**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.

#### **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.*

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**.

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

**Figure 8.** *Definition sketch of mechanical equilibrium of cohesive material on a horizontal base.*

linear and equal to *p gz* = ∆ρ , where *z* is the vertical distance under the mud surface profile and Δρ is the density difference with supernatant water.

The resulting horizontal driving force follows from the integration of the pressure profiles as: *F gh h <sup>p</sup>* =∆ ∆ ρ . The friction force at the bottom is τ = ∆ τ *F x <sup>y</sup>* Mechanical equilibrium gives τ ρ = ∆ <sup>d</sup> d *<sup>y</sup> <sup>h</sup> gh x* . Integration from the toe (x = 0) of the mud profile gives a square-root (sqrt) profile for layer thickness τ ρ <sup>=</sup> <sup>∆</sup> <sup>2</sup> *<sup>y</sup> h x g* .

### **4.3 Emden outer harbour mud rheology**

#### *4.3.1 Flow curve*

Rheology of fluid mud from the Emden Outer Harbour is reported in [35–37]. 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**.

### *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

#### **Figure 9.**

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

#### **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.*

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.

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 *Rheology, Rheometry and Wall Slip DOI: http://dx.doi.org/10.5772/intechopen.108048*

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].
