Franklin M. Orr

Onset of convection in a gravitationally unstable diffusive boundary layer in porous media

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 200m and 0.3 m.

Pendular rings between solids: meniscus properties and capillary force

The Laplace–Young equation is solved for axisymmetric menisci, analytically in terms of elliptic integrals for all possible types of pendular rings and liquid bridges when the effect of gravity is negligible, numerically for selected other cases in order to assess gravitys effect. Meniscus shapes, mean curvatures, areas and enclosed volumes are reported, as are capillary forces. It is shown that capillary attraction may become capillary repulsion when wetting is imperfect. The special configurations of vanishing capillary force and of zero mean curvature are treated. The range of utility of the convenient ‘circle approximation’ is evaluated.

Gravity currents with residual trapping

Motivated by geological carbon dioxide (CO 2 ) 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 CO 2 storage in aquifers lacking structural closure, if CO 2 is injected far enough from the outcrop of the aquifer.

Onshore Geologic Storage of CO2

The possibility that substantial quantities of CO2 can be injected into subsurface porous rock formations has been investigated sufficiently to show that pore space available to contain the CO2 is abundant. Multiple rock types and physical mechanisms can be used to trap the CO2 indefinitely. With careful site selection and operations, leakage to the near-surface region can be avoided. The next step is to test these injection processes at the scale of a large power plant.

A New Model of Trapping and Relative Permeability Hysteresis for All Wettability Characteristics

The complex physics of multiphase flow in porous media are usually modeled at the field scale using Darcy-type formulations. The key descriptors of such models are the relative permeabilities to each of the flowing phases. It is well known that, whenever the fluid saturations undergo a cyclic process, relative permeabilities display hysteresis effects. In this paper, we investigate hysteresis in the relative permeability of the hydrocarbon phase in a two-phase system. We propose a new model of trapping and waterflood relative permeability, which is applicable for the entire range of rock wettability conditions. The proposed formulation overcomes some of the limitations of existing trapping and relative permeability models. The new model is validated by means of pore-network simulation of primary drainage and waterflooding. We study the dependence of trapped (residual) hydrocarbon saturation and waterflood relative permeability on several fluid/rock properties, most notably the wettability and the initial water saturation. The new model is able to capture two key features of the observed behavior: (1) nonmonotonicity of the initial-residual curves, which implies that waterflood relative permeabilities cross; and (2) convexity of the waterflood relative permeability curves for oil-wet media caused by layer flow of oil.

Low IFT drainage and imbibition

Experiments were performed in four cores to investigate the effects of interfacial tension (IFT) changes and phase density difference on drainage and imbibition of oil/water/alcohol mixtures. In imbibition experiments, in which the cores were initially saturated with oil, both total recovery and the rates of recovery increased when IFT and density differences were reduced, despite reduced capillary and gravity driving forces. Time scale analysis shows that a transition from capillary-dominated flow to gravity-dominated flow as IFT is reduced, accounts for the observations. At high ratios of capillary to gravity forces (hight inverse Bond numbers, NB−1) the flow is capillary dominated and counercurrent. At low NB−1, gravity segregation dominates the flow. When the initial wetting phase saturation is zero, snap-off residual oil droplets are partially suppressed for NB−1 < 1.0. When an initial oil saturation is present, however, entrapment of residual oil is not suppressed for NB−1 as low as 0.04. In free drainage experiments, in which a core filled with an equilibrated wetting phase was surrounded by an oil phase, the wetting phase was completely retained in the core for NB−1 > 1.0. For NB−1 < 1.0, however, gravity forces were sufficient to overcome capillary entry pressures and free damage occurred. Significant reductions in oil saturation (60%) were observed for NB−1 < 0.1. The potential for recovery of oil from fractured reservoirs based on free drainage during gas injection is discussed. Critical scaling shows that a multicontact miscible gas injection process can be used to create the low value of NB−1 required. Estimates of NB−1 for fractured reservoirs indicate that use of gravity drainage is possible even for low-permeability reservoirs as long as fractures exist with vertical connections of sufficient height.

CO2 capture and storage: are we ready?

Options for capture and storage of CO2 that would otherwise be released into the atmosphere by combustion of fossil fuels are considered. This paper assesses whether CO2 can be captured, whether sufficient potential capacity exists for storage in geologic formations, describes physical mechanisms that can prevent escape of the CO2 from the subsurface, delineates methods for monitoring the movement of CO2 in the subsurface and for detecting leaks, and describes field experience with CO2 injection. While much remains to be learned about the design of specific storage projects, the current state of knowledge of carbon capture and storage is sufficient to permit testing at the scale of large power plants.

Use of carbon dioxide in enhanced oil recovery.

Large volumes of oil will remain in U.S. oil reservoirs when standard recovery methods have been completed. Supercritical carbon dioxide can be used to recover part of that remaining oil. If carbon dioxide is dense enough, it extracts hydrocarbons from the oil to make a mixture miscible with crude oil. Such a mixture can recover 95 percent of the oil in controlled laboratory flow settings. Heterogeneity of reservoir rocks and the low viscosity of carbon dioxide reduce the fraction of oil recovered in projects to lower but still significant levels. With the construction of three pipelines to carry naturally occurring carbon dioxide from Colorado and New Mexico to Permian basin oil fields, large-scale implementation of enhanced oil recovery by carbon dioxide flooding is now beginning.

Static drop on an inclined plate: Analysis by the finite element method

Abstract The shapes of drops of given volume and density in contact with an inclined plate over a circular wetted area are found from the Young—Laplace equation by means of the finite element method. Contact angle variations around the contact circles at mechanical equilibrium are determined, and the maximum advanced and minimum receded contact angles are calculated as functions of plate inclination and drop volume. The largest inclination at which a drop of given volume can remain static is found. An approximate force balance used by previous investigators is evaluated and Larkins one-parameter family of solutions that satisfy no simple boundary condition is discussed.

Gravity currents in horizontal porous layers : transition from early to late self-similarity

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 ∝ t 1/2 . 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 ∝ t 1/3 . 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 CO 2 , 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 ∝ t 1/2 to x ∝ t 1/3 .