3 results
Gravity currents with residual trapping
- M. A. HESSE, F. M. ORR, JR, H. A. TCHELEPI
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- Journal:
- Journal of Fluid Mechanics / Volume 611 / 25 September 2008
- Published online by Cambridge University Press:
- 25 September 2008, pp. 35-60
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Motivated by geological carbon dioxide (CO2) storage, we present a vertical-equilibrium sharp-interface model for the migration of immiscible gravity currents with constant residual trapping in a two-dimensional confined aquifer. The residual acts as a loss term that reduces the current volume continuously. In the limit of a horizontal aquifer, the interface shape is self-similar at early and at late times. The spreading of the current and the decay of its volume are governed by power-laws. At early times the exponent of the scaling law is independent of the residual, but at late times it decreases with increasing loss. Owing to the self-similar nature of the current the volume does not become zero, and the current continues to spread. In the hyperbolic limit, the leading edge of the current is given by a rarefaction and the trailing edge by a shock. In the presence of residual trapping, the current volume is reduced to zero in finite time. Expressions for the up-dip migration distance and the final migration time are obtained. Comparison with numerical results shows that the hyperbolic limit is a good approximation for currents with large mobility ratios even far from the hyperbolic limit. In gently sloping aquifers, the current evolution is divided into an initial near-parabolic stage, with power-law decrease of volume, and a later near-hyperbolic stage, characterized by a rapid decay of the plume volume. Our results suggest that the efficient residual trapping in dipping aquifers may allow CO2 storage in aquifers lacking structural closure, if CO2 is injected far enough from the outcrop of the aquifer.
Gravity currents in horizontal porous layers: transition from early to late self-similarity
- M. A. HESSE, H. A. TCHELEPI, B. J. CANTWEL, F. M. ORR, Jr
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- Journal:
- Journal of Fluid Mechanics / Volume 577 / 25 April 2007
- Published online by Cambridge University Press:
- 19 April 2007, pp. 363-383
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We investigate the evolution of a finite release of fluid into an infinite, two-dimensional, horizontal, porous slab saturated with a fluid of different density and viscosity. The vertical boundaries of the slab are impermeable and the released fluid spreads as a gravity current along a horizontal boundary. At early times the released fluid fills the entire height of the layer, and the governing equation admits a self-similar solution that is a function of the viscosity ratio between the two fluids. This early similarity solution describes a tilting interface with tips propagating as x ∝ t1/2. At late times the released fluid has spread along the boundary and the height of the current is much smaller than the thickness of the layer. The governing equation simplifies and admits a different similarity solution that is independent of the viscosity ratio. This late similarity solution describes a point release of fluid in a semi-infinite porous half-space, where the tip of the interface propagates as x ∝ t1/3. The same simplification of the governing equation occurs if the viscosity of the released fluid is much higher than the viscosity of the ambient fluid. We have obtained an expression for the time when the solution transitions from the early to the late self-similar regime. The transition time increases monotonically with increasing viscosity ratio. The transition period during which the solution is not self-similar also increases monotonically with increasing viscosity ratio, for mobility ratios larger than unity. Numerical computations describing the full evolution of the governing equation show good agreement with the theoretical results. Estimates of the spreading of injected fluids over long times are important for geological storage of CO2, and for the migration of pollutants in aquifers. In all cases it is important to be able to anticipate when the spreading regime transitions from x ∝ t1/2 to x ∝ t1/3.
Onset of convection in a gravitationally unstable diffusive boundary layer in porous media
- A. RIAZ, M. HESSE, H. A. TCHELEPI, F. M. ORR
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- Journal:
- Journal of Fluid Mechanics / Volume 548 / 10 February 2006
- Published online by Cambridge University Press:
- 01 February 2006, pp. 87-111
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We present a linear stability analysis of density-driven miscible flow in porous media in the context of carbon dioxide sequestration in saline aquifers. Carbon dioxide dissolution into the underlying brine leads to a local density increase that results in a gravitational instability. The physical phenomenon is analogous to the thermal convective instability in a semi-infinite domain, owing to a step change in temperature at the boundary. The critical time for the onset of convection in such problems has not been determined accurately by previous studies. We present a solution, based on the dominant mode of the self-similar diffusion operator, which can accurately predict the critical time and the associated unstable wavenumber. This approach is used to explain the instability mechanisms of the critical time and the long-wave cutoff in a semi-infinite domain. The dominant mode solution, however, is valid only for a small parameter range. We extend the analysis by employing the quasi-steady-state approximation (QSSA) which provides accurate solutions in the self-similar coordinate system. For large times, both the maximum growth rate and the most dangerous mode decay as $t^{1/4}$. The long-wave and the short-wave cutoff modes scale as $t^{1/5}$ and $t^{4/5}$, respectively. The instability problem is also analysed in the nonlinear regime by high-accuracy direct numerical simulations. The nonlinear simulations at short times show good agreement with the linear stability predictions. At later times, macroscopic fingers display intense nonlinear interactions that significantly influence both the front propagation speed and the overall mixing rate. A dimensional analysis for typical aquifers shows that for a permeability variation of 1—3000 mD, the critical time can vary from 2000 yrs to about 10 days while the critical wavelength can be between 200 m and 0.3 m.