An example: to find a lazy wave riser shape

Background

In deepwater oil and gas production, that transport production fluids from subsea trees, located thousands of feet below the water surface, to the surface platform. Similarly, in offshore wind development using floating facilities, cables and umbilicals hang from the vessel to transfer power and signals between the surface and seabed.

A lazy wave riser or cable/umbilical is an attractive configuration for these applications. A steel lazy wave riser (SLWR), equipped with distributed buoyancy in its lower section, reduces the top tension and minimizes payload on the vessel. Additionally, it decouples vessel motion from the riser’s touch-down zone.

For designers, understanding the riser configuration and top tension is crucial for flow assurance assessments, detailed riser analysis, and material takeoff. However, this often raises the question: Is there an efficient way to determine these parameters?

The Solution

the catenary shape of a riser can be mathematically expressed using a hyperbolic cosine function. Based on this equation, we have developed a SLWR configuration calculation module that provides a fast and accurate solution for determining the riser configuration.

This module offers a streamlined method to quickly generate the riser configuration, delivering essential parameters for design and analysis. Below, we demonstrate how a lazy wave riser configuration is created using this tool.

Inputs

The inputs required for determining the lazy wave configuration are illustrated in the riser sketch, as shown in Figure 1.

To generate a lazy wave riser shape, only nine key inputs are required in the following exampe:

• Hang-off depth: 4200.0 ft
• Hang-off angle: 12.0 deg
• Riser submerged weight: 80.0 lb/ft
• Buoyancy section submerged weight: -80.0 lb/ft
• Length from top hang-off to buoyancy starting location: 4500 ft
• Length of buoyed section: 1400 ft
• Vessel Offset (optional): 420 ft
• Pipe outer diameter (optional): 12 in
• Riser pipe Young's modulus (optional): 29,000 ksi

These inputs are entered into the fields shown in Figure 2.

After the "Calculate" button is clicked, the SLWR configuration is generated.

SLWR Configuration

The SLWR configuration was generated for three vessel positions, with the key parameters summarized in Table 1. The following metrics were calculated:

• Riser Hang-off Angle: The angle at which the riser is attached to the vessel.
• Span Length: The total curved length of the riser.
• Horizontal Length: The riser projected horizontal distance on seabed.
• Bend Radius at the Riser Touch-Down Point (TDP): The curvature radius of the riser at the point it touches the seabed.
• Riser TDP Movement: The horizontal shift of the TDP due to vessel motion.
• Top and Bottom Tensions: The riser’s tension at the hang-off point and TDP, respectively.

Additionally, the riser gain was defined as the span length minus the sum of the riser’s horizontal length and the vessel offset. For the calculation to be valid, the difference in gain length between the vessel’s offset position and its neutral position must remain under 10 feet. This calculation is suitable for vessel offset up to 14% of the water depth and a riser hang-off angle ranging from 8 and 20 degrees.

Table 1: Lazy Wave Riser Configuration

Parameter	Vessel at Neutral Position (No Offset)	Vessel Offset 420.0-ft to Riser Slack/Near Direction	Vessel Offset 420.0-ft to Riser Taut/Far Direction
Riser Hang-Off Angle theta (deg)	12.00	8.66	17.31
Riser Total Span Length L16 (ft)	6948.2	6845.1	7168.3
Riser Total Horizontal Length X (ft)	4451.5	3928.3	5091.6
Riser Gain (ft)	2496.8	2496.8	2496.8
Riser TDP Movement (ft)	0.0	103.1	-220.1
Riser Section Min Bend Radius (ft)	881.7	616.3	1361.4
Buoyancy Section Min Bend Radius (ft)	-881.7	-616.3	-1361.4
Riser Top/Hang-Off Tension (kips)	339.3	327.3	366.0
Riser Bottom/Seabed Tension (kips)	70.5	49.3	108.9
Seabed Clearance at Riser TDP - Node 2 (ft)	841	725	986
Buoy Clearance at Right End - Node 3 (ft)	908	874	992
Buoy Clearance at Left End - Node 5 (ft)	488	512	499
Buoy Height (ft)	420	362	493
Buoy End Slope	30.0%	25.9%	35.2%

The riser catenary shapes are illustrated in Figure 3 for the vessel in three different positions. When the riser hang-off moves 420.0 ft toward the slack (or near) direction, the riser TDP shifts 103.1 ft closer to the vessel. Conversely, when the riser hang-off moves 420.0 ft toward the taut (or far) direction, the riser TDP shifts 220.1 ft away from the vessel.

The riser configuration, zoomed in on the bottom section, is shown in Figure 4.

The riser bending stress along the length (from touch down to hang-off) is presented in Figure 5. If the focus is more on the bend radius for a flexible riser or umbilical, the pipe's radius of curvature will be provided instead.

Is the calculation using the hyperbolic formula accurate for predicting the riser configuration under uniform weight? To verify the accuracy of the lazy wave riser shape calculation, a detailed Finite Element Analysis (FEA) was performed. The results of this verification are presented in the next section.

Verification

The lazy wave riser shape calculation is based on the hyperbolic cosine function under uniform gravity, which does not account for the pipe’s stiffness. For deepwater risers and cables over long distances, the pipe is sufficiently flexible, and the contribution from pipe stiffness is minimal.

A Finite Element (FE) model was created for the same riser. The model utilized 1,220 beam elements to simulate the riser, extending from the hang-off point to 3,000 feet beyond the touchdown. The primary difference between the FE model and the hyperbolic equation is that the FE model incorporates the pipe's axial, bending, and torsional stiffness. Finite Element Analysis (FEA) was performed to determine the riser configuration.

The riser configuration obtained from FEA was then compared to the configuration generated by the module, which is based solely on the hyperbolic function. The comparison results are presented in Table 2.

Table 2: SLWR Configuration Comparison

Parameter	vessel Offset Position
	No Offset			420 ft Near			420 ft Far
	Module	FEA	Diff.	Module	FEA	Diff.	Module	FEA	Diff.
Riser Hang-Off Angle (deg)	12.00	12.00	0.0%	8.7	8.6	0.6%	17.3	17.3	0.3%
Riser Total Span Length (ft)	6948	6969	-0.3%	6845	6876	-0.5%	7168	7182	-0.2%
Riser Total Horizontal Length (ft)	4452	4471	-0.4%	3928	3958	-0.8%	5092	5104	-0.2%
Seabed Clearance at Riser TDP - Node 2 (ft)	841	840	0.1%	725	724	0.1%	986	985	0.1%
Buoy Clearance at Right End - Node 3 (ft)	908	907	0.2%	874	873	0.2%	992	991	0.2%
Buoy Clearance at Left End - Node 5 (ft)	488	487	0.1%	512	512	0.0%	499	498	0.2%
Riser Top/Hang-Off Tension (kips)	339.3	339.1	0.0%	327.3	327.2	0.0%	366.0	365.8	0.0%
Riser Bottom/Seabed Tension (kips)	70.5	70.2	0.4%	49.3	48.8	0.99%	108.9	108.5	0.3%

For the vessel in the neutral position and far offset position, the maximum difference between the Module (hyperbolic calculation) and FEA results is less than 0.5%. When the vessel moves to the near offset position, the maximum difference between the two methods is less than 1%.

he bending stress (at the pipe's outer fiber) along the riser was also calculated using FEA. The stress profiles obtained from the Module and FEA are presented in Figure 3. While the stress calculation from the Module does not capture the stress peaks at the two inflection points - located on either side of the buoyancy section where the riser curve transitions from concave up to concave down or vice versa - it aligns well with the FEA results in other regions. This demonstrates that, outside the inflection points, the stress calculated using the riser curvature in the Module is reasonably accurate.

Three bending stress peaks were identified along the riser, progressing from left to right: the touch-down point (TDP), the hog bend point within the buoyancy section, and the sag bend point. The maximum bending stresses at these three locations are presented and compared in Table 3.

Table 2: Maximum Bending Stress at SLWR Lower Section

Vessel Position	Maximum Bending Stress (ksi)
	Touch Down Point 6		Hog Band in Buoyancy Section		Sag Bend Point 2
	Module	FEA	Module	FEA	Module	FEA
Neutral Position (No Offset)	16.44	15.93	16.44	16.44	16.44	16.46
Offset to Near Direction	23.53	22.04	23.53	23.45	23.53	23.45
Offset to Far Direction	10.65	10.53	10.65	10.64	10.65	10.64

The peak stresses calculated using this module align well with the FEA results. At the touch-down point (TDP), the module assumes the same radius of curvature as at the sag bend point. This assumption may lead to an overestimation of the stress at the touch-down point.

Conclusion

Compared to FEA, the hyperbolic theory-based module offers a highly effective solution with sufficient accuracy for determining a lazy wave riser shape. In practical project execution, however, there is no need to use FEA to validate the hyperbolic equations. Instead, the results generated by this module serve as direct inputs for riser configuration, material takeoff for pipes and buoyancy elements, as well as for verifying riser FEA results during the concept development, FEED, and detailed engineering stages. You can log in or sign up to access this claculation module.