The Kinetics of Polymer Adsorption in Transport Through Porous Media: Basic Theory

1. Introduction

1.1. Background on Polymer Adsorption - Equilibrium and Kinetic

Polymer adsorption onto a solid substrate is important in a number of industrial processes such as water treatment, mineral processing and in enhanced oil recovery (EOR) (Sorbie 1991; Fleer et al. 1993; Srebnik 2003; Tadros 2013; Zhu et al. 2021). Figure 1 shows a schematic outline of a bulk polymer adsorption measurement for a polymer solution of initial (t = 0) concentration, c_o (mg/L). The polymer solution volume is V (in L), and the mass of substrate is m (g). As the polymer adsorbs on the solid substrate, the corresponding polymer solution concentration declines until after sufficient time it reaches an equilibrium value, denoted c_eq in Figure 1. By simple material balance, this corresponds to an equilibrium polymer adsorption level, Γ (mg/g) given by Eq. 1, as follows:

$$\mathrm{\Gamma }\left(c_{eq}\right)={\displaystyle \frac{V(c_0-c_{eq})}{m}}$$

(1)

A single experiment of this type will, for a given well-defined system*, give a single point, $\mathrm{\Gamma } vs. c_{eq}$ on the polymer/substrate equilibrium adsorption isotherm; *i.e. at a given set of conditions - temperature (T), solution composition, pH, specific mineral, specific polymer etc. By carrying out this process for a number of initial polymer concentrations, then a series of points on the equilibrium adsorption isotherm can be built up. In most cases, it is possible to fit one of the common analytical forms of the adsorption isotherm to this data, e.g. a Langmuir or Freundlich form. A large number of possible alternative analytical forms of the adsorption isotherm are summarised in recent reviews (Wang and Guo 2020; Danat et al. 2026). Many of the analytical adsorption isotherms have some theoretical basis, often based on the assumed kinetics of how the equilibrium is reached, and this is the case for both Langmuir and Freundlich isotherms, as discussed below. However, the various analytical forms of adsorption isotherms can also be viewed and utilised simply as empirical forms that fit the polymer/substrate equilibrium adsorption data(Wang and Guo 2020; Danat et al. 2026). From the perspective of data fitting, more flexible isotherm forms may be used, such as the extended Langmuir (Gdanski and Funkhouser 2005) and the hybrid Langmuir/Freundlich model (Sips 1948).

Any polymer/solvent/substrate system will come to equilibrium at some time, and such experiments can be used to construct the equilibrium the adsorption isotherm, Γ(c), which is a function of polymer concentration, c. Furthermore, for many polymer adsorption applications, the kinetics of the polymer/substrate adsorption process can be very important; i.e. how long it takes for equilibrium adsorption to be reached. The literature on polymer adsorption for many applications is vast, and here we will consider mainly the kinetics of the adsorption process, with a special focus on adsorption within porous media, i.e. during flow through sand mineral packs or rock cores.

Whether in bulk solution or in a porous medium, the adsorption process for polymers at the mineral surface is considerably more complex than that for the adsorption of small molecular species (Gramain and Myard 1981; Sorbie 1991; Fleer et al. 1993; Tadros 2013). For a high molecular weight flexible chain molecule in aqueous solution (e.g. hydrolysed polyacrylamide (HPAM) in water), adsorption at the surface of any substrate involves several stages, as follows:

(i)

Diffusion to the surface: polymer transport to the adsorbing surface by diffusion of the high molecular weight polymer molecule is relatively “slow” compared with smaller molecular species, since D₀ ~ 1 x 10^-12 to 1 x 10^-13 m²/s (D₀ is the molecular diffusion coefficient);

(ii)

Molecular anchoring: surface contact and initial attachment of the macromolecule at one or two polymer segments giving an initial “anchoring” of the polymer chain, with the remainder of the flexible chain being in the solvent;

(iii)

Polymer chain surface conformation: several adjustments in chain conformation then take place including the attachment of further chain segments to the surface (known as “trains”), which form “loops” of the now bound polymer chain in the solvent and “tails” of polymer at each end of the polymer chain, also in the solution. This is described by statistical mechanical lattice theory which has an extensive literature (Scheutjens and Fleer 1979, 1980; Fleer and Scheutjens 1982; Fleer et al. 1993);

(iv)

Additional polymer surface “crowding”: simultaneously, as more polymer molecules arrive on the surface, a “crowding” effect occurs and the already adsorbed molecules may adjust their configuration by forming larger solution loops, which has the physical effect of extending the adsorbed polymer layer thickness; and

(v)

Relaxation to equilibrium: the gradual relaxation of the flexible chain polymer molecule and its neighbours on the surface occurs, thus reaching a (dynamic) equilibrium surface conformation of adsorbed polymer loops, trains and tails.

The thermodynamic driving force for the polymer-surface adsorption process is the energy of attachment of the (many) polymer trains on the surface (usually by displacing surface bound solvent molecules), which must counterbalance the loss of conformational entropy of the free long chain in solution. A further complication arises in the adsorption of polymers due to their polydispersity, i.e. the wide molecular weight distribution of the polymer (Stuart et al. 1980). The smaller molecules with much higher diffusion coefficients adsorb much faster than the higher molecular weight components, however these larger species will gradually displace the lower molecular weight species, which again may not be a rapid process. A feature of this entire adsorption process is that it may take from hours to several days to reach full equilibrium of the system even in a well-mixed system; this is well described in Tadros (2013) and (Fleer et al. 1993). Even in their studies of HPAM adsorption on Millipore filters, (Gramain and Myard 1981) found that it took 3 or 4 days for the adsorption process to come to equilibrium.

In the light of the discussion above, we must consider whether the kinetic measurements of polymer adsorption rates made in bulk static adsorption tests carried out in a beaker, as shown in Figure 1, are relevant to the adsorption process in porous media. For this beaker-scale case, diffusion from the bulk liquid to the sold surface within the granular solid mass may require a substantially exaggerated time compared to that for liquid within a porous medium. Within a pack or rock core the maximum distance for diffusion will generally be less than 50 µm. A flexible coil polymer, such as HPAM, of about 20 MDa with a D_o between 10^-12 m²/s and 10^-13 m²/s would take between ~0.17 hours and ~ 1.7 hours1 to travel a distance l = 50 mm, according to the Einstein-Schmoluchowski formula, t = l ²/(2dD₀) (d is the dimension, assumed 2). And this arrival time might be increased if the polymer solution itself were concentrated (by polymer molecular entanglement) and viscous effects (which reduce D_o). However, this is not the longest time in the polymer adsorption process, stages (ii) –(iv) will generally take much longer than this (Tadros 2013).

Recent work (Silva et al. 2023; Silva et al. 2024) conducted using the method illustrated in Figure 1 has indicated that for certain water-soluble polymers, such as high molecular weight partially hydrolysed polyacrylamide (HPAM), the time scale to reach adsorption equilibrium may, under some circumstances, be several days. In this experimental work, the system was occasionally stirred to assist the adsorption process, but the caution expressed above should be kept in mind, since we might expect such timescales to be lower within porous media. This has an obvious effect on exactly how long a time period that the adsorption experiment should be carried out if the correct equilibrium adsorption is to be achieved.

1.2. Dynamic Polymer Adsorption in Porous Media

For cases of polymer flow through porous media, a longer timescale to reach equilibrium adsorption also has implications on how polymer core floods are carried out, in particular on the chosen volumetric flow rate, q; more precisely, we show that it is the fluid pore velocity, n = q/(Af) which should be considered2. If polymer adsorption onto a porous medium, such as a sandstone rock or a sand pack, is being measured, then the residence time in this experiment will be relevant since it may be too short to reach equilibrium. The fluid residence time, T_r, is given by T_r = V_p/q, where V_p is the core pore volume. If equilibrium is assumed (incorrectly) in such a system when making the adsorption estimates, then erroneous results may be obtained.

We now consider the relevant literature on the adsorption of polymers in dynamic (flowing) mineral pack and core floods. To measure the level of adsorption (Γ_max) in such a system, the method is quite different from the bulk adsorption beaker test illustrated in Figure 1, as discussed below in the context of an actual example.

Many dynamic core and pack flooding polymer adsorption experiments have been presented in the literature and an early review of these has been presented (Sorbie 1991). However, some of the more recent results find polymer production profiles that show a puzzling type of “tailing” in their effluents (Wang et al. 2020; Seright and Wang 2023a, b). The cause of this effluent behaviour has been uncertain in the interpretation of these experiments, and it has been attributed to various effects such as bridging adsorption (Seright and Wang 2023a) and a dual time scale for equilibrium adsorption onto quartz versus clays (Seright and Wang 2023b). Indeed, kinetic effects have also been suspected in the interpretation of this work.

During extensive dynamic-retention (i.e. coreflood or mineral pack) experiments associated with the Milne Point polymer flood, Seright and Wang (2023b, 2023a) observed this “tailing phenomenon” when the clays, illite or kaolinite, were present in quartz-based sands. Figure 2 and Figure 3 show this tailing behaviour as a function of injection rate and other conditions for an HPAM polymer of Mw = 18-million-g/mol (M Da) in glass bead packs containing 9% of added illite.

In Figure 3, the dimensionless effluent concentration of polymer (c/c₀) shows a good correlation with the “exposure parameter”, L_p, which is given by Eq. 2:

L_p = (t-t_b)u C^-0.5

(2)

Where t is total time since the start of polymer injection, t_b is time at polymer arrival at the end of the core, u is superficial or Darcy velocity, and C is polymer concentration. To account for the tailing phenomenon in the above two figures, Seright and Wang (2023b) speculated that there was a different (slower) time or “exposure” constant for polymer adsorption onto illite versus onto quartz or glass. The timescale for polymer injection ranged from 2.7 hours (for the 12.4-ft/d experiment) to 108 hours (for the 0.31-ft/d experiment). These flow velocities (as Darcy velocities, u = q/A, where A is the cross-sectional area of the pack) correspond to residence times of ~0.19 hours (12.4 ft/day) and ~7.7 hours (0.31 ft/day)

As is the general practice in polymer core flooding, the experimental results in Figure 2 are used to measure the level of polymer adsorption in the core. The adsorption for input polymer concentration c₀ is proportional to the dimensionless area, A, between the tracer (the no illite curve in Figure 2), and the polymer effluent, for a given case. The mass of adsorbed polymer is calculated as, m_ads = A.PV(in L).c₀ (mg), which leads to an adsorption level at c₀ of Γ = m_ads/M_core (mg/g) where M_core is the mass of the core (in g). Thus, it appears that the adsorption in Figure 2 is “rate dependent”, i.e. the adsorption level, Γ, has a different value at each flow rate. With this interpretation, Figure 2 indicates that polymer adsorption is higher at lower flow rates.

Other workers, such as Idahosa et al. (2016) have also observed this effect, even mentioning “rate dependent polymer adsorption”, in the title of their paper. In this paper, we note that at the higher flow rates the fluid residence time in the core is only ~2 hours.

Mousapour et al. (2024) presented a recent series of polymer floods where the levels of adsorption were measured as described by measuring A difference between polymer and tracer effluents, but again low fluid residence times were used.

There is an issue with all of the above dynamic (core flow) measurements of polymer adsorption, and in most similar studies in the literature. In particular, this effluent difference (A) method is based on an important assumption; it is assumed that the adsorbing system is at equilibrium. At equilibrium, when the effluent polymer concentration reaching the input level (c₀), then the adsorbed level (at c₀) is unequivocally known (to some experimental accuracy). However, if the system is not at equilibrium, the material balance simply tells us how much polymer (mass) is left in the core, but it does not necessarily mean that it is all adsorbed. At any value of effluent polymer concentration for this latter case, the in situ polymer may be in the process of adsorbing; and therefore, a lower value of adsorption level Γ will be estimated.

1.3. Aims and Objectives of This Paper

As noted above, kinetic effects have been suspected as being at least part of the explanation of polymer effluent behaviour in pack and core floods (Seright and Wang 2023b). In this work, it is shown how such effects might arise and how they might explain this polymer effluent tailing in polymer transport through porous rock cores or packed beds. The findings of this paper are then used to propose improved experimental procedures for both interpreting and quantifying polymer adsorption in the presence of kinetic adsorption effects. Also, procedures are proposed for proving whether the observed polymer effluent effects are actually due to kinetic phenomena or other possible causes.

The 3 main objectives of this paper are:

(i)

To present a polymer transport and kinetic adsorption model which is appropriate for modelling polymer core and pack floods, including and accurately modelling the physics and kinetics of polymer adsorption described in this paper;

(ii)

To apply this model to predict the qualitative behaviour of polymer core flood effluents in comparative equilibrium and non-equilibrium (kinetic) cases and to compare these results to experimental results published in the literature, and indeed to explain these observed results;

(iii)

To use our results to propose improved experimental procedures for both interpreting and quantifying polymer adsorption in the presence of non-equilibrium (kinetic) effects. Clear procedural methods are presented for demonstrating that the observed polymer effluent effects are due to kinetic phenomena, as we claim.

2. The Freundlich Isotherm – Equilibrium and Kinetic Aspects

2.1. Origins of the Equilibrium Freundlich Model

In this work, we will assume a Freundlich adsorption isotherm for the polymer/substrate equilibrium system. One feature of the Freundlich form that makes it appropriate for polymer adsorption is that it has a very steeply rising isotherm at concentrations just above c =0. We note that the same work has been carried for the Langmuir isotherm and the final conclusions and predictions are very similar to those from this paper, so the precise analytical equation of the isotherm is not of major importance. The equilibrium Freundlich isotherm, as a simple empirical form, is given by:

$$\mathrm{\Gamma }(c)=\alpha .c^{\beta }$$

(3)

where a and b in Eq. 3 are constant for a given set of conditions as listed above. The units used in this work are such that polymer concentration, c, is in mg/L, Γ is in mg/g, giving units for a as mg^(1-b)L^bg^-1 and b is dimensionless. However, like many adsorption isotherms, the Freundlich model is based on an underlying view of the kinetics of the adsorption and desorption processes, and how this arises is outlined in Figure 4, where the schematic graphic relates specifically to the local process at pore/mineral grain scale.

Figure 4 shows the assumed adsorption and desorption rate laws in the Freundlich model. The rate of adsorption from the bulk fluid is given as R_a and the rate of desorption from the solid is denoted R_d. The dimensions of the associated rate constants are as follows: k_a^* has units (mg/L)^(1-b).(time units)^-1 such that R_a will be (mg/L).(time units)^-1, and the dimensions of k_d^* are (time units)^-1, such that R_d will be (mg/g).(time units)^-1. Thus, to convert to common units of mass/time in the material balance, then the modified rate laws assumptions are:

$$\begin{array}{c}\mathrm{Adsorption} \mathrm{rate}: {R^{\prime }}_a=V{k}_a^{*}{C}^{\beta } \mathrm{where} \mathrm{this} \mathrm{will} \mathrm{be} \mathrm{in} \mathrm{units} \mathrm{of} \mathrm{mass}/\mathrm{time} (\mathrm{e}.\mathrm{g}. \mathrm{mg}/\mathrm{hour})\hfill \\ \mathrm{Desorption} \mathrm{rate}: {R^{\prime }}_d=m{k}_d^{*}\mathrm{\Gamma } \mathrm{also} \mathrm{in} \mathrm{the} \mathrm{same} \mathrm{units} \mathrm{of}, \mathrm{mass}/\mathrm{time}\hfill \end{array}$$

As shown in Figure 4, at equilibrium then ${R^{\prime }}_a={R^{\prime }}_d$ and therefore, it is clear that in general:

$$\mathrm{\Gamma }(c)=\left({\displaystyle \frac{V.k_a^{*}}{m.k_d^{*}}}\right)c^{\beta }$$

(4)

And hence the constant, a, in the Freundlich model is given by

$$\alpha =\left({\displaystyle \frac{V.k_a^{*}}{m.k_d^{*}}}\right)$$

(5)

It may appear unusual here to have the factor (V/m) as part of the Freundlich constant a, but this is of particular use in the work presented here, and Eq. 3 still applies with this value of a for any V/m ratio, as we will show. In the derivation of the Freundlich isotherm in Figure 2 we could have directly equated the quantities R_a and R_d; $R_a=R_d$implies that $k_a^{*}{c}^{\beta }=k_d^{*}\mathrm{\Gamma }$ which would lead to the “primitive” form of the Freundlich adsorption isotherm as ${\mathrm{\Gamma }}^{\prime }=\left(k_a^{*}/k_d^{*}\right)c^{\beta }={\alpha }^{\prime }{c}^{\beta }$, where the ${\alpha }^{\prime }$constant is given by ${\alpha }^{\prime }=\left(k_a^{*}/k_d^{*}\right)$; a closer examination of the quantity ${\mathrm{\Gamma }}^{\prime }$ shows that it has units of concentration (shown below). This makes no reference to the (V/m) ratio, but this form is not a proper adsorption isotherm since the units must incorporate implied m and V since it does not give the adsorption in mg/g, as required. The relation between the form used in this work and this “primitive” form can be seen when we use this equation to calculate both the equilibrium adsorption and concentration levels, i.e. Γ_eq and c_eq, as shown in Figure 1, where we know the actual isotherm, i.e. $\mathrm{\Gamma }=\alpha c^{\beta }$. In such a calculation, we know, m, V and the initial polymer concentration, c₀ and we must find both Γ_eq and c_eq. The construction to perform this calculation is shown schematically in Figure 5. Clearly, the mass of polymer initially in the system is Vc_o (in mg), and at equilibrium this mass is distributed to between the adsorbed polymer (mΓ_eq) and the polymer in solution (Vc_eq), and mass conservation requires that Eq. 6 applies:

$$\mathrm{Mass} \mathrm{of} \mathrm{polymer} \mathrm{in} \mathrm{system}=V{c}_0=m{\mathrm{\Gamma }}_{eq}(c_{eq})+V.c_{eq}$$

(6)

However, since we know the equation for the Freundlich isotherm and the values of the a and b parameters, then to find c_eq we must solve the equation:

$${\mathrm{F}(\mathrm{c}}_{eq})=m{\mathrm{\Gamma }}_{eq}(c_{eq})+V.c_{eq}-V{c}_0=0$$

(7)

Dividing through Eq. 7 by V and using Eq. A5 gives that:

$${\mathrm{F}(\mathrm{c}}_{eq})={\alpha }^{\prime }{c}_{eq}^{\beta }+c_{eq}-c_0=0$$

(8)

where the first term in Eq. 8 is the primitive form ${\mathrm{\Gamma }}^{\prime }={\alpha }^{\prime }{c}^{\beta }$, and this can be used to determine the equilibrium concentration (c_eq) arising from any initial c₀ value by numerically solving Eq. 8 for c_eq. The corresponding adsorption level, Γ (in mg/g), is then found by calculating $\mathrm{\Gamma }=\left(m/V\right){\alpha }^{\prime }{c}_{eq}^{\beta }=\alpha c_{eq}^{\beta }$.

2.2. The Full Kinetic Adsorption Model

The full kinetic model is now considered describing the kinetics of the change in the (increasing) adsorption level, Γ, and that of the (decreasing) solution concentration level. Clearly, Γ increases as c decreases until equilibrium is reached at Γ = Γ_eq and c = c_eq. These 2 equations are of course closely related and written as follows:

$$\begin{array}{l} \mathrm{adsorption} \text{-}\mathrm{desorption}\\ \left({\displaystyle \frac{d\mathrm{\Gamma }}{dt}}\right)=+{\displaystyle \frac{k_a^{*}V{c}^{\beta }}{m}} -{\displaystyle \frac{k_d^{*}m\mathrm{\Gamma }}{m}} ; \Rightarrow \left({\displaystyle \frac{d\mathrm{\Gamma }}{dt}}\right)=+\left({\displaystyle \frac{k_a^{*}V{c}^{\beta }}{m}} \right)-k_d^{*}\mathrm{\Gamma }\end{array}$$

(9)

$$\left({\displaystyle \frac{dc}{dt}}\right)=-{\displaystyle \frac{k_a^{*}V{c}^{\beta }}{V}}+{\displaystyle \frac{k_d^{*}m\mathrm{\Gamma }}{V}} ; \Rightarrow \left({\displaystyle \frac{dc}{dt}}\right)=-k_a^{*}{c}^{\beta }+\left({\displaystyle \frac{k_d^{*}m\mathrm{\Gamma }}{V}}\right)$$

(10)

These coupled ordinary differential equations (ODEs) are not soluble analytically, except under simplified conditions as shown below. However, to illustrate how they operate, a simple numerical example is presented using the model data in Table 1. Using this data, Eqs. 9 and 10 can be integrated numerically to obtain the profiles of Γ vs. t and C vs. t, as shown in Figure 6. From the numerical integration of the full kinetic equations above, the final equilibrium level of adsorption was found to be Γ_eq = 0.11380 mg/g and the concentration was C_eq = 84.82593 mg/L3. This can be checked by substituting the value of c_eq into the Freundlich model with a = 0.0375, b= 0.25 (in the units of Table 1); this gives exactly Γ_eq = 0.11380 mg/g, as required.

2.3. The Simplified Kinetic Freundlich Model

As a simplification, consider only the adsorption term only in the kinetics of polymer adsorption, as follows (Eq. 9):

$$\begin{array}{l} \mathrm{adsorption} \mathrm{desorption}\\ \left({\displaystyle \frac{d\mathrm{\Gamma }}{dt}}\right)=+\left({\displaystyle \frac{k_a^{*}V{c}^{\beta }}{m}} \right)-k_d^{*}\mathrm{\Gamma }\end{array}$$

(11)

Clearly, the problem with solving this kinetic equation is that there is a “c” in the adsorption term on the right hand side of the equation This concentration term is changing (reducing) with t, as shown in the results for the full kinetic model in Figure 6 above. But suppose we make the further assumption that concentration, c, is set to $c=c_0$, i.e. the maximum or initial input concentration This would mean that the term $\left({\displaystyle \frac{k_a^{*}V{c}_0^{\beta }}{m}} \right)$ is no longer changing but is a constant. From Eq. 5, this term is clearly:

$$\left({\displaystyle \frac{k_a^{*}V{c}_0^{\beta }}{m}} \right)=k_d^{*}\alpha c_0^{\beta }=k_d^{*}{\mathrm{\Gamma }}_{\max }$$

(12)

where ${\mathrm{\Gamma }}_{\max }$ is the highest level of adsorption that can be reached if c = c₀, and this allows us to write Eq. 11, as follows:

$$\left({\displaystyle \frac{d\mathrm{\Gamma }}{dt}}\right)= k_d^{*}{\mathrm{\Gamma }}_{max}-k_d^{*}\mathrm{\Gamma }=k_d^{*}\left({\mathrm{\Gamma }}_{max}- \mathrm{\Gamma }\right)$$

(13)

This simple ODE is actually a standard form which has a well-known solution, which is easily shown to be:

$$\mathrm{\Gamma }(t)={\mathrm{\Gamma }}_{\max }\left(1-e^{-k_d^{*}t}\right)$$

(14)

The full system of equations shown in Section 2.2 above had a final polymer concentration of c_eq = 84.83 mg/L giving a final adsorption level of Γ_eq = 0.114 mg/g. In the approximate model where we set c= 100 ppm (i.e. the initial fixed concentration, c₀), then Γ_max = 0.119 mg/g (using the Freundlich parameters in Table 1). These final adsorption levels are quite close and we may therefore ask; how do the results of the approximation in Eq. 14 compare with the full kinetics given in Section 2.2 above (i.e. the numerical integration of Eqs. 9 and 10)? The two methods (1. the full kinetic and 2. the approx. model) are compared in Figure 7 for the data given in Table 1 above. The agreement, as shown in Figure 7 (with kd* = 0.01 hour^-1), is seen to be very good, and probably much less than experimental error in polymer assay.

3. The Polymer Transport and Kinetic Adsorption Equations

Polymer transport through porous media in 1D including kinetic adsorption using the Freundlich based model is described by the modified convection-dispersion equations as follows (Lake 1989; Sorbie 1991):

$$\left({\displaystyle \frac{\partial c}{\partial t}}\right)=D\left({\displaystyle \frac{{\partial }^{2}c}{\partial x^2}}\right)-v\left({\displaystyle \frac{\partial c}{\partial x}}\right)-{\displaystyle \frac{\rho }{\varphi }}\left({\displaystyle \frac{\partial \mathrm{\Gamma }}{\partial t}}\right)$$

(15)

$$\left({\displaystyle \frac{\partial \mathrm{\Gamma }}{\partial t}}\right)=\left({\displaystyle \frac{V{k}_a}{m}}\right)c^{\beta }-k_d\mathrm{\Gamma }$$

(16)

where D is the dispersion coefficient, n is the pore velocity of the fluid ($v=q/(\varphi A)$), q is the volumetric flow rate, f is the porosity and A is the cross-sectional area of the 1D system of length, L (a core or sand pack, for example), r is the rock density and the other quantities associated with polymer adsorption are described previously. Eqs. 15 and 16 are in the laboratory units introduced earlier in this paper.

The complete corresponding set (to Eqs. 15 and 16) of dimensionless equations for polymer transport and kinetic adsorption, derived in Appendix A, is as follows:

$$\left({\displaystyle \frac{\partial C}{\partial T}}\right)={\displaystyle \frac{1}{N_{Pe}}}\left({\displaystyle \frac{{\partial }^{2}C}{\partial X^2}}\right)-\left({\displaystyle \frac{\partial C}{\partial X}}\right)-N_{Ad}\left({\displaystyle \frac{\partial \tilde{\mathrm{\Gamma }}}{\partial T}}\right)$$

(17)

and

$$\left({\displaystyle \frac{\partial \tilde{\mathrm{\Gamma }}}{\partial T}}\right)=N_{Da\_A}\left[C^{\beta }-\tilde{\mathrm{\Gamma }}\right]$$

(18)

where $C={\displaystyle \frac{c}{c_o}};X={\displaystyle \frac{x}{L}};T={\displaystyle \frac{v.t}{L}};\tilde{\mathrm{\Gamma }}={\displaystyle \frac{\mathrm{\Gamma }}{{\mathrm{\Gamma }}_{\max }}}$, are the dimensionless concentration, length, time and adsorption level, respectively, and $0\le C\le 1 \mathrm{and} 0\le \tilde{\mathrm{\Gamma }}\le 1$. The 3 dimensionless groups governing the system are the Peclet number, $N_{Pe}$, the Adsorption number, $N_{Ad}$, and the adsorption Damköhler number, $N_{Da\_A}$, which are given by the expressions in Eq. 19 below (see Appendix A):

$$N_{Pe}={\displaystyle \frac{vL}{D}};\hspace{1em}{N}_{Ad}=\left({\displaystyle \frac{\rho {\mathrm{\Gamma }}_{\max }}{\varphi c_o}}\right)\hspace{1em}\mathrm{and}\hspace{1em}{N}_{Da\_A}=\left({\displaystyle \frac{L{k}_d^{*}}{v}}\right)$$

(19)

Since the transport is convection dominated, the Peclet number is generally relatively high ($N_{Pe}$> 80) and is not of great importance in this system; we have checked this by simulation, but no further details are given here. However, the other two numbers, the Adsorption number, $N_{Ad}$, and the adsorption Damköhler number, $N_{Da\_A}$, are both very important, as will be demonstrated below. $N_{Ad}$ quantifies the degree of retardation of the polymer front for a system at equilibrium adsorption, and is often called the retardation factor, R. $N_{Da\_A}$ expresses the balance between the residence time of the polymer in the system (i.e. $L/v$) to the timescale of kinetic adsorption (i.e., $1/k_d^{*}$). As expected, long residence times (low velocities) and fast adsorption kinetics (i.e. high values of $N_{Da\_A}$) favour equilibrium, or near equilibrium, conditions; whereas low residence times (high velocities) and slow adsorption times (i.e. low values of $N_{Da\_A}$) lead to highly non-equilibrium kinetic dominated behaviour.

We again stress, that although the model and results presented here are based on the underlying equilibrium and kinetic assumptions of the Freundlich adsorption model, the qualitative finding are true for any “normal” (concave down) analytical adsorption isotherm. Indeed, we take as a common assumption that the equilibrium polymer adsorption isotherm, is steeply rising at low concentrations, is concave downwards and has a maximum adsorption level or a plateau adsorption level (Γ_max). We also note (but do not strictly assume) that this plateau adsorption level may occur at a relatively “low” polymer concentration level of c ≈ 50 – 500 mg/L. Numerical simulation examples will be presented to illustrate the extent to which the variation in where the plateau occurs impacts the main results and findings of this work. The transport and kinetic equations (Eqs. 15 and 6; or Eqs. 17 and 18) are solved numerically using standard finite difference methods

4. Numerical Results for Polymer Transport for Equilibrium and Kinetic Polymer Adsorption

In order to achieve the objectives outlined in Section 1 above, we present a series of simulations addressing various aspects of the polymer transport/kinetic problem.

4.1. Context of the Numerical Examples and Exact $N_{Da\_A}$ Scaling

The entire range of possible results which can arise for any mathematical model of a given polymer transport and kinetic adsorption model, in a 1D core system (e.g. a sandstone core or mineral pack) range from equilibrium adsorption (as $N_{Da\_A}\to \infty$), to effectively no adsorption (as $N_{Da\_A}\to 0$), the latter case being described as “tracer like” behaviour. When the system is at equilibrium, the adsorption rate is very high relative to the polymer fluid residence time. Otherwise, the adsorption rate is either comparable to or long compared with the residence time. To systematically vary the adsorption Damköhler number, there are two exactly scaled equivalent choices (demonstrated below and in Appendix A):

(i)

we can choose a fixed flow velocity (n) and then systematically vary the adsorption rate constant $k_d^{*}$ (remembering to change $k_a^{*}$ in exact proportion to keep a constant), or

(ii)

we may take a constant kinetic rate, $k_d^{*}$, and then vary the flow rate (n) from a very low to a high value.

Numerical examples are presented below which demonstrate that these equivalent cases scale exactly.

The work in this paper was stimulated by two relatively recent sets of experimental observation:

• To investigate whether the effluent “tailing” behaviour measured in the experiments of Seright and coworkers was due to the kinetic adsorption behaviour of the polymer (Wang et al. 2020; Seright and Wang 2023a, b) or possibly some other effect; and
• To study the consequences of the recent observations by Silva and coworkers that HPAM polymer adsorption at lower temperatures (T ~30^oC) could take relatively long times to reach equilibrium for sand in beakers; i.e. several days (Silva et al. 2023; Silva et al. 2024). Again, we caution that beaker tests may give exaggerated (i.e. too long) experimental adsorption time scales (i.e. too low values of k_d*), for the reasons discussed above.

To construct numerical examples that are relevant to recent experimental results, we took one of the typical observations of Silva et al (2023; 2024) based on static bulk adsorption tests, as shown in Figure 8. This figure presents a plot of HPAM adsorption measurements (shown ●) over time (in hours) onto quartz sand at T = 31^oC, and this data is matched by fitting Eq. 14 to it, in which Γ_max = 98mg/g and $k_d^{*}$ = 0.01 hour^-1. The match is not perfect, but it gives the correct order of magnitude of the $k_d^{*}$and this is all we require for our simulations. To context this value of the adsorption rate ($k_d^{*}$ = 0.01 hour^-1), recall that the adsorption “half-life” (i.e. the time to get to dimensionless adsorption $\tilde{\mathrm{\Gamma }}$ = (Γ/Γ_max) = 0.5) is given by ${\tau }_{1/2}=-\ln \left(0.5\right)/k_d^{*}$ ≈ 69 hours (~2.9 days), and to reach $\tilde{\mathrm{\Gamma }}$ = 0.95, the${\tau }_{0.95}$ ≈ 300 hours (~12.5 days).

The polymer adsorption literature does note that the whole adsorption process, including stages (i) – (v) briefly reviewed in Section 1, may take “hours to days” (Tadros 2013). However, it is reasonable to ask for the specific results here: how much of the long time (~300 hours) taken for polymer adsorption shown in Figure 8 is taken up with diffusion of the polymer through the bulk and then into the porous medium (an unconsolidated sand)? In other words, is the rate constant (k_d*) which we derive here too low at the value k_d* = 0.01 hour^-1? In fact, this precise number may indeed be too low to directly apply in a porous medium, but to be sure of this we would need to compare this result with experiments where full system stirring or agitation took place to ensure constant contact of the surface with the polymer solution (experiments in progress). However, using this number (k_d* = 0.01 hour^-1) does not affect our conclusions, since we demonstrate below that the results depend only on the adsorption Damköhler number (N_Da-A). This quantity is the ratio of the residence time of the polymer in the core, (L/n), to the timescale of the adsorption process, (1/k_d*). Thus, simulated polymer effluent results for kinetic polymer adsorption at any value of k_d1* and fluid velocity n₁, are the same at any other value such as k_d2* but for an adjusted flow velocity in the core, n₂ = n₁ . (k_d2*/ k_d1*), since they have the same value of N_Da-A. This exact scaling is demonstrated below.

4.2. The Numerical Examples

In this work, a 1D core flood is simulated with the physical properties listed in the first two columns of Table 2. The numerical grid used for the simulations is given in the next two columns, and the base case specifications of the polymer properties are given in the final two columns. Note that the manner in which the polymer input data is presented implies that we input the kinetic adsorption parameters $k_a^{*} and k_d^{*}$ and then calculate the value of the Freundlich constant, a. In practice, however, we actually first decide on the equilibrium Freundlich isotherm parameters (a and b in Eq. 2), with units such that if polymer concentration, c, is in mg/L, then the adsorption level is in mg/g of rock. The quantity $k_d^{*}$ is chosen based on kinetic adsorption experiments as shown in Figure 8, and then the corresponding $k_a^{*}$ quantity is calculated using Eq. 5 as $k_a^{*}=\left({\displaystyle \frac{\alpha m.k_d^{*}}{V}}\right)$. For the parameters in Table 2, the base case Freundlich isotherm, Γ₁, is shown in Figure 9(a), and two other more steeply rising isotherms, Γ₂ and Γ₃, used in later sensitivities, are also shown in Figure 9(b).

As noted above, a complete set of cases spanning the entire range of physical possibilities from equilibrium adsorption (N_Da-A → ∞) to almost “tracer like” behaviour (N_Da-A → 0) can be constructed by choosing a single rate parameter (k_d* ) and then varying the fluid volumetric flow rate (q) (i.e. the velocity, n = q/(Af)). The range of examples for which results are presented below, is listed in Table 3, where the types are behaviour listed from “near equilibrium” to “almost tracer like” are anticipated and will be explained as the results are discussed below.

4.3. Continuous Injection Polymer Equilibrium and Kinetic Adsorption Cases

The first set of numerical results is shown for continuous injection of several PV (6 – 12) of polymer solution at a concentration of c₀ = 1000 mg/L into the core. Continuous injection simulations for all of the cases in Table 3 have been carried out.

“End Member” Cases – Equilibrium and Tracer: Before presenting the range of non-equilibrium cases, we first consider the “end member” cases of equilibrium adsorption (N_Da-A →∞) and almost no adsorption (N_Da-A → 0; tracer like behaviour). The effluent curves for these two cases are shown in Figure 10. An analytical prediction explained in Appendix A, is that the retardation (R = N_Ad-A) of the equilibrium polymer is R = 1.96 (PV), and this is what is calculated numerically (see Figure 10). A second point to note is that the equilibrium polymer breakthrough effluent is somewhat steeper or sharper than the tracer effluent curve. This behaviour is well known as the concave downwards adsorption isotherm is self-sharpening (p20 ff. Sorbie 1991). Since we have just referred to these two simulations as “end member” cases, we may expect that all the other “intermediate” effluent behaviours might show effluents lying between these two cases, but this is certainly not the case as is now shown.

The Entire Range of Polymer Effluent Behaviours: Figure 11 presents a selection of calculations covering the whole range of possible kinetic adsorption behaviour between equilibrium adsorption and tracer like cases. This figure shows a very wide range of behaviours including many cases where the long “tailing” of the effluent goes well outside the domain bounded by the equilibrium and tracer cases in Figure 10. To unravel this complexity, we will now consider subsets of these cases under the headings:

o 

Near-equilibrium cases at high values of N_Da-A;

o 

Transitional kinetics cases;

o 

Highly non-equilibrium cases (show the long-tailing behaviour quite clearly);

o 

Almost tracer-like cases (showing a plateau behaviour), N_Da-A→ 0.

Near Equilibrium Polymer Effluents (Cases 1 – 4; Table 3): The near-equilibrium cases are shown in Figure 12(a) plotted over the entire flood (0 – 6 PV), and in Figure 12(b) plotted over the narrow range of the flood front (2.7 – 3.3 PV). Although a clear trend is visible over these four cases as the system moves gradually away from the full equilibrium cases, in practice from an experimentally measured polymer concentration effluent viewpoint, these cases could probably not be distinguished analytically.

The Transitional Kinetic Polymer Effluents (Cases 5 – 8; Table 3): The transitional kinetics cases are shown in Figure 13 plotted over the entire flood (0 – 10 PV). These cases would be quite distinct from the equilibrium effluent, especially Case 8 which shows very significant “tailing” where the effluent does not reach the input polymer concentration (c₀) until over 10 PV of injection. Cases 6 – 8 show that the non-equilibrium effect causes the breakthrough time to become rather earlier, moving quite close to the tracer breakthrough time (see Figure 10). This feature will be even more marked as the system moves further from equilibrium in the following cases.

Highly Non-Equilibrium Polymer Effluent Cases (Cases 9 – 12; Table 3): The highly non-equilibrium polymer adsorption kinetic cases are shown in Figure 14 plotted over the entire flood (0 – 12 PV). These cases are characterised by showing early breakthrough which actually closely tracks the tracer breakthrough. This is an important prediction/finding since it may lead the experimentalist to believe that adsorption (often identified by polymer retardation) is very low, when it is in fact not low, it is simply a non-equilibrium effect, and the system has not yet reached its (relatively high) equilibrium adsorption value. A second important prediction, which is best seen in Cases 11 and 12, is that the polymer effluent shows a very long flat plateau-like region below the input polymer concentration (c₀ = 1000 mg/L) for many PV. In fact, it can be seen from the numerics that the effluent polymer concentration in this “plateau” is actually increasing very slowly (by only a few mg/L per PV injected) and this could not be detected analytically. Experimentally, it may be possible to detect analytically that the effluent is below the input concentration. However, when this has actually been observed in the laboratory, it has sometimes been interpreted as being an error of analysis and the plateau has been re-normalized to the input concentration. Such a “correction” may compound the error in the measurement of polymer adsorption when kinetic effects are present.

Highly Non-Equilibrium Almost Tracer-Like Polymer Effluent Cases (Cases 12 – 17; Table 3): These highly non-equilibrium almost tracer-like polymer adsorption kinetic cases are shown in Figure 15 plotted over the entire flood (0 – 10 PV). The scale of polymer concentration in Figure 16 is from c = 700 -1050 mg/L, and it can be seen that the group of Cases 14 – 17 all show plateau behaviour which gradually approaches (but does not quite reach) the input polymer concentration (c₀ = 1000 mg/L). The lower concentration part of the effluent for these cases almost coincides with the tracer profile. Figure 17 shows a more expanded view of Cases 14 – 17 for concentration between 900 – 1005 mg/L, where the actual value of the plateau level at 10 PV of polymer injection is listed on the figure. Although a very clear pattern of behaviour is seen in these calculations, it is doubtful if current polymer analytical techniques could distinguish a concentration value in this range from the full input concentration (1000 mg/L).

Overview of the Flow Regimes Due to Non-Equilibrium Effects: The results in this section show the range of polymer effluent behaviours which can arise due to non-equilibrium adsorption processes. Polymer effluent behaviour has already been reported in the literature which qualitatively resembles some of the various simulated effluents in this work (Wang et al. 2020; Seright and Wang 2023a, b).

The four approximate regimes that can occur are summarised in Figure 17 where the adsorption Damköhler number is shown as a function of volumetric flow rate, q (strictly n = q/(Af)). The four regimes are described as follows:

o 

Near-equilibrium cases with R ≈ N_Ad ($\infty >N_{Da\_A}>\sim 25$)

o 

Transitional kinetics cases ($~5>N_{Da\_A}>\sim 0.5$)

o 

Highly non-equilibrium – with significant “tailing” ($~0.25>N_{Da\_A}>\sim 0.025$)

o 

Near tracer-like - showing plateau behaviour with c < c₀ ($~0.013>N_{Da\_A}>0$)

4.4. Scaling Behaviour of Kinetic Adsorption & Transport

The results in the Appendix show that the system is governed by three dimensionless numbers, viz. (i) the Peclet number, N_Pe, which is of minor importance in these calculations; (ii) the Adsorption number, N_Ad, which is essentially the equilibrium retardation factor (R), as shown in Figure 10, and (iii) the Adsorption Damköhler number, N_Da-A, which is a measure of how close or far from equilibrium adsorption the system is located. Thus, for the same Peclet and Adsorption numbers – i.e. the same adsorption isotherm and core properties – then systems with the same adsorption Damköhler number are exactly similar. Since, $N_{Da\_A}=\left({\displaystyle \frac{L{k}_d^{*}}{v}}\right)$, then if the ratio $k_d^{*}/\nu$ is constant between any two cases, these two simulations should be exactly identical. This allowed us to take just one k_d* above and vary the flow rate to span all the possibilities from equilibrium through kinetic to almost no adsorption (i.e. all N_Da-A), as has already been carried out in Section 4.1, Section 4.2 and Section 4.3 above. This leads to Figure 18 which plots rate constant, k_d* vs. fluid velocity, n, showing lines of constant N_Da-A = 0.005 (close to equilibrium), 0.507, 2 and 40 (far from equilibrium). Systems having the k_d*/n values on any of these constant N_Da-A lines are essentially identical, and to illustrate this, calculations for the three points (denoted ■) on the N_Da-A = 0.507 line are shown in Figure 17.

Figure 19 shows the simulated effluent for Case 8 (q = 0.4 cm³/hour) in Table 3 (for which N_Da-A = 0.507) – the other two scaled cases are taken by halving and doubling both the n and k_d* values such that the N_Da-A value is preserved. Clearly, these effluents are identical (to ~6 decimal places), thus demonstrating the exact scaling of these systems.

4.5. The Effect of “Shut-Ins” on the Polymer Effluents

The clear hypothesis from the simulations presented above in Section 4.3 is that the form of the polymer effluent profile is strongly affect by how close to, or far from, equilibrium the system is (the value of N_Da-A). This work implies that the strong “tailing” observed in polymer core adsorption experiments is due to non-equilibrium effects. This may or may not be the case, but the central question is: what predictions can be made – or how should these polymer displacement experiments be performed – such that we can confirm or refute this hypothesis?

For example, it was noted above that in the case of non-equilibrium behaviour where plateau levels below the input value were observed (e.g. Cases 11 - 17) or where there is considerable tailing, then it may not be certain that this is due to kinetic effects on the adsorption. In fact, the role of the kinetics can be shown by predicting what would happen when we stop the flow and then restart; i.e. we perform a “shut-in” in the core flooding sequence. Clearly, if the system is not at equilibrium, then on a shut-in, it will re-equilibrate to some extent and there will be some resultant change in the effluent profiles. This would not happen if the entire flow and adsorption process were at full adsorption equilibrium; in this latter case, nothing should be observed.

To test this theoretically, we performed the core flood simulations including such a shut-in period. Injection of the polymer is simulated for a given period of time until sometime after breakthrough at the outlet and then the flow is stopped (n =0) for a period of time. During this shut-in time, according to the governing equations (Eqs. 11 and 12), no transport occurs but the adsorption kinetics (Eq. 12) continues and in the long term would reach local equilibrium.

The results from two simulations of this type are presented in Figure 20 and Figure 21 based on Cases 9 and 12 in Table 3. In each case 4 PV of polymer (again at c_o = 1000 mg/L) was injected followed by a 40-hour shut-in period, after which the flow is resumed at the same flow rate, and results in both cases are compared with the original polymer effluent profiles with continuous injection (no shut-in). In both cases, a very clear drop in the effluent polymer concentration is observed, and as may be expected, the drop is largest in the Case 12 example since this is furthest from equilibrium (i.e. it has the larger N_Da-A). It is also discernible in both Figure 20 and Figure 21 that the dip in effluent polymer concentration in both cases lasts for approximately 1 PV and is of a magnitude which should be easily detectable experimentally. After this dip, the polymer concentration rises to a level just above the continuous injection case. Again, this may be expected qualitatively, but the magnitude of the polymer concentration increase following the shut-in would be difficult to detect experimentally using conventional polymer assay techniques.

Experimentally, there is no particular difficulty in carrying out multiple shut-in stages in a polymer core or mineral pack flood. The effect of two shut-ins is shown for Case 10 in Figure 22. In this case, both shut-in stages show a clear response in the polymer effluent dip since the system is quite far from equilibrium, although this will depend on the system.

4.6. The Effect of the Shape of the Equilibrium Adsorption Isotherm

The results presented in this paper so far have been based on the example of the Freundlich equilibrium adsorption isotherm represented by Eq. 2 and shown in Figure 9(a); which also appears as Γ₁(c) in Figure 9(b). Any other polymer adsorption isotherm of this type (i.e. concave downwards, monotonic with a plateau region), such as the Langmuir isotherm would obviously give the same value of Adsorption number, N_Ad, if it had the same Γ_max at c_o. In addition, it would be expected to behave in a qualitatively similar manner, and this has been checked for the Langmuir isotherm (results not presented here). However, the question arises: does the shape of the equilibrium isotherm have any unexpected interaction with the kinetics, as described in this work? In particular, is there an observable effect if the isotherm is a very steeply rising “stripping” isotherm as shown for the 2 cases, Γ₂(c) and Γ₃(c), in Figure 9(b)? The formulae for the three cases are shown inset on Figure 9(b).

From the analysis of the dimensionless flow equations, Eqs. A12 and A13 (Appendix A), it can be seen that the three dimensionless parameters ($N_{Pe}, N_{Ad} \mathrm{and} N_{Da\_A}$) are all the same, but the Freundlich exponent b differs between the 3 cases (Eq. A13); b = 0.25 (base case, Γ₁), 0.04 (Γ₂) and 0.01 (Γ₃). The latter case (0.01) is close to an assumption that polymer retention/adsorption is almost independent of polymer concentration which has been a common assumption among some porous media researchers (Seright and Wang 2023a). A direct comparison between the effluent profiles for the three forms of the adsorption isotherm is shown in Figure 23(a) for the equilibrium case and in Figure 23(b) for the non-equilibrium Case 9 (q = 1 cm³/hour), the far from equilibrium Case 12 (q = 10 cm³/hour) was also simulated but the three isotherm cases gave virtually identical results, and it is not shown here. The equilibrium cases in Figure 23(a) show that the stripping isotherm (Γ₃) is slightly sharper than the base case (Γ₁), and the other case (Γ₂) lies very close to the Γ₃ case. Figure 23(b) shows the most deviation of the two stripping isotherms (Γ₂ and Γ₃) from the base case (Γ₁) for the range of non-equilibrium cases simulated. This demonstrates fairly conclusively that the range of effluent behaviours observed is principally due to the effect of the adsorption kinetics, and it has little dependence on the precise form of the equilibrium adsorption isotherm.

5. Summary and Conclusions

In this paper, a model for the transport and kinetic adsorption of polymer flowing through porous media is developed and used to both, (i) understand some of the observed behaviour of the polymer effluent profiles from core floods, and (ii) to make some testable experimental predictions. It is assumed that the underlying equilibrium adsorption isotherm, Γ(c), is of Freundlich form, and hence the kinetic expressions used to describe polymer adsorption and desorption are consistent with this form. However, it is shown that the precise form of the adsorption isotherm is not of major importance and any equilibrium form, such as the Langmuir or Sips equations, leads to virtually identical conclusions. The main issue is that the adsorption isotherm is monotonic, concave downwards, shows steeply rising behaviour at low concentration and has a fixed maximum adsorption level at the injected polymer concentration, c₀, or has a plateau in the adsorption at higher polymer concentrations (not essential).

The coupled set of two equations (Eqs. 15 and 16) for polymer transport and the kinetic adsorption of polymer governs the transport and kinetic adsorption process. The dimensionless form of these equations (Eqs. 17 and 18) show that the system behaviour entirely depends on three dimensionless numbers, viz. the Peclet number, $N_{Pe}$, the Adsorption number, $N_{Ad}$, and the adsorption) Damköhler number, $N_{Da\_A}$. Only the latter two are of major importance in this work. The Adsorption number, $N_{Ad}$, is identical to the quantity R (R = $N_{Ad}$) which is the amount that polymer effluent is retarded behind the tracer in an equilibrium adsorption system. The adsorption Damköhler number, $N_{Da\_A}=\left(L{k}_d^{*}/\nu \right)$, describes the ratio of the fluid residence time in the system, (L/n) to the timescale of the adsorption process (1/k_d*). As $N_{Da\_A}\to \infty$ , the system approaches equilibrium; that is where the adsorption level on the porous medium and in the fluid locally instantaneously obey the equilibrium adsorption isotherm. As $N_{Da\_A}\to 0$, the system tends toward non-adsorbing “tracer-like” behaviour, in that the polymer breakthrough curve is almost the same as the non-adsorbing tracer effluent curve.

The main conclusions of this work are as follows:

(i)

When simulating a displacement in a polymer adsorption core flood, the polymer transport and kinetic adsorption model shows a wide range of polymer effluent regimes, which can be broadly classified as:

o 

Near-equilibrium cases with R ≈ N_Ad ($\infty >N_{Da\_A}>\sim 25$)

o 

Transitional kinetics cases ($~5>N_{Da\_A}>\sim 0.5$)

o 

Highly non-equilibrium – with significant “tailing” ($~0.25>N_{Da\_A}>\sim 0.025$)

o 

Near tracer-like - showing plateau behaviour with c < c₀ ($~0.013>N_{Da\_A}>0$)

(ii)

From the fact that the three dimensionless groups listed above control the entire transport and kinetic polymer adsorption process, it is predicted that identical results can be achieved either by (a) taking a fixed kinetic rate (k_d* – and hence k_a*) and varying the flow rate, i.e. q or n= q/(Af), or (b) by fixing the injection rate and varying the rate of adsorption (i.e. k_d* – and k_a*) such that N_Da-A is preserved. This scaling is represented in Figure 18, and example calculations are given to confirm this exact scaling (Figure 19).

(iii)

The model polymer effluent predictions are qualitatively similar to experimental core flood results appearing in the literature. A direct simulation of this data to match some of these results is currently in progress.

(iv)

If the kinetic adsorption effect is the correct explanation of the “tailing” and “plateau” behaviour in the polymer effluent profiles in experimental core floods, then we demonstrate how this can be established. It is shown that by performing a “shut-in” (i.e. reducing the injection rate to q = 0) for a time of period of some hours (~ 5 to 40 hours), depending on the timescale of the adsorption process, then a dip in polymer concentration will be observed in the measured effluent. Preliminary polymer adsorption core flood experiments have confirmed this finding (Silva et al. 2024).

(v)

The shape of the polymer adsorption isotherm (see Figure 9) was radically changed from the gradually rising case Freundlich function (Γ₁) to successively sharper “stripping” isotherms (Γ₂ and Γ₃); R = N_Ad was identical for all 3 cases. This had a small effect on the equilibrium breakthrough polymer effluent profiles, where the stripping cases gave a slightly sharper breakthrough profile for the more stripping isotherm cases (Γ₂ and Γ₃). However, this had a very small effect on the polymer effluent profiles in the non-equilibrium region and corresponding cases had almost identical profiles for Γ₁ and Γ₂ /Γ₃. This demonstrates that the exact form of the equilibrium adsorption isotherm has little effect on the non-equilibrium polymer effluent profiles.

In future work, we intend to apply the above principles to test whether they are sufficient to explain the dynamic retention experiments of Seright and Wang (2023a, 2023b) and other observations in the literature. Preliminary results are optimistic and will be presented in a forthcoming paper.