.4668.86 3.47693.7456 7.8.6 Surge response at 9.86 Surge response.8.6.4..8.6.4...38.4.34.8.84.86.36.4.4.8.58.6 99 49 99 49 99 Wavelenght 8
International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 44 Figure 5: Surge motion response of an FPSO on beam waves. Table 3: Sway response at (following waves) Period frequency ω Wavelength sway response 7.88574.78449 78.537. 7.6.85789.6655 9.565. 8.3.746988.55799.396. 8.7.7644.5786.93. 9..6839.46495 3.78.35.6.3844 6.497.5.6.58496.345 8.566.8.563636.37686 93.9.6.4.54386.95783 8.66.7.9.58.745 6.997.5.56667.66944 3.7596.9.3.5465.548 4.448..6.4963.46 54.45.8.9.486.3996 66.676.4 The sway response of an FPSO on following swell wave 3 Sway response.5.8.4.5..9.5.5.35..5.8.6.7... 7 7 7 3 Wavelenght Figure 6: sway motion response at angle of attack. 8
International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 45 From the above figure (Fig. 6) the plot interpretation is that at a wave length of m the sway response at attack angle, the sway response will be.m. Table 4: Sway response at 9 (Beam waves) Period (s) frequency ω Wavelength Sway response 7.88574.78449 78.537.6 7.6.85789.6655 9.565.6 8.3.746988.55799.396.6 8.7.7644.5786.93.34 9..6839.46495 3.78.567.6.3844 6.497.45.6.58496.345 8.566.4536.563636.37686 93.9.4.4.54386.95783 8.66.4374.9.58.745 6.997.8.56667.66944 3.7596.458.3.5465.548 4.448.944.6.4963.46 54.45.96.9.486.3996 66.676 3.888 Sway Sway response at 9 4.5 4 3.888 3.5 3.96.5.944.458.5.8.567.34.45.4536.4.4374.5.6.6.6 66 6 66 6 66 36 Wavelenght 8
International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 46 Figure 7: sway motion response at 9 angle of attack. The above plot shows the relationship between the sway displacements and the wave length obtained from encounter frequency and period. Table 5: Heave response at angle of attack. Period (s) Frequency (Hz) ω Wavelength Heave response.6.3844 6.497.4.59654.355399 73.36.9.6.58496.345 8.566.5.5.5393.966.933.7.45.49799.47996 48.39.5 3..469697.65 79.9. 3.9.44643.98955 39.685.37 4.4.43556.85378 33.939.8 4.9.467.7345 355.774.4 5.43333.7844 36.569.5 5.56.398458.58768 387.9864.69 6.3875.556 4.393.55 7.36476.33 463.7.8 8.344444.864 59.9.4 3 Heave response.5.4 Heave response.5.5.8.5.69.55.4.8.5.9.5.7..37 6 6 36 46 wavelenght Figure 8: Heave motion response of FPSO at angle of attack. 8
International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 47 Table 6: Heave response at 9 angle of attack Period (s) Frequency ω Wavelength Heave response.6.3844 6.497.4.59654.355399 73.36.99.6.58496.345 8.566.65.5.5393.966.933.87.45.49799.47996 48.39.65 3..469697.65 79.9.3 3.9.44643.98955 39.685.47 4.4.43556.85378 33.939.99 4.9.467.7345 355.774.55 5.43333.7844 36.569.67 5.56.398458.58768 387.9864.859 6.3875.556 4.393.75 7.36476.33 463.7.98 8.344444.864 59.9.64 Heave response of an FPSO at 9 3.64 Heave response.5.5.65.99.98.859.67.75.55.47.5.99.65.87.3 6 6 36 46 56 Wavelenght 8
International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 48 Figure 9: Heave motion response of FPSO at 9 angle of attack. 3. Reduction of swell wave impact on the structure. The effect of swell wave on FPSOs would be reduced by employing the principle of added mass, maintaining proper ballasting of tanks during discharge and finally building of swell wave breakers around the structure. 4 Conclusion From the results obtained in table -6, it is obvious that the response of the FPSO is minimal when the wave attack angle is less than 9 with the least response obtained at the position of the following angle however it becomes maximum at the heading wave direction. From the response diagram as found in figure 4-9, the degree of response of the structure for surge and sway yields a greater response when the angle of attack is, whereas the yaw response is greater when the angle of attack is at 9 respectively. 5 References [] Zhao, C., Bai, Y., & Shin Y., (). Extreme Response and Fatigue Damages for FPSO Structural Analysis, Proc th ISOPE Conference, ISOPE, Stavanger, Vol., pp. 3-38. [] Kelvin, E., George, Z.F., Michel O., Sylvie, V.I. (4). West African swell project (WASP). Nigeria [3] Newman J.N., (967). The drift force and moment on ship in waves. J.Ship Research., 5-6. [4] Chakrabarti, S.K. (987). Hydrodynamics of offshore structures, Computational Mechanics Publications, Springer-Verlag. 34(54). [5] Jensen, J., Mansour A. & Olsen A. (4): Estimation of ship motions using closed-form expressions, Ocean Engineering, 3, 6-85. [6] Ardhuin, F., Reilly, W.C., Herbers, T.H., & Jessen, P.F., (3). Swell transformation across the continental shelf. Attenuation and directional broadening. Oceanography. Manchester City. [7] Aryawan, I. & Stewart, G. (8). Response Based Motion Analysis Methods for a Turret Moored FPSO. Marine Operations Specialty Symposium. 398-4, ISBN 978-98-8-6-4. [8] Kawahara, Y., Maekawa, K. & Ikeda, Y., (). A Simple Prediction Formula of Roll Damping of Conventional Cargo Ships on the Basis of Ikeda s Method and Its Limitation. Fluid Mechanics 8
International Journal of Scientific & Engineering Research Volume 9, Issue, February-8 49 and Its Applications, Vol. 97, p465-486. Nomenclature Symbol ω k μ ξ φ Ɛ λ Ů σ t g d ρ β A εj Interpretation Frequency Wave number angle of attack Wave elevation Phase angle Phase gap Magnification factor Velocity potential Phillip s constant Time Gravity depth Density of salt water. Angle between the wave propagation direction and the x-axis. amplitude of the sea spectrum characterizing sea state phase angle which is considered randomly 8