Thermal inertia and reversing buoyancy in flow in porous media
Thierry Menand, Alan Raw, and Andrew W. Woods BP Institute for Multiphase Flow, University of Cambridge, Cambridge, UK
Received 17 September 2002; revised 4 December 2002; accepted 17 January 2003; published 21 March 2003. [1] The displacement of fluids through porous rocks is
fundamental for the recharge of geothermal and hydrocarbon reservoirs [Grant et al., 1982; Lake, 1989], for contaminant dispersal through the groundwater [Bear, 1972] and in controlling mineral reactions in permeable rocks [Phillips, 1991]. In many cases, the buoyancy force associated with density differences between the formation fluid and the displacing fluid controls the rate and pattern of flow through the permeable rock [Phillips, 1991; Barenblatt, 1996; Turcotte and Schubert, 2002]. Here, using new laboratory experiments, we establish that a striking range of different flow patterns may develop depending on whether this density contrast is associated with differences in temperature and/or composition between the two fluids. Owing to the effects of thermal inertia in a porous rock, thermal fronts lag behind compositional fronts [Woods and Fitzgerald, 1993; Turcotte and Schubert, 2002], so that two zones of different density develop in the region flooded with injected fluid. This can lead to increasing, decreasing or even reversing buoyancy in the injected liquid; in the latter case it may then form a double-flood front, spreading along both the upper and lower boundary of the rock. Recognition of these different flow regimes is key for predicting sweep efficiency and dispersal patterns in natural and engineered flows, and offers new opportunities for the enhanced recovery of natural resources in porous rocks. INDEX TERMS: 1832 Hydrology: Groundwater transport; 1884 Hydrology: Water supply; 1829 Hydrology: Groundwater hydrology; 3665 Mineralogy and Petrology: Mineral occurrences and deposits. Citation: Menand, T., A. Raw, and A. W. Woods, Thermal inertia and reversing buoyancy in flow in porous media, Geophys. Res. Lett., 30(6), 1291, doi:10.1029/2002GL016294, 2003.
1. Introduction
[2] When a fluid of temperature T + T and composition C + C migrates through a porous layer initially saturated with fluid of temperature T and composition C, thermal and compositional fronts develop across which the properties of the injected fluid adjusts to that of the formation fluid. The compositional front travels with the interstitial speed, u/f, where u is the transport (Darcy) velocity andf the porosity of the matrix. However, heat is transferred between the invading fluid and the rock matrix, causing the thermal front to travel with the slower speed u where = (rCp)liq/(rCp) O(1) is the ratio of the specific heat of the liquid to the average specific heat of the liquid and porous matrix [Woods and Fitzgerald, 1993; Barenblatt, 1996; Turcotte and Schubert, 2002]. This separation of the two fronts results in three regions of different density (Figure 1a).
Near the source, the fluid retains the temperature and composition of the injected solution r(T + T, C + C ) (region a); ahead of the thermal front, but still within the injected solution, the fluid temperature has adjusted to that of the rock, and so the density has value r(T, C + C ) (region b); finally, in the original formation fluid, the density has value r(T, C ) (region c). These three regimes may be clearly seen in a simple draining experiment (Figure 1b) in which a fresh, hot aqueous solution, dyed in red, drains at a constant rate through a fine-bead pack initially saturated with a cold, saline aqueous solution. The succes-sive positions of both the fluid front (compositional signal) and the temperature front (thermal signal) have been meas-ured during the experiment (Figures 1b and 1c) and are in very good accord with the theoretical prediction.
2. Impact of the Thermal Inertia on Buoyancy Driven Flows in Porous Media
[3] In flows driven by the gravitational force associated with the difference in density between the injected and original fluid, the spatial decoupling of the thermal and compositional signals in the migrating fluid is crucial. To illustrate this we present three laboratory experiments (A, B and C) in which a dyed aqueous solution was injected from a point source at the base of a bead pack initially saturated with an aqueous solution of different density. In all experi-ments, the injected fluid is approximately 0.5% less dense than the fluid originally in the bead pack. However, the injected fluid had three different temperatures and compo-sitions denoted by points A (T = 0C, C = 0.75 wt%), B (T = 19C, C = 0) and C (T = 33.5C, C = 0.75 wt%) in Figure 2. As the three injected fluids migrate through the porous layer, the temperature of the injected fluid adjusts to that of the porous layer and hence their densities evolved in very different ways (Figure 2). Fluid A remains at the same temperature and hence density; fluid B cools to the temperature of the bead pack and the density adjusts to the density of the formation fluid; fluid C cools until its temperature matches that of the formation, at which point it has become dense relative to the formation fluid. These different density structures lead to very different flooding patterns as shown in Figures 3a to 3c.
[4] In case A, fluid of the same temperature as the fluid in the bead pack, but smaller composition, was injected from below. This produced a buoyant plume with no change in density of the source fluid (Figure 2, point A). As the flow evolved, there was a small amount of mixing in the head of the flow, but subsequently a sharp, nearly parallel-sided plume developed (Figure 3a). In case B, fluid of the same composition as the fluid in the bead pack, but higher temperature, was injected from below. As this fluid migrated upwards, the thermal front lagged behind the
GEOPHYSICAL RESEARCH LETTERS, VOL. 30, NO. 6, 1291, doi:10.1029/2002GL016294, 2003
Copyright 2003 by the American Geophysical Union. 0094-8276/03/2002GL016294
injected fluid so that neutrally buoyant fluid was supplied to the head of the plume (Figure 2, point B). This neutrally buoyant fluid then displaced the formation fluid radially outwards (Figure 3b), leading to a much larger head structure than in the saline experiment (Figure 3a). Mean-while, the continuing injection of hot fluid heated up the porous layer to form a focused hot plume behind the head. Heat transferred from this plume to the surrounding porous matrix produced a zone of heated ambient fluid beside the source. This ambient fluid also ascended, producing the clear halo around the hot plume of injected fluid (Figure 3b). In case C, the injected fluid was hot but relatively saline compared to the formation fluid. As it displaced the for-mation fluid and cooled down, the density of the injected fluid actually became larger than the formation fluid, owing to the greater initial composition (Figure 2, point C). This change in the sign of the density of the injected fluid led to a more complex flow pattern. Initially, the injected fluid was of smaller density than the formation fluid, and so it ascended upwards. However, once it had cooled down, the density difference became controlled by the
composi-tional difference, so that the injected fluid became relatively dense. It then descended back to the base of the bead pack, and spread laterally as a relatively dense compositional gravity current (Figure 3c). The continuing injection pro-duced a growing fountain of hot fluid above the source and supplied the laterally spreading dense current.
[5] In the case of reversing buoyancy, a fourth, different flow regime, case D, may develop and this is shown in Figure 3d. Here, relatively hot but saline fluid (Figure 2, point D) was injected from the top of the bead pack rather than the base of the bead pack. Initially, the relatively buoyant fluid spread laterally along the upper surface of the formation. However, as the fluid migrated through the Figure 2. A Temperature-Composition phase diagram for an aqueous solution of NaCl illustrating curves of constant density,r1<r2<r3. The different initial conditions for the experiments A-D are shown by solid circles. Arrows indicate how the density of the injected fluid evolves as it moves through the bead pack and its temperature adjusts to that of bead pack.
Figure 1. (opposite) (a) Diagram illustrating the spatial distribution of (i) temperature, T; (ii) composition, C and (iii) density,r as fluid of one temperature and composition is injected into a bead pack saturated with fluid of different temperature and composition. The spatial decoupling of the thermal and fluid fronts creates three regions in which the fluid has different density: near the source (region a), between the thermal and compositional fronts (region b) and in the original formation fluid (region c). (b) Photographs from an experiment in which hot, fresh water dyed red drained downwards at a constant rate through a bead pack initially saturated with cold, saline water. The bead pack has porosity 0.4 ± 0.01, has dimensions 15 cm wide, 19 cm high, 1 cm deep, and is made of ballotini 425 – 600mm in diameter. The specific heat ratio for this experiment including the thermal mass of the bead pack and cell walls, which are in equilibrium with the fluid, is 1.7. The flow rate was 0.3 cm3s 1. The advancing front of fresh water is shown by the red dye front, while the yellow horizontal line on the liquid crystal strip illustrates the leading edge of the thermal front. (c) The location of the dye front and the thermal front measured during the experiment are indicated by circles and squares respectively. The solid line indicates the location of the dye front as predicted by a pure volume displacement, indicating that there is little dispersive mixing across the compositional front. The dashed line indicates the location of the thermal front ut. Both are in excellent accord with the data.
hot thermal front and cooled, the greater composition of the injected fluid caused it to become dense relative to the formation (Figure 2, point D), and the fluid sank to the base of the layer. Here, a compositionally-driven dense current spread out along the base of the layer. Meanwhile, the continuing injection of fluid led to a gradual lateral spreading of the hot thermal front along the upper surface of the porous layer, thereby forming a double-flood front (Figure 3d).
3. Conclusion
[6] This rich variety of flow patterns resulting from the separation of the thermal and compositional fronts may be key for accurate prediction of (i) the dispersal of contam-inants injected into permeable sub-surface rocks, especially in evaluating risks of sub-surface storage of nuclear waste; (ii) the dispersal of reacting fluids over geological time-scales, which may control the spatial distribution of diagenetic reactions and hence mineral deposits; and, (iii) the sweep pattern in high permeability geothermal or hydrocarbon reservoirs in which there are often significant temperature and compositional contrasts between the injected and formation water. These complex thermo-sol-utal transport effects introduce new challenges for theoret-ical models of dispersion and transport, since the flow structures identified herein are controlled by thermal boun-dary layers located in the interior of the flow domain and which are therefore hard to resolve accurately. However, the fundamental process of buoyancy reversal can lead to injected fluids spreading along both the upper and lower boundaries of a permeable rock; as with our experiments. If the injected fluid is engineered appropriately, this could result in substantially enhanced oil recovery from high temperature sub-surface hydrocarbon reservoirs or layered reaction zones within a reacting permeable matrix.
[7] Acknowledgments. This work was supported by the BP Institute for Multiphase Flow. T. Menand was also supported by a Newton Trust Fellowship.
References
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Bear, J., Dynamics of Fluids in Porous Media, 764 pp., Dover, Mineola, N.Y., 1972.
Grant, M. A., I. G. Donaldson, and P. F. Bixley, Geothermal Reservoir Engineering, 369 pp., Academic, San Diego, Calif., 1982.
Lake, L. W., Enhanced Oil Recovery, 550 pp., Prentice-Hall, Old Tappan, N.J., 1989.
Phillips, O. M., Flow and Reactions in Permeable Rocks, 285 pp., Cam-bridge Univ. Press, New York, 1991.
Turcotte, D. L., and G. Schubert, Geodynamics, 456 pp., Cambridge Univ. Press, New York, 2002.
Woods, A. W., and S. D. Fitzgerald, The vaporization of a liquid front moving through a hot porous rock, J. Fluid Mech., 251, 563 – 579, 1993.
T. Menand, A. Raw, and A. W. Woods, BP Institute for Multiphase Flow, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, U.K. (thierry@bpi.cam.ac.uk; alan@bpi.cam.ac.uk; andy@bpi. cam.ac.uk)
Figure 3. Four sequences of photographs illustrate the flow patterns that arise when fluid of one temperature and composition, dyed red, is continuously injected from a point source into a porous layer saturated with fluid of different temperature and composition. The initial conditions are shown in Figure 2. In the experiments, darker dye is added to the red injected fluid after some time to reveal the evolving flow pattern. (a) The fluid injected at the base of the bead pack has smaller composition than the formation fluid, but the same temperature; (b) The fluid injected at the base of the bead pack has the same composition, but is hotter that the formation fluid; (c) The fluid injected at the base of the bead pack has greater composition but is hotter than the formation fluid, but so that initially it is of smaller density; (d) The fluid injected at the top of the bead pack has greater composition but is hotter than the formation fluid so that, as in (c), it is initially of smaller density.
