Straining calculation

Straining, the physical  filtration of particles in soils, is modelled similarly to attachment as an irreversible, first-order loss. There is thus a straining rate coefficient, but contrary to the case of attachment, this coefficient decreases as a function of depth. The idea is that particles will be less frequently found in dead-end pores if it is the case that overall, particles are being transported to deeper and deeper layers in the soil.

 \(k_{straining}=\Psi k\)
\(\Psi = (\frac{d_{50 + z}}{d_{50}})^{-\beta }\)


\(k_{straining}\) is the straining rate constant and \(\Psi\) is the depth-dependent straining rate coefficient. \(k\) expresses the pseudo-first order rate of the interaction itself. This coefficient depends on the distance \(z\) from the origin/injection point of the nanomaterials in the porous medium and also on the average aggregate (collector) diameter \(d_{50}\). \(\beta\) is  an empirical factor expressing the intensity of this depth dependence.




Straining is usually assumed to occur at the same time as other processes such as attachment. There is thus an assumption of a "second type of interaction site" where straining occurs. The straining rate constant itself, however, is a calibration constant fitted to column outflow experiments. Moreover, b is assumed equal to 0.43.

Used in 



 \(\theta \frac{dC}{dt}=v\theta \frac{dC}{dz}\)

\(-\theta (\sum D )\frac{d^{2}C}{(dz)^{2}}\)

\(-\Psi k_{straining} - k_{att}\)

Soil transport calculation 



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Consult the NanoFASE Library to see abstracts of these deliverable reports:

NanoFASE Report D7.2 Soil property - NM fate relationships 

Bradford, S.A., et al., Modeling Colloid Attachment, Straining, and Exclusion in Saturated Porous Media. Environmental Science & Technology, 2003. 37(10): p. 2242-2250





  Geert Cornelis

  Swedish University of Agricultural Sciences (SLU)