**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.

**Keywords:** Rheometry, wall slip, viscometry, particulate slurries, clay aggregates, non-Newtonian, fluid mud, lutocline

### **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

#### *Advances in Slurry Technology*

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 densificationrelated 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

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

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:


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**:

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

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

#### **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.*

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].

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 semianalytical 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:


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).

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

#### **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].
