# Exchange Flow of Oil and Seawater in Leaking Subsea Pipelines

by Peter Carr, EPConsult LLC, Houston

Risk analyses for subsea oil pipelines invariably include an evaluation of the likelihood of oil spills of different sizes. The resulting relationship between oil spill frequency and oil spill volume forms the basis for the assessment of environmental and ecological risks, for regulatory approval, for oil spill contingency planning, and for the establishment of insurance premiums. This EPConsult White Paper focuses on one of the challenges involved in accurately calculating the oil spill volume for a given leak scenario.

## Accurate Calculation of Oil Spill Volumes

There are several mechanisms that will drive oil out of a leaking subsea crude oil pipeline. For pipelines that operate at a pressure above the hydrostatic pressure of the sea (the most usual case and the only one that will be considered in this White Paper), the discharge is driven initially by the excess of the internal pressure over the external pressure. On detecting the leak, the operator will shut down the pipeline by stopping pumps and closing isolation valves. Residual pressure will then drive out a further quantity of oil (the pipeline contracts and the oil expands as the residual pressure dissipates). The pipeline then arrives at a condition where the internal pressure at the leak orifice equilibrates with the hydrostatic pressure of the sea. At this stage, the pipeline is still oil-filled. It is often erroneously assumed that the oil stops leaking out at this time. The reality is that leakage continues due to the density difference between seawater and oil. The heavier seawater intrudes into the pipeline via the lower part of the leak orifice, and the lighter oil is displaced from the pipeline via the upper part of the leak orifice. Since the discharge of oil from an isolated pipeline is from a closed volume, the outflow of oil is matched by an equal inflow of water.

The intruding water flows along the bottom of the pipeline as a gravity current, as shown schematically in Figure 1, until the moving water front encounters either the end of the pipeline or a natural trap – a location where there is sufficient local increase in pipeline elevation to prevent the water front traveling further. At the end of the pipeline, or at a trap, the water front is reflected. The oil leakage then continues until the entire pipeline segment is full of water, and all the oil has been displaced from the segment, apart from any small quantity that may be left behind clinging to the pipe walls.

Many risk assessments of oil pipelines fail to recognize that the oil/water exchange flow process exists. This lack of recognition extends to current US government guidance. The US Minerals Management Service (MMS) has published a guideline (Ref. 1) on how to calculate the worst case discharge from an oil pipeline that takes no account of gravity driven exchange flow. A computer program published by the MMS (Ref. 2, 3) for evaluating oil discharge volumes also takes no account of the phenomenon.

The first phase of oil spillage, when discharge is due to differential pressure, involves high rates of discharge but usually only for a short duration (i.e. until the pipeline is shut down and for a short time subsequently while the residual pressure dissipates).

The second phase of spillage, when discharge is due to gravity-driven oil/seawater exchange flow, involves lower rates of oil discharge but (typically) much longer durations. The oil volume discharged may be much larger in the second phase than in the first, so accurate analysis of the second phase (the oil/seawater exchange phase) is important.

Little research appears to have been done on this second phase of spillage, i.e. on the oil/seawater exchange process. The most relevant work dates from the 1970’s and 1980’s (Ref. 4-7). There is an enormous volume of work on other density-driven processes (opening of lock gates, mixing of saltwater and fresh water in estuaries, gravity currents in the atmosphere, lakes, rivers, and oceans, etc.) but the results cannot be directly applied to subsea pipeline leakage problems. An excellent general description of gravity currents has been provided by Simpson (Ref. 8).

## Modeling Exchange Flow

The maximum possible oil spill due to exchange flow can easily be calculated based on hydrostatics and pipeline geometry. Figure 2 shows a typical situation. A rupture is postulated in a sloping section of pipeline. In the final stable equilibrium configuration, after all flow processes have subsided, the seawater will reach to the top of the leak orifice. This means that water will reach point A on the uphill section. Figure 2 is of course drawn to an exaggerated vertical scale. The horizontal distance between the leak orifice and point A may be large. Turning attention to the downhill slope shown in the figure, the intruding water will not be stopped by minor upwards undulations such as at point B. However, at a significant upwards undulation as occurs between points C and D, the water inflow will be trapped. It can be seen by inspection that the requirement for an effective intrusion trap is a local increase in elevation of one pipe diameter or more.

Intrusion traps may arise naturally due to seabed undulations or they may be created by design as shown in Figure 3. To create a trap for a pipeline on a horizontal seabed, the pipeline should be lowered (dimension H in the figure) by one pipe diameter plus an allowance for construction tolerances. On a sloping seabed, the same geometry may be used but H should be further increased to compensate for the slope.

Spill volumes based on hydrostatics and pipeline geometry take no account of the possibility of operator intervention. For a small leak, it may take years for the maximum possible spill volume to be realized and it is unrealistic to assume that the operator would take no action over such a long time scale. For large leaks and steeply inclined pipelines, the time to total oil displacement may be measured in hours or days and it may be a difficult challenge for the operator to mount an effective response. Therefore, it is important not only to be able to calculate the maximum possible spill volume, but also the speed of the exchange process, so that intervention measures can be properly planned.

Methods for modeling the time-dependent characteristics of exchange flow in subsea pipelines have been developed by Kranenburg and Vegt (Ref. 4, 5), and Fannelöp (Ref. 6, 7). Experimental corroboration of the analytical methods has been limited. Fannelöp’s approach is entirely theoretical while Kranenburg and Vegt verified their methodology only for 2-inch pipelines. In 2-inch pipelines, oil and water flows are mainly laminar, while in larger pipelines turbulence plays an important role.

EPConsult has tentatively extended Kranenburg and Vegt’s approach to account for turbulent flow in larger diameter pipelines. There is a need for experimental work on exchange flow in pipelines of larger diameter in order to confirm and refine the analytical methods. The two available models (Kranenburg and Vegt, and Fannelöp) both contain “early time” and “late time” solutions for the exchange process. The early time solution assumes the fluids have zero viscosity. Both models adopt the same early time solution apart from a small difference in a multiplying coefficient. Both early time solutions also coincide with a formula originally developed by Benjamin (Ref. 9). The early time solution gives an upper bound to the discharge rate. The late time solutions consider viscous effects (friction and mixing) as the seawater gravity flow intrudes deeper into the pipeline, displacing the oil.

A detailed description of the models is outside the scope of this paper. Briefly, Fannelöp treats the outflow of oil as a standard pipe flow problem with a constant pressure gradient and a specified Fanning friction factor. Kranenburg and Vegt, on the other hand, provided a system of differential equations based on conservation of mass and momentum. Kranenburg and Vegt presented a solution of these equations by means of a transformation of variables, called a “similarity transform”. This solution was limited to laminar flow conditions. Finding it difficult to extend Kranenburg and Vegt’s similarity transform solution to include turbulence, EPConsult developed a direct solution to the underlying system of equations based on the method of finite differences. This required dividing both the pipeline length coordinate and the time coordinate into small intervals and marching in both directions in the finite difference scheme. Within this solution, it was possible to make adjustments to account for flow in the laminar, critical, transitional, and turbulent regimes.

## Examples

The following examples are based on the model by Kranenburg and Vegt, as modified by EPConsult to account for non-laminar flow regimes.

Figure 4 shows the volume of oil predicted to be discharged versus time for a full bore rupture of a pipeline of 1.0-m internal diameter on a horizontal seabed. Figure 5 presents similar information for a pipeline of 0.5-m internal diameter. In both examples, the oil is assumed to be Arab medium with a density of 873 kg/m^{3} and a viscosity of 13.8 x 10^{-6} m^{2}/s. Comparing the graphs, it may be seen that the volume of oil discharged at a given time is not proportional to the pipe cross-sectional areas but disproportionately larger for the larger pipe. This is because an intruding water front travels faster in larger diameter pipelines. For the 1.0-m diameter pipe, the water front is predicted to travel 32-km in 10 days and, for the 0.5-m pipe, 15 km in 10 days. Ruptures of large diameter pipelines are very rare. For a 50-mm leak in either of the example pipelines, the discharge rate due to exchange flow is calculated as only 55 bbl/day.

Calculations such as these may be useful when developing oil spill contingency plans and operator intervention measures.

## Conclusions

Exchange flow of oil and seawater must be taken into account when calculating potential oil spill volumes from leaking subsea pipelines, if accurate results are to be achieved.

The maximum possible leak volume due to exchange flow can be predicted easily from hydrostatics and knowledge of the pipeline profile. However, knowledge of the speed of the exchange flow process is also important because it allows an operator to plan and design intervention measures to suppress the discharge before the maximum possible leak volume has been realized.

Analytical methods have been developed for predicting the speed of the exchange flow process, however, there is a need for further experimental research to validate these methods.

The potential spill volume due to exchange flow can be reduced by creating intrusion traps (low points) at intervals along a pipeline route. On an undulating seabed, traps may occur naturally.

Well-planned operator interventions following a leak can also reduce the spill volume by interrupting the exchange flow process. Operators should consider including intrusion traps in the pipeline design and intervention measures in the spill response plan.

## References

- U.S. Department of the Interior, Minerals Management Service. Pipeline oil spill volume estimator. Available at: http://www.mms.gov/tarprojects/390/PocketGuidejune_19.pdf accessed on November 27, 2008.
- Computer program. MMS Worst Case Discharge v. 1.0 (beta). SINTEF Applied Chemistry/Well Flow, 2002. Available from U.S. Department of the Interior, Minerals Management Service. U.S. Department of the Interior, Minerals Management Service.
- Pipeline oil spill volume estimation model (POSVEM). Available at: http://www.mms.gov/tarprojects/390.htm, accessed on November 27, 2008.
- Kranenburg, C. Exchange flow of oil and seawater in a ruptured submarine pipeline. Applied Ocean Research, 1984, Vol. 6, No. 1.
- Kranenburg C and Vegt E. Leakage from ruptured submarine oil pipeline. Journal of Transportation Engineering, Vol. 111, No. 5, ASCE, Sept. 1985.
- Fannelöp TK. Flow processes and leak rates associated with broken underwater oil pipelines. Norwegian Maritime Research, 1977; 5:6-13.
- Fannelöp TK. Fluid mechanics for industrial safety and environmental protection. Vol. 3. Elsevier, 1993.
- Simpson, JE. Gravity currents in the environment and the laboratory. Cambridge University Press, 2nd ed., 1997.
- Benjamin TB. Journal of Fluid Mechanics, 1968; 31, part 2:209-248.