1 Introduction

Primary cementing is the process by which oil and gas wells are sealed during construction. The seal is achieved by placing cement into a narrow gap formed between the drilled borehole and the outside of a steel casing (or liner) that is placed in the well. The cement not only seals the well hydraulically, preventing fluids from migrating axially along the wellbore between fluid-bearing zones, but also provides mechanical support, resisting geo-mechanical stresses. Failure to achieve proper zonal isolation can have significant effects on well productivity, as well as potential environmental and health consequences [1].

Fig. 1
figure 1

Schematic of the well and of the variables that describe the eccentric annular geometry

Full size image

The process of placing the cement slurry into the narrow annular gap gives rise to a wide range of challenging fluid displacement problems, the understanding of which affects our ability to design wells of high integrity. This paper studies one class of flows commonly found, i.e., turbulent flows.

The primary cementing process proceeds as follows; see Fig. 1 (top). A new section of the well is drilled. The drillpipe is removed from the wellbore, leaving drilling mud inside the wellbore. A steel tube (casing or liner) is inserted into the wellbore, typically leaving an annular gap of \(\approx \) 2 cm. The tubing is inserted in sections of length \(\approx \) 10 m each, threaded together so that cemented sections can extend 100–1000 m. The so-called centralizers are fitted to the outside of the tube, to prevent the heavy steel tubing from slumping to the lower side of the wellbore. However, even in (nominally) vertical wells it is common that the annulus is eccentric and this is especially true in inclined and horizontal wells. With the steel casing in place and drilling mud on the inside and outside, the operation begins. First, the drilling mud is conditioned by circulating around the flow path. Next a sequence of fluids are circulated down the inside of the casing and returning up the outside of the annulus. Preflushes (washes or spacer fluids) are followed by one or more cement slurries. The fluid volumes are designed so that the cement slurries fill the annular space to be cemented. Drilling mud follows the final cement slurry to be pumped and the operations end with the cement slurry held in the annulus (with a valve system) and allowed to hydrate (i.e. set) over a period of many hours.

With reference to Fig. 1 (top), it can be seen that the completed well often has a telescopic arrangement of casings and liners.Footnote 1 Thus, the operation is repeated more than once on most wells. Typical, well inner diameters can start at anything up to 50 cm and can end as small as 10 cm in the producing zone. As the well proceeds deeper, pressures in fluid-bearing zones of the rock formation increase and must be balanced by the wellbore pressures in order to maintain primary well control. In deep or long wells the frictional component of the pressure can become significant. The need to control frictional pressures and these natural changes in annular geometry mean that the flow regimes experienced when cementing the surface casing (at the top) may be quite different from those found when cementing the production casing. As an example, in a long horizontal well drilled through a weak formation, one is likely to pump the fluids in laminar flow. Where adequate pump capacity is present and where there are no risks of either an influx nor fracturing the well, turbulent flow might be preferred.

This brings us to one of the main operational questions, which we will not succeed in answering here: is it preferable to cement a well in turbulent or laminar flow? There is a perception that turbulent regimes should be used where possible; see [1]. However, the actual evidence for this appears scant. For example, [2] is an often-cited study which does show improved displacement efficiencies. However, this study was performed in an era before our understanding of laminar displacements that evolved.

Laminar displacements allow fluid rheology and density differences to become part of the flow design. From the 1970s to early 1990s, a number of rule-based systems evolved to guide laminar mud displacement, e.g. [3,4,5]. For the past 10–20 years the industry has been able to access model-based simulators [6, 7] and analyses [8, 9] that have been shown to improve on the rule-based systems [10]. Such simulators are now actively used in cementing case studies, where they compare favourably with post-placement logging of the wells, e.g. [11,12,13,14]. Thus, early studies such as [2] have conclusions that may not be valid when instead comparing with modern designs for laminar displacement flows. Often operational or geological circumstances dictate the flow regime, so that it is rarely feasible in a field setting to cement one well in laminar flow and an identical adjacent well in turbulent flow. Herein lies the value of model-based simulations and laboratory experiments, where one can isolate individual effects. This paper derives the type of leading-order model tools that can be used to make such comparisons, over the scale of the well.

Our motivation for studying primary cementing stems from the important consequences of operational failure. Leakage during the primary cementing operation can lead to gas pockets and channels, that compromise the well. A number of other defects may arise either during the cementing of a well, or afterwards during cement hydration, that allow the well to leak later. The most common defects include: (a) residual mud channelling; (b) wet micro-annulus; (c) mixing/contamination of the slurry. Examples of these features can be found in [15]. Residual mud channelling is where the yield stress of the mud holds it in place (typically) on the narrow side of the annulus as preflushes and cement slurry by-pass. It is a bulk flow feature predicted well by simple mechanical arguments and more sophisticated models; see e.g. [6, 16]. Wet micro-annulus is a local mechanical effect, where the displacing fluid does not generate sufficient shear stress to mobilize the mud at the wall. To some extent, this is predictable in model flows, e.g. [17,18,19,20], but can be further complicated with mud dehydration and uneven wellbores. Mixing (and consequent contamination) occur in different scenarios: downward displacement within the casing (e.g. [21, 22]); fluid instabilities in laminar annular flows (e.g. [10, 23, 24]); turbulent annular displacement flows. In combination with mud channels, micro-annuli or even mud pockets left behind in irregular wellbores (e.g. washouts), residual drilling fluid can be partially eroded/dispersed in a passing slurry and continue to contaminate cement over long lengths.

While the above mechanisms are flow related and reflect our own interest, not all wellbore leakage has a fluid-mechanical cause. A good overview is given in [1], e.g. geomechanical stresses/deformation and casing de-bonding (=dry micro-annulus). The net summary is that many wells leak and this is common worldwide, although there is wide variation in cement evaluation techniques, regulatory practice, collection of (and access to) well data, and definition of leakage severity. For example, Dusseault et al. [25] state that over 550,000 wells have been drilled in Canada and collates survey data that show between 5–20% of wells with some form of leakage. The interested reader is also referred to [26, 27], which analyze different leakage factors for significant datasets.

This paper concentrates on model development. Although we target turbulent flows, we also incorporate previous models for laminar flow regimes, e.g. [6], so that mixed flow regimes can be modelled. Flow regime depends on local geometry and fluid properties as well as the overall imposed flow rate. It is relatively common within the annulus that one fluid can be fully turbulent (e.g. a chemical wash or low-viscous spacer) while others are laminar. Indeed, this also can occur on a single section of the annulus, e.g. turbulent on the wide side, laminar or even static on the narrow side, (see later and [1]). Furthermore, although some fluids can be strongly turbulent, the more viscous fluids (muds, viscous spacers and slurries) are often only in weakly turbulent, transitional or laminar regimes. These flow regimes have become more prevalent in recent decades as extended and horizontal wells, require reduced flow rates to control friction pressures.

The organization of the paper is as follows: In Sect. 2, we derive an elliptic equation that models the stream function, for a given concentration field. Section 3 develops an asymptotic analysis to find the equation governing the leading-order evolution of the fluid concentrations. The model is summarized in Sect. 4, where we present examples of displacement flows with the increasing levels of complexity. The paper is closed in Sect. 5 with conclusions. Two appendices present, firstly, an overview of turbulent closure models used, and secondly, variational analysis that is preliminary to qualitative study, e.g. existence and uniqueness of weak solutions.

2 Modelling turbulent displacement flows

As in [6], a cylindrical coordinate system \(({\hat{r}},\theta ,\hat{\xi })\) is used to describe the well geometry: \(\hat{\xi }\) measures distance along the central axis of the casing \({\hat{r}} = 0\) which is assumed to be inclined to the vertical with angle \(\beta (\hat{\xi })\). The local cross section of the well, outside the casing, is assumed to be that of an eccentric annulus, with inner radius \({\hat{r}}_i(\hat{\xi })\), equal to the outer radius of the casing and outer radius \({\hat{r}}_o(\hat{\xi })\) equal to the inner radius of the hole (or previous casing). At each depth \(\hat{\xi }\), the mean radius \({\hat{r}}_a(\hat{\xi })\) and the mean half-gap width \({\hat{d}}(\hat{\xi })\) are defined by

$$\begin{aligned} {\hat{r}}_a(\hat{\xi }) \equiv \frac{1}{2} \left[ {\hat{r}}_o(\hat{\xi }) + {\hat{r}}_i(\hat{\xi })\right] , \quad {\hat{d}}(\hat{\xi }) \equiv \frac{1}{2} \left[ {\hat{r}}_o(\hat{\xi }) - {\hat{r}}_i(\hat{\xi })\right] . \end{aligned}$$
(1)

Inner and outer radii, and the distance, \({\hat{e}}(\hat{\xi })\), between the two centres of the cylinders are given, see Fig. 1 (lower). It is assumed that \({\hat{e}}(\hat{\xi }) < 2{\hat{d}}(\hat{\xi })\) (the cylinders do not touch), and that variations in the geometry with \(\hat{\xi }\) are slow. For simplicity, it is also assumed that the narrow side of the annulus will be found on the lower side of the well and that the casing remains stationary. Both these assumptions can be relaxed with care, e.g. [28, 29].

The annular displacement is modelled as a concentration dependent multi-fluid flow that is fully turbulent and incompressible. The fluid constituents contribute to the mixture density \(\hat{\rho }\) and to the rheological properties of the mixture. The latter are used in closure expressions that define the deviatoric stress tensor \(\hat{\tau }_{ij}\). In primary cementing, a sequence of K fluids is pumped around the flow path, from the bottom of the annulus to the top. A typical sequence would be mud, wash, spacer, lead slurry, tail slurry and mud. When circulating, drilling muds, spacer fluids and cement slurries are predominantly shear thinning, nonlinearly viscous, and inelastic, often also with a significant yield stress. Washes are Newtonian fluids. In general, each constituent fluid can be effectively modelled as a Herschel-Bulkley fluid. We shall address evolution of the mixture concentrations at length in Sect. 3; however, for the present, we assume that all fluid properties are approximated effectively by closures that depend on the local mean fluid concentrations: both ensemble averaged and averaged across the annular gap.

We adopt the usual Reynolds decomposition for turbulent flows, into mean and fluctuating parts. Because the flows will be time-varying as the displacement proceeds, the mean part (denoted with an overbar below) is interpreted as an ensemble average. The velocity and pressure are \(\hat{\mathbf { u}} = ({\hat{u}},{\hat{v}},{\hat{w}})\) and \({\hat{p}}\), respectively. In the local coordinate system, the Reynolds-decomposed Navier–Stokes system is

$$\begin{aligned} \frac{\partial }{\partial {\hat{t}}} \left[ \hat{\rho }\bar{{\hat{u}}}\right] + \hat{\mathbf { \nabla }}\cdot \left[ \hat{\rho }\bar{{\hat{u}}} \bar{\hat{\mathbf {u}}} \right]= & {} \frac{1}{{\hat{r}}} \frac{\partial }{\partial {\hat{r}}} \left[ {\hat{r}}\bar{\hat{\tau }}_{{\hat{r}}{\hat{r}}}\right] + \frac{1}{{\hat{r}}} \frac{\partial }{\partial \theta } \bar{\hat{\tau }}_{{\hat{r}}\theta } + \frac{\partial }{\partial \hat{\xi }}\hat{\tau }_{{\hat{r}}\hat{\xi }} - \frac{\bar{\hat{\tau }}_{\theta \theta }}{{\hat{r}}} - \frac{\partial \bar{{\hat{p}}}}{\partial {\hat{r}}} + \hat{\rho } {\hat{g}}_{{\hat{r}}} \nonumber \\&+\frac{1}{{\hat{r}}} \frac{\partial }{\partial {\hat{r}}} \left[ {\hat{r}}\hat{\tau }^t_{{\hat{r}}{\hat{r}}}\right] + \frac{1}{{\hat{r}}} \frac{\partial }{\partial \theta } \hat{\tau }^t_{{\hat{r}}\theta } + \frac{\partial }{\partial \hat{\xi }}\hat{\tau }^t_{{\hat{r}}\hat{\xi }} - \frac{\hat{\tau }^t_{\theta \theta }}{{\hat{r}}}, \end{aligned}$$
(2)
$$\begin{aligned} \frac{\partial }{\partial {\hat{t}}} \left[ \hat{\rho }\bar{{\hat{v}}}\right] + \hat{\mathbf { \nabla }}\cdot \left[ \hat{\rho }\bar{{\hat{v}}} \bar{\hat{\mathbf {u}}} \right]= & {} \frac{1}{{\hat{r}}^2} \frac{\partial }{\partial {\hat{r}}} \left[ {\hat{r}}^2 \bar{\hat{\tau }}_{\theta {\hat{r}}}\right] + \frac{1}{{\hat{r}}} \frac{\partial }{\partial \theta } \bar{\hat{\tau }}_{\theta \theta } + \frac{\partial }{\partial \hat{\xi }}\bar{\hat{\tau }}_{\theta \hat{\xi }} - \frac{1}{{\hat{r}}} \frac{\partial \bar{{\hat{p}}}}{\partial \theta } + \hat{\rho } {\hat{g}}_{\theta } \nonumber \\&+\frac{1}{{\hat{r}}^2} \frac{\partial }{\partial {\hat{r}}} \left[ {\hat{r}}^2 \hat{\tau }^t_{\theta {\hat{r}}}\right] + \frac{1}{{\hat{r}}} \frac{\partial }{\partial \theta } \hat{\tau }^t_{\theta \theta } + \frac{\partial }{\partial \hat{\xi }}\hat{\tau }^t_{\theta \hat{\xi }} \end{aligned}$$
(3)
$$\begin{aligned} \frac{\partial }{\partial {\hat{t}}} \left[ \hat{\rho }\bar{{\hat{w}}}\right] + \hat{\mathbf { \nabla }}\cdot \left[ \hat{\rho }\bar{{\hat{w}}} \bar{\hat{\mathbf {u}}} \right]= & {} \frac{1}{{\hat{r}}} \frac{\partial }{\partial {\hat{r}}} \left[ {\hat{r}}\bar{\hat{\tau }}_{\hat{\xi }{\hat{r}}}\right] + \frac{1}{{\hat{r}}} \frac{\partial }{\partial \theta } \bar{\hat{\tau }}_{\hat{\xi }\theta } + \frac{\partial }{\partial \hat{\xi }}\bar{\hat{\tau }}_{\hat{\xi }\hat{\xi }} - \frac{\partial \bar{{\hat{p}}}}{\partial \hat{\xi }} + \hat{\rho } {\hat{g}}_{\hat{\xi }} \nonumber \\&+\frac{1}{{\hat{r}}} \frac{\partial }{\partial {\hat{r}}} \left[ {\hat{r}}\hat{\tau }^t_{\hat{\xi }{\hat{r}}}\right] + \frac{1}{{\hat{r}}} \frac{\partial }{\partial \theta } \hat{\tau }^t_{\hat{\xi }\theta } + \frac{\partial }{\partial \hat{\xi }}\hat{\tau }^t_{\hat{\xi }\hat{\xi }} , \end{aligned}$$
(4)
$$\begin{aligned} 0= & {} \frac{1}{{\hat{r}}} \frac{\partial }{\partial {\hat{r}}} \left[ {\hat{r}} \bar{{\hat{u}}}\right] + \frac{1}{{\hat{r}}} \frac{\partial \bar{{\hat{v}}}}{\partial \theta } + \frac{\partial \bar{{\hat{w}}}}{\partial \hat{\xi }} . \end{aligned}$$
(5)

The components \(\bar{\hat{\tau }}_{ij}\) are those that come from ensemble averages of the viscous stress tensor (which itself is nonlinear). The components \(\hat{\tau }^t_{ij}\) are the turbulent (Reynolds) stresses, resulting from the fluctuating components of the velocity field \(\hat{\mathbf {u}}'\), defined as

$$\begin{aligned} \hat{\tau }^t_{ij} = -\hat{\rho } \overline{{\hat{u}}'_i {\hat{u}}'_j}. \end{aligned}$$
(6)

The gravitational acceleration, \(\hat{\mathbf { g}} = ({\hat{g}}_{{\hat{r}}},{\hat{g}}_{\theta },{\hat{g}}_{\hat{\xi }})\), is given by

$$\begin{aligned} {\hat{g}}_{{\hat{r}}} = -{\hat{g}} \sin \beta (\hat{\xi }) \cos \theta , \quad {\hat{g}}_{\theta } = {\hat{g}} \sin \beta (\hat{\xi }) \sin \theta , \quad {\hat{g}}_{\hat{\xi }} = -{\hat{g}} \cos \beta (\hat{\xi }) , \end{aligned}$$
(7)

where \({\hat{g}} = 9.81\,\hbox {m/s}^2\).

2.1 Scaling and simplification

We wish to reduce our model to something more tractable than (2)–(5), by exploiting the aspect ratio of the annulus. The annulus geometry is typically long and narrow, with the typical gap half-width (\(\sim \)1 cm) being much smaller than a typical azimuthal distance (\(\sim \)30 cm), which in turn is much smaller than a typical length of the annulus (\(\sim \)500 m).

Following [6], let the mean radius (\({\hat{r}}_{a,0}\)), the local and global aspect ratios \((\delta (\hat{\xi })\) and \(\delta _0\) respectively) be defined by

$$\begin{aligned} {\hat{r}}_{a,0} = \frac{1}{{\hat{Z}}} \int _{\hat{\xi }_{bh}}^{\hat{\xi }_{tz}} {\hat{r}}_{a}(\hat{\xi })~\mathrm {d}\hat{\xi }, \quad \delta (\hat{\xi }) = \frac{{\hat{d}}(\hat{\xi })}{{\hat{r}}_{a}(\hat{\xi })}, \quad \delta _{0} = \frac{1}{{\hat{Z}}} \int _{\hat{\xi }_{bh}}^{\hat{\xi }_{tz}}\delta (\hat{\xi })~\mathrm {d}\hat{\xi }. \end{aligned}$$
(8)

where \({\hat{Z}}\) is the length of the zone of the well to be cemented, extending upwards from bottom hole, \(\hat{\xi }_{bh}\), to the top of the zone, \(\hat{\xi }_{tz}\). Scaled axial and azimuthal coordinates \(\xi \) and \(\phi \) are then

$$\begin{aligned} \xi = \frac{\hat{\xi } - \hat{\xi }_{bh}}{\pi {\hat{r}}_{a,0}}, \quad \phi = \frac{\theta }{\pi }. \end{aligned}$$
(9)

In each cross section, we define the local average radius, \(r=r_a(\xi )\), and local annulus eccentricity, \(e(\xi )\), by

$$\begin{aligned} r_a(\xi ) = \frac{{\hat{r}}_{a}(\hat{\xi })}{ {\hat{r}}_{a,0} }, \quad e(\xi ) = \frac{{\hat{e}}(\hat{\xi })}{2{\hat{d}}(\hat{\xi })}. \end{aligned}$$
(10)

The centreline of the annular gap is at \({\hat{r}} = {\hat{r}}_{a,0} r_a(\xi ) r_c(\phi ,\xi )\). The radial coordinate is scaled relative to the distance from the centreline of the annulus, as follows:

$$\begin{aligned} y = \frac{{\hat{r}} - {\hat{r}}_{a,0} r_a(\xi )r_c(\phi ,\xi )}{{\hat{r}}_{a,0} \delta _0}, \end{aligned}$$
(11)

i.e. y is a local annular gap coordinate.

We assume a narrow annulus approximation: \(\delta (\xi ) \sim \delta _0 \ll \pi \); noting that \(\delta _0/\pi \), denotes the ratio of radial (gap) length-scales to azimuthal length-scale. To leading order in \(\delta _0/\pi \), we have \(r_c(\phi ,\xi ) \sim \)1 and find that the inner and outer walls are at \(y = \mp H(\phi ,\xi )\), where

$$\begin{aligned} H(\phi ,\xi ) = \frac{\delta (\xi )r_a(\xi )[1 + e(\xi )\cos \pi \phi ]}{\delta _0} . \end{aligned}$$
(12)

More complex geometries are readily accommodated by specifying any \(H(\phi ,\xi )\) of O(1) that varies slowly with \(\xi \), e.g. \(H(\phi -\phi _0(\xi ),\xi )\) with H as above, retains the eccentric annular shape but shifts the wide side of the annulus to \(\phi = \phi _0(\xi )\). Helically varying well eccentricity, elliptic cross sections and irregular washouts are each fairly common deviations away from (12).

Time is scaled with an advective timescale: \(\pi {\hat{r}}_{a,0}/\hat{\bar{W}}\) where \(\hat{\bar{W}}\) is representative of a mean axial velocity. For simplicity, we assume that the fluids are pumped following a schedule of pump rates: \({\hat{Q}}_\mathrm{pump}({\hat{t}})\), typically a step function. The pump schedule is used to define a representative flow rate \({\hat{Q}}_0\), e.g. the maximum flow rate. The mean axial velocity is \(\hat{\bar{W}} = {\hat{Q}}_0/{\hat{A}}_{0}\), where \({\hat{A}}_{0} = 4 \pi \delta _{0} [{\hat{r}}_{a,0}]^2\) is a typical cross-sectional area of the annulus.

We now consider the relative sizes of the different terms in (2)–(5), with a view to simplification. Given that \(\delta _{0} /\pi \ll 1\), we may assume that the dominant components of mean velocity will be in the \((\phi ,\xi )\)-directions, scaling approximately with \(\hat{\bar{W}}\). The incompressibility condition (5) suggests that the radial component of mean velocity scales with \(\delta _{0} \hat{\bar{W}}/\pi \). Therefore, we can see that the acceleration and inertial terms on the left-hand side of (2), (3) and (4) have respective sizes:

$$\begin{aligned} \frac{\delta _{0}}{\pi }\frac{\hat{\rho } \hat{\bar{W}}^2}{\pi {\hat{r}}_{a,0}},~~~~~~ \frac{\hat{\rho } \hat{\bar{W}}^2}{\pi {\hat{r}}_{a,0}},~~~~~~\frac{\hat{\rho } \hat{\bar{W}}^2}{\pi {\hat{r}}_{a,0}} . \end{aligned}$$

Simplifying the stress terms on the right-hand side of (2), (3) and (4) is less straightforward. A number of authors have computed turbulent flows of Newtonian fluids in uniform eccentric annuli and calculated the Reynolds stresses, e.g. [30,31,32,33]. Here insofar as we are concerned, the main point is that each component of \(\hat{\tau }^t_{ij}\) has similar magnitude, because the fluctuating velocity is inherently three-dimensional. In considering the viscous stresses, we need to consider two regions separately: the turbulent core of the annular flow and the wall region. In the turbulent core, the mean viscous stresses \(\bar{\hat{\tau }}_{ij}\), are the average of the viscous stress and not the viscous stress of the averaged strain rate. Although the strain rate associated with the mean velocity can be estimated using scaling arguments, the fluctuating velocity also contributes to each component of the strain rate and indeed this contribution may be dominant in the core. Nonlinearity of the constitutive relations together with the probabilistic nature of the strain rate tensor prevents any easy simplification. A nice discussion may be found in [34, Chaps. 4 and 5]. All that we assume here is that the viscous stress components in the core all have similar magnitude, say \(\bar{\hat{\tau }}_{ij} \sim \hat{\tau }_{\nu ,0}\) for some viscous stress scale \(\hat{\tau }_{\nu ,0}\).

Conventionally, the Reynolds stresses in the turbulent core have similar size to the wall shear stresses (which will vary azimuthally in an eccentric annulus). In the wall region, the Reynolds stresses vanish, and the viscous stresses increase over a thin layer to match the wall shear stress. In this wall region, it is possible to estimate relative sizes of the different shear rates and approximate the flow. Thus, both turbulent and viscous stresses have roles to play in describing these flows. This is particularly true in cementing flows which are generally not highly turbulent, with the exception of low viscosity washes. For the present, we adopt a stress scale \(\hat{\tau }_0\) that we apply to mean turbulent and viscous stresses, and discuss the order of magnitudes of dimensionless terms in different flow regimes later.

Next we consider the gravitational terms. It is common to exploit density differences in creating buoyancy forces to aid in displacing the in-situ drilling mud. To capture this aspect, we scale all densities with the density \(\hat{\rho }_1\) of the first fluid in the pumping sequence, i.e. the in-situ drilling mud, and subtract the hydrostatic pressure from \(\bar{{\hat{p}}}\). It is assumed that the pressure remaining balances the dominant stress gradients: \(\bar{{\hat{p}}} = \bar{{\hat{p}}}_{bh}({\hat{t}}) + \hat{\rho }_1 \mathbf {g}\cdot \mathbf {x} + (\pi \hat{\tau }_0 / \delta _0 ) \bar{p}\), where \(\bar{{\hat{p}}}_{bh}({\hat{t}})\) denotes the bottom-hole pressure.

We now substitute the above variables into (2), (3) and (4) and divide through by the largest dimensional scales, to give the following.

$$\begin{aligned} \underbrace{O\left( \frac{\delta _0^3}{\pi ^3} \frac{\hat{\rho } \hat{\bar{W}}^2 }{ \hat{\tau }_0} \right) }_{IT}= & {} - \frac{\partial \bar{p}}{\partial y} + \underbrace{O\left( \frac{\delta _0}{\pi } \frac{\rho -1}{Fr^2} \right) }_{BT} + \underbrace{O\left( \frac{\delta _0}{\pi } \right) }_{ST} + \underbrace{O\left( \frac{\delta _0}{\pi } \right) }_{CT} , \end{aligned}$$
(13)
$$\begin{aligned} \underbrace{O\left( \frac{\delta _0}{\pi } \frac{\hat{\rho } \hat{\bar{W}}^2 }{ \hat{\tau }_0} \right) }_{IT}= & {} - \frac{1}{r_a}\frac{\partial \bar{p}}{\partial \phi } + \frac{(\rho -1) \sin \beta \sin \pi \phi }{Fr^2} + \frac{\partial }{\partial y} [\tau ^t_{\phi y}+\bar{\tau }_{\phi y} ] + \underbrace{O\left( \frac{\delta _0}{\pi } \right) }_{ST} + \underbrace{O\left( \frac{\delta _0}{\pi } \right) }_{CT}, \end{aligned}$$
(14)
$$\begin{aligned} \underbrace{O\left( \frac{\delta _0}{\pi } \frac{\hat{\rho } \hat{\bar{W}}^2 }{ \hat{\tau }_0} \right) }_{IT}= & {} - \frac{\partial \bar{p}}{\partial \xi } - \frac{(\rho -1) \cos \beta }{Fr^2} + \frac{\partial }{\partial y} [\tau ^t_{\xi y} + \bar{\tau }_{\xi y}], + \underbrace{O\left( \frac{\delta _0}{\pi } \right) }_{ST} + \underbrace{O\left( \frac{\delta _0}{\pi } \right) }_{CT}, \end{aligned}$$
(15)
$$\begin{aligned} 0= & {} \frac{\partial \bar{u}}{\partial y} + \frac{1}{r_a}\frac{\partial \bar{v}}{\partial \phi } + \frac{\partial \bar{w}}{\partial \xi } + \underbrace{O\left( \frac{\delta _0}{\pi } \right) }_{CT}, \end{aligned}$$
(16)

where \(\rho \) is the scaled density and Fr is the Froude number:

$$\begin{aligned} Fr = \sqrt{\frac{\hat{\tau }_{0}}{ \hat{\rho }_1 {\hat{g}} \delta _0 {\hat{r}}_{a,0} }} . \end{aligned}$$

Our leading-order model is the narrow gap limit, \(\delta _0 /\pi \rightarrow 0\), with other parameters fixed. The different terms that will vanish in this limit are identified by the under-braces as follows: IT denote the leading-order inertial terms; BT denotes the next order buoyancy terms; ST denote the next order stress terms (turbulent and viscous); CT denote terms associated with curvature.

If the flow is fully turbulent, the stress scale \(\hat{\tau }_0\) is the wall shear stress, so the ratio \(\hat{\rho } \hat{\bar{W}}^2 / \hat{\tau }_0 \) that appears in the IT under-braces is effectively \(\sim 1/f_f\), for a representative turbulent friction factor \(f_f\). Although multiplied by \(\delta _0 /\pi \) and formally vanishing in the narrow gap limit, these are likely to be the next largest terms. In the case that the flow is laminar, the Reynolds stress terms vanish, and the expression \(\hat{\rho } \hat{\bar{W}}^2 / \hat{\tau }_0 \) becomes effectively the Reynolds number, since in this case \(\hat{\tau }_0\) is correctly interpreted as a viscous stress scale. Taking \(\delta _0 /\pi \rightarrow 0\) leads to the laminar displacement model of [6].

The curvature terms arise both due to replacing the r-derivativesmarizewith y-derivatives (changing 1 / r to \(1/r_a + O(\delta _0 /\pi )\)), and due to slow variations in \(\xi \) as the well trajectory changes (assumed of \(O(\delta _0 /\pi )\)). The next order stress terms are also only \(O(\delta _0 /\pi )\) smaller in the turbulent flow: from scaling, using the geometric aspect ratio, i.e. the partial derivatives are smaller in the \((\phi ,\xi )\)-plane than with respect to y. If the flow were laminar, then the Reynolds stresses would vanish and scaling arguments can be used to estimate the size of the viscous stresses; the next largest stresses appear at \(O((\delta _0 /\pi )^2)\).

To summarize, if we consider the next order terms we see that there is a proliferation of terms at order \(\delta _0 /\pi \): buoyancy terms, inertial, stress and curvature. The main point here is that to include the next order of terms in \(\delta _0 /\pi \) is prohibitively complex. On the other hand, considering the formal narrow gap limit, \(\delta _0 /\pi \rightarrow 0\), although the scaling arguments are different the leading-order equations are similar to those of [6].

2.2 Narrow gap approximation

Proceeding with the narrow gap approximation, we take \(\delta _0 /\pi \rightarrow 0\) in (13)–(16). To eliminate \(\bar{u}\), we integrate (16) across the gap width, using conditions of no-slip at the annulus walls:

$$\begin{aligned} \frac{\partial }{\partial \phi }[ 2H \bar{\bar{v}} ] + \frac{\partial }{\partial \xi } [ 2r_a H \bar{\bar{w}} ] = 0, \end{aligned}$$
(17)

where

$$\begin{aligned} \bar{\bar{v}}(\phi ,\xi ,t) = \frac{1}{2H} \int _{-H}^{ H } \bar{v} ~\mathrm {d} y, \quad \bar{\bar{w}}(\phi ,\xi ,t) = \frac{1}{2H} \int _{-H}^{ H } \bar{w} ~\mathrm {d} y. \end{aligned}$$
(18)

Equation (17) is satisfied using a stream function:

$$\begin{aligned} 2r_a H \bar{\bar{w}} = \frac{\partial \varPsi }{\partial \phi }, \quad 2H \bar{\bar{v}} = - \frac{\partial \varPsi }{\partial \xi }. \end{aligned}$$
(19)

For later convenience, we introduce 2D annular divergence and gradient operators as follows:

$$\begin{aligned} \mathbf {\nabla }_a \cdot \mathbf {q} = \frac{1}{r_a} \frac{\partial q_\phi }{\partial \phi }+\frac{\partial q_\xi }{\partial \xi }, ~~~~~~~~~~ \nabla _a q = \left( \frac{1}{r_a} \frac{\partial q}{\partial \phi },\frac{\partial q}{\partial \xi } \right) . \end{aligned}$$

Turning now to the momentum balance, from (13) we see the pressure is independent of y, as is \(\rho \) (see below in Sect. 3). Equations (14) and (15) may be integrated across the annular gap, assuming symmetry at \(y=0\) for this leading-order approximation:

$$\begin{aligned} \left( \tau ^t_{\phi y} + \bar{\tau }_{\phi y} , \tau ^t_{\xi y} + \bar{\tau }_{\xi y} \right)= & {} y \left( \frac{1}{r_a}\frac{\partial \bar{p}}{\partial \phi } - \frac{(\rho -1) \sin \beta \sin \pi \phi }{Fr^2}, \frac{\partial \bar{p}}{\partial \xi } + \frac{(\rho -1) \cos \beta }{Fr^2} \right) . \end{aligned}$$
(20)

The leading-order stresses are only non-zero in the direction of the modified pressure gradient. Viewed in the \((\phi ,\xi )\)-plane, this is a one-dimensional (1D) turbulent shear flow through a channel of width \(2H(\phi ,\xi )\), driven in the direction of

$$\begin{aligned} - \left( \frac{1}{r_a}\frac{\partial \bar{p}}{\partial \phi } - \frac{(\rho -1) \sin \beta \sin \pi \phi }{Fr^2}, \frac{\partial \bar{p}}{\partial \xi } + \frac{(\rho -1) \cos \beta }{Fr^2} \right) , \end{aligned}$$

which is therefore also the direction of the streamlines, say \(\mathbf {e}_s\):

$$\begin{aligned} \mathbf {e}_s = \frac{(\bar{\bar{v}},\bar{\bar{w}})}{\sqrt{\bar{\bar{v}}_0^2 + \bar{\bar{w}}_0^2}} = \frac{1}{|\nabla _a \varPsi |}\left( - \frac{\partial \varPsi }{\partial \xi },\frac{1}{r_a} \frac{\partial \varPsi }{\partial \phi } \right) . \end{aligned}$$
(21)

The integrated momentum balance (20) can now be resolved along the streamlines, in the \(\mathbf {e}_s\) direction:

$$\begin{aligned} \tau ^t_{s y} + \bar{\tau }_{s y} = -\frac{y}{H} \tau _w, \end{aligned}$$
(22)

where the dimensionless wall shear stress \(\tau _w\) is:

$$\begin{aligned} \tau _w = H \left| \left( \frac{1}{r_a}\frac{\partial \bar{p}}{\partial \phi } - \frac{(\rho -1) \sin \beta \sin \pi \phi }{Fr^2}, \frac{\partial \bar{p}}{\partial \xi } + \frac{(\rho -1) \cos \beta }{Fr^2} \right) \right| . \end{aligned}$$
(23)

In [35], we solved (22) to give an expression for the leading-order turbulent mean velocity for a Herschel–Bulkley fluid, based on the phenomenological approach of Metzner–Reed–Dodge [36,37,38]. For the purposes of our model, we simply note that [35] provides a closure expression \(\tau _w = \tau _w(|\nabla _a \varPsi |;\phi ,\xi ,t)\). The local dependency \((\phi ,\xi )\) explicitly reflects the local geometric variables, and the fluid rheology is represented implicitly with dependency \((\phi ,\xi ,t)\) as the fluids are displaced; see Appendix A.

Combining (21) and (23), we have

$$\begin{aligned} \frac{1}{|\nabla _a \varPsi |}\left( - \frac{\partial \varPsi }{\partial \xi },\frac{1}{r_a} \frac{\partial \varPsi }{\partial \phi } \right) = - \frac{H}{\tau _w} \left( \frac{1}{r_a}\frac{\partial \bar{p}}{\partial \phi } - \frac{(\rho -1) \sin \beta \sin \pi \phi }{Fr^2}, \frac{\partial \bar{p}}{\partial \xi } + \frac{(\rho -1) \cos \beta }{Fr^2} \right) , \end{aligned}$$
(24)

which can be rearranged and cross-differentiated to eliminate the pressure:

$$\begin{aligned} \mathbf {\nabla }_a \cdot [\mathbf {S} + \mathbf {b}] = 0 \end{aligned}$$
(25)

in which

$$\begin{aligned} \mathbf {S} = \frac{r_a \tau _w(| \nabla _a \varPsi |) }{H|\nabla _a \varPsi |} \nabla _a \varPsi ~~~\text {and}~~~\mathbf {b} = \frac{r_a \left( \rho -1 \right) }{Fr^2 } \left( \cos \beta , \sin \pi \phi \sin \beta \right) . \end{aligned}$$
(26)

The term \(\mathbf {\nabla }_a \cdot \mathbf {S}\) in (25) is a quasilinear elliptic operator on \(\varPsi \) and the term \(\mathbf {\nabla }_a \cdot \mathbf {b}\) provides a source term that is driven by the spatial gradients of the buoyancy vector \(\mathbf {b}\). Note that (25) contains no time derivatives: time enters only via (i) boundary data, e.g. if the flow rate changes; (ii) through the fluid concentrations, which affect both fluid rheology and buoyancy.

Note that we have still not fixed the shear stress scale \(\hat{\tau }_0\), and this is not particularly important as all terms in (25) are scaled with \(\hat{\tau }_0\). Although this paper has targeted turbulent flows in terms of the novel contribution to modelling of primary cementing, laminar flows are more prevalent. The scale \(\hat{\rho }_1 \hat{\bar{W}}^2\) also over-estimates the turbulent stresses (by using the mean velocity and not a friction velocity), and these in turn are larger than the typical laminar viscous stresses. In addition, the objective has been to develop a model capable of dealing with mixed flow regimes. Consequently, we define a \(\hat{\tau }_0\) that is relevant to the laminar viscous stresses:

$$\begin{aligned} \hat{\tau }_0 = \max _{k} \left\{ \hat{\tau }_{k,Y} + \hat{\kappa }_k \hat{\dot{\gamma }}_0^{n_k} \right\} ,~~~~~~ \hat{\dot{\gamma }}_0 = \frac{3 \hat{\bar{W}}}{\delta _0 {\hat{r}}_{a,0}}, \end{aligned}$$
(27)

with \(\hat{\tau }_{k,Y}, \hat{\kappa }_k\), and \(n_k\), respectively the yield stress, consistency and power-law index of fluid k in the pumped sequence. Using this scale we define the functions \(\tau _w(| \nabla _a \varPsi |)\) and \(\mathbf {S}\), by using the closure expressions in [35], as outlined in Appendix A.

2.2.1 Boundary conditions

The elliptic second order equation (25) determines the stream function, and hence gap-averaged velocity, at each time. It requires suitable boundary conditions in order to be solved. The annular domain has been reduced via the scaling to rectangular domain \(\varOmega \), representing the unwrapped gap-averaged annulus. At each timestep it is necessary to specify suitable boundary conditions on \(\partial \varOmega \) in order to solve (25). We suppose that \(\partial \varOmega \) can be split into \(\partial \varOmega _{\varPsi }\) and \(\partial \varOmega _{S}\):

$$\begin{aligned}&\varPsi = \varPsi _b, ~~~~ (\phi ,\xi ) \in \partial \varOmega _{\varPsi }, \end{aligned}$$
(28)
$$\begin{aligned}&\mathbf {S}\cdot \mathbf {n} = f, ~~~~ (\phi ,\xi ) \in \partial \varOmega _{S}, \end{aligned}$$
(29)

where \(\varPsi _b\) and f are specified boundary data. The conditions are explained below.

First, in the azimuthal direction if the geometry is fixed so that the narrow side of the annulus is the lowest side, then one may simplify the model by assuming that the flow is symmetric in \(\phi \) along both wide and narrow sides:

$$\begin{aligned} \varPsi (0,\xi ,t)= & {} 0, \end{aligned}$$
(30)
$$\begin{aligned} \varPsi (1,\xi ,t)= & {} 2 Q(t), \end{aligned}$$
(31)

where Q(t) is the dimensionless flow rate (and \(\varOmega = [0,1] \times [0,Z]\)). Note that by rearranging (24), \(\mathbf {S}\) can also be expressed in terms of the pressure gradients as

$$\begin{aligned} \mathbf {S} = \left( -r_a\frac{\partial \bar{p}}{\partial \xi } - \frac{r_a(\rho -1) \cos \beta }{Fr^2}, \frac{\partial \bar{p}}{\partial \phi } - \frac{r_a(\rho -1) \sin \beta \sin \pi \phi }{Fr^2} \right) . \end{aligned}$$

Thus, the symmetry condition \(\bar{\bar{v}} = 0\), which gives \(S_{\xi } = 0\), also implies that \(\frac{\partial \bar{p}}{\partial \phi } = 0\).

On the other hand, suppose we consider a full annulus, with no symmetry imposed at wide and narrow sides (\(\varOmega = [0,2] \times [0,Z]\)). Then an alternate to (30) and (31) would be

$$\begin{aligned} \varPsi (\phi +2,\xi ,t) = \varPsi (\phi ,\xi ,t) + 4 Q(t), \end{aligned}$$
(32)

fixing only the total flow rate. In using (32), if one wanted to work with the pressure, the pressure would be 2-periodic in \(\phi \).

Secondly, for the end conditions, following [9], we might expect to impose Dirichlet conditions at the inflow, \(\xi = 0\):

$$\begin{aligned} \varPsi (\phi ,0,t)= & {} \varPsi _0(\phi ,t) , \end{aligned}$$
(33)

e.g. a uniform inflow velocity can be specified, reflecting the fact of some kind of entry/development region, following local mixing as the fluids enter the annulus. Similarly, at the outflow (\(\xi = Z\)), it might be appropriate to assume a fully developed flow profile:

$$\begin{aligned} \varPsi (\phi ,Z,t)= & {} \varPsi _Z(\phi ,t). \end{aligned}$$
(34)

The fully developed profile above of course needs to be specified. The natural way to do this is by neglecting \(\xi \)-derivatives in (25), due to the length of the annulus, which leads to

$$\begin{aligned} \frac{\partial }{\partial \phi } [S_{\phi } + b_{\phi }] = 0 , \end{aligned}$$

which in turn implies that \(\frac{\partial \bar{p}}{\partial \xi }\) is independent of \(\phi \). The fully developed profile would be found computationally by decreasing \(\frac{\partial \bar{p}}{\partial \xi }\), which increases the axial velocity at each \(\phi \) (hence the flowrate), until the net imposed flowrate through the exit section is attained. Although this appears a convoluted procedure, it is straightforward numerically.

As an alternative to (33) and (34), one might impose Neumann conditions (\(\frac{\partial \varPsi }{\partial \xi }= 0\)) at each end of the annulus, e.g. as in [10]. Note that the Neumann condition, implies that \(S_{\xi } = 0\), which specifies \(\mathbf {S}\cdot \mathbf {n}\) on the boundary with outward normal \(\mathbf {n}\), i.e. a condition of type (29). Depending on the density gradients at inflow and outflow we might choose to specify \(\mathbf {S}\cdot \mathbf {n}\) in terms of the buoyancy \(\mathbf {b}\cdot \mathbf {n}\).

3 Mass transport

We now turn to transport of the different fluids along the annulus, assuming that at each time the elliptic problem (25) can be solved to give the gap-averaged velocity field. This suggests a similar model reduction will be appropriate for the different fluids and now proceed to derive this.

The concentrations of each individual fluid component \(c_k\) are modelled by an advection–diffusion equation:

$$\begin{aligned} \frac{\partial c_k }{\partial {\hat{t}}} + \frac{1}{{\hat{r}}} \frac{\partial }{\partial {\hat{r}}} [{\hat{r}}{\hat{u}}c_k] + \frac{1}{{\hat{r}}} \frac{\partial }{\partial \theta } [{\hat{v}}c_k]+ \frac{\partial }{\partial \hat{\xi }} [{\hat{w}} c_k] = \hat{\mathbf { \nabla }} \cdot [{\hat{D}}_{k,m} \hat{\nabla } c_k], \end{aligned}$$
(35)

where \(\sum _{k=1}^{K} c_k = 1\) and \({\hat{D}}_{k,m}\) represents the molecular diffusivity of species k within the mixture. For the turbulent flow, we apply the usual Reynolds decomposition and introduce the closure:

$$\begin{aligned} -\overline{\hat{\mathbf {u}}'c'_k} = {\hat{D}}_{t} \hat{\nabla } \bar{c}_k, \end{aligned}$$
(36)

where \({\hat{D}}_{t}\) is the turbulent diffusivity of species k (assumed the same for each species). Equation (35) becomes

$$\begin{aligned} \frac{\partial \bar{c}_k }{\partial {\hat{t}}} + \frac{1}{{\hat{r}}} \frac{\partial }{\partial {\hat{r}}} [{\hat{r}}\bar{{\hat{u}}}\bar{c}_k] + \frac{1}{{\hat{r}}} \frac{\partial }{\partial \theta } [\bar{{\hat{v}}}\bar{c}_k]+ \frac{\partial }{\partial \hat{\xi }} \left[ \bar{{\hat{w}}} \bar{c}_k] = \hat{\mathbf { \nabla }} \cdot [({\hat{D}}_{t}+{\hat{D}}_{k,m}) \hat{\nabla } \overline{c}_k\right] . \end{aligned}$$
(37)

We now apply the scaling introduced earlier. We anticipate that the main diffusive term is \({\hat{D}}_{t}\) which will scale with the local gap width and friction velocity, but here introduce a global scale: \(\delta _{0}{\hat{r}}_{a,0} \hat{\bar{W}} \) for the purposes of simplification. The scaled system is

$$\begin{aligned} \frac{\delta _{0}}{\pi } \left( \frac{\partial \bar{c}_k }{\partial t} + \frac{\partial }{\partial y} [\bar{u}\bar{c}_k] + \frac{1}{r_a} \frac{\partial }{\partial \phi } [\bar{v}\bar{c}_k]+ \frac{\partial }{\partial \xi } [\bar{w} \bar{c}_k] \right)= & {} \frac{\partial }{\partial y} \left[ D_k \frac{\partial \bar{c}_k }{\partial y} \right] + \left( \frac{\delta _{0}}{\pi } \right) ^2 \mathbf {\nabla _a} \cdot [D_k \nabla _a \bar{c}_k ] \end{aligned}$$
(38)

with \(D_k = D_t + 1/Pe_k\) and \(Pe_k = \delta _{0}{\hat{r}}_{a,0} \hat{\bar{W}}/{\hat{D}}_{k,m}, D_t = {\hat{D}}_{t}/[\delta _{0}{\hat{r}}_{a,0} \hat{\bar{W}}]\). On the left-hand side of (38), we have neglected terms that come from approximating geometry/curvature effects, which are \(O(\delta /\pi )\) smaller than those considered. On the right-hand side, we have also neglected terms of \(O(\delta /\pi )\) that come from approximating the radial diffusion term.

Eliminating these curvature/geometry terms only, while retaining the other terms may appear questionable as a perturbation procedure. However, note that the intention is to include the leading-order effects of all physically relevant transport processes. It is evident that (38) represents a singular perturbation, in which the leading-order concentration will be constant across the annular gap (see below). We thus retain the first order advective component on the left-hand side as this is responsible both for advection of the mean concentration and dispersive effects, within the plane of the narrow annulus. We also wish to evaluate the balance between turbulent diffusion and dispersion within the \((\phi ,\xi )\)-plane and consequently retain the diffusive terms in \((\phi ,\xi )\) directions. Lastly, although curvature may effect cross-gap diffusion the leading-order effect is included in the order 1 terms.

3.1 Dispersion effects

We look for a perturbation approximation to (38) in terms of the parameter \(\delta _0 /\pi \ll 1\). The velocity is assumed to have the following form

$$\begin{aligned} \bar{\mathbf {u}}= & {} \bar{\bar{\mathbf {u}}}_0 + \tilde{\mathbf {u}}_0 + \left( \frac{\delta _0}{\pi }\right) \tilde{\mathbf {u}}_1 + \left( \frac{\delta _0}{\pi }\right) ^2 \tilde{\mathbf {u}}_2 +\cdots , \end{aligned}$$
(39)
$$\begin{aligned} 0= & {} \int _{-H}^H \tilde{\mathbf {u}}_j~\mathrm {d} y, ~~~~j=0,1,2\ldots , \end{aligned}$$
(40)

i.e. the velocity (which we recall is anyway ensemble averaged) is decomposed into a gap-averaged component \(\bar{\bar{\mathbf {u}}}_0\) and successive components at each order that describe the y-variation. We assume that \(\bar{\bar{\mathbf {u}}}_0 = (0,\bar{\bar{v}},\bar{\bar{w}})\), as defined in (18), and that \({\tilde{u}}_0 = 0\), i.e. the radial component of velocity only arises at the first order. Similarly we write

$$\begin{aligned} \bar{c}_k= & {} \bar{c}_{k,0} + \left( \frac{\delta _0}{\pi }\right) \bar{c}_{k,1} + \left( \frac{\delta _0}{\pi }\right) ^2 \bar{c}_{k,2} +\cdots , \end{aligned}$$
(41)
$$\begin{aligned} 0= & {} \int _{-H}^H \bar{c}_{k,j} ~\mathrm {d} y, ~~~~j=1,2\ldots , \end{aligned}$$
(42)

Here following [39, 40], we use method of multiple timescales suggested by [41] in which we assume that the variables respond on both the advective time t and on a slower timescale \(T = \left( \delta _0/\pi \right) t\), where we expect diffusive effects to come into play. Note that no-slip boundary conditions are satisfied by \(\bar{\mathbf {u}}\) at the walls, where also the diffusive fluxes of \(\bar{c}_k\) are zero (Neumann condition).

Substituting these expressions into (38), we find that at leading-order \(\bar{c}_{k,0}\) is independent of y, as we have already assumed previously in analysing the momentum balance. From (42) we interpret \(\bar{c}_{k,0}\) as the gap-averaged mean concentration. The first-order equations are as follows:

$$\begin{aligned} \left( \frac{\partial \bar{c}_{k,0} }{\partial t} + \frac{\partial }{\partial y} [{\tilde{u}}_0\bar{c}_{k,0}] + \mathbf {\nabla }_a \cdot [\bar{c}_{k,0}(\bar{\bar{v}}+{\tilde{v}}_0,\bar{\bar{w}}+{\tilde{w}}_0)] \right)= & {} \frac{\partial }{\partial y} \left[ D_{k,0} \frac{\partial \bar{c}_{k,1} }{\partial y} \right] . \end{aligned}$$
(43)

Integrating across the channel shows that

$$\begin{aligned} \frac{\partial \bar{c}_{k,0} }{\partial t} + (\bar{\bar{v}},\bar{\bar{w}})\cdot \nabla _a \bar{c}_{k,0}= & {} 0 \end{aligned}$$
(44)

and on substituting back into (43) and using the continuity equation

$$\begin{aligned} \frac{\partial }{\partial y} \left[ D_{k,0} \frac{\partial \bar{c}_{k,1} }{\partial y} \right]= & {} ({\tilde{v}}_0,{\tilde{w}}_0)\cdot \nabla _a \bar{c}_{k,0}, ~~ \Rightarrow ~~ \frac{\partial \bar{c}_{k,1} }{\partial y} = \frac{\nabla _a \bar{c}_{k,0} \cdot \displaystyle { \left( \int _{-H}^y ({\tilde{v}}_0,{\tilde{w}}_0) ~\mathrm {d}y' \right) } }{D_{k,0} } . \end{aligned}$$
(45)

The expression (44) says that on the advective timescale t the leading-order concentration is simply advected along the streamlines.

To understand evolution on the slow timescale T, we move to a frame of reference moving along the gap-averaged streamlines. The coordinates (s, n) align locally with the directions \(\mathbf {e}_s\) and \(\mathbf {e}_n\): tangential and normal to the streamlines, respectively. The gap-averaged speed in the direction of the streamline is denoted \(\bar{\bar{s}}_0 = \sqrt{\bar{\bar{v}}_0^2 + \bar{\bar{w}}_0^2} = |\nabla _a \varPsi |/(2H)\). To further integrate (45), recall that in analysing the momentum balance in the previous section, we have shown that the leading-order turbulent velocity is in the direction of the pressure gradient. Therefore, the two vectors \((\bar{\bar{v}}_0,\bar{\bar{w}}_0) \) and \(({\tilde{v}}_0, {\tilde{w}}_0)\) are parallel and only in direction \(\mathbf {e}_s\), i.e.

$$\begin{aligned} (\bar{\bar{v}},\bar{\bar{w}})\cdot \nabla _a \bar{c}_{k,0} = \bar{\bar{s}}_0 \frac{\partial \bar{c}_{k,0}}{\partial s} ~~~\text {and}~~~ ({\tilde{v}}_0 ,{\tilde{w}}_0)\cdot \nabla _a \bar{c}_{k,0} = {\tilde{s}}_0 \frac{\partial \bar{c}_{k,0}}{\partial s} , \end{aligned}$$

where \({\tilde{s}}_0(y)\) gives the variation in the mean speed across the narrow gap. Substituting into (43) and integrating the first order terms, we get

$$\begin{aligned} \bar{c}_{k,1} = \bar{c}_{k,1}(-H) + \frac{\partial \bar{c}_0}{\partial s} \int _{-H}^y \frac{1}{D_{k,0}} \int _{-H}^{y'} {\tilde{s}}_0 \,\, \mathrm{d}y'' \mathrm{d}y' = \bar{c}_{k,1}(-H) + \frac{\partial \bar{c}_0}{\partial s} k(y) \end{aligned}$$
(46)

in which

$$\begin{aligned} k(y) = \int _{-H}^y \frac{1}{D_{k,0}} \int _{-H}^{y'} {\tilde{s}}_0\,\, \mathrm{d}y'' \mathrm{d}y' . \end{aligned}$$

By construction, \(\bar{c}_{k,1}(-H) = - \bar{k} \frac{\partial \bar{c}_{k,0}}{\partial s} \) where \({\bar{k}}\) is the average of k(y) across the gap. Thus, \(\bar{c}_{k,1}\) is expressed in terms of \(\frac{\partial \bar{c}_{k,0}}{\partial s} \) and the distribution of velocity across the gap. Before proceeding, note that the leading-order velocity is based on the narrow channel approximation, which leads to an even function: \({\tilde{s}}_0(y)\) is symmetric about \(y=0\). Also since

$$\begin{aligned} 0 = \int _{-H}^{H} {\tilde{s}}_0(y)\,\, \mathrm{d}y = 2 \int _{0}^{H} {\tilde{s}}_0(y) \,\, \mathrm{d}y = 2 \int _{-H}^{0} {\tilde{s}}_0(y)\,\, \mathrm{d}y, \end{aligned}$$

we may write

$$\begin{aligned} k(y) = \int _{-H}^y \frac{1}{D_{k,0}} \int _{0}^{y'} {\tilde{s}}_0\,\, \mathrm{d}y'' \mathrm{d}y' , \end{aligned}$$

and note that the integral of \({\tilde{s}}_0(y)\) will be an odd function. The function \(D_{k,0}(y)\) is also defined by the leading order velocity and can be assumed to be an even function. The integrand above is therefore also an odd function. From this, we may conclude that k(y) is an even function and that \(k(-H) = k(H) = 0\). Similarly, \(\bar{c}_{k,1}\) is an even function of y.

At the next order of asymptotic expansion, in the moving frame of reference, we collect terms of \(\mathcal {O}(\left( \delta _0/\pi \right) ^2)\):

$$\begin{aligned} \frac{\partial {\bar{c}}_{k,0}}{\partial T}= & {} - \mathbf {\nabla }_a \cdot \left[ {\bar{c}}_{k,1}({\tilde{v}}_0 , {\tilde{w}}_0 )\right] + \mathbf {\nabla _a}.(D_{k,0} \nabla _a {\bar{c}}_{k,0}) \nonumber \\&- \frac{\partial {\bar{c}}_{k,1}}{\partial t} - \bar{\bar{s}}_0 \frac{\partial {\bar{c}}_{k,1}}{\partial s} - \frac{\partial }{\partial y} ({\tilde{u}}_1 {\bar{c}}_{k,0}) - \mathbf {\nabla }_a \cdot \left[ {\bar{c}}_{k,0}({\tilde{v}}_1,{\tilde{w}}_1)\right] \nonumber \\&+ \frac{\partial }{\partial y} \left( D_{k,0} \frac{\partial {\bar{c}}_{k,2}}{\partial y} \right) + \frac{\partial }{\partial y} \left( D_{k,1} \frac{\partial {\bar{c}}_{k,1}}{\partial y} \right) + \frac{\partial }{\partial y} \left( D_{k,2} \frac{\partial {\bar{c}}_{k,0}}{\partial y} \right) . \end{aligned}$$
(47)

We integrate (47) across the gap width. The terms in the first line of (47) do not vanish. In the second line, the first two terms are linear in quantities that integrate to zero. For the last two terms, we use the incompressibility of \(\tilde{\mathbf {u}}_1\):

$$\begin{aligned} \frac{\partial }{\partial y} ({\tilde{u}}_1 {\bar{c}}_{k,0}) + \mathbf {\nabla }_a \cdot \left[ {\bar{c}}_{k,0}({\tilde{v}}_1,{\tilde{w}}_1)\right] = \tilde{\mathbf {u}}_1 \cdot \nabla {\bar{c}}_{k,0} = \nabla _a {\bar{c}}_{k,0}\cdot ({\tilde{v}}_1,{\tilde{w}}_1) . \end{aligned}$$

These terms now integrate to zero across the gap. In the third line, the terms vanish as there is no flux through the walls.

On substituting from (46), we see that the slow time evolution of \({\bar{c}}_{k,0}\) in the frame of reference moving along the streamline is governed by

$$\begin{aligned} 2H \frac{\partial {\bar{c}}_{k,0}}{\partial T}= & {} - \int _{-H}^H \frac{\partial }{\partial s} \left[ {\tilde{s}}_0 (k(y) - \bar{k}) \frac{\partial \bar{c}_{k,0}}{\partial s} \right] ~\mathrm{d} y + \int _{-H}^H \mathbf {\nabla _a}.(D_{k,0} \nabla _a {\bar{c}}_{k,0})~\mathrm{d} y . \end{aligned}$$
(48)

The first term on the right-hand side of (48) is the Taylor dispersion term, which we write as follows:

$$\begin{aligned} \int _{-H}^H \frac{\partial }{\partial s} \left[ {\tilde{s}}_0 (k - \bar{k}) \frac{\partial \bar{c}_{k,0}}{\partial s} \right] ~\mathrm{d} y= & {} \frac{\partial }{\partial s} \left[ \left( \int _{-H}^H {\tilde{s}}_0 k ~\mathrm{d} y \right) \frac{\partial \bar{c}_{k,0}}{\partial s} \right] + 2\frac{\partial H}{\partial s} \frac{\partial \bar{c}_{k,0}}{\partial s} [ {\tilde{s}}_0(H) \bar{k}] \\= & {} -\frac{\partial }{\partial s} \left[ 2H D_T \frac{\partial \bar{c}_{k,0}}{\partial s} \right] + 2\frac{\partial H}{\partial s} \frac{\partial \bar{c}_{k,0}}{\partial s} [ {\tilde{s}}_0(H) \bar{k}], \\ D_T= & {} \frac{1}{2H}\int _{-H}^{H} \frac{1}{D_{k,0}(y)} \left( \int _{-H}^{y} {\tilde{s}}_0(y') \mathrm{d}y'\right) ^2 \mathrm{d}y . \end{aligned}$$

The second term on the right-hand side of (48) reflects the average effect of the diffusivity \(D_{k,0}\). At \(y = \pm H\), the turbulent term vanishes, leaving only a negligible molecular contribution \(1/Pe \ll 1\). Therefore, we can write

$$\begin{aligned} \int _{-H}^H \mathbf {\nabla _a}\cdot (D_{k,0} \nabla _a {\bar{c}}_{k,0})~\mathrm{d} y= & {} \mathbf {\nabla _a} \cdot \int _{-H}^H(D_{k,0} \nabla _a {\bar{c}}_{k,0})~\mathrm{d} y \\&-\nabla _a H \cdot \nabla _a {\bar{c}}_{k,0} [D_{k,0}(H) + D_{k,0}(-H)] \\= & {} \mathbf {\nabla _a} \cdot [ 2H \bar{D} \nabla _a {\bar{c}}_{k,0}] + O(1/Pe), \\ \bar{D}= & {} \frac{1}{2H} \int _{-H}^H D_{k,0}~\mathrm{d} y . \end{aligned}$$

Combining the above expressions with (44), reverting back to the single timescale t and transforming back into the fixed frame of reference, we arrive at the following equation for the evolution of the leading-order concentration:

$$\begin{aligned} \frac{\partial c_{k,0}}{\partial t}= & {} - (\bar{\bar{v}}_0,\bar{\bar{w}}_0) \cdot \mathbf {\nabla }_a c_{k,0} + \frac{\delta _0}{\pi } \left( \frac{1}{2H}\mathbf {e}_s \cdot \nabla _a \left[ (2H D_T) \mathbf {e}_s \cdot \nabla _a c_{k,0}\right] \right. \nonumber \\&\left. - \frac{\bar{k}{\tilde{s}}_0(H)}{H} (\mathbf {e}_s \cdot \nabla _a H)(\mathbf {e}_s \cdot \nabla _a c_{k,0} ) + \frac{1}{2H} \mathbf {\nabla _a}\cdot \left[ 2H \bar{D} \nabla _a {\bar{c}}_{k,0}\right] \right) . \end{aligned}$$
(49)

Equation (49) describes how the leading-order concentrations of fluid k change. The right-hand side has four terms. Firstly, we have advection with the mean flow. Secondly, we have a pure Taylor-dispersion term, which we can see takes the form of an anisotropic diffusivity, i.e. only along the streamlines (in direction \(\mathbf {e}_s\)). The third term results from variations in width of the annulus. The fourth term on the right-hand side gives the averaged effect of the diffusivity. In [35], we have modelled the velocity profiles for the flow along a uniform plane channel, i.e. \(\bar{\bar{s}}_0 + {\tilde{s}}_0(y)\) and have used this, together with estimates of the turbulent diffusivity, to compute the Taylor dispersivity \(D_T\). In general, it is found that \(\bar{D} \ll D_T\). In highly turbulent flows, \(D_T\) decreases, but still remains two orders of magnitude larger than \(\bar{D}\). Thus, it is the first two dispersive terms that are of most interest.

We may extract the variation of H from the first term and rewrite (49) as

$$\begin{aligned} \frac{\partial c_{k,0}}{\partial t}= & {} - (\bar{\bar{v}}_0,\bar{\bar{w}}_0) \cdot \mathbf {\nabla }_a c_{k,0} + \frac{\delta _0}{\pi } \left( \mathbf {e}_s \cdot \nabla _a [D_T \mathbf {e}_s \cdot \nabla _a c_{k,0} ] + \frac{1}{2H} \mathbf {\nabla _a}\cdot [ 2H \bar{D} \nabla _a {\bar{c}}_{k,0}] \right) \nonumber \\&+ \frac{\delta _0}{\pi } (\mathbf {e}_s \cdot \nabla _a H)(\mathbf {e}_s \cdot \nabla _a c_{k,0} ) \frac{D_T - D_T^*}{H}, \end{aligned}$$
(50)

where

$$\begin{aligned} D_T^* = \bar{k}{\tilde{s}}_0(H) = - \frac{{\tilde{s}}_0(H)}{2H}\int _{-H}^{H} \frac{y}{D_{k,0}(y)} \int _{-H}^{y} {\tilde{s}}_0(y') \mathrm{d}y' \mathrm{d}y . \end{aligned}$$
(51)

We see that nominally \(D_T^*\) has the same size as \(D_T\). Note that the last term in (50) will vanish when the annulus is concentric. Also we should note that in most displacement flows through long annuli the streamlines are pseudo-parallel to the \(\xi \)-axis for most of the annulus and H generally varies slowly with \(\xi \), so this last term is mostly insignificant. However, in interfacial regions between two displacing fluids we often see azimuthal velocities of a similar size to the axial velocity. These regions are of course also where the main Taylor dispersion term is active. Therefore, the interplay between these two terms needs investigating. Local closure expressions for \(D_T, D_T^*\) and \(\bar{D}\), in terms of the expressions derived in [35], are given later in Appendix A.5.

3.2 Boundary conditions

Boundary conditions for (50) are generally that \(c_{k,0}\) is specified at the inflow to the annulus, either from a pump schedule or from coupling with a predictive model of the downwards displacement flow in the casing. Along the sides of the annulus, either a symmetry condition is imposed, i.e. \(\frac{\partial c_{k,0}}{\partial \phi } = 0\), or potentially a periodicity condition in the case that the full annulus is resolved and no symmetry is assumed. At the outflow, we generally assume that \(\frac{\partial c_{k,0}}{\partial \xi } = 0\), although in cases of large density differences we may have counter-current flows with denser fluids entering the annulus and need to specify accordingly.

4 Annular displacement flow model

To summarize the derivations of the previous 2 sections, leading-order annular displacement flows (across mixed flow regimes) are governed by the coupled system:

$$\begin{aligned} \mathbf {\nabla }_a \cdot [\mathbf {S} + \mathbf {b}]= & {} 0 \end{aligned}$$
(52)
$$\begin{aligned} \frac{1}{2H} \nabla _a \varPsi= & {} (\bar{\bar{w}}_0,-\bar{\bar{v}}_0), \end{aligned}$$
(53)
$$\begin{aligned} \frac{\partial c_{k,0}}{\partial t} + (\bar{\bar{v}}_0,\bar{\bar{w}}_0) \cdot \mathbf {\nabla }_a c_{k,0}= & {} \frac{\delta _0}{\pi } \left( \mathbf {e}_s \cdot \nabla _a \left[ D_T \mathbf {e}_s \cdot \nabla _a c_{k,0} \right] + \frac{1}{2H} \mathbf {\nabla _a}\cdot \left[ 2H \bar{D} \nabla _a {\bar{c}}_{k,0}\right] \right) \nonumber \\&+ \frac{\delta _0}{\pi } (\mathbf {e}_s \cdot \nabla _a H)\left( \mathbf {e}_s \cdot \nabla _a c_{k,0}\right) \frac{D_T - D_T^*}{H} \end{aligned}$$
(54)

with associated boundary conditions. Due to the lengthy derivation, we shall defer full analysis of this system to be future paper. Here we focus on presenting some example simulations that indicate the process complexity (Sect. 4.2) and give some basic estimates for diffusion/dispersion effects.

4.1 Computational overview

For brevity, we do not present algorithmic details here, as the method is very similar to that used in [7]. The computational domain is rectangular, and we use a second-order finite difference method on a staggered mesh: representing the fluid concentrations at cell centres and the stream function at the gridpoints (corners). We have used 30 mesh cells in the azimuthal direction and 100 cells along the annulus. At each timestep, the stream function is found by solving (52) using an augmented Lagrangian procedure as in [7]. This algorithm is slow to converge and not motivated by (or needed for) the turbulent regime, i.e. many faster methods could be employed. However, mixed flow regimes are common in primary cementing, and this algorithm resolves those parts of the flow where fluids are immobile due to the yield stress. Hence, we retain this feature at the expense of speed.

The concentration enters the stream function equation through the buoyancy field \(\mathbf {b}\) and through the local fluid properties. Having found the stream, we construct the velocity field from (53) and advance the fluid concentrations in time by solving (54). The algorithm employed for the advective parts of (54) is the FCT scheme, as in [7], with the diffusive and dispersive terms discretized explicitly using central differences. For the non-diffusive problem, a concentric annulus with positive density difference leads to an analytical solution for the interface shape [9], which has been used as a validation problem.

4.2 Example displacement scenarios

To explain different features of our model, here we present a few displacement examples. We focus these examples at annulus dimensions that are more typical of a laboratory flow loop setting, with \({\hat{r}}_{i} = 6.5\) cm, \({\hat{r}}_{o} = 9\) cm, \({\hat{Z}} = 20\) m. The annulus will here be assumed vertical \(\beta = 0\). More realistic well dimensions are saved for our later study in which we systematically explore turbulent and mixed regime effects. For all the results presented, we have restricted the simulation to only one half of the annulus—assuming symmetry at the wide and narrow sides.

Example 1

In the first example, we probe the effect of dispersion. Fluid 1 has properties: \({\hat{\rho }}_1 = 1100\hbox { kg/m}^3, {\hat{\kappa }}_1 = 0.002\hbox { Pa.s}^n, n_1=0.9, \hat{\tau }_{Y,1} =0\) Pa. The displacing fluid has identical properties. The annulus has eccentricity \(e=0.5\) but is otherwise uniform, and the flow rate is constant (\(Q = 0.01\hbox { m}^3/\hbox {s}\)). In this scenario as the fluids are identical, we expect a stream function that varies only with \(\phi \) (parallel streamlines), i.e. the flow is fully developed and velocity is only in the axial direction. Figure. 2 presents a colourmap of the fluid concentrations as they advance along the annulus: fluid 1 (red) is displaced by fluid 2 (blue). The eccentricity leads to faster flows on the wider side (W) than on the narrow side (N). The interface constantly elongates as a consequence. In Fig. 2a, we have set to zero all terms on the right-hand side of (54). In Fig. 2b, we included Taylor dispersion and the turbulent diffusivity (first and second terms on the right-hand side of (54). Finally, Fig. 2c solves (54) in its complete form.

Fig. 2
figure 2

Turbulent displacement in an eccentric annulus: \(e=0.5\). Colour maps of the concentration are shown in dimensionless coordinate system \((\phi ,\xi )\) at different dimensionless times as the front progresses along the annulus. White lines are streamlines with spacing \(\varDelta \varPsi =0.25\). a No dispersion or diffusivity (\(\bar{D} = D_T = D_T^* = 0\)); b only Taylor dispersion (\(D_T^* = D_T\)); c both dispersive terms included. Fluid 1 (red) properties: \({\hat{\rho }}_1 = 1100\hbox { kg/m}^3, {\hat{\kappa }}_1 = 0.002 \hbox { Pa.s}^n, n_1=0.9, \hat{\tau }_{Y,1} =0\) Pa; fluid 2 is identical (blue). (Color figure online)

Full size image

Figure 2a essentially shows the performance of the FCT scheme in advecting the front. Numerical smearing (diffusion/dispersion) of the front is present but is kept to a few cells in width. For this flow in particular (1D), there are no secondary flows that can advectively mix intermediate concentrations. The front is sharpened by mesh refinement of course, at the expense of longer computational times. As commented and illustrated in Fig. 6, turbulent diffusivity is at least two orders of magnitude smaller than Taylor dispersion. Including the diffusivity terms here has little effect as shown in Fig. 2a; numerical diffusion/dispersion has a larger effect on these mesh sizes.

Figure 2b shows a significant effect of \(D_T\), in comparison to Fig. 2a. The front spreads most on the wide side, thickening to about 2 m by the end of the annulus. We observe that there is no discernible difference between Figs. 2b, c, although the term \(D_T^*\) is in fact larger than \(D_T\) (see Fig. 6). This illustrates the points made earlier regarding the last term in (54). Here the streamlines are parallel, so \(\mathbf {e}_s \cdot \nabla _a c_{k,0}\) is only in the \(\xi \) direction. Equally, H does not vary with \(\xi \) so that \(\mathbf {e}_s \cdot \nabla _a H = 0\).

Fig. 3
figure 3

a, b Mixed regime displacement flows in an eccentric annulus \(e=0.5\) at \({\hat{Q}}\) =0.01 m\(^3\)/s. White lines are streamlines with \(\varDelta \varPsi =0.25\). The last 2 panels on the right show the flow regime: white, grey and black regions are turbulent, transitional and laminar, respectively. a Unsteady displacement with properties: fluid 1 (red), \({\hat{\rho }}_1 = 1300\hbox { kg/m}^2, {\hat{\kappa }}_1 = 0.002\hbox { Pa.s}^n, n_1=0.8, \hat{\tau }_{Y,1} =0\hbox { Pa}\); and fluid 2 (blue), \({\hat{\rho }}_2 = 1200\hbox { kg/m}^3, {\hat{\kappa }}_2 = 0.004\hbox { Pa.s}^n, n_2=0.9, \hat{\tau }_{Y,2} =1\) Pa. b Steady displacement with properties: fluid 1 (red), \({\hat{\rho }}_1 = 1100\hbox { kg/m}^3, {\hat{\kappa }}_1 = 0.002\hbox { Pa.s}^n, n_1=0.9, \hat{\tau }_{Y,1} =0\) Pa; and fluid 2 (blue), \({\hat{\rho }}_2 = 1300\hbox { kg/m}^3, {\hat{\kappa }}_2 = 0.01\hbox { Pa.s}^n, n_1=0.7, \hat{\tau }_{Y,1} =1\) Pa. c A fully turbulent unsteady displacement with the same fluids as case b, but now \({\hat{Q}}\) = 0.05 m\(^3\)/s. (Color figure online)

Full size image

Example 2

Our second example explores mixed flow regimes and buoyancy effects, again in an eccentric annulus (\(e = 0.5\)). In Fig.  3, fluid 1 (red) has properties: \({\hat{\rho }}_1 = 1300\hbox { kg/m}^3, {\hat{\kappa }}_1 = 0.002 \hbox { Pa.s}^n, n_1=0.8, \hat{\tau }_{Y,1} =0\) Pa, and fluid 2 (blue) has properties: \({\hat{\rho }}_2 = 1200\hbox { kg/m}^3, {\hat{\kappa }}_2 = 0.004\hbox { Pa.s}^n, n_1=0.9, \hat{\tau }_{Y,1} =1\) Pa. The fluids are pumped at \(Q = 0.01\hbox { m}^3/\hbox {s}\). The fluid rheologies are not very different to example 1, but there is a significant (unstable) density difference. Figure 3a is an extreme example of an unsteady displacement. There is an unfavourable density difference but favourable rheology difference between the displaced and displacing fluids. However, this rheology difference is apparently ineffective as we observe that fluid two rapidly channels up the wide side of the annulus by-passing fluid 1. The flow regimes are shown in the last two panels of the figure, at 2 times late in the displacement (white, grey and black regions are turbulent, transitional and laminar, respectively). We observe ahead of the front that fluid 1 is turbulent all around the annulus. Primarily turbulent flows are found behind the front (in both fluids). This strange flow configuration results because fluid 1 is in fact flowing backwards against the displacement direction, on the narrow side behind the displacement front.

The second row of this example shows another mixed regime displacement. The fluid rheologies are only slightly changed from Fig. 3a, but the density difference is reversed, i.e. fluid 2 is heavier than fluid 1. We see a displacement front that is approximately steady (apart from dispersion at the interface) and which effectively removes fluid 1 from the annulus. This reinforces the impression that the rheological difference between the two fluids is largely irrelevant: the steady displacement results from the stabilizing density difference, whereas that in Fig.  3a was unsteady. In this example, mixed regime flows are found behind the advancing displacement front. Due to the eccentricity and the yield stress in fluid 2, the flow ranges from fully turbulent through transitional to a laminar regime as we move azimuthally.

Figure 3a, b gives the impression that in turbulent displacements, buoyancy effects play a key role. This runs slightly counter to the intuitive notion that turbulent flows are effective in spreading momentum azimuthally. To explore this, we further increase the flow rate to \({\hat{Q}} = 0.05\hbox { m}^3/\hbox {s}\) in Fig. 3c, keeping the fluid properties the same as in b. Remarkably, the more turbulent flow appears to displace less effectively. We observe that the front advances and elongates along the wide side of the annulus. The concentration profiles are similar to those of Example 1.

Intuitively what has happened here is that the turbulent stresses have increased (approximately quadratically) with the increasing flow rate and now dominate the stabilizing buoyancy terms. At this point, neither buoyancy nor rheology is relevant in comparison to the turbulent stresses. In this highly turbulent limit, the asymptotic growth of \(\tau _w\) (see Fig. 5a) is largely independent of n. Thus, we expect a similar distribution of velocity around the annulus in both fluids and observe that the frontal evolution is primarily advective and unidirectional.

Example 3

In our final example, we explore the effects of dispersion in a more complex displacement scenario. Here fluid 1 (red) is displaced by sequentially pumping fluid 2 (green) and then fluid 3 (blue), with properties: \({\hat{\rho }}_1 = 1300\hbox { kg/m}^3, {\hat{\kappa }}_1 = 0.002\hbox { Pa.s}^n, n_1=0.8, \hat{\tau }_{Y,1} =0\hbox { Pa}; {\hat{\rho }}_2 = 1100\hbox { kg/m}^3, {\hat{\kappa }}_2 = 0.05\hbox { Pa.s}^n, n_2=1, \hat{\tau }_{Y,2} =1\hbox { Pa}; {\hat{\rho }}_3 = 1300\hbox { kg/m}^3, {\hat{\kappa }}_3 = 0.002\hbox { Pa.s}^n, n_3=0.8, \hat{\tau }_{Y,3} =0\) Pa. Thus, the unstable density difference is expected to destabilize the front between fluids 1 and 2, whereas the stable density difference between fluids 2 and 3 is expected to stabilize the front between fluids 2 and 3. The annulus is now concentric.

Fig. 4
figure 4

Turbulent displacement in a concentric annulus with 3 fluids. White lines are streamlines with spacing \(\varDelta \varPsi =0.25\). Fluid 1 (red): \({\hat{\rho }}_1 = 1300\hbox { kg/m}^3, {\hat{\kappa }}_1 = 0.002\hbox { Pa.s}^n, n_1=0.8, \hat{\tau }_{Y,1} =0\) Pa; fluid 2(green): \({\hat{\rho }}_2 = 1100\hbox { kg/m}^3, {\hat{\kappa }}_2 = 0.05\hbox { Pa.s}^n, n_2=1, \hat{\tau }_{Y,2} =1\) Pa; fluid 3 (blue): \({\hat{\rho }}_3 = 1300\hbox { kg/m}^3, {\hat{\kappa }}_3 = 0.002\hbox { Pa.s}^n, n_3=0.8, \hat{\tau }_{Y,3} =0\) Pa. a \(Q=0.01\,\hbox {m}^3/\hbox {s}\); b \(Q=0.05\,\hbox {m}^3/\hbox {s}\). (Color figure online)

Full size image

Figure 4a shows that what we expect does take place. The flows are turbulent and since the annulus is concentric we might hope for a steady uniform front to advance. The density difference between fluids 1 and 2 however promotes instability. Note that the system is mathematically analogous to a 2D miscible porous media displacement, with a nonlinear flow law. We expect the adverse density gradient to be vulnerable to fingering-type instabilities at the diffuse front, and once these start asymmetries will grow as we observe. Indeed, the initial instability grows and we see that the unstable fluid 2 eventually advects ahead of fluid 3, i.e. as a large buoyant droplet. Presumably, if the annulus was mildly eccentric this would influence the azimuthal position where the instabilities first arise and how they develop. The second interface has no fingering instability between fluids 2 and 3 and due to the positive buoyancy is stabilizing.

As Fig. 3c suggests, here we might expect that if we increase flow rate, then the flow loses its sensitivity to density difference and the interface between fluids 1 and 2 may become stable. In Fig. 4b, we test this idea. We increase the flow rate to \(Q=0.05\,\hbox {m}^3/\hbox {s}\) and observe that this stabilizes the interface between fluids 1 and 2 in spite of their unsteady density difference. Unlike example 2, the annulus is now concentric so that the geometry does not itself have any bias, and we appear to have a steady (but dispersive) displacement.

4.3 Estimates of diffusion/dispersion in turbulent displacements

The example simulations of the previous section reveal some interesting physical features that we may now understand better via an order of magnitude analysis. Firstly, having determined that the advective contributions to Taylor dispersion have little effect in regular annuli, we can focus on the first and second terms on the right-hand side of (54), i.e. Taylor dispersion and mean turbulent diffusivity. Regarding the mean turbulent diffusivity, a rudimentary analysis suggests that the turbulent diffusivity scales with the gap width times the friction velocity, which in our variables corresponds to

$$\begin{aligned} \hat{\bar{D}} \sim {\hat{W}}_0 = \delta _0 \sqrt{f_f} {\hat{r}}_{a,0} \hat{\bar{W}} |\nabla _a \varPsi |, \end{aligned}$$
(55)

where \(f_f\) is the friction factor (see Sect. 1). Thus, the time to diffuse across the annular gap is approximately

$$\begin{aligned} {\hat{t}}_\mathrm{gap} \sim \frac{[H \delta _0 {\hat{r}}_{a,0}]^2}{\hat{\bar{D}}} \sim \frac{\delta _0 {\hat{r}}_{a,0} 2H }{ \sqrt{f_f} \hat{\bar{W}} (|\nabla _a \varPsi |/2H)} . \end{aligned}$$

Physically, this is at most a few seconds in any turbulent section of a typical cemented annulus, which is of course why the Taylor dispersion regime is always relevant. Note though that (55) is only applicable to turbulent parts of the annulus. Secondly, note that as \(\delta _0/\pi \rightarrow 0, {\hat{t}}_{gap} \rightarrow 0\) and thus formally, in the limit of our narrow-gap model, this mixing is effectively infinite.

Secondly, we have seen that in many situations displacement fronts tend to elongate as the displacement proceeds. Thus, we may ask instead how long it takes for turbulent diffusivity to diffuse azimuthally:

$$\begin{aligned} {\hat{t}}_\mathrm{azimuth} \sim \frac{[\pi {\hat{r}}_{a,0}]^2}{\hat{\bar{D}}} \sim \frac{\pi ^2 {\hat{r}}_{a,0}}{\delta _0 \sqrt{f_f} \hat{\bar{W}}} . \end{aligned}$$

Evidently \({\hat{t}}_\mathrm{azimuth} \rightarrow \infty \) as \(\delta _0/\pi \rightarrow 0\). More generally this estimate could take many hundreds of seconds in a cemented well. Whether this is relevant depends on the time taken to cement a well segment of length \({\hat{Z}}, {\hat{t}}_\mathrm{cem}={\hat{Z}}/\hat{\bar{W}}\). Therefore, azimuthal mixing takes place approximately when \({\hat{Z}}/\hat{\bar{W}} \gtrsim {\hat{t}}_\mathrm{azimuth}\), i.e.

$$\begin{aligned} \frac{\pi ^2 {\hat{r}}_{a,0}}{\delta _0 \sqrt{f_f} {\hat{Z}} } \lesssim 1 . \end{aligned}$$

Lengths of several hundred metres are required for equality above and we conclude that azimuthal turbulent diffusion is only likely to occur in long cemented sections. The fraction of the azimuthal distance over which turbulent diffusivity spreads across the interface is roughly

$$\begin{aligned} \frac{ \sqrt{\hat{\bar{D}} {\hat{Z}}/\hat{\bar{W}}}}{\pi {\hat{r}}_{a,0}} = \sqrt{\frac{\delta _0 \sqrt{f_f}{\hat{Z}}}{\pi ^2 {\hat{r}}_{a,0}}} . \end{aligned}$$

It seems that the direct effects of turbulent diffusivity may be significant in long wells, in diffusing the front azimuthally. However, such conclusions are premature and require a more extensive study. Note, for e.g., that the estimates made above are really based on a model scenario in which 2 parallel streams of fluid enter the annulus axially and diffuse azimuthally. Instead displacement flows are dynamic and additional timescales enter this type of estimate, e.g. the development time of the front, in order to elongate, and the axial extent of the elongated interface.

In the streamwise direction, as we have seen, Taylor dispersion will dominate. In dimensional terms, the Taylor dispersivity gives a streamwise diffusivity of size: \(\delta _0 \hat{\bar{W}} {\hat{r}}_{a,0} D_T\), with \(D_T\) of order 1. We see that after a time \({\hat{Z}}/\hat{\bar{W}}\) Taylor dispersion has typically spread the displacement front a distance \(\sqrt{\delta _0 {\hat{Z}} {\hat{r}}_{a,0} D_T}\), which is typically in the range of \(1-20\) m. Again, such estimates are crude and rely on the streamlines being pseudo-parallel to the annulus axis. Situations such as the steady displacement of Fig. 3b require better understanding, as a steady displacement in an eccentric annulus implies that the streamlines are not axial in the vicinity of the displacement front.

5 Conclusions

In this paper, we have derived a practical leading-order model for simulating 2D turbulent and mixed regime displacement flows, as encountered in the process of primary cementing. The model presented complements that derived in [6] which was focused at laminar displacement flows. Indeed this model is included in that developed here, as the description for laminar displacements. However, the leading-order turbulent model has two main differences with [6].

First, the treatment of the momentum equations differs. Unlike the laminar flow, the turbulent Reynolds stress components all have similar size. The leading-order flow is a turbulent shear flow, but only due to differential scaling of the lengths. At the next order of approximation many more unknowns enter the model. Having derived the turbulent shear flow the analysis is similar to [6] in developing the field equations for the stream function (or pressure), but with the closure expressions coming from turbulent-flow hydraulics.

The second principal difference comes in the treatment of the fluid concentrations. First we note that the assumed decoupling of averaged concentration from velocity, in the advective part of (54), is more valid here than for the laminar model in [6]. Second, following our analysis, we find that turbulent flows are governed by complex diffusive and dispersive transport processes (absent in the laminar flows). The largest effect is Taylor dispersion, which diffuses only along the streamlines. The gap-averaged turbulent diffusivity acts isotropically but appears to be relevant only in sufficiently long wells. We have also derived terms that describe the influence of annulus gap variations on dispersion, again in the streamwise direction. These terms collectively are complex to understand, and in this paper, we have restricted ourselves to a few sample computations. For the laminar displacement flow, only molecular diffusion should be present, which results in smaller effects than those here.

The examples presented show only a few (of many) interesting effects, based on simulations of displacement flows along a small lab-scale annulus. As expected, rheology plays only a minor role in these turbulent flows, but density differences remain vitally important it appears. We have observed both steady and unsteady displacements (i.e. those in which the front elongates along the annulus), similar to in laminar displacements, but now with diffuse frontal regions. However, although phenomenologically similar the underlying physical conditions under which these flows arise are not yet studied or understood. This paper has primarily focused on developing the modelling tools, and our future research will be directed towards analysing these flows in depth.