Rheology, Rheometry and Wall Slip

Abstract

In diverse resource, processing and dredging applications wall slip occurs. In hydraulic transport of highly concentrated particulate mixtures, wall slip can be beneficial as it may substantially reduce hydraulic gradients. In other occasions, for instance in rheometry, wall slip may obscure rheology. Rheometric wall slip is not specific to industrial slurries and appears in natural (fluid) mud as well, mostly found in harbours and estuaries. In natural (fluid) muds, in contrary to industrial muds, coarse solids are absent. However, similarly, (clay) colloids govern their non-Newtonian flow characteristics. It is exciting to see that wall slip does not only occur in the case of dispersed coarse materials but also in the absence of those. In this chapter, we elaborate on wall slip in some existing resource industry rheometry data and compare them with typical recent results of fluid mud rheology. Moreover, measurement of a (stationary) fluid mud’s longitudinal profile in a harbour basin is used to examine consequences of utilising slippage data. We finally evaluate measuring element usage and implementation of rheology in calculation methods.

1. Introduction

Rheology is the study of the flow behaviour of materials. Rheometry is the means of quantifying flow (rheological) characteristics of materials. Rheology strongly depends on the fabric of the material and how that changes with time, stress and shear rate. In rheology, the flow behaviour of a given material is visualised by a flow curve in a diagram that depicts the relationship between shear stress and shear rate.

Roto- and capillary-viscometers are standard tools used to carry out rheometry. Often flow (rheological) characteristics of a material are described by a set of parameters, so-called rheological parameters, such as various yield stresses and viscosity. These parameters are important to the characterisation and the modelling of the flow of clay-rich mixtures (consisting of fine and coarse solids) and are often used as a basis in the design and optimisation of soft sediment management plans and processes, amongst many, such as pipe flow, deposition flows, mixing and flocculation processes, emplacements, and the flow in other unit process operations. In these applications, when it comes to initiating flow, the static yield stress (SYS) is a key governing parameter to be determined. Dynamic yield stress (DYS) and viscous properties are, on the other hand, important to the study of fluid flow. The yield points are defined later in this chapter. Rheology and rheometry can also be relevant in the study of stagnant conditions of soft sediments. For instance, static yield stress governs the critical state of stagnant material not only in tailings storage facilities but also in harbour basins.

Ideally, rheometry results should be independent of applied measuring elements. However, in practice, it is often seen that different measurement elements result in different rheological values. The difference may be originated by: i) no or wrong correction factor when it comes to conversion of machine (i.e. rheometer) shear rate to physical shear rate; ii) an ill-defined rheological protocol and iii) occurrence of wall slippage during rheometry. Particularly, wall slip of mixtures in smooth walled measuring geometries can be deceiving. Wall slip occurs when the composition of the mixture at the wall is not exactly the same as in the interior of the mixture [1]. One way to detect the occurrence of wall slip in rheometry is to compare the flow curves of a given material obtained from various geometrical dimensions (pipe/capillary diameter, or sheared gap size). In the presence of wall slip, the rheological results from different geometrical dimensions would differ (shifted flow curves) (Mooney (1931)) [2].

In long-distance pipe flow of highly concentrated mixtures, wall slip may also occur and potentially constitutes a convenient and energy-efficient way of transportation (Goosen and Patterson (2014)) [3, 4]. In the emplacement of industrial tailings, there are many occasions where laminar flow prevails. In the simplest form, lubrication type of shallow flow models (Coussot and Proust (1996)) [5] can be applied for predicting tailings deposition behaviour on beaches in tailings storage facilities [6, 7]. For constant properties, the analytical outcome of such a theory, for a slowly advancing mass or a 1-D deposit on a horizontal base, is a square-root (SQRT) thickness vs. distance longitudinal profile. Assuming critical state conditions of a slowly advancing mass, the shear stress at the bottom equals yield stress [6]. There exist comparable theories for material on mildly sloping inclines, the frontal shape is very similar, but further up the slope, an equilibrium depth is reached. For Newtonian fluid on inclines, see [8]. For stagnant material at a critical state, see [9]. The shape of the frontal profile for other rheological models falls in between. If incorrect yield stresses are applied in such theories, the calculation of deposit depth or flow depth in combination with velocity is strongly affected. In reverse, the desktop Bostwick rheometer, which releases a small mass of material to an inclined channel, uses a similar principle to quantify the strength (i.e. yield stress) of material. For tailings rheology, an upscaled apparatus was developed by [10].

Tailings generally demonstrate time-varying strength properties. This may originate from progressing formation of clay aggregates at constant porewater content known as thixotropy or consolidation (loss of porewater and hence densification-related strength increase). Time/distance varying properties can be included in a numerical version of a lubrication model, but this does not really help to verify the relation between strength (yield stress) and the geometry of an emplacement, which we are pursuing in this chapter.

Rheometry of industrial materials can be difficult because of their inherent time dependency of strength, as well as their dewatering and shear settling of solids. Moreover, materials may have experienced treatment, e.g. by polymers, hence increasing difficulties to properly assess their flow properties. We examined a somewhat simpler material: fluid mud in harbours and estuaries where sand has already settled elsewhere, and the mud predominantly consists of colloidal clays. Rheological properties of fluid mud appear of a comparable magnitude to those of industrial materials where similarly, the colloids are responsible for non-Newtonian rheological properties and where similarly wall slip signatures are observed in rheometry data. In one of the recent fluid mud researches, wall slip is described as two-step yielding (e.g. Shakeel et al. 2020 [11]).

Since rheological characterisation cannot be seen without considering its application, this chapter elaborates on wall slip in pipelines. Some existing industry rheometry data is revisited and recast in formats that allow a more straightforward comparison with the latest measurements that reveal similar slippage. Typical recent results of fluid mud rheology, affected by and checked for wall slip, are presented in the format of flow curves and viscosity curves. Referring to a prototype measurement of a fluid mud’s longitudinal profile in a harbour basin, eventually a proxy for emplacements, we examine the consequences of utilising slippage. We finally evaluate measuring element usage and application of rheology in computational fluid dynamics (CFD) and analytical methods.

The present contribution may also connect readers to a number of classical papers/manuscripts, pointing out similarities with recent data and sharing findings.

2. Rheological approach and wall slip experiences

2.1 Rheology in the plastic regime

In rheology, one needs to distinguish between conditions below and above the yield point. Below the yield point materials depict viscoelastic behaviour with no to limited flow (in other words, plastic deformation). In this domain, the viscoelastic behaviour of materials is characterised by shear modulus and loss modulus. For our engineering application, we investigate rheology at and above the yield point where materials experience plastic deformation or, in other words, flow. In this regime, the material’s viscosity generally reduces with increasing shear rate and thus are characterised as a yield-shear-thinning material.

The existence, definition and determination of yield points in non-Newtonian materials are debated in the literature [12]. The yield point is defined as the lowest shear stress value above which material will act as a fluid and below which it behaves like a very soft solid matter. This definition is subjective, because the boundary between the fluid-like and solid-like state is not discrete but continuous. Many materials have time- and shear-dependent properties, and under different applied rheometry protocols, they give (somewhat) other yield points. Moreover, transition direction (i.e. from fluid-like to solid-like or vice versa) may occur at different stress levels. This is why in materials with time-dependent strength (thixotropic), at least two different yield points can be distinguished. Engineering applications may require knowledge of various yield points, depending on the application. Principal yield points are as follows:

• The static yield stress (SYS): defined as the minimum stress required for initiating the (shear) flow in a stagnant material under stress.

• The dynamic yield stress (DYS): defined as the minimum stress required for maintaining a given material in flow.

Time-independent materials have unique flow curves. However, the flow behaviour of time-dependent materials may vary depending on their shear and resting history, resulting in different flow curves. Our materials are easily remoulded, and recovery is relatively slow. The section of the flow curve where the shear rate returns to zero after experiencing high shear rates is hence called the remoulded flow curve. Equilibrium conditions may be achieved under continuous shearing at discrete shear rates. By plotting those equilibrium conditions (shear rate-shear stress pairs), an equilibrium flow curve (EFC) is obtained. Remoulded and equilibrium flow curves are amenable for description by mathematical time-independent models such as Bingham, Hershel-Bulkley, Worrall-Tuliani and Oswald-DeWaele power law.

For modelling purposes of flow, segregation and settling, it is preferred to have rheological information of the carrier fluid and rheology of the mixture [13]. The current state of the art stands where we may be satisfied if we can quantify one of these in detail and relate it to the other via single-point tests, for instance, through detailed measurements without coarse to determine the rheology of the carrier fluid as a baseline, supplemented by vane yield stress measurement of mixtures with different ratios of fines and coarse, in case of model materials similar to [14], or vice versa conducting vane-type rheology of mixtures, utilising vane-in-cup method (Barnes and Carnali 1990 [15]) or the vane-in-bucket method (Fisher et al. 2007 [16]) in combination with removal of coarse to determine the colloid’s carrier fluid rheology.

2.2 Wall slip

2.2.1 Origin/nature of wall slip

In a non-Newtonian mixture under motion (flow), the velocity appears not always to reduce to zero at the wall because the mixture may have a subtle difference in composition here. Wall slip in a pipe may occur in two different ways, schematised in Figure 1a,b:

1. for the same flow rate: the velocity profile flattens, with slippage at the wall.

2. for the same pressure gradient along a pipeline segment: the velocity profile offsets in downflow direction, with slip at the wall.

Figure 1.

Wall slip (a, b) in a pipe and (c, d) in Couette shear flow. Wall slip is demarcated by the dotted red line adjacent to solid line representing the wall.

Both these conditions boil down to the same state where there is a mismatch between flow velocity, pressure drop and rheology of the bulk of the material.

In concentric cylinder rheometry, a proxy for Couette shear flow, wall slip is imagined to occur at the inner cylinder as the shear stresses are highest there. Upon increasing the angular velocity, slippage at both walls (i.e. at the outer surface of the bob and at the inner wall of the cup) may be speculated, as shown in Figure 1d.

2.2.2 Industrial flows: bulk slipping of non-Newtonian mixtures

In hydraulic pipe transport of highly concentrated non-Newtonian mixtures, the lower the water content of slurries, the higher their strength, and therefore, higher flow velocities are needed to keep the flow in turbulence to counteract the settling of particles (on this, a practical transition criterion is given by [17]).

Figure 2 depicts measured and calculated deposition velocities in a series of different pipe diameters [3, 4]. Increasing the density, the non-Newtonian properties increase and laminar flow is found for velocities below the red curve. Beyond 1580 kg/m³, the flow velocity needs to be above the red line for turbulence to keep the material suspended, but beyond 1650 kg/m³ the material slides as one bulk and laminar flow at low velocities is possible here. If the material would not slide as one bulk, internal shear would lead to settling of the solids [18].

Figure 2.

Observed deposition conditions and stagnant material for different pipe diameters, after [3, 4].

The onset of this wall slipping phenomenon is hard to predict. To achieve sliding flow condition, material needs to be self-supportive. For instance, coarse should not settle in unsheared mixture (a condition referred to as gelled bed by [19]; or freely settling concentration by [3, 4, 20]).

Bulk slipping is also experienced in the transport of cement, shotcrete and fresh grouts [21]. In the construction industry, this allows the pumping of highly concentrated (thick) mixtures over significant distances both horizontally and vertically.

To control thickener operation, it is often not sufficient to rely only on underflow density. Hence, rheological properties of the material are also required. One reason is that the yield stress is found to correlate with the compressive strength of thickened tailings (Green and Boger (1997 [22])), governing the thickening process. As a new advancement, industrial-scale online pipe viscometers are built to measure tailings rheology for controlling of thickener’s operation, e.g. Chryss et al. 2019 [23] and Boomsma et al. 2022 [24]. These viscometers have multiple measurement units with different tube diameters to enable correction for wall slip.

2.2.3 Industrial flows: wall slip in the presence of water as carrier fluid

In the case of Newtonian carriers, like water, rheometry is difficult because of fast settling solids. In that case, accumulated experience with pipe flow serves as a basis for detecting indirect evidence of wall slip. Two distinct occasions at which wall slip is likely, in the presence of water as the carrier fluid in hydraulic pipeline transport of a mixture, are as follows:

Particles experience off-the-wall forces and therefore tend to move towards the centre of the flow, leaving a fluid-rich layer adjacent to the wall. Bartosik & Shook (1995) [25, 26] experienced such flow conditions in vertical pipe flows.

If solid particles do not fit within the laminar sublayer of turbulent flow, wall friction is found to not obey homogenous mixture theory enabling semi-analytical mathematical modelling of wall slip in these specific conditions [27]. In homogenous mixtures, the components are uniformly distributed throughout the mixture. In other words, only one phase of matter exists in a homogeneous mixture model.

2.2.4 Rheometric wall slip

Checking for wall slip in rheometry data can be conducted by comparing the rheometry data obtained from conducting:

• capillary rheometry using different capillary pipe diameters;

• concentric cylinder bob-cup (BC) rheometry using different annular gap widths;

• parallel plate (PP) rheometry using different gap (standoff) distances;

• rough/grooved/serrated/splined/profiled geometries;

• vane testing (as a radical check for wall slip).

In case of wall slip, the unprocessed flow curves of smaller gaps are expected to be shifted somewhat towards higher shear rates compared to larger gap. In other words, when wall slip occurs, a given shear stress value is achieved at a smaller shear rate with increasing gap size. Mind that, in the case of concentric cylinder rotoviscometry, a Couette-inverse transformation might need to be applied first for larger gap size because of non-uniform shearing across the gap.

An effective way to verify for occurrence of wall slip in rotoviscometry with well-defined shear rates is to additionally conduct tests with a vane (Barnes and Carnali 1990 [15], Boger et al. 2008 [28, 29], Buscall et al. 1993 [30]) and compare the results, as shown in Figure 3. Unrealistic steep branches in flow curve plots have been revealed accordingly at low shear rates (Figure 3a). We experience that the wall slip affected BC shear stresses can be a factor 5 lower than of those measured by vane (Section 3).

Figure 3.

Vane eradicates wall slip (a,b,c,d). Mine stope fill [28, 29] and nickle limonite [28]. Plotted in original [29] log-log coordinates (a), viscosity plot (b) and lin-lin coordinates (c, d).

2.3 Inverse Couette problem (non-uniform shear rate in gap)

Shear stresses and shear rate that are outputted by a rotoviscometer are derived from measured torque and rotational velocity. When either one, or both, is non-uniform within the mixture in the element, the rotoviscometer’s conversion is based on Newtonian fluid conditions. Users may need to apply corrections in post-processing if deemed necessary. A complication with concentric cylinder elements is that the shear stress varies with radial position per conservation of torque with radial position. Hence, a non-ideal Couette flow is tested, and the shear rate across the annular gap is a function of the non-Newtonian properties of the fluid itself. Therefore, it is best to use small gaps. For wider gaps, needed for instance when there are coarse in the mixture, this constitutes the so-called inverse Couette problem: how to calculate the shear rate at measured shear stress [31]?

2.4 Mooney correction for wall slip

In capillary rheometry, it is possible to detect wall slip and to apply a correction to quantify the true rheology of the bulk of the material by applying tubes of different diameter [2]. The method can also be applied to sheared gaps of different sizes. In using a series of different gaps sizes or pipe diameters, it is assumed that the wall slip velocity is identical for the same shear stress. Since the minute thickness of the wall slip layer, consisting of base fluid, is difficult to predict, it remains necessary to rely on rheometry for quantification of wall slip.

2.5 Viscosity regularisation

Yield pseudo-plastic models (e.g. Bingham model) fitted to rheological measurements do not quantify stresses below the yield stress. Fluid flow calculation methods nevertheless need info in this regime: see the central plug in Bingham pipe flow where shear stresses are lower than the yield stress. A CFD model usually solves the flow relying on viscosity. Without intervention, the viscosity would unacceptably rise to infinity at a shear rate of zero. This needs to be limited in the model by providing shear stress values below the yield stress. This is achieved by regularisation of viscosity at low shear rates [32].

3. Detailed rheometry of mixtures without coarse

3.1 Material

Detailed rheology measurements are possible only when the coarse fraction is removed from the mixture. An alternative is to revert to a system where natural segregation has already removed the sand fraction. Natural fluid muds constitute such a system and as mentioned earlier it also appears to display wall slip signature. Results of rheometry [33] conducted on natural fluid mud from Beerkanaal, the port of Rotterdam, The Netherlands, with an initial density of 1263 kg/m³ will be presented. Earlier measurements of the fluid mud from the port of Hamburg and Emden are also analysed [11, 34, 35, 36, 37]. The yield stress of natural fluid muds is considered decisive for navigable depth in ports and waterways.

3.2 Configuration and protocol

Configurations with well-defined shear rates such as concentric bob-cup (BC) cylinders are preferred in rheometry. However, as already mentioned, a BC system is prone to slippage: following Boger [29] on usage of vane, slippage appears also to be radically prevented in a vane-cup (VC) configuration, interestingly. The shear rate in a VC, on the other hand, has to be defined.

Two standard rheometry protocols, namely controlled shear rate (CSR) and controlled shear stress (CSS), are applied and compared to investigate the occurrence of wall slip in BC, PP and VC configurations. The vane is applied with 1 mm bottom clearance. Table 1 provides information on the applied measuring elements and testing protocols. Tests are conducted by Haake Mars 1. The basis protocol consists of a linear ramp-up of either shear rate or shear stress during a certain time, followed by a constant phase and a ramp-down phase.

Fig. ref	Element	CSR [1/s]	CSS [Pa]	Ramp-up [s]	Const [s]	Ramp-down [s]
4	BC	0 to 100	0 to 120	180	60	180
4	VC	0 to 116	0 to 120	180	60	180
5	BC	0 to 100	n.a.	180	60	180
5	gBC	0 to 100	n.a.	240	60	240
5	VC	0 to 580	n.a.	240	90	240
6	PP	n.a.	0 to 50	50	n.a.	n.a.
6	VC	n.a.	0 to 50	50	n.a.	n.a.
6	BC	n.a.	0 to 50	50	n.a	n.a.
9	BC	0.1 to 90	n.a.	n.a.	7*35	n.a.

Table 1.

Applied measuring elements and protocols (B = bob CC25DIN, gBC = grooved bob CC25Din, C = cup CC25 DIN, V = vane FL22, PP = parallel plate P35).

3.3 Results and discussion

3.3.1 Wall slip signature in bob-cup (BC) configuration

A definite difference is found between tests conducted in BC and VC. Figure 4a depicts the result of CSR and CSS tests in a BC configuration. At low shear rates (< 10 [1/s]), the ramp-down part of the curve crosses the ramp-up part in both CSR and CSS tests. This is because the measured shear stresses during the initial part of the ramp-up are underestimated, a typical wall slip signature in a BC configuration, compared with Figure 3. Figure 4b shows the result of CSR and CSS tests in a VC configuration where no wall slip is observed. The viscometer’s outputted rotational velocity has been transformed to an equivalent shear rate at the rim of a virtual cylinder encompassing the vane, similar to the BC configuration, to allow comparisons. The shear stress-shear rate relationships over the ramp-down (remoulded) phase are in excellent agreement.

Figure 4.

Example of rheometric results on fluid mud from Beerkanaal, port of Rotterdam, The Netherlands, flow curve (a, c) and viscosity plot (b, d) with different elements and protocols.

3.3.2 Quantification of static and dynamic yield stress

The static yield stress is retrieved from CSS tests in the VC configuration: the shear stress at the point where the linear part of the ramp-up curve leaves the vertical axis and deflects to the non-linear part (at 45 Pa, Figure 4b). At this point, the pronounced wall slip also stops (Figure 4a): the shear stress has reached the yield stress of the unsheared material in the middle of the gap. The dynamic yield stress can be obtained from both CSR and CSS methods in the VC configuration: the intercept of the ramp-down curve and the vertical axis. In the BC configuration, the dynamic yield stress is underestimated by a factor 3 (comparing Figure 4a and b) and the static yield stress by a factor 5.

Here, we see that viscosity plots on the log-log scale are magnifying yielding at low shear rates but are also magnifying artefacts when the material transitions from an elastic state to plastic. Other utilised elements such as grooved bob or parallel plate elements may reveal more.

3.3.3 Slippage with rough/profiled elements

Utilising rough measuring elements is a well-known remedy to reduce wall slip, but it may not solve the wall slip problem entirely. We experienced that a rough element may only shift the wall slip problem to lower shear rates, as shown in Figure 5. Further roughening measures might be necessary to fully eliminate the wall slip.

Figure 5.

Grooved bob (GBC) rheometry results compared to smooth bob-cup (BC) and vane-in-cup (VC), fluid mud from Beerkanaal, the port of Rotterdam, The Netherlands.

Figure 5 shows the rheometry results of three different elements/protocols on the same mud. CSR ramping mode is applied. We see that the BC (blue) starts slipping at about 10 Pa, reaching a maximum shear stress at 40 Pa (end of steep slippage branch), completing a loop and returning finally back to 10 Pa, paralleling with the wall slip of the ramp-up. The vane performs differently; it peaks at 55 Pa and merges with the up-ramp of the BC. Upon return, it only bends down a little before cessation of shear. The grooved bob (green) slips at 15 Pa and follows a course between the CSR bob and the CSR vane results. This grooved bob result is positioned similarly with respect to BC in CSS measurements conducted by [11].

3.3.4 Gap variation parallel plate (PP) elements

A way to investigate wall slip and enable correction (e.g. Mooney transformation) is by conducting a rheometry method using a series of different gaps. This is easily done with a parallel plate (PP) measuring element. Figure 6 includes such exercise.

Figure 6.

From viscosity plot to flow curve, fluid mud from the port of Hamburg, Germany: a): Data presentation after [11] and b): Transformed into flow curve, including Mooney wall slip correction and comparison with BC and VC.

This figure illustrates a combination of rheometry data of Shakeel et al. [11] that contributed to their rejection of wall slip, while in contrary revealing wall slip upon closer inspection of the data from the PP element. These data are very similar to a PP measurement presented in Ewoldt et al. (2015) [38] on Nivea lotion. When repeated with sandpaper glued to the plates, their result was similar to our vane result for >1 [1/s].

The PP outputs higher shear stress, calculated at the rim of the element utilising Newtonian fluid theory, than the other elements. In reality, with non-Newtonian shear-thinning fluids, a higher torque is measured for the same shear stress at the rim: the stresses in the centre part of the plates are comparably higher; hence, the outputted calculated shear stress is higher than the true shear stress at the rim, hence a tendency for higher shear stresses.

The shear rate of the vane is calculated as described in Section 3.3.2 and is hence comparable to the BC element: an associated shear rate in a Newtonian fluid. Despite the virtual wider annular gap of the vane, no Couette inverse method has been applied to convert to the actual shear rate in the non-Newtonian fluid. If such correction would have been applied, the steep branch of the vane would have shifted somewhat to the right, in other words to higher shear rates.

Usage of a too small gap size may also play a role, increasing shear stresses because of confinement. Yan et al. (2009) [39] show rheometry results where the yield stress increases for axial gaps smaller than 0.2 mm. It is conjectured that this occurs when the plate distance becomes smaller than the correlation distance of the structure of the fluid. Hence, jamming effects emerge. So, the higher shear stresses for the 1 mm gap, for shear rates <5 [1/s], indicate that the unsheared clay aggregates have a size of 1 to 2 mm.

Figure 6 shows that the commencement of wall slip with a parallel plate system is postponed with respect to a concentric cylinder. This could be due to non-uniform conditions over the plate’s surface, and hence total slip commences when yield stresses are exceeded at most of its surface.

Perfect alignment should not be expected. We applied the Mooney wall slip correction procedure to the PP result of [11], despite that shear stress and slippage velocities are not uniform in a PP element, compromising the basic assumption of the Mooney transformation. Also, the bottom clearance of the vane is smaller than its virtual annular gap. Hence, it is conceivable that data for <0.6 [1/s] is influenced by local failure here.

As we see in Figure 6, the Mooney wall slip correction procedure projects data back to lower shear rates, close to vane results. This implies that, practically speaking, a vane is a better element to address the internal strength properties of mixtures in rheometry.

4. The pitfall of using wall slip affected rheometry data: a case study example

4.1 Emden outer harbour

Fluid mud, such as at the bottom of harbours and estuaries, constitutes a simplified composition compared to slurries in mineral, resources, and processing industries which often contain a substantial amount of coarse settling solids.

In this subsection, we examine rheological measurements carried out on fluid mud from the Emden Outer Harbour and their importance to the lutocline (= the fluid mud’s longitudinal profile) based on mechanical equilibrium.

The ecological-economical and maritime problem of continuous fines settlement in harbour basins is in Emden approached through a sediment managing scheme consisting of recirculation of accumulated fluid mud over a trailing suction hopper dredger (TSHD) to prevent densification into stiff material, through which vessels cannot sail.

As mentioned earlier, thixotropic strength gain during rest increases the static yield stress beyond the dynamic yield stress. After conditioning by the TSHD, or after vessels manoeuvred through the mud, we can assume the strength state of the mud layer to be somewhere between the static and dynamic yield stress and apparently high enough to prevent the mud from flowing to the river Ems fairway.

Reported rheological measurements on Emden Harbour mud are comparable with the BC data in Figures 4–6. In the following, skipping problematic wall slip conditions in the existing data, the yield stress is again estimated by drawing the tangent to the flow curve for >10 [1/s], hence quantifying yielding at the intersection with the vertical axis.

Multibeam echo sounder measurements conducted in the harbour of Emden contain a 15-kHz profile showing a nearly horizontal reflection line indicating a strong transition at the base of the mud, as shown in Figure 7.

Figure 7.

Fluid mud profile in outer harbour Emden (after [34, 35]) and superimposed calculated lutocline for excess density 123 kg/m³ and 10 Pa yield stress.

At a higher frequency, i.e. 210 kHz, a curved profile is found that represents the surface of the fluid mud layer (lutocline): [34, 35]. At about 1 km from the fairway of the river Ems, this layer has grown to a thickness of about 3 m. The 210-kHz profiles show a typical SQRT profile, associated with a stagnant slump or slowly advancing mass of material of uniform density and strength. The SQRT profile can be used as a validation case for yield stress measurement (rheometry data) of the fluid mud of this harbour, which will be discussed further in this section.

4.2 Mechanical equilibrium of longitudinal mud profile

Examples of models for laminar deposition profiles applied in deposition modelling are published by [6, 7]. The essential mechanical equilibrium of such models is repeated here and sketched in Figure 8. A density variation with depth can be schematised into a homogenous density. Static vertical pressure profiles are then linear and equal to $p=\Delta \rho gz$, where z is the vertical distance under the mud surface profile and Δρ is the density difference with supernatant water.

Figure 8.

Definition sketch of mechanical equilibrium of cohesive material on a horizontal base.

The resulting horizontal driving force follows from the integration of the pressure profiles as: $F_p=\Delta \rho gh\Delta h$. The friction force at the bottom is $F_{\tau }={\tau }_y\Delta x$ Mechanical equilibrium gives ${\tau }_y=\Delta \rho gh\frac{\text{d}h}{\text{d}x}$. Integration from the toe (x = 0) of the mud profile gives a square-root (sqrt) profile for layer thickness $h=\sqrt{\frac{2{\tau }_y}{\Delta \rho g}x}$.

4.3 Emden outer harbour mud rheology

4.3.1 Flow curve

Rheology of fluid mud from the Emden Outer Harbour is reported in [35, 36, 37]. These measurements are conducted using a smooth BC and applying a shear stress ramp-up (CSS) as well as a controlled shear rates (CSR) protocol. Equilibrium flow curves obtained from the latter protocol are shown in Figure 9.

Figure 9.

Equilibrium flow curves of fluid mud from the Emden outer harbour, Germany, after [36]. Dotted lines: Extrapolation to zero shear rate.

4.3.2 Yield stress

In transitioning from soft soil condition to flow, two yield conditions are distinguished in fluid mud research. One is at the first sign of yielding, when the viscosity strongly declines, and a second condition is originally termed as ‘maximum fluidization’, Wurpts and Torn (2005) [35]. Similar is applied by [36, 40]. The yield stress at maximum fluidisation is determined by drawing a line through the flow curve beyond 10 [1/s] and extrapolating it towards the vertical axis. Since the data in Figure 9 represents equilibrium conditions, the intercept of this fitted tangent line with the vertical axis is however close to the static yield stress. A similar yield stress, but probably a bit higher, is found by Shakeel (2022) [41] via viscosity plots of up-ramped CSS and calls it fluidic yield stress (probably very close to ‘maximum fluidisation’). Yield stress data of Emden mud, collected from diverse sources, are plotted against their corresponding solid concentrations in Figure 10a. As can be seen, the collected data forms two data groups. One group organises itself around a lower diagonal (first sign of yielding, pseudo or slip-yield stress, [1]), and the other groups around an upper nearly parallel line. A factor 4 difference in yield stress is found between these two data groups.

Figure 10.

a) Lower and upper yield stresses of Emden outer harbour samples collected. b) Measured vertical SSC profile and calculated shear and yield stresses.

The data of [35] extends to higher solids concentrations. There is also some supportive data in an earlier paper by [34] on maximum fluidisation, where a profiled bob element was used, and the simpler rotoviscometer could only be operated in CSR mode. This data lines up a bit above extrapolated upper yield stress data.

4.4 Application of longitudinal mud profile theory

To ensure mechanical stability, the yield stress of the mud should at least exceed the local shear stress. Suspended solids concentrations (SSCs) in the Emden Outer harbour are in the range of 105 g/l to 400 g/l [37]. Without counter force by the muds’ yield stress, the material may flow out of the harbour basin. The mean excess density of the fluid mud is 123 kg/m³, which is calculated on basis of a measured SSC profile, a water density of 1015 kg/m³ (=average of low tide and high tide) and a solids density of 2636 kg/m³ ([42] Dollard mud). Referring to Figure 7, a bottom yield stress of 10 Pa suffices to keep the circulated fluid mud at its position respecting the shape of longitudinal profile.

We though also have to evaluate if stability conditions along the entire depth are satisfied. Figure 10b shows that the lower yield stress values, which we earlier showed to originate from rheometric wall slip, are not compatible with the muds’s conditions in the upper half of the profile. The fluid mud in Emden Outer Harbour would not be able to stay in place and have to run off to the fairway of the river Ems according to wall slip affected rheometry data, which is not the case.

5. Evaluation and takeaways

5.1 Application of rheological characterisation

In using rheology in calculation methods, the entire shear stress-shear rate relation needs to be carefully quantified. Primarily, the remoulded/equilibrium flow curve is needed. Only in exceptional cases where the material is set in motion, un-remoulded conditions, as obtained with CSS ramp-up applied for Hamburg and Emden mud earlier, apply. So, up- and down-ramping (CSS or CSR) is necessary to capture the remoulded/equilibrium conditions. Depending on the application, we need the rheology as a function of fine and coarse constituents.

There exists the risk that wall slip data is mistaken for actual rheology. The rheometric wall slip branch appears akin to viscosity regularisation. Wall slip is, though, situated at relatively higher shear rates, and it does not extend down to a shear stress of zero. In natural fluid muds, this branch appears to be taken for real in some approaches. In CFD modelling of the Elbe-Dollard area [43], the rheological fit (Worrall Tuliani [44]) is laid through this branch, and upon implementation in a CFD code, the viscosity measured at yielding is applied down to zero shear stress. At low shear rates relevant to the fluid muds, the shear stresses are hence underestimated (as a consequence the material flows too fast for acting shear stresses). A similar approach is proposed by Shakeel [41], where the wall slip branch is followed with a fit extending down to the origin. However, in application to field conditions, the first step in yielding (or commencement of wall slip) seems ultimately be skipped [45].

For CFD modelling, we need to have some viscosity (regularisation) at low shear rates. It is advised to have high viscosities here and at least follow the vane curve, but not the plate-plate, and certainly not the bob-cup concentric Couette results.

Implementations where particularly vane-type remoulded (Figure 4) branches are followed (fitted by Bingham & viscosity regularisation) are the TUDflow3D model in application to Water Injection Dredging of fluid mud [46, 47] for a schematised keel of vessels sailing through fluid mud. The authors were involved in associated rheometry. Bingham type of models [13], with added viscosity regularisation, are employed in the modelling of tailings deposition in the Delft3Ds model [48], and rheology is augmented by coarse solids. This model cannot be fitted to rheometric wall slip branches, and we never considered doing such.

5.2 Element usage

Given the complexity of mixtures, it is no surprise that manufacturers of rheometers offer a plurality of geometrical shapes for measuring elements. The search for good methods to circumvent problems arising from wall slip can be traced back for decades. Industries handling mixtures with high concentrations of solids often apply vane techniques. The vane-in-cup method and vane-in-bucket method have been developed at the end of the previous century and are since then applied in several studies, e.g. in oil sands [49]. We have found that the vane also reveals wall slip in a material of which one would not expect it at first sight. It is therefore that we advocate the usage of vane for non-Newtonian slurries with high concentrations of solids and recommend bob-cup in the absence of solids, but with a check for wall slip by utilising vane and act according to findings.

Concentric cylinder testing has the advantage that if settling occurs, it does not immediately jeopardise the measurement, contrary to horizontally orientated elements. The Couette inversion problem in concentric cylinder testing is mostly circumvented by applying thin gaps. This is, however, not always possible with solids-laden coarse mixtures and/or when flocs and/or aggregated structure are significantly large. Jamming and other confinement effects may occur, and the mixtures are also prone to wall slippage. In the case of grouts and mortars, dedicated testing devices have been built [50]. For solids-laden settling mixtures, vane testing/verification should be highly prioritised, without the need to revert to dedicated built devices.

Fluid muds are best first tested by concentric cylinder elements because of their good definition of shear rates and ability to accommodate small amounts of sand. Next is to verify for wall slip by vane testing. If samples contain too much sand, it is advised to test by vane and consider applying a transformation method depending on the size of the container. It should also be noted that rotoviscometers may output rotational velocity [rad/s] for vane instead of the shear rate [1/s] as is the case for other elements. Outputted viscosity is then based on rotational velocity.

The present tests reveal significant differences at low shear rates and coincidence at shear rates beyond wall slip. If segregation, i.e. the settling of sand, is essential to the flow problem, it is advised to test with sand (for a short duration, no extensive testing because sand settles under shearing), but also to test the material without sand in a concentric cylinder or similar. In the case of settling sand, the carrier fluid rheology determines the sand settling rate, and the mixture rheology determines the fluid flow. The same considerations regarding element usage apply if excessive wall slip may be expected, like with flocculated materials [49].

6. Conclusion

Wall slip of particulate systems is well known and is rather a rule than an exception. Here, we are encountering it also in a clay-only system, like natural fluid mud (with a sporadic settling solid, if any). The aggregate structure might similarly be a bit different at the wall and apparently also slips at low shear rates.

With complex mixtures, it is often not possible to apply the ideal measuring geometry: wider gaps are necessary and different (surface) geometries. In that case, direct outcomes of rotoviscometer need to be post-processed for rheological characterisation and/or vane testing is necessary complementary to testing with elements with better-defined shear rates.

Exploring in detail the low shear behaviour of yield shear thinning fluids by plotting the results as viscosity graphs accentuates measuring difficulties; care should be exercised not to interpret as (bulk) rheology.

It is unknown what is causing the wall slip of fluid muds, but whatever the cause, the first sign of yielding is not the static yield stress for application in rheological calculations.

Since wall slip of mixtures may constitute an efficient way of transport (pumping of fresh mortar is based on that), it would be worth systematically investigating if/how wall slip in rotoviscometers compares with wall slip in capillary viscometers, at least in the absence of coarse solids.

Acknowledgments

Thanks to Port of Rotterdam, Hamburg Port Authority and TKI for financing measurements such as described in Section 3, Pavan Goda MSc for measurements on Beerkanaal mud and Dr. Dennis Oberrecht of NLKWN Niedersachsen, for kindly providing the Batchelor reports of Franz and Oberrecht containing original rheometric data.