## 2 Mathematical Model  
### Colloidal Particle Transport in Saturated Porous Media  
Three main mechanisms can be taken into account for colloidal particle-sand interaction during the transport of particles in saturated porous media: straining, attachment, and detachment [12; 18]. Straining is the retention mechanism of particles caused by a relatively large particle size compared to the size of pore space [19; 20]. Attachment is another retention mechanism of particles induced by the net attraction energy between the colloidal particles and sand, typically explained by the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory [21; 22]. The detachment mechanism represents the separation of retained colloidal particles from sand induced by fluid drag forces [23; 24]. The one-dimensional continuum scale advection-dispersion-type governing equation with consideration of the above three main mechanisms can be expressed as:  
\[\frac{\partial(nC_{c})}{\partial t}=-q\frac{\partial(nC_{c})}{\partial z}+ \frac{\partial}{\partial z}\left(nD_{c}\frac{\partial C_{c}}{\partial z} \right)-nk_{att}\psi_{att}C_{c}+\rho_{b}k_{det}S_{att}-nk_{str}\psi_{str}C_{c}, \tag{1}\]  
\[\rho_{b}\frac{\partial S_{c}}{\partial t}=nk_{att}\psi_{att}C_{c}-\rho_{b}k_{ det}S_{att}+nk_{str}\psi_{str}C_{c}, \tag{2}\]  
where \(n\) (-) is the porosity, \(C_{c}\) (M L\({}^{-3}\)) is the particle concentration in aqueous phases, t (T) is the time, \(q\) (L T\({}^{-1}\)) is the [PERSON]'s velocity, \(z\) (L) is the depth, \(D_{c}\) (L\({}^{2}\) T) is the dispersion coefficient of particles, \(\rho_{b}\) is the bulk density of sand (M L\({}^{-3}\)), and \(k_{att}\) (T\({}^{-1}\)), \(k_{det}\) (T\({}^{-1}\)) and \(k_{str}\) (T\({}^{-1}\)) are the first-order coefficients accounting for attachment, detachment, and straining, respectively. \(\psi_{att}\) and \(\psi_{str}\) are the dimensionless functions for colloidal particle attachment and straining, respectively, and S\({}_{c}\) (M M\({}^{-1}\)) is the solid-phase concentration of retained colloidal particles (\(S_{c}=S_{att}+S_{str}\), where S\({}_{att}\) (M M\({}^{-1}\)) and S\({}_{str}\) (M M\({}^{-1}\)) are the solid-phase concentrations of attached and strained colloidal particles). \(\psi_{att}\)[25; 26], \(\psi_{str}\)[12], \(k_{att}\)[27], and \(k_{str}\)[12] in Equation (1) can be expressed as functions of the radius of colloidal particle (\(r_{c}\)) and the radius of sand (\(r_{s}\)) (see details in [17]):  
\[\psi_{att}(r_{c},r_{s})=1-\frac{S_{att}}{S_{\max}(r_{c},r_{s})}, \tag{3}\]  
\[\psi_{str}=\left(\frac{d_{s50}+z}{d_{s50}}\right)^{-\beta}, \tag{4}\]\[k_{str}(r_{c},r_{s})=4.495\bigg{(}\frac{r_{c}}{r_{s}}\bigg{)}^{1.42}, \tag{5}\]  
\[k_{att}(r_{c},r_{s})=\frac{3(1-\theta_{w})}{2d_{s50}}\alpha(r_{c})\eta_{0}(r_{c}, r_{s})v_{s}, \tag{6}\]  
where \(S_{\max}\) (M M\({}^{-1}\)) is the maximum attached mass of colloidal particles per unit mass of sand, \(d_{s50}\) (L) is the median size of sand, \(\beta\) (-) is a fitting parameter (\(\beta\) = 0.43, see [12]), \(\alpha\) (-) is the attachment efficiency, \(\eta_{0}\) (-) is the single-collector collision efficiency [28; 29], and \(v_{s}\) (L T\({}^{-1}\)) is the average seepage velocity. Note that the \(S_{\max}\) adopted in this paper was determined by the equation presented in [18], derived from the geometry of spherical colloidal particles and sand.