The dredged slurry used in this study was taken from a construction site in Dongtou District, Wenzhou City, Zhejiang Province, China. The properties of the slurry sample are listed in Table 1 . The liquid and plastic limits of the slurry were 53% and 32%, respectively. The particle size distribution of the slurry is shown in Figure 1 .

As the variation in pixel intensity is crucial in PIV analysis (White et al., 2003; Stanier and White, 2013; Stanier et al., 2016), tracer particles must be used to create artificial textures when performing geotechnical tests on fine-grained soils of uniform color (Take and Bolton, 2011; Kim et al., 2017). Black-dyed sand was used as tracer particles in this PIV analysis (Bauer et al., 2016; Kwak et al., 2020; Wang et al., 2021), which has been proven a good representativeness (>93%) in the movement of particles within the slurry (Wang et al., 2021). Since the tracer particles were randomly sprayed on the contact of the Perspex sheet and slurry sample, they had little effect on the slurry improvement process due to their small amount and high permeability (Wang et al., 2019). Moreover, as the PIV algorithm (Stanier et al., 2016) used in this study will update the new ‘reference’ image during the calculation process when the correlation coefficient for one of the subsets contravenes user-defined thresholds. Thus, a small amount of the tracer particles mixed into the slurry during the consolidation has no significant influence on the results.

The integral PVD used in this study had dimensions of 100 ± 3 mm (width) × 4 mm (thickness) × 310 mm (height). The tensile strength of the PVD was >2 kN/10 cm, the equivalent pore diameter was 75 μm, and the permeability coefficient was > 5×10^-3 cm/s.

The testing apparatus consists of three parts: a model testing box, a vacuum system, and a data and image acquisition system.

As shown in Figure 2 , a cuboid aluminium model testing box with an internal dimension of 870 mm (length) × 160 mm (width) × 620 mm (height) was made for the laboratory tests. Silicon grease was smeared onto the inner wall of the case to reduce the friction between the soil and the model walls (Sun et al., 2022b). The front wall of the testing box consisted of a Perspex window with a thickness of 45 mm, in which control points for optimizing the photogrammetric transformation parameters were set (White et al., 2003). To minimize friction between the Perspex window and slurry sample, the window was polished and then flanged to the testing box (Sun et al., 2022b). The top of the box was sealed with a polyvinyl chloride (PVC) membrane to ensure airtightness during vacuum preloading. The PVD was attached to the Perspex sheet and fixed to a custom-made iron bracket to prevent bending during the tests. Model tests with a single PVD and dual PVDs were carried out to compare the effects of PVD layout under vacuum preloading. In in-situ practice, the PVDs layout patterns commonly follow a square or triangular arrangement, with spacing ranging from 0.6–1.5 m (Chu et al., 2000; Tang and Shang, 2000; Bergado et al., 2002; Wang et al., 2019). Thus, the dual PVDs model test with a spacing of 0.6 m was conducted. The single-PVD model test was used to simulate the slurry under a large spacing between PVDs. The PVDs were connected to the vacuum system via a vacuum tube and geosynthetic cap. As shown in Figures 2C, D , the slurry displacements were tracked by PIV through observation windows 1 (OW1) and 2 (OW2; dimensions 300 × 180 mm) in the Perspex sheet on the right side of the PVD.

As shown in Figures 2A, B , the vacuum system was mainly composed of a vacuum pump, water separation bottle and vacuum tubes. The water separation bottle separated the airflow from the water flow. The water separation bottle was connected to the vacuum pump by vacuum tubes to apply vacuum pressure. Based on the in-situ applications of vacuum preloading tests, the 80 kPa vacuum pressure was adopted as the design value for the model test.

The data and image acquisition system consisted of five pore water pressure transducers (PPTs), an electronic scale, a light emitting diode (LED) light source, and a complementary metal oxide semiconductor (CMOS) camera. As shown in Figures 2C, D , the PPTs (P1–P5) were fixed at positions 25, 50, 100, 200, and 300 mm away from the PVD and 60 mm from the top of the PVD to record the variation in pore water pressure along the horizontal direction during the tests. The electronic scale was placed at the bottom of the water separation bottle to record the water discharged mass during vacuum preloading. The CMOS camera (18 megapixels) was placed on the front side of the model testing box to record the slurry sample’s displacement during consolidation. The LED light source was placed between the box and camera to illuminate the observation windows. Figure 3 shows a typical captured photo demonstrating that the tracer particles can be distinguished from the light-gray slurry, ensuring the PIV analysis. In the PIV analysis, the photographs were divided into thousands of slurry elements (shown in Figure 3 ) for correlation calculation (White et al., 2003).

The primary procedures of the test are as follows, more details of the operation of the PIV test are given in Sun et al. (2022b):

(i) Check the airtightness of the test apparatus. (ii) Spray tracer particles onto the Perspex sheet using a powder spray ball. (iii) Connect the PVD to the geosynthetic cap and seal the connection to reduce vacuum pressure loss. (iv) Fix the PVDs on a custom-made iron bracket, which was then fixed to the bottom of the testing box. (v) Connect the geosynthetic cap, water separation bottle and vacuum pump through the vacuum tube and seal each interface with clamps to reduce vacuum pressure loss. (vi) Install the PPTs and Perspex sheet into the testing box. (vii) Place slurry into the model case slowly and place one layer of PVC membrane on top of the dredged slurry. Seal the model testing box using a flange connection. (viii) Adjust the camera and light source in front of the model testing box and take 1 image each minute. (ix) Switch on the vacuum pump and record the pore water pressure and water discharge. (x) Stop the vacuum preloading test after 340 h, when the pore water pressure no longer decreases, and the water discharged mass remains stable.

Figure 4 shows the variations in water discharged mass (m) and water mass discharged rate (q) versus consolidation time (t) for the single PVD model test (test C1) and the dual PVDs model test (test C2). Based on the m curves in Figure 3 , it can be observed that the increased PVD resulted in a higher m for test C2, with m values of 15.94 kg at t=340 h, which is approximately 24% higher than test C1. Furthermore, due to the increased PVD in test C2, the q was significantly higher than test C1 at t=0 h, with q=0.237 kg/h and 0.637 kg/h for test C1 and C2, respectively. Notably, q tended to be the same at 80 h in both tests, which means that the clogging stabilized. Based on the q curves, the consolidation process can be divided into three stages: Stage I (0–10 h), where clogging developed rapidly, resulting in a rapid decrease in q. In stage II (10–80 h), the formation velocity of the clogging was relatively slow, resulting in q decreasing slowly. In stage III (80–340 h), the clogging remained stable and q was similar in both tests.

Figure 5 shows the pore water pressure (U) dissipation law, in which Figures 5A, B show how U dissipated with consolidation time at different positions, with the three stages defined above marked with dashed lines. Figure 5C plots the U distribution along the horizontal direction at the end of each stage.

As shown in Figure 5 , the U dissipation rate at P1 (25 mm away from the PVD, x=25 mm) was much faster than that at other positions within stage I. Since the vacuum pressure did not propagate to the far field, the U in test C2 at P1 was not affected by the right PVD, thus resulting in a similar lateral distribution of U within stage I (see Figure 5C ). In stage II, the U at P2 (x=50 mm) decreased significantly as the vacuum pressure was transmitted to the far field. Notably, as shown in Figure 5C , the U dissipation in test C2 was greater than that in test C1 within stage II, and test C1 had a higher vacuum pressure loss (i.e., the slope of the curve) from P1 to P2. This indicates that the adjacent PVD enhanced the consolidation efficiency of dredged slurry, reducing the clogging between P1 and P2. Within stage III, U dissipated further, and the U dissipation curves at P3, P4 and P5 (100, 200 and 300 mm away from the PVD, respectively) were approximated in both tests. As shown in Figure 5C , at t=340 h in C2, the U dissipation decreased slowly along the lateral direction and the uneven consolidation phenomenon was reduced, while in C1, the U dissipation at P1–P2 were significantly greater than those at P3–P5. This indicates that the effect of an adjacent PVD can relieve the vacuum pressure loss near the PVD during stage III, thus improving the vacuum preloading effect.

The horizontal and vertical displacements of each slurry element were calculated, respectively. Figure 6 shows the horizontal displacement (d _x) field of the slurry during the three stages, where the negative d _x represents the slurry move toward to the PVD. Due to the significant settlement of the slurry during the consolidation, some elements defined in the PIV technique move outside the calculation field while some undefined elements move in the calculation field, resulting in a blank area.

As shown in Figures 6A, B , the d _x fields in the range of x=0–225 mm of the two tests were roughly the same in stage I. Based on the vacuum pressure transition characteristics, the displacement field can be divided into two zones by the d _x. Zone I is a consolidated zone where the slurry has been consolidated, and the d _x=-2–0 mm. Zone II is a transitional zone, where the |d _x|>2 mm. It is worth noting that in test C2, a horizontal displacement contour with d _x=0 mm appeared in the range of x=250–300 mm, while the slurry still moved toward the PVD at the same position in test C1. This is because the vacuum pressure from the two PVDs in C2 reached equilibrium at x=250–300 mm. This finding is consistent with the test results reported by Wang et al. (2019), which indicated that the d _x in the midpoint between two PVDs is zero. The difference between this study and Wang et al. (2019) may be attributed to the difference in vacuum pressure loss in the two PVDs. Figures 6C, D show the variations in d _x for the two tests in stage II. In stage II, as the vacuum pressure propagated further in the slurry, the d _x of the far-field slurry increased further, and the maximum horizontal displacement moved away from the PVD. It is worth noting that zone I was further enlarged in stage II, and the width of zone I was larger in test C1 than in test C2. As reported by Sun et al. (2022a); Sun et al. (2022b), the dredged slurry near the PVD rapidly moves towards the PVDs under the vacuum pressure gradient, resulting in the formation of a clogging zone. A larger Zone I indicates a thicker thickness of the clogging zone, which implies that the adjacent PVD can effectively reduce the thickness of the clogging zone around the PVD, thereby reducing vacuum pressure loss and improving the consolidation efficiency of the dredged slurry (as shown in Figure 5 ). Figures 6E, F show the variations in d _x for both tests in stage III. Within stage III, the vacuum pressure loss due to the clogging zone caused a significant decrease in d _x. Furthermore, in test C2, the decrease in d _x of the slurry is also due to the horizontal vacuum stress offset by the vacuum pressure transmitted from the right PVD.

To further compare the differences in d _x between the two tests, the mean horizontal displacement ( d ¯ x ) distributions of the slurry within each stage were calculated. Figure 7 shows the d ¯ x with error bars representing ± 1 standard deviation in each stage.

As shown in Figure 7A , since the vacuum pressure transmitting from the right PVD did not reach x=0–225 mm, the d ¯ x distribution and standard deviation of the slurry in this range were approximately the same in stage I. Within x=225–300 mm, the vacuum pressures from the two PVDs canceled each other out, resulting in a slightly larger standard deviation of d _x in test C2 than in C1, indicating a more scattered d _x of the dredged slurry. In addition, the slopes of each curve reflect the degree of lateral strain experienced by the slurry element. The positive slope indicates that the element was compressed horizontally, while the negative slope signifies the element was under extension. And the higher negative slope in x=225–300 mm means the slurry underwent a larger extension (see below). As shown in Figure 7B , the d ¯ x of C1 was larger than that of C1 in x=0–75 mm, and the d ¯ x of C2 was much smaller than that of C1 in x=75–263 mm. The larger d ¯ x of C2 in x=0–75 mm indicates that higher consolidation effect during this stage. Furthermore, as the vacuum pressure transmitted from the PVD was partially canceled by the vacuum pressure from the adjacent PVD, it is noteworthy that the d _x distribution of the slurry in C2 was more scattered, especially in x=75–263 mm. In Figure 7C , the d ¯ x and standard deviation in x=0–114 mm in test C2 was larger than in C1, indicating that the slurry near the left PVD in test C2 was further consolidated. In addition, the d ¯ x in x=114–300 mm was much greater in test C1 than in C2 in stage III. Differing from stage II, the d _x in C1 was more discrete, which can be explained by the vacuum stress from the right PVD acting on the slurry element having stabilized, leading to a lower and more uniform d _x value.

Figure 8 shows the vertical displacement (d _y) of the displacement vector of dredged slurry within the three stages. A positive d _y value represents vertical settlement of the slurry. As shown in Figures 8A, B , a non-uniform settlement was observed near the PVD, characterized by smaller d _y near the PVD and larger d _y in the far field. Moreover, owing to the limited propagation of vacuum pressure, d _y of test C1 demonstrates an initial increase followed by a subsequent decrease along the x. Additionally, in test C2, the slurry vertical settlement superimposed in the far field, resulting in a “W”-shaped distribution of d _y contour lines. Figures 8C, D show the d _y fields of the slurry during stage II in the two tests. In stage II, the d _y was developed toward the far field and downward. In addition, the d _y of the slurry corresponding to C2 was superimposed by the effect of the two PVDs, resulting in d _y being much larger than that of C1. Figures 8E, F show the d _y field in stage III corresponding to C1 and C2. In stage III, d _y was significantly reduced due to the stabilization of clogging, which also reduced the difference in d _y between the two tests.

Similarly, the mean vertical displacements ( d ¯ y ) and standard deviations of the vertical displacements of the slurry along the horizontal direction within each stage were calculated. As shown in Figure 9A , the d ¯ y near the PVD (x≈0 mm) corresponding to tests C1 and C2 were approximated. Since the vacuum pressure did not propagate to the far field, the slurry element underwent the same stress state, resulting in similar standard deviations in d ¯ y in tests C1 and C2. As shown in Figure 9B , in the range of x=0–28 mm, the slurry in test C1 trended to be stable, whereas the slurry in C2 still underwent a small settlement. As vacuum pressure propagated to the far field, the d ¯ y difference between the two tests increased further. In addition, d ¯ y in test C1 decreased at x=220–300 mm, which indicates that the vacuum pressure was significantly reduced. Moreover, the standard deviation in d ¯ y in stage II was greater in test C2 than in C1. This is because the vertical non-uniform vacuum stress applied to the slurry element between the two PVDs made d _y of the slurry more discrete. Figure 9C shows the d ¯ y distributions of two tests in stage III. The differences in the d ¯ y and standard deviation in both tests were reduced due to the loss of vacuum pressure. As dual PVDs reduce the clogging zone near the PVD, the slurry near the PVD further settled.

The GeoPIV-RG algorithm (Stanier et al., 2016) was adopted to convert the horizontal and vertical displacements to horizontal strain and vertical strain, respectively. Natural strains were used, as is conventional for large-strain behavior with positive compression (White and Bolton, 2004; Stanier et al., 2016).

Figure 10 shows the mean horizontal strain ( ϵ ¯ x ) profiles of the three stages in the two tests. Within stage I, the slurry elements near the PVD (x=0–28 mm) underwent severe horizontal compression ( ϵ ¯ x ≈19%). In the range of x=0–50 mm, the horizontal vacuum stress acting on the slurry element was greater than the stress of the element maintaining the at-rest state, which led to the slurry being compressed horizontally. As the vacuum pressure attenuated along the lateral direction, ϵ ¯ x gradually decreased to 0 and was converted to horizontal extension. Within x=0–225 mm, the ϵ ¯ x distribution was approximately the same for both tests. The horizontal extension strain of C2 was greater than that of C1 in the x=225–300 mm range due to the effect of the two PVDs. In stage II, as the slurry elements near the PVD consolidated in stage I rapidly, the ϵ ¯ x in x=0–28 mm were low. In addition, the ϵ ¯ x of test C2 in this range was larger than that in test C1, indicating a longer consolidation stage of slurry near the PVD. As the horizontal vacuum stress transmitted from the PVD was partially cancelled, the ϵ ¯ x values in the range of x=28–100 mm were less than those in test C1. In stage III, the horizontal compression range in C1 increased further to x≈225 mm, while that in C2 was only x=0–115 mm, and the ϵ ¯ x of test C2 in the range was only about 3%.

Figure 11 shows the mean vertical strain ( ϵ ¯ y ) profiles of during the three stages in the two tests. In stage I, ϵ ¯ y was about zero at x=0 mm and gradually increased with x. In test C2, ϵ ¯ y was greater than in C1 due to the effect of the two PVDs. In stage III, the slurry elements in the far field compressed further vertically under the vertical vacuum stress, and ϵ ¯ y was much greater than in stage I. In stage III, ϵ ¯ y was slightly reduced compared with stage II. Compared with the ϵ ¯ x values, it can be inferred that the slurry in test C2 was mainly compressed vertically in stage III.

To compare the degree of uneven consolidation of slurry along the horizontal direction under different PVD layout, the volume strains of slurry elements at each stage in tests C1 and C2 were calculated, and the mean volume strains ( ϵ ¯ vol ) are shown in Figure 12 . As shown in Figure 12A , slurry in both tests C1 and C2 experienced rapid compression near the PVD (x=0–28 mm), with ϵ ¯ vol ≈19%. In the range of x=28–300 mm, the ϵ ¯ vol of slurry element in tests C1 and C2 was relatively similar. This is attributed to the fact that the vacuum pressure was not transmitted to the far field within this range, resulting in minimal stress difference among the slurry elements. Figure 12B indicates that during stage II, slurry units within the range of x=100–300 mm experienced greater volume compression strains under the influence of dual PVDs, leading to larger dissipation of pore water pressure within this range (as shown in Figure 5C ). As shown in Figure 12C , the ϵ ¯ vol of slurry element in test C2 were greater than those in test C1. This implies that under the influence of dual PVDs, slurry elements in test C2 underwent greater volume compression after the stabilization of the clogging zone, with compression mainly occurring vertically (as shown in Figure 11C ). This indicates that test C2 achieved better consolidation effects for the slurry.

The clogging zone has been widely reported (Tang et al., 2010; Bao et al., 2014; Xu et al., 2020; Sun et al., 2022a). The development of clogging zone thickness in the presence of one or two PVDs was investigated. As mentioned above, the formation of clogging zone was related to the horizontal displacement of the slurry. As per (Sun et al., 2022b), a contour representing zero horizontal strain was defined as the boundary of the clogging zone. The average thickness of the clogging zone over the whole field was used to characterize the thickness versus time.

Figure 13 shows the variation in clogging zone thickness (R) with consolidation time. In stage I, R was roughly the same in both tests. In early stage II (t=10–25 h), the clogging zone thicknesses were approximately equal because the vacuum pressure from the right PVD did not transfer to the observation field. When the vacuum pressure was transmitted to the far field, the horizontal vacuum pressure applied to the slurry elements was offset by the vacuum pressure transmitted from the right PVD, making R greater in test C1 than in C2 at 20–80 h. Since the development of clogging zone thickness stabilized at about 140 h, in which the water discharge rate and the pore water pressure dissipation rate tend to a constant, the later variations in R are not shown in Figure 11 . The fitted curves are listed below:

The two tests revealed different consolidation characteristics under different PVD layout. These are summarized as follows:

In stage I, the slurry near the PVD was rapidly consolidated by horizontal compression under vacuum pressure. Since the vacuum pressure transmitted from the right PVD did not reach this vicinity, the dual PVD had less influence on the consolidation of the slurry near the PVD in this stage. Thus, there was a similar pore pressure dissipation distribution ( Figure 5 ). Besides, the accumulation of vertical vacuum stress and the offset of horizontal vacuum stress led to a larger vertical displacement and strain in the far-field slurry under the dual PVD condition.

In stage II, as the vacuum pressure in test C2 was transmitted from one PVD to the other PVD, the vacuum stresses between the two PVDs were superimposed in the vertical direction, while they offset each other in the horizontal direction. This resulted in a further reduction in the horizontal migration of slurry in test C2 during stage II, reducing the clogging and the thickness of clogging zone in this test. As a result, the slurry near the PVD further consolidated during this stage II, leading to larger pore water pressure dissipation of slurry.

In stage III, the slurry deformation became gradually stabilized and the displacement and strain were much smaller than in stage II, resulting in slow rates of water drainage and pore water pressure dissipation ( Figures 4 and 5 ). The consolidation patterns of the slurry exhibit certain differences during stage III. In test C1, the displacement field of the slurry can be divided into two zones: a zone near the PVD characterized by horizontal compression and vertical compression, and a far-field zone characterized by vertical compression and horizontal compression. On the other hand, in test C2, the deformation pattern of the slurry is predominantly characterized by vertical compression and horizontal stretching, with only a small amount of horizontal compression strain observed near the PVD. Furthermore, the less vacuum pressure loss in test C2 resulted in higher volume compression of the slurry, thereby enhancing the consolidation effect of the slurry.

This paper provided equations describing the relationship between clogging zone thickness and time that can be used in consolidation calculations that consider the clogging effect. As indicated by Liu et al. (2021) the thickness of the clogging zone is vital important in calculating the consolidation process of dredged slurries improved by vacuum preloading. It can thus make the calculation more practical when considering the gradual increase in the radius of the clogging zone. A direct and accessible method can be used for applying the equation in consolidation calculation, in which the effective zone around a PVD can be divided into a clogging zone and a normal zone in the consolidation model. As shown by Eq. (1) or (2), the radius of the clogging zone can be assumed as a function of time with other basic assumptions in the large or small strain consolidation theory reserved.