Erratum to: Generalized Solution for 1-D Non-Newtonian Flow in a Porous Domain due to an Instantaneous Mass Injection

Fig. 1

2 Total sensitivity indices $(TSI)$ of each random parameter ($n,\,K,\,C_{p},\,\phi )$ with respect to the front position $L(T)$ versus time $T$ for cylindrical geometry ($d=2$), $S=1$, $M_0 =1$, $r_c =1$; 3 As (2) but total and partial variances $(V)$; 4 As (2) but with respect to the excess pressure $\Delta P$ versus radial distance $R$ and $T=1$; 5 As (4) but total and partial variances

1 Erratum to: Transp Porous Med (2012) 93:63–77 DOI 10.1007/s11242-012-9944-9

A review of the above work has revealed the following errors listed below.

• (i) In the expression of the mobility ratio $k/{\mu _\mathrm{ef}}$, the factor $(3+n)$ is wrongly written for the factor $(3n+1)$; hence the correct formulations of Eqs. (4) and (15) should read:

  $$\begin{aligned} \frac{k}{\mu _\mathrm{ef}}=\frac{1}{2H}\left({\frac{n\phi }{3n+1}} \right)^{n}\left(\frac{8k}{\phi }\right)^{(1+n)/2}, \end{aligned}$$

  (4)
  $$\begin{aligned} \chi _n =\frac{8^{(1+n)/2}}{2}\left({\frac{n}{3n+1}} \right)^{n}. \end{aligned}$$

  (15)

• (ii) There is a missing minus sign before the pressure gradient within parentheses on the r.h.s. of Eq. (5); the correct formulation is

  $$\begin{aligned} \frac{\partial ^{2}p}{\partial r^{2}}+\frac{(d-1)n}{r}\frac{\partial p}{\partial r}=n\left({\phi \cdot c_0+c_p}\right)\left({\frac{\mu _\mathrm{ef}}{k}} \right)^{1/n}\left({-\frac{\partial p}{\partial r}} \right)^{{(n-1)}/n}\frac{\partial p}{\partial t}; \end{aligned}$$

  (5)

  this does not affect further developments which are based on the correct dimensionless version (14).

• (iii) There is a missing factor $K^{(n+1)/2n}$ in the definition of dimensionless velocity $V$; hence the correct versions of Eqs. (13) and (31) are:

  $$\begin{aligned} V=\frac{\chi _n^{1/n}}{\phi ^{{\left({1-n}\right)}/2n}}K^{(n+1)/2n}\left({-\frac{\partial P}{\partial R}} \right)^{1/n}, \end{aligned}$$

  (13)
  $$\begin{aligned} V\!=\!\frac{\chi _n^{1/n}}{\phi ^{(1-n)/2n}}K^{(n+1)/2n} \left({\frac{1\!-\!n}{1\!+\!n}A\beta } \right)^{1/(1-n)}T^{-{[(1+d)]}/[1+n+d(1-n)]}\eta (\eta _{1}^{1+n}-\eta ^{1+n})^{1/(1-n)}. \end{aligned}$$

  (31)

• (iv) The exponent of dimensionless permeability $K$ in Eq. (16) is $(1+n)/2n$ and not $(1-n)/2n$; the correct version is

  $$\begin{aligned} A=(r_{c} \phi +C_{p})\frac{\phi ^{(1-n)/2n}}{\chi _{n}^{1/n} K^{(1+n)/2n}}. \end{aligned}$$

  (16)

As a consequence of corrections (i) and mainly of (iv), Figs. 2–5 in Di Federico and Ciriello (2012) need amendment. The revised version of the figures, drawn for dimensionless injected mass $M_0=1$, is shown below. It is seen that the overall trends discussed in Di Federico and Ciriello (2012) remain valid, albeit with a larger influence of permeability on the variance of front position and pressure and, correspondingly, a higher total sensitivity index for permeability.

Moreover, the reference to Di Federico et al. (2010) in Di Federico and Ciriello (2012) contains a typographical error in page number. That reference should read as indicated in the reference section of this erratum.