Erratum to: Generalized Solution for 1-D Non-Newtonian Flow in a Porous Domain due to an Instantaneous Mass Injection
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Erratum to: Generalized Solution for 1-D Non-Newtonian Flow in a Porous Domain due to an Instantaneous Mass Injection Vittorio Di Federico · Valentina Ciriello
Published online: 22 February 2013 © Springer Science+Business Media Dordrecht 2013
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/μef , the factor (3 + n) is wrongly written for the factor (3n + 1); hence the correct formulations of Eqs. (4) and (15) should read: n (1+n)/2 nφ 8k k 1 = , (4) μef 2H 3n + 1 φ χn =
8(1+n)/2 2
n 3n + 1
n .
(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 μef 1/n ∂2 p ∂ p (n−1)/n ∂ p (d − 1)n ∂ p = n φ · c0 + c p ; (5) − + ∂r 2 r ∂r k ∂r ∂t this does not affect further developments which are based on the correct dimensionless version (14).
The online version of the original article can be found under doi:10.1007/s11242-012-9944-9. V. Di Federico (B)· V. Ciriello Dipartimento di Ingegneria Civile, Chimica, Ambientale e dei Materiali (DICAM), Università di Bologna, Bologna, Italy e-mail: [email protected] V. Ciriello e-mail: [email protected]
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V. Di Federico, V. Ciriello
Fig. 2–5 2 Total sensitivity indices (T S I ) of each random parameter (n, K , C p , φ) with respect to the front position L(T ) versus time T for cylindrical geometry (d = 2), S = 1, M0 = 1, rc = 1; 3 As (2) but total and partial variances (V ); 4 As (2) but with respect to the excess pressure P versus radial distance R and T = 1; 5 As (4) but total and partial variances
(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: 1/n ∂ P 1/n χn V = (1−n)/2n K (n+1)/2n − , (13) ∂R φ 1/n
V=
χn
φ
K (n+1)/2n (1−n)/2n
1−n Aβ 1+n
1/(1−n)
T −[(1+d)]/[1+n+d(1−n)] η(η11+n − η1+n )1/(1−n) . (31)
(iv) The exponent of dimensionless permeability K in Eq. (16) is (1 + n)/2n and not (1 − n)/2n; the correct version is A = (rc φ + C p )
φ (1−n)/2n 1/n χn K (1+n)/2n
.
(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 M0 = 1, is shown below. It is seen that the overall trends discussed in
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Generalized Solution for 1-D Non-Newtonian Flow
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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.
References Di Federico, V., Ciriello, V.: Generalized solution for
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