Primary cementing of oil and gas wells in turbulent and mixed regimes

Abstract

We present a detailed derivation of a practical two-dimensional model for turbulent and mixed regimes in narrow annular displacement flows, such as are found during the primary cementing of oil and gas wells. Such mixed cross regimes, including those in which different regimes exist in the same annular cross section, are relatively common in primary cementing. The modelling approach considers scaling based on the disparity of length-scales, which allows a narrow-gap averaging approach to be effective. With respect to the momentum equations, the leading-order equations correspond to a turbulent shear flow in the direction of the modified pressure gradient. This leads to a nonlinear elliptic problem that is the natural extension of the laminar displacement model in Bittleston et al. (J Eng Math 43:229–253, 2002). The mass transport equations that model the miscible displacement are however quite different. To leading-order turbulence effectively mixes the fluids. Changes in concentrations within the annular gap arise due to the combined effects of advection with the mean flow, anisotropic Taylor dispersion (along the streamlines) and turbulent diffusivity. The diffusive and dispersive effects are modelled for fully turbulent and transitional flows following Maleki and Frigaard (J Non-Newt Fluid Mech 235:1–19, 2016). The model derived allows the investigation of different well geometries and inclinations, pum** sequences and fluid rheologies, all of which can have importance. A number of computed examples are presented with the aim of demonstrating the complexity of turbulent displacements.

Appendices

Appendix A: The wall shear stress closure

Here we outline the closure relationship that defines the dimensionless $\tau _w(|\nabla _a \varPsi |;\phi ,\xi ,t)$, locally in the annulus, and hence

$$\begin{aligned} \mathbf {S} = \frac{r_a \tau _w(| \nabla _a \varPsi |) }{H|\nabla _a \varPsi |} \nabla _a \varPsi . \end{aligned}$$

we follow the methods in [35], where the flow of a Herschel–Bulkley fluid along a narrow channel is studied in laminar, transitional, and turbulent regimes.

We start by reconstructing the dimensional variables, which are then scaled following [35]. Given the shear stress scale $\hat{\tau }_0$, the dimensional wall shear stress is $\hat{\tau }_w = \hat{\tau }_0 \tau _w$. The dimensional mean speed (averaged across the local gap) and the dimensional local annular gap are given as

$$\begin{aligned} {\hat{W}}_0 = \hat{\bar{W}} \frac{1}{2H} |\nabla _a \varPsi | , ~~~~~~ 2 {\hat{H}} = 2 H \delta _0 {\hat{r}}_{a,0} \end{aligned}$$

(56)

recall that $\hat{\bar{W}}$ is the velocity scale for the entire annulus. We also assume that, according to the concentrations of the fluids at $(\phi ,\xi ,t)$, we may construct the dimensional local mixture density $\hat{\rho }$ and the rheological parameters $\hat{\tau }_{Y}, \hat{\kappa }, n$.

Now, following [35], we define

$$\begin{aligned} \hat{\dot{\gamma }}_{N} = \frac{6 {\hat{W}}_0}{2{\hat{H}}}, ~~~~~~ \hat{\kappa }_p = \hat{\kappa } \left( \frac{2n+1}{3n}\right) ^n. \end{aligned}$$

(57)

The power-law Reynolds number is

$$\begin{aligned} Re_{p} = \frac{6 \hat{\rho } {\hat{W}}_{0}^2}{ \hat{\kappa }_p (\hat{\dot{\gamma }}_{N})^{n}} . \end{aligned}$$

(58)

The Hedström number and scaled wall shear stress are

$$\begin{aligned} He = \hat{\tau }_{Y} \left( \frac{\hat{\rho }^n (2{\hat{H}})^{2n}}{\hat{\kappa }_p^{2}} \right) ^{1/(2-n)}, ~~~~ H_w = \hat{\tau }_{w} \left( \frac{\hat{\rho }^n (2{\hat{H}})^{2n}}{\hat{\kappa }_p^{2}} \right) ^{1/(2-n)} . \end{aligned}$$

(59)

The procedure in [35] gives a detailed description of the map** from $Re_p$ to $H_w$ and vice versa, which is parameterized by (n, He). Observe that $H_w \propto \tau _w$ and $Re_p \propto |\nabla _a \varPsi |^{2-n}$, so the map** $H_w \leftrightarrow Re_p$ defines our closure relation. Figure 5 shows an example of this map** for different n at $He = 100$. The sensitivity to He is not extreme. We now outline the methodology in the different regimes.

1.1 A. 1 Laminar flows

For laminar regime the map** $H_w \leftrightarrow Re_p$ (i.e. ${\hat{\tau _w}} \leftrightarrow {\hat{W}}_0$) is

$$\begin{aligned} \frac{(6 Re_{p})^{n/(2-n)}}{H_w} = E\left( n,\frac{He}{H_w}\right) , \end{aligned}$$

(60)

where

$$\begin{aligned} E(n,y_Y) = \left( 1-y_Y\right) ^{(n+1)}\left( \frac{n}{n+1}y_Y+1\right) ^n , \end{aligned}$$

(61)

i.e. $y_Y = He/H_w = \tau _Y / \tau _w$, which is the dimensionless width of the plug region. This nonlinear relationship is resolved to give $H_w(Re_p)$ and hence to define $\mathbf {S}$.

It is noted that $\mathbf {S}/|\nabla _a \varPsi |$ is singular as $|\nabla _a \varPsi | \rightarrow 0$, which is the limit where the yield stress of the fluid is not exceeded at the walls of the channel and $\tau _w < \tau _Y$ is indeterminate. In the laminar displacement model of [6, 7], the vector $\mathbf {S}$ is defined explicitly to reflect this yielding phenomenon:

$$\begin{aligned}&\mathbf {S} = \left[ \frac{r_a \chi (| \nabla _a \varPsi |) }{|\nabla _a \varPsi |} + \frac{r_a \tau _Y }{H|\nabla _a \varPsi |} \right] \nabla _a \varPsi ~~\Leftrightarrow ~~ |\mathbf {S}| > \frac{r_a \tau _Y }{H}, \end{aligned}$$

(62)

$$\begin{aligned}&| \nabla _a \varPsi | = 0 ~~\Leftrightarrow ~~ |\mathbf {S}| \le \frac{r_a \tau _Y }{H}. \end{aligned}$$

(63)

To connect the model derivation here with that of [6, 7], note that the function $\chi (| \nabla _a \varPsi |)$ is simply

$$\begin{aligned} \chi (| \nabla _a \varPsi |) = \frac{\tau _w(| \nabla _a \varPsi |) - \tau _Y}{H} , \end{aligned}$$

(64)

where $\tau _w$ is defined implicitly from (60).

The function $\chi (| \nabla _a \varPsi |)$ increases strictly monotonically (as does $\tau _w(| \nabla _a \varPsi |)$). We can examine the limits of (60), both as $H_w \rightarrow \infty$ and as $H_w \rightarrow He$ (yield limit). For the latter, we find

$$\begin{aligned} Re_p^{n/(2-n)} \sim [H_w - He]^{n+1} ~~\Rightarrow ~~ \chi \sim | \nabla _a \varPsi |^{n/(n+1)} ; \end{aligned}$$

see Fig. 5b. As $H_w \rightarrow \infty$, we find $Re_p^{n/(2-n)} \sim H_w$, i.e. $\chi \sim | \nabla _a \varPsi |^n$, reflecting the shear-thinning behaviour. These limiting behaviours agree with those in [7], where the laminar model is analysed in more depth. The difference here though is that the limit $H_w \rightarrow \infty$ is not physically attained in the laminar regime: we transition to turbulent flow.

1.2 A. 2 Fully turbulent flows

For the fully turbulent regime, in [35] we use the Dodge–Metzner relation, which translates into the following equation defining $H_w \leftrightarrow Re_p$:

$$\begin{aligned} Re_p = H_w ^{1-\frac{n}{2}} 6^{1-n} 2 ^{1-\frac{n}{2}} \left[ \frac{4.0}{{n'}^{0.75}} \log \left( 6^{1-n'} 2 ^{1-\frac{n'}{2}} E^ {\frac{n'}{n}} H_w^{\frac{n'}{n}- \frac{n'}{2}} \right) -\frac{0.395}{{n'}^{1.2}} \right] ^{2-n} , \end{aligned}$$

(65)

where the generalized power-law index $n'(n,y_Y)$ is

$$\begin{aligned} n^\prime (n,y_Y) = n(1-y_Y) \frac{ny_Y+n+1}{2n^2y_Y^2+2ny_Y +n+1} \end{aligned}$$

(66)

for $y_Y= He/H_w$, which represents the dimensionless (effective laminar) plug width. Equation (65) must be solved iteratively for $H_w$ if $Re_p$ is specified, but defines $Re_p$ explicitly if $H_w$ is specified.

The function $\tau _w(|\nabla _a \varPsi |)$ is found to increase monotonically in the turbulent regime. Considering He fixed (the rheology) and taking $H_w \rightarrow \infty$, we find $y_Y \rightarrow 0, n^{\prime } \rightarrow n$ and $E \rightarrow 1$. Thus, we find that

$$\begin{aligned} Re_p \sim H_w ^{1-\frac{n}{2}} \log H_w ~~\Rightarrow ~~ | \nabla _a \varPsi | \sim \sqrt{\tau _w} \left[ \log \tau _w \right] ^{1/(2-n)} , \end{aligned}$$

as $\tau _w \rightarrow \infty$. Thus, $\tau _w$ grows slightly less fast than $| \nabla _a \varPsi |^2$, which would be the expectation in a fully rough turbulent regime, and the rheological dependency on n is minimal (in the exponent of the $\log$ term only), as would also be expected. Thus, we see essentially parallel curves in Fig. 5a at large $H_w$, independent of n.

1.3 A. 3 Transitional regimes

The transitional regime occurs between two critical values of the Metzner-Reed Reynolds number, $Re_{MR,1}(n,He/H_w)$ and $Re_{MR,2}(n,He/H_w)$ [35]. $Re_{MR,1}$ is the Reynolds number at which the flow is not laminar any more and $Re_{MR,2}$ is the Reynolds number at which the flow is fully turbulent. Between these values an interpolation is used based on the log of the friction factor. This results in a monotone variation in $H_w$ vs $Re_p$, as seen in Fig. 5. $Re_{MR,1}(n,He/H_w)$ and $Re_{MR,2}(n,He/H_w)$ can be straightforwardly mapped into $H_{w,1}$ and $H_{w,2}$ that bound shear stress for transitional flows.

1.4 A. 4 Friction factor

In passing, we note that in hydraulics, it is common to express closures in terms of a friction factor $f_f$, defined as the ratio of wall shear stress to inertial stress scale. This can of course be done here, leading to

$$\begin{aligned} \mathbf {S} = \frac{r_a \rho f_f \big | \nabla _a \varPsi \big | }{ 8 H^3} \nabla _a \varPsi . \end{aligned}$$

In [35], the relevant expressions for $f_f$ are given. For simpler fluids, such as power-law fluids, $f_f$ is straightforward to evaluate across all regimes, and the Metzner–Reed Reynolds number has proven to be a useful tool. For more complex fluids, the Metzner–Reed Reynolds number is not explicitly defined in terms of the process variables and $f_f$ is just another level of algebraic complexity that obscures the essential map** between local mean velocity ($Re_p$) and wall shear stress ($H_w$). The only advantage apparent to us in using $f_f$ for Herschel–Bulkley fluids is a degree of familiarity with the friction-factor concept.

Fig. 5

The closure $H_w(Re_p)$ from [35], showing asymptotic behaviour: a $H_w \rightarrow \infty$; b $H_w \rightarrow He$. The closure is plotted for $He = 100$ and $n = 0.2,~0.4,~0.6,~0.8,~1$: green—laminar; red—transitional; black— turbulent. (Color figure online)

Fig. 6

Variation of $\bar{D}_{1D}$(blue lines), $D_{T,1D}$(black lines) and $D_{T,1D}^*$(red lines) with wall shear stress for $n=0.2,0.4,0.6,0.8$ and 1

1.5 A. 5 Evaluating the Taylor dispersion terms

Similar to Sect. 1, we can find the averaged turbulent diffusivity and Taylor dispersion coefficients (${\bar{D}}, D_T, D_T^*$) using the method introduced in [35]. We first construct the dimensional parameters, e.g.

$$\begin{aligned} {\hat{D}}_T = D_T \delta _0 {\hat{r}}_a \hat{{\bar{W}}}_0, \end{aligned}$$

and then rescale it using the scaling defined in [35]. We eventually find

$$\begin{aligned} D_T = \frac{1}{2} |\nabla _a \varPsi | D_{T,1D}, \end{aligned}$$

(67)

where $D_{T,1D}$ is the dispersion coefficient obtained assuming a locally 1D channel flow [35]. Similar relations can be derived for $\bar{D}$ and $D_T^*$, i.e. multiplying the 1D results from [35] by $|\nabla _a \varPsi |/2$.

It is worthwhile to compare the values of $\bar{D}_{1D},~D_{T,1D}$ and $D_{T,1D}^*$ (or equivalently $\bar{D},~D_{T}$ and $D_{T}^*$). Figure 6 plots turbulent diffusivity and dispersion coefficients as a function of wall shear stress for fully turbulent flows. $H_{w,1}$ and $H_{w,2}$ are defined in Sect.1. As Fig. 6 shows, ${\bar{D}}$ is 2–3 orders of magnitude smaller than $D_T$. This is a typical feature of turbulent Taylor dispersion [42]. In addition, $D_T^*$ is almost always larger than $D_T$. This is interesting, although the results computed so far have not revealed where these terms become important.

Appendix B: Variational principles

Equation (25) is an elliptic second-order equation, in which time evolution enters only via the fluid concentrations (see Sect. 3) or via flow rate changes. Here, we develop the variational theory relevant to solving (25) in a rectangle $\varOmega$ with boundary $\partial \varOmega _{\varPsi } \bigcup \partial \varOmega _{S}$, under which conditions (28) and (29), respectively are satisfied. We regard any suitably smooth $\tilde{\varPsi }$ as an admissible stream function provided that (28) is satisfied. Similarly, $\tilde{\mathbf {S}}$ will be regarded as admissible provided that

$$\begin{aligned} \mathbf {\nabla }_a \cdot [\tilde{\mathbf {S}} + \mathbf {b}] = 0 , \end{aligned}$$

and that (29) is satisfied. The following statements are easily proven using Green’s theorem in the plane:

• For any admissible $\tilde{\varPsi }$ & $\tilde{\mathbf {S}}$:

  $$\begin{aligned} 0 = \int _{\varOmega } \tilde{\varPsi } \mathbf {\nabla }_a \cdot \mathbf {b} - \nabla _a \tilde{\varPsi } \cdot \tilde{\mathbf {S}}~\mathrm{d}\varOmega + \int _{\partial \varOmega _{\varPsi }} \varPsi _b \tilde{\mathbf {S}}\cdot \mathbf {n}~\mathrm{d}s + \int _{\partial \varOmega _{S}} \tilde{\varPsi } f ~\mathrm{d}s. \end{aligned}$$

  (68)

• For $\varPsi$ & ${\mathbf {S}}$ that solve (25) with boundary conditions (28) and (29):

  $$\begin{aligned} 0 = \int _{\varOmega } {\varPsi } \mathbf {\nabla }_a \cdot \mathbf {b} - \nabla _a {\varPsi } \cdot {\mathbf {S}}~\mathrm{d}\varOmega + \int _{\partial \varOmega _{\varPsi }} \varPsi _b {\mathbf {S}}\cdot \mathbf {n}~\mathrm{d}s + \int _{\partial \varOmega _{S}} {\varPsi } f ~\mathrm{d}s . \end{aligned}$$

  (69)

• For the solution $\varPsi$ & ${\mathbf {S}}$, and any other admissible $\tilde{\varPsi }$:

  $$\begin{aligned} 0 = \int _{\varOmega } [\tilde{\varPsi } - \varPsi ] \mathbf {\nabla }_a \cdot \mathbf {b} - [\nabla _a \tilde{\varPsi } - \nabla _a \varPsi ] \cdot \mathbf {S}~\mathrm{d}\varOmega + \int _{\partial \varOmega _{S}} [\tilde{\varPsi }- \varPsi ] f ~\mathrm{d}s . \end{aligned}$$

  (70)

• For the solution $\varPsi$ & ${\mathbf {S}}$, and any other admissible $\tilde{\mathbf {S}}$:

  $$\begin{aligned} \int _{\varOmega } \nabla _a \varPsi \cdot [\tilde{\mathbf {S}} - \mathbf {S}] ~\mathrm{d}\varOmega = \int _{\partial \varOmega _{\varPsi }} \varPsi _b [\tilde{\mathbf {S}}-\mathbf {S}]\cdot \mathbf {n}~\mathrm{d}s . \end{aligned}$$

  (71)

Now we consider the closure relationship defining $\mathbf {S}$, which is outlined in Appendix A. Provided that $| \nabla _a \varPsi | > 0$ or equivalently $|\mathbf {S}| > r_a \tau _Y /H$, we can write this as

$$\begin{aligned} |\mathbf {S}|(| \nabla _a \varPsi |) = \frac{r_a}{H} \tau _w(| \nabla _a \varPsi |) = r_a \chi (| \nabla _a \varPsi |) + \frac{r_a \tau _Y }{H}. \end{aligned}$$

(72)

The function $\chi (| \nabla _a \varPsi |)$ represents the contribution to the modified pressure gradient that is surplus to that needed to yield the fluid locally. It is continuous and strictly monotone. As the flow transitions through regimes, from laminar through to turbulent, the gradient of $\chi$ is continuous within any flow regime (but discontinuous when the flow transitions between regimes). Recall that $\chi (| \nabla _a \varPsi |)$ also has a local dependency on $(\phi ,\xi ,t)$ through the local geometry and fluid concentrations present. However, in general we may represent $|\mathbf {S}|$ graphically (at any $(\phi ,\xi ,t)$) as in Fig. 7a.

Fig. 7

a $|\mathbf {S}|(| \nabla _a \varPsi |)$; b $| \nabla _a \varPsi |(|\mathbf {S}|)$. The shaded areas contribute to the dissipation and potential functions. The shaded areas sum to $|\mathbf {S}|| \nabla _a \varPsi |$, used to establish Lemma 3

1.1 B. 1 Stream function and pressure potential functionals

The stream function potential functional $J(\tilde{\varPsi })$ is defined as

$$\begin{aligned} J(\tilde{\varPsi })= & {} \int _{\varOmega } \int _0^{| \nabla _a \tilde{\varPsi } |} |\mathbf {S}|(x) ~\mathrm{d}x - \tilde{\varPsi } \mathbf {\nabla }_a \cdot \mathbf {b} ~\mathrm{d}\varOmega - \int _{\partial \varOmega _{S}} \tilde{\varPsi } f ~\mathrm{d}s , \end{aligned}$$

(73)

which has the following property.

Lemma 1

The solution $\varPsi$ minimizes $J(\tilde{\varPsi })$ over all admissible $\tilde{\varPsi }$.

Proof

We look at:

$$\begin{aligned} J(\tilde{\varPsi })- J({\varPsi })= & {} \int _{\varOmega } \left( \int ^{| \nabla _a \tilde{\varPsi } |}_{| \nabla _a \varPsi |} |\mathbf {S}|(x) ~\mathrm{d}x \right) - (\tilde{\varPsi } - \varPsi ) \mathbf {\nabla }_a \cdot \mathbf {b} ~\mathrm{d}\varOmega - \int _{\partial \varOmega _{S}} (\tilde{\varPsi } - \varPsi ) f ~\mathrm{d}s\\= & {} \int _{\varOmega } \int ^{| \nabla _a \tilde{\varPsi } |}_{| \nabla _a \varPsi |} |\mathbf {S}|(x) ~\mathrm{d}x ~\mathrm{d}\varOmega - \int _{\varOmega } [\nabla _a \tilde{\varPsi } - \nabla _a \varPsi ] \cdot \mathbf {S}(| \nabla _a \varPsi |) ~\mathrm{d}\varOmega \\\ge & {} \int _{\varOmega } \int ^{| \nabla _a \tilde{\varPsi } |}_{| \nabla _a \varPsi |} |\mathbf {S}|(x) ~\mathrm{d}x ~\mathrm{d}\varOmega - \int _{\varOmega } (|\nabla _a \tilde{\varPsi }| - |\nabla _a \varPsi | ) | \mathbf {S}|(| \nabla _a \varPsi |) ~\mathrm{d}\varOmega \\= & {} \int _{\varOmega } \int ^{| \nabla _a \tilde{\varPsi } |}_{| \nabla _a \varPsi |} (|\mathbf {S}|(x) - | \mathbf {S}|(| \nabla _a \varPsi |))~\mathrm{d}x \ge 0 . \end{aligned}$$

We have used (70) and then the Cauchy–Schwarz inequality above. In this last expression, note that $|\mathbf {S}|(x) > | \mathbf {S}|(| \nabla _a \varPsi |)$ whenever $x > | \nabla _a \varPsi |$ due to monotonicity. Thus, the sign of the integrant changes according to the limits and the integral is always positive. $\square$

The minimization of $J(\tilde{\varPsi })$ can also be expressed as a variational inequality, which is the basis of the augmented Lagrangian method used. Considering now Fig. 7b, we can define the function $| \nabla _a \varPsi |(|\mathbf {S}|)$ by effectively inverting $|\mathbf {S}|(| \nabla _a \varPsi |)$, as illustrated, i.e.

$$\begin{aligned} | \nabla _a \varPsi |(|\mathbf {S}|) = \left\{ \begin{array}{lcl} |\mathbf {S}|^{-1}(| \nabla _a \varPsi |), &{} ~~&{} |\mathbf {S}| > \frac{r_a \tau _Y}{H}, \\ 0, &{} &{} |\mathbf {S}| \le \frac{r_a \tau _Y}{H} . \end{array} \right. \end{aligned}$$

We now define the pressure potential function $K(\tilde{\mathbf {S}})$ for any admissible $\tilde{\mathbf {S}}$ as follows.

$$\begin{aligned} K(\tilde{\mathbf {S}}) = - \int _{\varOmega } \int _{\frac{r_a \tau _Y}{H}}^{|\tilde{\mathbf {S}}|} | \nabla _a \varPsi |(y)~\mathrm{d}y ~\mathrm{d}\varOmega + \int _{\partial \varOmega _\varPsi } \varPsi _b \tilde{\mathbf {S}} \cdot \mathbf {n}~\mathrm{d}s. \end{aligned}$$

(74)

Analogous to Lemma 1, we have the following.

Lemma 2

The solution $\mathbf {S}$ maximizes $K(\tilde{\mathbf {S}})$ over all admissible $\tilde{\mathbf {S}}$.

Proof

We look at

$$\begin{aligned} K(\mathbf {S}) - K(\tilde{\mathbf {S}})= & {} \int _{\varOmega } \int _{|\mathbf {S}|}^{|\tilde{\mathbf {S}}|} | \nabla _a \varPsi |(y)~\mathrm{d}y ~\mathrm{d}\varOmega + \int _{\partial \varOmega _\varPsi } \varPsi _b [\mathbf {S} - \tilde{\mathbf {S}} ] \cdot \mathbf {n}~\mathrm{d}s\\= & {} \int _{\varOmega } \int _{|\mathbf {S}|}^{|\tilde{\mathbf {S}}|} | \nabla _a \varPsi |(y)~\mathrm{d}y ~\mathrm{d}\varOmega - \int _{\varOmega } \nabla _a \varPsi \cdot [\tilde{\mathbf {S}} - \mathbf {S}] ~\mathrm{d}\varOmega \\\ge & {} \int _{\varOmega } \int _{|\mathbf {S}|}^{|\tilde{\mathbf {S}}|} | \nabla _a \varPsi |(y)~\mathrm{d}y ~\mathrm{d}\varOmega - \int _{\varOmega } |\nabla _a \varPsi | (|\tilde{\mathbf {S}}| - |\mathbf {S}|) ~\mathrm{d}\varOmega \\= & {} \int _{\varOmega } \int _{|\mathbf {S}|}^{|\tilde{\mathbf {S}}|} (| \nabla _a \varPsi |(y) - |\nabla _a \varPsi | )~\mathrm{d}y ~\mathrm{d}\varOmega \ge 0 . \end{aligned}$$

Here we have used (71) and then the Cauchy–Schwarz inequality. In the last expression, note that if $|\tilde{\mathbf {S}}| > | \mathbf {S}|$ then $| \nabla _a \varPsi |(y) > | \nabla _a \varPsi |$ due to monotonicity; similarly when $|\tilde{\mathbf {S}}| < | \mathbf {S}|$. Thus, the sign of the integrand changes according to the limits and the integral is always positive. $\square$

Finally, since the shaded areas in Fig. 7a, b, sum to give $|\mathbf {S}| |\nabla _a \varPsi |$, we have

$$\begin{aligned} \int _{\varOmega } |\mathbf {S}| |\nabla _a \varPsi | ~\mathrm{d}\varOmega \int _{\varOmega } \int _0^{| \nabla _a \varPsi |} |\mathbf {S}|(x) ~\mathrm{d}x ~\mathrm{d}\varOmega + \int _{\varOmega } \int _{\frac{r_a \tau _Y}{H}}^{|\mathbf {S}|} | \nabla _a \varPsi |(y)~\mathrm{d}y ~\mathrm{d}\varOmega , \end{aligned}$$

which can be combined with (69). In combination with the above-mentioned minimization and maximization principles, we have the following minimax principle:

Lemma 3

The solution pair $(\mathbf {S}, \varPsi )$ satisfy

$$\begin{aligned} K(\tilde{\mathbf {S}}) \le K(\mathbf {S}) = J(\varPsi ) \le J(\tilde{\varPsi }) , \end{aligned}$$

for all admissible $\tilde{\mathbf {S}}$ and $\tilde{\varPsi }$.

In the porous media context, similar variational principles are used to describe nonlinear filtration, e.g. [43, 44]. The first (integral) terms in both $J(\cdot )$ and $K(\cdot )$ are referred to as dissipation potentials. In the porous media context, one is often more concerned with determining the pressure field, and a stream function formulation is restrictive in only applying to 2D flows. Thus, typically $K(\cdot )$ is referred to as the primal potential and $J(\cdot )$ as dual potential. Here however, we treat Lemma 1 as the primal principle as it leads to a unique stream function (see below). Note that the terminology dissipation results from (69) which is essentially a mechanical energy balance, equating the dissipation within the system to the work done by buoyancy forces and by the boundary terms.

1.2 B. 2 Existence and uniqueness

Lemma 1 is the basis of an existence and uniqueness result. Firstly, note that $J(\tilde{\varPsi })$ can be split as follows:

$$\begin{aligned} J(\tilde{\varPsi })= & {} \int _{\varOmega } r_a \int _0^{| \nabla _a \tilde{\varPsi } |} \chi (x) ~\mathrm{d}x ~\mathrm{d}\varOmega + \int _{\varOmega } \frac{r_a \tau _Y}{H} | \nabla _a \tilde{\varPsi } | ~\mathrm{d}\varOmega - \int _{\varOmega } \tilde{\varPsi } \mathbf {\nabla }_a \cdot \mathbf {b} ~\mathrm{d}\varOmega - \int _{\partial \varOmega _{S}} \tilde{\varPsi } f ~\mathrm{d}s , \nonumber \\= & {} J_0(\tilde{\varPsi }) + J_1(\tilde{\varPsi }) - L(\tilde{\varPsi }). \end{aligned}$$

(75)

The functional $J_0$ is strictly convex as the integrand has second derivative equal to the derivative of $\chi$, which is a strictly monotone function. The functional $J_1$ containing the yield stress is convex, bit not strictly. Finally, L denotes the linear parts.

This problem structure is in a format where standard results may be applied (e.g. [45, Theorem 2.1, Chap. 5]), to guarantee the existence of a unique weak solution. The relevant function space is determined by the behaviour of $J_0$ as $||\tilde{\varPsi }|| \rightarrow \infty$. From the analysis in Appendix A, we see that in the fully turbulent regime $| \nabla _a \varPsi | \sim \sqrt{\tau _w} \left[ \log \tau _w \right] ^{1/(2-n)}$ as $| \nabla _a \varPsi | \rightarrow \infty$ , and therefore also

$$\begin{aligned} \chi \left[ \log \chi \right] ^{2/(2-n)} \sim | \nabla _a \varPsi |^2 . \end{aligned}$$

This suggests $\chi \gtrsim | \nabla _a \varPsi |^{2 - \epsilon }$ for any small $\epsilon > 0$ as $| \nabla _a \varPsi | \rightarrow \infty$, i.e. the $\log$ term is less significant than any power.

Proceeding now as in [10], we can infer that $\varPsi \in W^{1,3-\epsilon }(\varOmega )$, with further details specific to the boundary conditions to be considered. It is interesting to compare with the results for the purely laminar case considered in [10], where the growth of $\chi$ using only the laminar closure resulted in $\varPsi \in W^{1,1+n_{\min }}(\varOmega )$. It appears that the turbulent closure results in a smoother weak solution and a function space largely independent of the rheology.

1.3 B. 3 Pressure formulation and stress maximization

The maximization of $K(\tilde{\mathbf {S}})$ leads to an equality as the optimality condition. The resulting partial differential equation (for the pressure) does not, however, uniquely determine $\mathbf {S}$, where the flow is stationary. We may derive the pressure equation directly by reorganizing (22) to eliminate $\varPsi$ instead of p:

$$\begin{aligned} 0= & {} \mathbf {\nabla }_a \cdot \left[ r_a^2 \frac{|\nabla _a \varPsi |(|\mathbf {S}|) }{|\mathbf {S}|} \left( \nabla _a p + \mathbf {b}_p \right) \right] , \end{aligned}$$

(76)

$$\begin{aligned} \mathbf {b}_p= & {} \frac{\rho -1}{Fr^2}\left( -\sin \beta \sin \pi \phi , \cos \beta \right) , \end{aligned}$$

(77)

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

(78)

The function $|\nabla _a \varPsi |(|\mathbf {S}|)$ is qualitatively illustrated in Fig. 7b.