1. Introduction
Offshore Pipelines have been one of key offshore installations for the offshore oil and gas industry, due to the growth of development of marine resources. In the marine environments, the existence of a pipeline changes the flow pattern and promotes the ability of sediment transport, which further affects the scour process [
1,
2,
3]. The development of scour beneath pipelines can cause a free spanning pipeline that will cause further structural damage.
In general, sediment incipient motion occurs at the beginning stage of local scour around marine installations [
4]. Sediment threshold is not only important for the general problems of sediment transport and scour, but also impacts the scour protection and the design of mitigation measures in engineering. Hence, it has been extensively studied by scientists and engineers [
5]. Shields number originally proposed by Shields [
6] has been widely used as an index for the sediment to move from the static condition [
4].
Since Shields [
6], numerous modifications of conventional Shields number have been proposed, based on a series of laboratory tests and field measurements. Among these, Madsen and Grant [
7] determined the maximum bottom shear stress under an oscillatory fluid motion according to the wave friction factor concept [
8] and proposed the Shields criterion for oscillatory unsteady flow. Whitehouse [
9] and Chiew and Parker [
10] investigated the threshold condition for the onset of cohesionless sediment transport on a sloping seabed. Later, Juez et al. [
11] proposed a new Shields number considering gravity currents over steep slopes and investigated the bed load transport at the action of gravity currents through both experimental and numerical models, based on their previous studies for bed load transport over the sloping river [
12,
13,
14]. However, only the basic forces (i.e., included submerged weight force, left and drag force) were included, and the seabed surface was assumed as an impermeable and rigid boundary in their studies, in which the seepage force was ignored.
In general, the seepage velocity in a seabed is rather small, compared with the free-stream velocity [
15]. Therefore, it has been considered as a minor effect on the flow fields. However, its associated change with the hydrodynamic force on the seabed surface could be visible for some cases with a structure [
16]. Hence, it is necessary to consider the impact of seepage to the sediment motion. For example, as pointed out by Cheng [
17] and Sumer et al. [
18,
19], seepage has significant effects on sediment incipient motion based on their experiments. Qi and Gao [
20] reported that the upward seepage brings the sand-bed more susceptible to scouring based on their wave flume experiments. In addition to the experimental studies, Niven [
21] theoretically examined the onset of sediment motion with upward seepage. Recently, Yang [
22] further proposed a formulas for sediment transport due to vertical flows.
To date, only a few studies have considered seepage on incipient sediment motion quantitatively. Cheng and Chiew [
23] considered the upward seepage in the force analysis for the threshold condition of sediment incipient motion and verified their theoretical model by a series of laboratory experiments. Recently, Guo et al. [
24] further proposed a new Shields number with the upward seepage, and integrated it into their numerical model (COMSOL model) to investigate the impact of upward seepage under an oscillatory flow. Later, Li et al. [
16] further included the upward seepage in their CFD scour model (OpenFOAM model). However, these studies only considered the upward seepage, i.e., one-dimensional. In fact, the real scientific problem should be either 2D or 3D.
The combination of scour and soil response has always been a major difficulty in the field of marine engineering [
25]. The purpose of this study is to make a preliminary connection of scour and soil response by considering seepage on sediment incipient motion, which would be further drawn into the process of scour in future research. By the way, the process of scour around marine structures would be more accurately predicted, which has a great effect on the scour protection and the design of mitigation measures.
In this paper, we proposed a new modified Shields number by considering 2D wave (current)-induced seepage. The paper is organized as follows: in
Section 2, the previous numerical model (PORO-FSSI-FOAM, [
26,
27]) for the wave/current induced soil response in a porous seabed in the vicinity of a submarine pipeline is outlined. In
Section 3, the new Shields number will be derived. Then, the effects of 2D seepage on sediment motion will be investigated. In
Section 4, comparisons between the new Shields number with the conventional Shields number will be presented through the parametric study. In
Section 5, the proposed model will be applied in the cases with a submarine pipeline in a trenched layer. Finally, the key finding will be summarized in
Section 6.
3. New Shields Number with Seepage Flow
In general, the onset of sediment motion will occur when the driving force acting on the particles is greater than the resistance. Based on the force balance of a soil particle, Shields [
6] defined the Shields number (
) as
where
and
are the unit weight of fluid and sediment, respectively;
is the mean grain size and
is the bed shear stress, which is determined by:
in which
is the friction velocity at the seabed surface, and
is the density of the fluid.
In this study, a non-cohesive sandy seabed is considered. Considering a coordinate system as shown in
Figure 8, the soil particles at the seabed surface are subjected to the following forces:
- (1)
In the z-direction: they are submerged weight (), lift force (), and seepage force in the z-direction ().
- (2)
In the x-direction: they are drag force () and seepage force in the x-direction ().
The equilibrium equation of the sediment particles at the seabed surface can be expressed as
in which
f is the static friction coefficient, and
is the resistant force in the
x-direction.
Herein, all soil particles are assumed to be spheres with a uniform diameter of
, then the submerged weight, lift force, and drag force in (
14) can be written as:
where
and
are the lift and drag coefficients, respectively, and
is the flow velocity approaching to the particle at the seabed surface.
Substituting (
15)–(
17) into (
14), the following relation can be obtained:
in which
is the bed shear stress at the critical condition,
is the Reynolds number;
is the ratio of velocity of the particles and friction velocity [
5].
The left side of (
18) is the new Shields number modified of two-dimensional seepage. Compared with (
12), the extra part of
reflects the impact of two-dimensional seepage. In order to calculate the extra part, seepage force should be calculated firstly, which would be obtained above.
Considering the soil-fluid mixture of the seabed, the seepage within a porous medium can be decomposed into the
x- and
z-directions and expressed as [
34,
39,
40],
in which
is the soil porosity and
is the pore-pressure in the seabed.
It is clear from the right-hand-side of (
14) that the resultant force in the
x- direction is the same as the horizontal seepage force (
) and drag force (
). Therefore, the sign must be consistent when they have the same direction and vice versa. However, the sign of
is always positive as seen from (
17), and its direction is the same as the flow. For that,
is introduced to adjust the sign of
to realize that
is positive when it has the same direction with
and
is opposite when it has the opposite direction with
. However,
can be used only when
is not equal to zero and the condition when
should be discussed separately, which is due to the fact that
is meaningless as the denominator and can not be a parameter to adjust the sign of
. For the special condition, drag force is zero, and there is only
in the horizontal direction. Hence, only
should be considered.
Now, the horizontal seepage force (
) is rearranged as
Here, the scaling factors
(
i =
x,
z,
) are introduced to represent the impact of seepage to the modified Shields number:
Now, we introduce the modified Shields numbers,
and
, to represent the impact of seepage:
in which the conventional Shields number(
) given in (
12) and the modified Shields number with vertical seepage given in (
25) [
24]. Note that
was used to replace the particle size of soil and uniform soil was considered in the above derivation of the modified Shields number. That is, we ignored the influence of wide graded of soil on the modified Shields number in practical situations. This is an important point to further investigate the influence of the widely grade of soil on the modified Shields number in the future.
Now, we further consider the influence of seepage on a sloping bed. In a sloping seabed, a coordinate system has a parallel to the slope as the
-direction and perpendicular to the slope as the
-direction, as shown in
Figure 9. The force balance on a sloping bed is presented in the figure.
Based on the force balance, the equilibrium equation in the
-direction can be written as:
Substituting (
15)–(
17) into (
27), the following relation can be obtained:
With the similar process, the seepage components in the
and
can be expressed as:
Herein, the scaling factors for a sloping seabed (
and
) are introduced below:
Similar to the case with a flat seabed, the scaling factor for a sloping seabed,
, is introduced to represent the influence of 2D seepage on the Shields number. Meanwhile, similar to (
25) and (
26), modified Shields numbers
and
on a sloping seabed can be written as:
Note that, when the angle of slop seabed is zero, i.e., a flat seabed, the results of modified Shields numbers
and
are identical to
and
in (
25) and (
26).
It is clear that the influence of the 2D seepage on sediment incipient motion depends on the scaling factor ( or ). The ratio, or ), is a magnification of the Shields number under the influence of 2D seepage in a flat seabed or sloping seabed. According to this, it is found that 2D seepage would promote the sediment incipient motion when and restrain the onset of sediment motion when . In addition, for the special case when , the seepage force is larger than the submerged weight force and the soil particle is in the suspension state, which is beyond the scope of our research.
Equations (
25), (
26), and (
32) used for the evaluation of the influence of 2D seepage on sediment incipient motion could further affect the evaluation of the process of scour and erosion of sediment by gravity currents under non negligible slopes, which are extremely significant in the design of marine structure. In practical offshore engineering, the scour protections are an extremely important part to ensure the stability of the superstructure [
41]. Hence, more accurate predictions of sediment initiation motion and scour can make the scour protections design more effective. It is obvious that the seepage promotes the onset of sediment when
, which means that the protective layer of scour design should be large enough. In addition, seepage restrains the onset of sediment, and the protective layer of scour design can be smaller when
.
Recently, two studies considered the seepage in the sediment motion or local scour. First, Guo et al. [
24] re-derived the Shields number with the vertical seepage. Second, Li et al. [
16] further considered the vertical seepage in their CFD scour model for a sloping seabed. However, their studies ignored the multidimensional of seepage. For that, the influence of horizontal is added in the paper. It is noted that the definition of
used in Li et al. [
16] was only considered the magnitude of the upward seepage. Since they used a 1D approach, they only focused on the case under wave troughs, where the seepage (
) is always upward. In this study, the whole cycle is considered, and the results are displayed below.
5. Engineering Application: A Buried Pipeline in a Trenched Pipeline
Nowadays, the trenched layer has been commonly used in marine oil and gas projects in order to prevent the damage of the pipeline. In the study, the case of a partially buried pipeline in a trenched layer is further considered to simulate the practical engineering application. Although this problem has been studied recently [
26,
46,
51], they mainly focused on the wave (current)-induced soil response and liquefaction near the pipeline but ignores the effect of seepage on the sediment motion. Hence, determining the influence of seepage on sediment motion around the half buried trenched pipeline has great engineering significance.
Herein, the influence of seepage on sediment motion around the trenched pipeline will be examined by the proposed model. The sketch of the numerical model with a trenched pipeline is displayed in
Figure 21, and detailed input data for the trench are listed in
Table 6. The other soil parameters are the same as the cases with the spanning pipeline above.
The distributions of corresponding enveloped scaling factors (
,
and
) around the trenched pipeline are plotted in
Figure 22. It is found that the
decrease is approaching the pipeline. However,
has a tremendous increase around the pipeline, which is due to the complex fluid environment and pore pressure. In addition,
and
are larger before the pipeline than the position behind the pipeline. Meanwhile,
also increases around the pipeline, which is mainly dependent on the horizontal seepage flow.
To explore the different effects of seepage on sediment incipient motion between a trenched layer and a flat layer,
Figure 23 compares
and
around a trenched pipeline and a semi buried pipeline in a flat layer. It is observed from the figure that both
and
in the trenched pipeline are larger than the pipeline in a flat layer, which are due to the faster flow velocity and larger pressure in the trench. However, the horizontal seepage has a greater contribution to the trenched pipeline than the pipeline in a flat layer by comparing the division of
and
.
The embedding methods of the trenched pipeline are multifarious. The full, half, and not buried trenched pipeline are widely used in the engineering environment, which corresponds to the case of
,
, and
in
Figure 24. Note that the different ranges of the
x-axis for the three cases are due to the different buried depths and the different lengths of the trench bottom. The full buried trenched pipeline hardly changes around the pipeline, which is due to the unaffected flow environment caused by the pipeline. However, the influence of seepage on the other cases becomes more significant around the pipeline.