Several simulations using HYDROTHERM (Hayba and Ingebritsen 1994) were carried out for comparison with both the finite core experiments and the SP flow similarity solutions. (Note: HYDROTHERM simulates separable phase groundwater flow only; it does not simulate two-phase homogeneous mixture flow.) Unfortunately pressures are constrained in HYDROTHERM to be no less than 0.5 bar so comparisons with the physical experiments are qualitative only.
Simulations were carried out on a one-dimensional core of length 14.5
em, Pr
=2000 kg m-3, ¢ = 0.18, K = 2 Wm-1 K-1 ,
Cr
= 1000 Jkg- 1 , and k = 1 X 10-12 m2. The core was assumed to be initially saturated with liquid water, at a pressure Pi = 1.1 bar and temperature Ti = Tsat (P) = 102.3°C. Constant temperature and pressure conditions of Pf = 1 bar and Tf = Tsat (P) = 99.6°C were placed on the depressurized end.When "no flux" boundary conditions were placed on the end opposite that which was depressurized, a boiling front was initiated at the depressurized end and progressed through the porous medium [see Figure 5.13 (a)] . As the system approached thermody namic equilibrium, increases in liquid saturation were seen [see Figure 5.13 (b)] . Cal culated velocities indicated that water was flowing back into the core replenishing the depleted zone. Such late time increases in liquid saturation indicate consistency in this aspect of the simulations and physical experiments. In the physical experiments this had a cooling effect on the region. In the event that such an effect were produced in hydrothermal eruptions, it would slow the eruption in progress, and any cooling effect on the area may explain long recovery times between subsequent eruptions.
5.5. Summary 1 03
When "end-water" was placed at the closed end of the porous medium, small-scale boiling was indicated almost immediately in this water (not shown) . This boiling, how ever, occurred in very small quantities in the simulations, much smaller than the NMR profiles indicated.
Good agreement between the simulation and the (SP) similarity solution was obtained up to the time at which the boiling front had progressed completely through the core. Late-time saturation increases are a feature of the finiteness of the core and are therefore not predicted by the similarity solution. (Note: This agreement for early time periods is true for simulations in which a relatively small pressure reduction was made at one end of the short core. The effects of the boundary conditions which are placed on the end opposite depressurization can be seen even during early time periods for large pressure drops.)
When constant temperature and pressure boundary conditions were placed on the end opposite that which was depressurized, we again saw the progression of a boiling front through the porous medium [see Figure 5.14 (a)]. As this boiling front moved through the core, heat was removed from the system [see Figure 5.14 (d), t < 20 s].
Once the boiling front had progressed completely through the porous medium, increases in liquid saturation were again seen, but water velocities indicate in this case these were due to water being fed from the end opposite that which was depressurized. A liquid resaturation front moved back through the core until the system eventually reached a steady state [see Figure 5.14 (b)].
Good agreement between the simulation and the (SP) similarity solution was again obtained up to the time at which the boiling front had progressed completely through the core.
5 . 5 Summary
In this chapter, a report is given on numerical experiments, physical experiments and computer simulations which have been conducted to investigate some aspects of transient boiling processes in porous media.
In Section 5.2, a comparison of the progression of boiling fronts under the differ ing flow regimes of HM and SP fluid flows was made. A one-dimensional semi-infinite mathematical model was solved for both cases. As the HM flow model is based on the assumption that the rapidity of motion (in hydrothermal eruptions) does not allow for separable phase flow to develop, it is interesting to note that the predictions of this flow regime are slower fluid mixture flow and a slower progression of the boiling front than predicted by the SP flow model. A comparison of results for the two flow cases also
1 0.9 0.8 1 0.9 0.8 c: .2 .2 c: 0.7 � '" 0; 60 s CI) :2 80 s '" cr :J 500 s "§ 0.6 '" 0.5 0; CI) "0 0.4 .; cr 0.3 :J 0.2 0 0 0 0 0 g g 0 g 0 0 0 0 0 0 0 tI> <J> 0. 1 (a) (b) 0 0 5 1 0 1 4.5 5 1 0 1 4.5 1 . 1 2 (c) 5 1 0 1 4.5 5 1 4.5 distance (cm) distance (cm)
Figure 5.13: Saturation, pressure and temperature curves for one-dimensional core. Con stant pressure and temperature conditions at x = 0 em, "no flux" boundary conditions at x = 14.5 em. (See text for details.)
5.5. Summary c: .S? � :::l (;j Vl " ':;
3"
0.3 0.2 0. 1 0 0 (a) 5 1 0 1 4 .5 (c) 5 1 0 1 4.5 distance (cm) 1 0 5 I 0.9 0.8 <: 0.7 .S? � 0.6 :::l 0.5 (;j Vl � 0.4 :::l cr 0.3 :J 0.2 0. 1 (b) 0 0 5 1 0 1 4.5 1 02.5 1 02.0 .= 1 0 1 .0 '" ... .., It 1 00.5 � (d) 99.5 0 5 1 0 1 4. 5 distance (cm)Figure
5.14:
Saturation, pressure and temperature curves for one-dimensional core. Con stant pressure and temperature conditions at both ends. (See text for details.)shows, due to the fact that the liquid and vapour must move at the same speed in HM flow, the liquid phase moves considerably faster and the vapour phase slightly slower in the HM flow case. Perhaps the most significant difference in the two flow cases, however, is that considerably more fluid boils in the HM flow case, particularly for smaller pressure drops across the core.
For both models of fluid mixture flow, investigation shows that the effects of thermal conductivity are negligible for k � 1 X 1 0-13 m2 . For cases in which the thermal con ductivity is negligible, the result of varying the permeability k on the solution is to scale the length of the boiling zone at any given time by
../k.
Also, for both fluid mixture flow models, the larger the porosity of the porous medium, the less boiling which occurs and the higher the final liquid saturation in the core. This difference in final liquid saturations is greater in SP flow as most of the fluid in the HM case boils regardless of the porosity. The results of the one-dimensional semi-infinite mathematical models were compared to results from physical experimental data. It was originally hoped that such a compar ison would allow for some conclusions to be made on whether HM flow or SP flow was likely to occur during early time periods in hydrothermal eruptions. However, more in vestigation is needed before such conclusions can be made. Because physical experiments were constrained to be finite, direct comparison of all aspects of results was not possible. Both numerical and physical experiments show a boiling front initiated at one end of the porous medium. However, the rate at which the front progressed through the core and boiling effects at the closed end of the core may be features of the finiteness of the physical experiment which were not predicted by the semi-infinite model. Differences in numerical and physical experimental configurations need to be resolved in order to gain quantitative agreement. Suggestions for future improvements to experimental work are given in Chapter 6.Numerical simulations using HYDROTHERM were conducted for comparison with both the physical experiment results and those from the SP similarity solution. Sim ulation calculations again show the progression of a boiling front through the porous medium. Some of the physical experiment effects due to the finiteness of the core (such as late time resaturation) which were not predicted by the semi-infinite similarity solu tion, were predicted by the computer simulations. Results of computer simulations and the SP similarity solution agree for "early" time periods, but the effects of the boundary conditions placed on the "closed" end of the finite core in the simulations do not allow for "late" time comparisons with the semi-infinite similarity solution.
C HAPT E R 6
Summary, Conclusions and Suggestions for Future Work