Analytic Model for DBF Under Multiple Particle Retention Mechanisms

Abstract

Discrepancies between classical model (CM) predictions and experimental data for deep bed filtration (DBF) have been reported by various authors. In order to understand these discrepancies, an analytic continuum model for DBF is proposed. In this model, a filter coefficient is attributed to each distinct retention mechanism (straining, diffusion, gravity interception, etc.). It was shown that these coefficients generally cannot be merged into an effective filter coefficient, as considered in the CM. Furthermore, the derived analytic solutions for the proposed model (PM) were applied for fitting experimental data, and a very good agreement between experimental data and PM predictions were obtained. Comparison of the obtained results with empirical correlations allowed identifying the dominant retention mechanisms. In addition, it was shown that the larger the ratio of particle to pore sizes, the more intensive the straining mechanism and the larger the discrepancies between experimental data and CM predictions. Finally, the CM and PM were compared via statistical analysis. The obtained $p$ values allow concluding that the PM should be preferred especially when straining plays an important role.

Appendix: Filter Coefficient for Straining

In this section, we derive a filter coefficient function related to straining based on the following stochastic model for particle retention and pore blocking kinetics (Sharma and Yortsos 1987; Santos and Bedrikovetsky 2006; Santos et al. 2008), respectively:

$$\begin{aligned}&\frac{\partial S}{\partial t}=\frac{1}{\Delta }\frac{\int _0^{r_\mathrm{s}} {r_\mathrm{p}^4 H \mathrm{d}r_\mathrm{p} } }{\int _0^\infty {r_\mathrm{p}^4 H \mathrm{d}r_\mathrm{p} } }UC\end{aligned}$$

(11)

$$\begin{aligned}&\frac{\partial H}{\partial t}=-\frac{1}{\Delta }\frac{r_\mathrm{p}^4 H}{\int _0^\infty {r_\mathrm{p}^4 H \mathrm{d}r_\mathrm{p} } }U\int \limits _{r_\mathrm{p} }^\infty C \mathrm{d}r \end{aligned}$$

(12)

where $H$ and $C$ represent pore radius ($r_\mathrm{p})$ and particle radius ($r_\mathrm{s})$ concentration distributions, respectively. In addition, $U$ is the Darcy’s velocity, and $\Delta$ is the distance between subsequent pore throats. In Eq. (11), $\left( {\int _0^{r_\mathrm{s} } {r_\mathrm{p}^4 H \mathrm{d}r_\mathrm{p} } } \right) \left( {\int _0^\infty {r_\mathrm{p}^4 H \mathrm{d}r_\mathrm{p} } } \right)^{-1}$ represents the flow fraction through pores smaller than the particle size $r_\mathrm{s}.$ Therefore, the above mentioned fraction represents the retention probability for particles with size $r_\mathrm{s}.$

Assuming that there are no suspended or deposited particles in the porous medium before the injection, we obtain the following initial and boundary conditions for the Eqs. (11) and (12):

$$\begin{aligned} \begin{array}{l} T=0:C=0; \quad S=0;\quad H=H_0 \\ X=0 :C=C_0 \\ \end{array} \end{aligned}$$

(13)

where $H_{0}$ is the initial pore concentration distribution, and $C_{0}$ is the injected particle concentration distribution.

Let us consider transport of monosized particles:

$$\begin{aligned} C\left( {r,x,t} \right)=c\left( {x,t} \right)\delta \left( {r-r_\mathrm{s} } \right) \end{aligned}$$

(14)

through a porous media with $N$ distinct pore sizes:

$$\begin{aligned} H\left( {r_\mathrm{p} ,x,t} \right)=h_1 \left( {x,t} \right)\delta \left( {r_\mathrm{p} -r_{\mathrm{p}1} } \right)+\cdots +h_N \left( {x,t} \right)\delta \left( {r_\mathrm{p} -r_{\mathrm{p}N} } \right) \end{aligned}$$

(15)

where $h_{i }$ is the concentration of pores with radius $r_{\mathrm{p}i },$ and $\delta$ is the Dirac’s delta function (see Fig. 5). In this case, substituting Eqs. (14) and (15) into Eqs. (11) and (12) results in

$$\begin{aligned} \frac{\partial \sigma }{\partial t}=\frac{1}{\Delta }\frac{\sum _{i=1}^n {r_{\mathrm{p},i}^4 h_i } }{\sum _{i=1}^N {r_{\mathrm{p},i}^4 h_i } }Uc\end{aligned}$$

(16)

$$\begin{aligned} \frac{\partial h_i }{\partial t}= \left\{ {\begin{array}{l} -\frac{1}{\Delta }\frac{h_i r_{\mathrm{p}i}^4 }{\sum _{i=1}^N {h_i r_{\mathrm{p}i}^4 } }Uc,\quad i\le n \\ 0, \quad i>n \\ \end{array}}\right. \end{aligned}$$

(17)

where $n$ defines the largest pore radius ($r_{\mathrm{p}n})$ smaller than the particle radius ($r_{\mathrm{s}}),$ see Fig. 5. In addition, $c$ and $\sigma$ represent the suspended and retained particle concentrations, respectively.

Fig. 5

Particle and pore concentration distributions

Comparing the traditional particle retention kinetics (second equation in system (1)) with the Eq. (16), we obtain

$$\begin{aligned} \lambda =\frac{\sum _{i=1}^n {h_i r_{\mathrm{p}i}^4 } }{\sum _{i=1}^N {h_i r_{\mathrm{p}i}^4 } }\frac{1}{\Delta } \end{aligned}$$

(18)

In addition, from Eq. (17), it follows that

$$\begin{aligned} \frac{\partial h_i }{\partial h_1 }=\left\{ {\begin{array}{l} -\frac{r_{\mathrm{p}i}^4 h_i }{r_{\mathrm{p}1}^4 h_1 },\quad i\le n \\ 0,\quad i>n \\ \end{array}} \right. \end{aligned}$$

(19)

The solution of Eq. (19) is given by

$$\begin{aligned} h_i \left( {h_1 } \right)=\left\{ {\begin{array}{l} h_{i,0} \left( {\frac{h_1 }{h_{i,0} }} \right)^{\left( {{r_{\mathrm{p}i} }/{r_{\mathrm{p}1} }} \right)^{4}}, \quad i\le n \\ h_{i,0} ,\quad i>n \\ \end{array}} \right. \end{aligned}$$

(20)

Moreover, the process of applying summation over all “$i$” in Eq. (20) and comparing the resulting equation with the Eq. (16) results in

$$\begin{aligned} \frac{\partial }{\partial t}\sum _{i=1}^n {h_i } =-\frac{\partial \sigma }{\partial t} \end{aligned}$$

(21)

Considering the initial conditions given by Eq. (13), the solution of Eq. (21) is given by

$$\begin{aligned} \sum _{i=1}^n {h_i } =h_0^\mathrm{s} -\sigma \end{aligned}$$

(22)

where

$$\begin{aligned} h_0^\mathrm{s} =\sum _{i=1}^n {h_{i,0} } \end{aligned}$$

(23)

Substituting Eq. (20) into Eq. (22), it follows that $h_{1}( \sigma$). Therefore, Eq. (18) can be rewritten as

$$\begin{aligned} \frac{\lambda _0 }{\lambda \left( \sigma \right)}=\left( {1+\frac{\sum _{i=n+1}^N {h_{i,0} r_{\mathrm{p}i}^4 } }{\sum _{i=1}^n {h_i \left( \sigma \right)r_{\mathrm{p}i}^4 } }} \right)\left( {\frac{\sum _{i=1}^n {h_{i,0} r_{\mathrm{p}i}^4 } }{\sum _{i=1}^N {h_{i,0} r_{\mathrm{p}i}^4 } }} \right) \end{aligned}$$

(24)

where

$$\begin{aligned} \lambda _0 =\frac{\sum _{i=1}^n {h_{i,0} r_{\mathrm{p}i}^4 } }{\sum _{i=1}^N {h_{i,0} r_{\mathrm{p}i}^4 } }\frac{1}{\Delta } \end{aligned}$$

(25)

Because all $h_{i}$ (with $i \le n)$ are decreasing functions, from Eq. (24) it follows that $\lambda$ is also a decreasing function of $\sigma .$ In addition, DBF occurs only if retention probability tends to zero (Santos and Barros 2010). Therefore,

$$\begin{aligned} \sum _{i=1}^n {h_i r_{\mathrm{p}i}^4 } <<\sum _{i=n+1}^N {h_i r_{\mathrm{p}i}^4 } <\sum _{i=1}^N {h_i r_{\mathrm{p}i}^4 } \end{aligned}$$

(26)

In this case, Eq. (24) tends to

$$\begin{aligned} \frac{\lambda \left( \sigma \right)}{\lambda _0 }=\frac{\sum _{i=1}^n {h_i \left( \sigma \right)r_{\mathrm{p}i}^4 } }{\sum _{i=1}^n {h_{i,0} r_{\mathrm{p}i}^4 } } \end{aligned}$$

(27)

Assuming that small pores can be represented by an unique pore radii, from Eqs. (27) and (22), it follows that

$$\begin{aligned} \frac{\lambda \left( \sigma \right)}{\lambda _0 }=1-\frac{\sigma }{h_0^\mathrm{s} } \end{aligned}$$

(28)