An Investigation of Polymer Mechanical Degradation in Radial Well Geometry

Abstract

It can be challenging to forecast polymer flooding performance at the field, in large part because of the complex non-Newtonian fluid rheology of polymer solutions. In this paper, we apply a model, previously developed to study linear core flooding experiments, to investigate polymer behaviour in radial flow near a vertical injector. The polymer system studied here is very common, HPAM in seawater. One key result is that a grid resolution on the order of millimetres is needed near the wellbore to accurately capture the well pressure, and the amount of mechanical degradation. We also demonstrate that for typical injection rates and permeabilities, the apparent shear thickening and mechanical degradation flow regimes are only relevant to consider within a few metres from the well. For the purposes of full field simulations, a pure shear thinning model should therefore suffice to describe fluid flow outside of the well grid blocks. Approximate analytical expressions are derived to test the numerical model. The steady-state molecular weight far away from the well is shown to scale as $\propto {Q^{-0.65}\cdot {k^{0.425}}}$, where Q is the injection flow rate, and k is permeability. This scaling makes it possible to collect simulated values onto a single curve and can be used to predict mechanical degradation under different conditions. The results are in broad agreement with observations made of polymer mechanical degradation at the Dalia field. For the case of linear flow, there is an additional length dependency of degradation. The model then predicts an approximate power-law scaling $M_{\mathrm{w}L}\propto {L^{\omega }}$, with $M_{\mathrm{w}L}$ being the model molecular weight at a distance L from the inlet, which is consistent with recent laboratory experiments.

Appendices

Appendices

1.1 Derivation of Steady-State Molecular Weight as a Function of Distance

For the molar polymer concentration, a mass balance applied to a control volume V with boundary A yields

$$\begin{aligned} \frac{\partial }{\partial {t}}\int _{V} \phi {C_{\mathrm{mol}}}{{\,\mathrm{d\!}\,}}{V} + \int _{A} C_{\mathrm{mol}}\mathbf {u_p}\cdot \hat{\mathbf {n}}{{\,\mathrm{d\!}\,}}{A} = \int _{V} \phi {{\mathscr {R}}(C_{\mathrm{mol}})}{{\,\mathrm{d\!}\,}}{V}, \end{aligned}$$

(12)

where $\hat{\mathbf {n}}$ is the outward-pointing unit normal vector to the surface element ${{\,\mathrm{d\!}\,}}{A}$, $\mathbf {u}$ is the Darcy velocity in vector form, and $\mathbf {u_p}=\mathbf {u}/(1-\hbox {IPV}_0)$. Dividing by $\hbox {IPV}_0$ ensures that the correct polymer concentration is transported across the boundary. In differential form, the above equation becomes

$$\begin{aligned} \frac{\partial {(\phi {C_{\mathrm{mol}}})}}{\partial {t}} = -\nabla \cdot ({\mathbf {u}_{p}C_{\mathrm{mol}}}) + \phi {\mathscr {R}}(C_{\mathrm{mol}}). \end{aligned}$$

(13)

The reaction term, given in units of pore volume concentration per time, is

$$\begin{aligned} {\mathscr {R}}(C_{\mathrm{mol}}) = f_{\mathrm{rup}}\cdot {C_{\mathrm{mol}}}. \end{aligned}$$

(14)

At steady-state, the volumetric concentration $C_{\mathrm{pol}}$ is constant throughout the radial model. For the molar concentration, we obtain

$$\begin{aligned} \nabla \cdot ({\mathbf {u}_{p}C_{\mathrm{mol}}}) = \phi \cdot {f_{\mathrm{rup}}}\cdot {C_{\mathrm{mol}}} \end{aligned}$$

(15)

For radially symmetric flow, we get

$$\begin{aligned} \frac{1}{r}\cdot \frac{\mathrm{d}}{\mathrm{d}r}(ru_{r}C_{\mathrm{mol}})=\phi \cdot {f_{\mathrm{rup}}} \cdot {C_{\mathrm{mol}}} \end{aligned}$$

(16)

Inserting $u_r=Q/(2\pi {r}h(1-\hbox {IPV}_0)$ yields

$$\begin{aligned} \frac{Q}{2\pi {h}r(1-\hbox {IPV}_0)}\cdot \frac{\mathrm{d}C_{mol}}{\mathrm{d}r} =\phi \cdot {f_{\mathrm{rup}}}\cdot {C_{\mathrm{mol}}}, \end{aligned}$$

(17)

and using that $M_\mathrm{w}=C_{\mathrm{pol}}/C_{\mathrm{mol}}$ we obtain

$$\begin{aligned} \frac{\mathrm{d}C_{\mathrm{mol}}}{\mathrm{d}r}=-\frac{C_{\mathrm{pol}}}{M_{\mathrm{w}}^2}\cdot \frac{\mathrm{d}M_\mathrm{w}}{\mathrm{d}r} =\frac{2\pi {h}r\phi (1-\hbox {IPV}_0)}{Q}\cdot {f_{\mathrm{rup}}}\cdot {C_{\mathrm{mol}}}, \end{aligned}$$

(18)

from which Eq. (8) follows. For the case of linear 1D geometry, we can repeat the above procedure to obtain an identical kind of formula,

$$\begin{aligned} \frac{\mathrm{d}M_\mathrm{w}}{\mathrm{d}x} = -\frac{A_{l}\phi (1-\hbox {IPV}_0)}{Q}\cdot {f_{\mathrm{rup}}}\cdot {M_\mathrm{w}}, \end{aligned}$$

(19)

where $A_l$ is the constant, cross-sectional area of the core.

1.2 Approximate Analytical Formulas for Degraded $M_\mathrm{w}$

Consider radial flow outwards from an injector at flow rate Q, in a homogeneous reservoir. Based on the derivations in the previous section, we see that at steady state, the molecular weight as a function of radial distance r must satisfy Eq. (8) in the main text, where $f_{\mathrm{rup}}$ is given by Eq. (6). The formula used for apparent viscosity in the simulation model is

$$\begin{aligned} \eta&=\eta _s + (\eta _{\mathrm{sh}}-\eta _s)\cdot {\eta _{\mathrm{elf}}} \nonumber \\&\approx {(\eta _{\mathrm{sh}}-\eta _s)\cdot {\eta _{\mathrm{elf}}}} \nonumber \\&=\eta _{s}\cdot \eta _{\mathrm{sp},\mathrm{sh}}\cdot \eta _{\mathrm{elf}}\nonumber \\&=\eta _{s}\cdot {\eta _{\mathrm{sp}0}}\cdot (1+(\lambda _{1}{\dot{\gamma }})^{x})^{-\frac{n}{x}} \cdot (1+(\lambda _{2}{\dot{\gamma }})^{x_2})^{\frac{m+n}{x_2}}, \end{aligned}$$

(20)

where $\eta _{\mathrm{sp},\mathrm{sh}}=\frac{\eta _{\mathrm{sh}}}{\eta _s}-1$ is the specific viscosity for the shear thinning part of the apparent viscosity, and $\eta _{\mathrm{elf}}$ is the elongational viscosity factor. In the second line, we have made an approximation by dropping the solvent viscosity term in the expression for $\eta$. The zero shear specific viscosity, $\eta _{\mathrm{sp}0}$, is calculated from a cubic polynomial in the product of intrinsic viscosity and polymer concentration,

$$\begin{aligned} \eta _{\mathrm{sp}0} = [\eta ]C_{\mathrm{p}}+k^{\prime }[\eta ]^{2}C_{\mathrm{p}}^2 + k^{\prime \prime }[\eta ]^{3}C_{\mathrm{p}}^3, \end{aligned}$$

(21)

with the intrinsic viscosity obtained from the molecular weight via the Mark–Houwink equation,

$$\begin{aligned}{}[\eta ]=K\cdot {M_{\mathrm{w}}^a}, \end{aligned}$$

(22)

for constants K and a. In terms of the introduced notation and the approximation introduced above, the viscosity term in (7) becomes

$$\begin{aligned} \eta ^{\alpha _d}\approx {\eta _{s}^{\alpha _d}}\cdot {\eta _{\mathrm{sp}0}}^{\alpha _d} \cdot (1+(\lambda _{1}{\dot{\gamma }})^{x})^{-\frac{n\alpha _d}{x}} \cdot (1+(\lambda _{2}{\dot{\gamma }})^{x_2})^{\frac{(m+n)\alpha _d}{x_2}}. \end{aligned}$$

(23)

Next, since in the degradation regime ${\dot{\gamma }}\gg {1}$, we assume that

$$\begin{aligned} (1+(\lambda _{1}{\dot{\gamma }})^{x})^{-\frac{n}{x}} \approx (\lambda _{1}{\dot{\gamma }})^{-n}, \end{aligned}$$

(24)

and

$$\begin{aligned} (1+(\lambda _{2}{\dot{\gamma }})^{x_2})^{-\frac{m+n}{x_2}} \approx (\lambda _{2}{\dot{\gamma }})^{m+n}. \end{aligned}$$

(25)

The expressions for n, $\lambda _1$, and $\lambda _2$ are given by, respectively, Eqs. 4, 21, and 30 in Lohne et al. (2017). We reproduce those equations here:

$$\begin{aligned} n=n(M_\mathrm{w})=1-\frac{1}{1+\left( a_{n}\hbox {KC}_{\mathrm{p}}M_{\mathrm{w}}^a\right) ^{b_n}}, \end{aligned}$$

(26)

and

$$\begin{aligned} \lambda _{1}=\lambda _{1}(M_\mathrm{w})=\lambda _{a}\cdot \frac{\eta _{s}\eta _{\mathrm{sp}0}(M_\mathrm{w})M_\mathrm{w}}{C_{\mathrm{p}}T}, \end{aligned}$$

(27)

and

$$\begin{aligned} \lambda _{2}=\lambda _{2}(M_\mathrm{w})=\frac{1}{N_{\mathrm{De}}^{\star }}\cdot \frac{3}{5R_g} \cdot \frac{\phi }{1-\phi }\cdot \frac{\eta _{s}KM_{\mathrm{w}}^{a+1}}{T}. \end{aligned}$$

(28)

In the degradation regime, the main contribution to the viscosity is from the shear thickening part. Hence, to make the analysis tractable we assume in the sequel that n is constant, i.e., independent of $M_\mathrm{w}$. With all these approximations, Eq. (23) reduces to

$$\begin{aligned} \eta ^{\alpha _d}&\approx {\eta _{s}^{\alpha _d}}\cdot {\eta _{\mathrm{sp}0}}^{\alpha _d} \cdot \lambda _{1}^{-n\alpha _d}\cdot \lambda _{2}^{(m+n)\alpha _d}\cdot {{\dot{\gamma }}^{m\alpha _d}} \nonumber \\&=\eta _{s}^{\alpha _d}\cdot {\eta _{\mathrm{sp}0}^{\alpha _{d}(1-n)}}\cdot \left( \frac{\lambda _{a}\eta _{s}M_\mathrm{w}}{C_{\mathrm{p}}T}\right) ^{-n\alpha _d} \cdot \left( \frac{3}{5N_{\mathrm{De}}^{\star }R_g}\cdot \frac{\phi }{1-\phi }\cdot \frac{\eta _{s}KM_{\mathrm{w}}^{1+a}}{T}\right) ^{(m+n)\alpha _d} \cdot {{\dot{\gamma }}^{m\alpha _d}}, \end{aligned}$$

(29)

Furthermore, we only include the first term in the expression for $\eta _{\mathrm{sp}0}$,

$$\begin{aligned} \eta _{\mathrm{sp}0}\approx {f_{1}(M_\mathrm{w})}\equiv {\hbox {KC}_{\mathrm{p}}M_{\mathrm{w}}^a}. \end{aligned}$$

(30)

The last two approximations are the boldest, especially Eq. (30), however as we shall see, without them it is not possible to integrate the degradation equation in terms of elementary functions. By combining Eqs. (3), (7), (8), (29) and (30), we obtain

$$\begin{aligned} \frac{\mathrm{d}M_\mathrm{w}}{\mathrm{d}r}&\approx {-\frac{2\pi {h}r\phi (1-\hbox {IPV}_0)}{Q}} \cdot \left( r_{\mathrm{deg}}\eta _{s}K{C_\mathrm{p}}\right) ^{\alpha _d} \cdot \sqrt{\frac{\phi }{2kC}}\nonumber \\&\quad \cdot \sqrt{R_{k}(1-\hbox {IPV}_0)} \cdot \left( \frac{\lambda _{a}\eta _{s}K}{T}\right) ^{-n\alpha _d}\nonumber \\&\quad \cdot \left( \frac{3}{5N_{\mathrm{De}}^{\star }R_g}\cdot \frac{\phi }{1-\phi }\cdot \frac{\eta _{s}K}{T}\right) ^{(m+n)\alpha _d} \cdot {M_{\mathrm{w}}^y}\cdot \left( \frac{\Omega }{r}\right) ^{\alpha _{d}(1+m)}, \end{aligned}$$

(31)

with y given by

$$\begin{aligned} y=1+\beta _d+\alpha _{d}(a(1+m)+m), \end{aligned}$$

(32)

and where we have defined $\Omega$ as the part of ${\dot{\gamma }}$ that does not depend on r. From Eq. (2), using that $u=Q/2\pi {r}h$, this means that

$$\begin{aligned} \Omega = \frac{4\alpha _{c}Q}{2\pi {h}\sqrt{8k{\phi }}}\cdot {\sqrt{\frac{R_\mathrm{k}}{1-\hbox {IPV}_0}}}. \end{aligned}$$

(33)

By collecting all terms other than $M_\mathrm{w}$ and r into a factor $\zeta$, it is seen that we approximate the original problem by the separable ODE

$$\begin{aligned} \frac{\mathrm{d}M_\mathrm{w}}{\mathrm{d}r}={-\zeta }\cdot {M_{\mathrm{w}}^y}\cdot {r^{\mathrm{w}}}, \end{aligned}$$

(34)

with $w=1-\alpha _{d}(1+m)$. Let $r_d$ be the radius beyond which there is no more, or negligible, mechanical degradation. By substituting ${\dot{\gamma }}=\Omega /r$, and integrating from $r_\mathrm{w}$ to $r_d$, we get

$$\begin{aligned} \int _{M_{\mathrm{w}0}}^{M_{\mathrm{wd}}} {M_{\mathrm{w}}^{-y}}{{\,\mathrm{d\!}\,}}{M_\mathrm{w}}=\Omega ^{1+w}\zeta \cdot \int _{{\dot{\gamma }}_{w}}^{{\dot{\gamma }}_d} {\frac{1}{{\dot{\gamma }}^{2+w}}}{{\,\mathrm{d\!}\,}}{{\dot{\gamma }}}, \end{aligned}$$

(35)

where $M_{\mathrm{wd}}=M_{\mathrm{w}}(r_d)$ is the steady-state molecular weight far away from the injection well, and $M_{\mathrm{w}0}$ is the initial molecular weight. From this, it immediately follows that

$$\begin{aligned} \frac{1}{1-y}\cdot \left( M_{\mathrm{wd}}^{1-y}-M_{\mathrm{w}0}^{1-y}\right) = \frac{-\Omega ^{1+w}\zeta }{1+w}\cdot \left( {\dot{\gamma }}_{d}^{-(1+w)}-{\dot{\gamma }}_{w}^{-(1+w)}\right) . \end{aligned}$$

(36)

By virtue of the definition of $r_d$, the shear rate ${\dot{\gamma }}_d$ will be negligible compared with ${\dot{\gamma }}_w$. Thus, as a final approximation, we will assume${\dot{\gamma }}_{d}^{-(1+w)}\approx {0}$. This is justified by comparing with the actual simulation results in the radial grid. Finally, by performing the necessary algebraic manipulations, one can show that

$$\begin{aligned} \frac{M_{\mathrm{wd}}}{M_{\mathrm{w}0}}\approx \frac{1}{(1+(y-1)\chi _r)^{\frac{1}{y-1}}}, \end{aligned}$$

(37)

where $\chi _r$ is the following complicated expression:

$$\begin{aligned} \chi _r = \chi _{0}\cdot {(1-\hbox {IPV}_0)\cdot {R_\mathrm{k}}}\cdot \frac{\phi ^{1+(m+n)\alpha _d}}{(1-\phi )^{(m+n)\alpha _d}} \cdot {C_{\mathrm{p}}^{\alpha _d}}\cdot {T^{-m\alpha _d}}\cdot {r_\mathrm{w}}\cdot {M_{\mathrm{w}0}^{y-1}} \cdot \frac{{\dot{\gamma }}_{w}^{\alpha _{d}(1+m)-1}}{k},\nonumber \\ \end{aligned}$$

(38)

with

$$\begin{aligned} \chi _0 =\frac{\alpha _c}{\sqrt{C}}\cdot \frac{1}{\alpha _{d}(1+m)-2} \cdot {r_{\mathrm{deg}}}^{\alpha _d} \cdot {\lambda _{a}}^{-n\alpha _d} \cdot \left( \frac{3}{5N_{\mathrm{De}}^{\star }{R_g}}\right) ^{(m+n)\alpha _d} \cdot (\eta _{s}K)^{\alpha _{d}(1+m)}.\qquad \end{aligned}$$

(39)

Table 2 Rock and fluid properties used as input parameters for the radial simulations

For the particular choice of model parameters used in this paper (Tables 2 and 3), we find that $\chi _{r}\propto {\frac{{\dot{\gamma }}_{w}^{6.5}}{k}}$. By inserting the definition of ${\dot{\gamma }}_w$, and collecting equal terms, one can further reduce (38) to

$$\begin{aligned} \chi _r= & {} \chi _{0}\cdot {\left( \frac{\alpha _c}{\sqrt{2}\pi }\right) ^{\alpha _{d}(1+m)-1}} \cdot {Q^{a_1}}\cdot {k^{a_2}}\cdot {T^{a_3}}\cdot {\phi ^{a_4}}\cdot {(1-\phi )^{a_5}}\nonumber \\&\cdot ~{(1-\hbox {IPV}_0)^{a_6}}\cdot {R_{k}^{a_7}}\cdot {r_{w}^{a_8}}\cdot {h^{a_9}} \cdot {M_{\mathrm{w}0}^{a_{10}}} \cdot {C_{\mathrm{p}}^{a_{11}}}, \end{aligned}$$

(40)

with the exponents given in Table 4. We remark that the temperature dependence is not fully captured by the T-term in Eq. (40), as we also have $\eta _s=\eta _{s}(T)$ and $K=K(T)$ in the term $\chi _0$. For very large $\chi _r$, Eq. (37) becomes

$$\begin{aligned} \frac{M_{\mathrm{wd}}}{M_{\mathrm{w}0}}\approx ((y-1)\cdot \chi _{r})^{-\frac{1}{y-1}} \end{aligned}$$

(41)

Table 3 Polymer properties used as input to the simulations

Table 4 Exponents appearing in Eq. (40)

1.3 Alternative Approximate Equations

Returning to Eq. (29), let us approximate $\eta _{sp0}$ in a different way than in the development of Eq. (37). Let R denote the ratio between the alternative formula and the original one, i.e.,

$$\begin{aligned} R\equiv \frac{\eta _{\mathrm{sp}0}}{f_{1}}=\frac{\eta _{\mathrm{sp}0}}{\hbox {KC}_{\mathrm{p}}M_{\mathrm{w}}^a}. \end{aligned}$$

(42)

Then, from Eq. (29) it is clear that the right-hand side of Eq. (34) must be multiplied by a factor $R^{\alpha _{d}(1-n)}$. Equivalently, when integrating the $M_\mathrm{w}$ and r terms, the integrand on the left-hand side of Eq. (35) must be multiplied by a factor $R^{\alpha _{d}(n-1)}$.

1.3.1 Power-Law

If the factor R is proportional to a power of $M_{\mathrm{w}}$, the integration can be performed in exactly the same way as before, and we end up with the same kind of formula as (37). The only difference is that the definition of $\chi _r$ must be modified with an extra prefactor, in addition to changing the exponent y. For instance, we can assume that $\eta _{\mathrm{sp}0}\approx {f_{3}}$, where $f_3$ is the third-order term in Eq. (21). An example of using the latter assumption is shown in the left plot of Fig. 4 (green curve).

1.3.2 Including Both the First and the Third-Order Term in $\eta _{\mathrm{sp}0}$

Another possibility is to improve the approximation of the cubic formula by only disregarding the quadratic term (which makes the smallest contribution). In this case, we get

$$\begin{aligned} R^{\alpha _{d}(1-n)}=\frac{f_1+f_3}{f_1}= 1+\frac{f_3}{f_1}=1+k^{\prime \prime }K^{2}C_{\mathrm{p}}^{2}M_{\mathrm{w}}^{2a}, \end{aligned}$$

resulting in the approximate equation

$$\begin{aligned} \int _{M_{\mathrm{w}0}}^{M_{\mathrm{wd}}}{(1+BM_{\mathrm{w}}^{2a})^{\alpha _{d}(n-1)} \cdot {}M_{\mathrm{w}}^{-y}}{{\,\mathrm{d\!}\,}}{M_\mathrm{w}}=\Omega ^{1+w}\zeta \cdot \int _{{\dot{\gamma }}_{w}}^{{\dot{\gamma }}_d} {\frac{1}{{\dot{\gamma }}^{2+w}}}{{\,\mathrm{d\!}\,}}{{\dot{\gamma }}}, \end{aligned}$$

(43)

with $B=k^{\prime \prime }K^{2}C_{\mathrm{p}}^{2}$. Substituting $u=B\cdot {M_{\mathrm{w}}^{2a}}$, we transform the integral on the left-hand side to

$$\begin{aligned} \frac{B^{\frac{y-1}{2a}}}{2a} \int _{u_0}^{u_d} (1+u)^{C}\cdot {u^D}{{\,\mathrm{d\!}\,}}{u} \end{aligned}$$

with $C=(n-1)\alpha _d$, $D=(1-y-2a)/2a$, $u_0=B\cdot {M_{\mathrm{w}0}^{2a}}$, and $u_d=B\cdot {M_{\mathrm{w}d}^{2a}}$. This definite integral may be expressed in terms of the Gaussian hypergeometric function $_{2}F_{1}$,

The left-hand side of Eq. (43) now becomes

If we denote the hypergeometric function evaluated at the upper and lower limits by, respectively, ${\mathscr {F}}_d$ and ${\mathscr {F}}_0$, we obtain the following approximate relationship:

$$\begin{aligned} M_{\mathrm{wd}}^{1-y}\cdot {{\mathscr {F}}_d}-M_{\mathrm{w}0}^{1-y}\cdot {{\mathscr {F}}_0}\approx \frac{(1-y)\Omega ^{1+w}\zeta }{1+w}\cdot {\dot{\gamma }}_{w}^{-(1+w)}. \end{aligned}$$

(44)

To isolate $M_{\mathrm{wd}}$ we need to compute the inverse of $_{2}F_{1}$, which we do by numerically solving the implicit Eq. (44).

1.4 Analytical Solution by Means of Numerical Integration

As documented in the main text, the use of Eq. (37) underestimates the amount of degradation. This is in large part due to neglecting the third-order term in Eq. (21). The assumption of a constant Carreau–Yasuda exponent n also explains a part of the discrepancy between simulation and calculation. This can be verified by performing numerical integration of Eq. (8). Tracing through the various definitions, one can show that the integral we need to solve is

$$\begin{aligned} \frac{\mathrm{d}M_\mathrm{w}}{\mathrm{d}r} = -\frac{2{\pi }h\phi (1-\hbox {IPV}_0)\Omega ^{\alpha _d}}{Q\sqrt{2C}}\cdot \sqrt{\frac{\phi }{k}} \cdot \sqrt{R_{k}(1-\hbox {IPV}_0)}\cdot (r_{\mathrm{deg}}\eta _s)^{\alpha _d}\cdot {r^{1-\alpha _d}}\cdot {\eta _{r}^{\alpha _d}} \cdot {M_{\mathrm{w}}^{1+\beta _d}}, \nonumber \\ \end{aligned}$$

(45)

where the relative apparent viscosity $\eta _r$ is given by

$$\begin{aligned} \eta _r = 1 + \eta _{\mathrm{sp}0}\cdot {(1+(\lambda _{1}{\dot{\gamma }})^{x})^{-n/x}}\cdot (1+(\lambda _{2}{\dot{\gamma }})^{x_2})^{(m+n)/x_2}. \end{aligned}$$

(46)

Numerical integration was performed using Python’s odeint function, which is essentially a wrapper to the LSODA solver in the FORTRAN library odepack. According to the online documentation (SciPy.org 2018), the integrator switches automatically between non-stiff (Adams) and stiff (backwards differentiation formulas, BDF) methods, depending on information available at the end of each integration step. See, e.g., Petzold (1983) for more details regarding this method.

1.5 Linear Geometry and Approximate Scaling Relationships

The procedure used to derive Eqs. (37) and (45) may also be applied to the case of linear 1D geometry by using Eq. (19). Let $M_{\mathrm{w}L}$ denote the steady-state molecular weight at $x=L$. By making the same assumptions as before, one can show that

$$\begin{aligned} \frac{M_{\mathrm{w}L}}{M_{\mathrm{w}0}}\approx \frac{1}{(1+(y-1)\chi _l)^{\frac{1}{y-1}}}, \end{aligned}$$

(47)

where $\chi _{_{l}}$ is similar to $\chi _r$ (not shown here). As a further test of the derived analytical expressions, we conducted a new series of simulations, this time in linear geometry at the core scale. For each simulation, the core permeability was chosen to be the same as in a corresponding radial case, and the flow rate was selected so that ${\dot{\gamma }}_{_{l}}={\dot{\gamma }}_w$, where ${\dot{\gamma }}_{_{l}}$ is the in situ shear rate in the core. The $L=7~\hbox {cm}$ core was discretized into 100 grid blocks, i.e., with a constant grid spacing $\Delta {x}=0.7~\hbox {mm}$. Figure 12 shows a comparison between the numerical integration results, and values obtained from the explicit first-order formula and the simulations. As discussed in the main text, the approximate formulas tend to overpredict $M_\mathrm{w}$. For the core scale simulations, very good agreement between simulation and numerical integration was obtained, the maximal relative error being less than 1%.

Fig. 12

Left plot: Simulated and calculated molecular weight at the exterior grid boundary, plotted versus the exact solution (numerical integration, Eq. (45)). The red triangles are obtained from the analytical formula, Eq. (37), while the blue circles represent the simulations. Right plot: Simulated and calculated molecular weight at the core effluent for a series of linear simulations, plotted versus the exact solutions

Note that under the assumptions used to derive Eq. (47), the model predicts an approximate power-law scaling for $M_\mathrm{w}$ as a function of distance in linear flow, all else being equal. Specifically, if the constant 1 can be neglected in the denominator, we have

$$\begin{aligned} \frac{M_{\mathrm{w}L}}{M_{\mathrm{w}0}}\propto {L^\omega }, \end{aligned}$$

(48)

with $\omega =1-y=-0.1$ in the present case. If approximating $\eta _{\mathrm{sp}0}$ with the third-order term rather than the first-order term (Sect. C.1), a slightly different exponent is obtained, but power-law scaling was seen for all considered cases in which degradation was substantial.