Erratum: (Computational and Applied Mathematics, Vol. 22, N. 3, pp. 359-395) Steam injection into water-saturated porous rock

Erratum: (Computational and Applied Mathematics, Vol. 22, N. 3, pp. 359-395)

Steam injection into water-saturated porous rock

W. Lambert^I; J. Bruining^II; D. Marchesin^I

^IInstituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, RJ, Brazil. E-mails: [e-mail]/ [e-mail]

^IIDietz Laboratory, Centre of Technical Geoscience, Mijnbouwstraat 120, 2628 RX Delft, The Netherlands. E-mail: [e-mail]

ABSTRACT

The Riemann problem for steam injection at boiling temperature into a porous medium saturated with water was solved in [1]. Here, we correct the Riemann solution for case III and redraw the speed diagrams 3.1, 3.4 and 4.5. We redraw also the solution in case III, Figure 4.4.

Mathematical subject classification: 76S05, 35L60, 35L67.

Key words: Porous medium, steamdrive, Riemann solution, balance equations, multiphase flow.

In [1], cases I and II are correct, but there are some mistakes that influence the solution in case III. The first relevant mistake is the statement that the saturation S_† maximizes v^SCF (Remark 11). The correct statement is:

Theorem 1. There are two saturation values that satisfy Eq_: (3.7) in [1] for each fixed T < T^b. The smallest S_† minimizes v^SCF. The other S_†† maximizes v^SCF, but it is irrelevant.

Moreover the solution is stable if we vary the left state, it follows that:

Corollary 1. In the limit as S_L tends to zero, the wave speed in the hot region given by Eq. (2.8) in [1] converges to zero, so the solution for the Riemann problem reduces a cooling discontinuity in the liquid water region.

From the above Theorem and Corollary, we see that the Figures 3.1 and 3.4 in [1] contain an error; we correct them in Figure 1.

The wave speed diagram in Figure 4.5 of [1] contains an error also. The correct diagrams are given in Figure 2. There are two diagrams because the saturation shock speed and the thermal shock are different for injected saturations larger than S_*. These diagrams are drawn out of scale for illustrative purposes, because the characteristic speed can be much larger than the other wave speeds. Figs. 3.1 and 3.4 in [1] contain an error also; those figures were summarized in Figure 4.5 in [1], which is corrected here in Figure 2. The Riemann solution for the cases I and II are correct, however the Riemann solution for case III contains an error. The mistake is the statement that the saturation shock speed is faster than v^SCF in cases I and II. The correct statement is that speed converges to zero if the water saturation at the left state tends to zero, as summarized in Corollary 1 above. The thermal wave speed is drawn in Figure 2.a) and in 2.b) we draw the saturation wave speed.

The solution diagram for case III given in Figure 4.4 of [1] must be modified. The strength of saturation shock tends to zero when the injection saturation tends to 1, while the speed of cooling discontinuity does not change. In Figure 3 we show the schematic solution for case III for three different injection saturations.

REFERENCES

[1] Bruining J., Marchesin D. and Van Duijn C.J. Steam Injection Into Water-Saturated Porous Rock. Computational and Applied Mathematics, 22(3) (2003), 359-395.

[2] Lambert W. Doctoral Thesis: IMPA, 2006, in preparation.

[3] Lambert W., Marchesin D. and Bruining J. The Riemann Solution of the balance equations for steam and water flow in a porous medium, submitted to Methods and Applications of Analysis 2005.

Received: 28/XII/02. Accepted: 18/VIII/03.

This work was supported in part by: CNPq scholarship 141573/2002-3, ANP/PRH-32 scholarship, CNPq 301532/2003-6, CNPq 450161/2004-8, FAPERJ E-26/152.163/2002 and IM-AGIMB.

#558/02.

Publication Dates

• Publication in this collection
  20 Apr 2006
• Date of issue
  Dec 2005