# Bibliography - Howard Stone

- Zheng, Zhong, Bo Guo, Ivan C. Christov, Michael Celia, and Howard Stone, 2015:
**Flow regimes for fluid injection into a confined porous medium** In , Cambridge University Press, **767**, doi:10.1017/jfm.2015.68 881–909

[ Abstract ]We report theoretical and numerical studies of the flow behavior when a fluid is injected into a confined porous medium saturated with another fluid of different density and viscosity. For a two-dimensional configuration with point source injection,
a nonlinear convection–diffusion equation is derived to describe the time evolution of the fluid–fluid interface. In the early time period, the fluid motion is mainly driven by the buoyancy force and the governing equation is reduced to a nonlinear diffusion equation with a well-known self-similar solution. In the late time period, the fluid
flow is mainly driven by the injection, and the governing equation is approximated by a nonlinear hyperbolic equation that determines the global spreading rate; a shock solution is obtained when the injected fluid is more viscous than the displaced fluid, whereas a rarefaction wave solution is found when the injected fluid is less viscous.
In the late time period, we also obtain analytical solutions including the diffusive term associated with the buoyancy effects (for an injected fluid with a viscosity higher than or equal to that of the displaced fluid), which provide the structure of the moving front. Numerical simulations of the convection–diffusion equation are
performed; the various analytical solutions are verified as appropriate asymptotic limits, and the transition processes between the individual limits are demonstrated. The flow behavior is summarized in a diagram with five distinct dynamical regimes:
a nonlinear diffusion regime, a transition regime, a travelling wave regime, an equal-viscosity regime, and a rarefaction regime.

- Christov, Ivan C., and Howard Stone, August 2014:
**Shear dispersion in dense granular flows**. *Granular Matter*, **16(4)**, doi:10.1007/s10035-014-0498-0 509-515

[ Abstract ]We formulate and solve a model problem of dispersion of dense granular materials in rapid shear flow down an incline. The effective dispersivity of the depth-averaged concentration of the dispersing powder is shown to vary as the Péclet number squared, as in classical Taylor-Aris dispersion of molecular solutes. An extension to generic shear profiles is presented, and possible applications to industrial and geological granular flows are noted.

- Zheng, Zhong, Ivan C. Christov, and Howard Stone, 2014:
**Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents**. *Journal of Fluid Mechanics*, Cambridge University Press, **747**, doi:10.1017/jfm.2014.148 218–246

[ Abstract ]We report experimental, theoretical and numerical results on the effects of horizontal heterogeneities on the propagation of viscous gravity currents. We use two geometries to highlight these effects: (a) a horizontal channel (or crack) whose gap thickness
varies as a power-law function of the streamwise coordinate; (b) a heterogeneous porous medium whose permeability and porosity have power-law variations. We demonstrate that two types of self-similar behaviours emerge as a result of horizontal
heterogeneity: (a) a first-kind self-similar solution is found using dimensional analysis (scaling) for viscous gravity currents that propagate away from the origin (a point of zero permeability); (b) a second-kind self-similar solution is found using a phase-plane
analysis for viscous gravity currents that propagate toward the origin. These theoretical predictions, obtained using the ideas of self-similar intermediate asymptotics, are compared with experimental results and numerical solutions of the governing partial
differential equation developed under the lubrication approximation. All three results are found to be in good agreement.

- Al-Housseiny, Talal, and Howard Stone, 2013:
**Controlling viscous fingering in tapered Hele-Shaw cells**. *Physics of Fluids*, **25**, doi:10.1063/1.4819317

[ Abstract ]We present a theoretical study of a variant of the classical viscous fingering instability, which occurs when a high viscosity fluid is displaced by a low viscosity fluid in a Hele-Shaw cell. In our system, the Hele-Shaw cell is tapered in the direction of fluid displacement. We consider two tapered Hele-Shaw geometries (rectilinear and radial), which have a constant depth gradient in the flow direction. We find that the presence of a depth gradient can alter the stability of the interface offering opportunities to control and tune fingering instabilities. In particular, the stability of the interface is now determined by both the viscosity contrast of the fluids and the ratio of the depth gradient to the capillary number of the system. We also demonstrate several applications of our analysis, including the inhibition of viscous fingering by controlling the injection flow rate in a radially tapered Hele-Shaw cell.

- Gor, Gennady Y., Howard Stone, and Jean Hervé Prévost, October 2013:
**Fracture Propagation Driven by Fluid Outflow from a Low-permeability Aquifer**. *Transport in Porous Media*, **100(1)**, doi:10.1007/s11242-013-0205-3 69-82

[ Abstract ]Deep saline aquifers are promising geological reservoirs for CO_{2} sequestration if they do not leak. The absence of leakage is provided by the caprock integrity. However, CO_{2} injection operations may change the geomechanical stresses and cause fracturing of the caprock. We present a model for the propagation of a fracture in the caprock driven by the outflow of fluid from a low-permeability aquifer. We show that to describe the fracture propagation, it is necessary to solve the pressure diffusion problem in the aquifer. We solve
the problem numerically for the two-dimensional domain and show that, after a relatively short time, the solution is close to that of one-dimensional problem, which can be solved analytically. We use the relations derived in the hydraulic fracture literature to relate the
width of the fracture to its length and the flux into it, which allows us to obtain an analytical expression for the fracture length as a function of time. Using these results we predict the propagation of a hypothetical fracture at the In Salah CO_{2} injection site to be as fast as a typical hydraulic fracture. We also show that the hydrostatic and geostatic effects cause the increase of the driving force for the fracture propagation and, therefore, our solution serves
as an estimate from below. Numerical estimates show that if a fracture appears, it is likely that it will become a pathway for CO_{2} leakage.

- Tsai, Peichun, K. Riesing, and Howard Stone, January 2013:
**Density-driven convection enhanced by an inclined boundary: implications for geological CO**_{2} storage. *Physical Review E*, American Physical Society, **87**,

[ Abstract ]We experimentally examine dissolution-generated, density-driven convection with an inclined boundary in both a Hele-Shaw cell and in a porous medium. The convection, manifested by descending, dense fingers, is generated by a diffusive mixing of two liquids at the interface. We investigate the dynamics, widths, and wavelengths of the fingers and characterize the global convective transport for a wide range of permeabilities and tilt angles of the boundaries. Our results have implications for CO_{2} storage in a saline aquifer when brine saturated with CO_{2} produces a heavier mixture, which may result in an enhanced mass transfer by convection. Our measurements reveal a further enhancement of convection with inclined boundaries, which suggests that sloping formations provide improved sites for CO_{2} storage.

- Zheng, Zhong, B. Soh, H.E. Huppert, and Howard Stone, February 2013:
**Fluid drainage from the edge of a porous reservoir** In , Cambridge University Press, **718**, doi:10.1017/jfm.2012.630 558-568

[ Abstract ]We report theoretical and experimental studies to describe buoyancy-driven fluid drainage from a porous medium for configurations where the fluid drains from an edge. We first study homogeneous porous systems. To investigate the influence of
heterogeneities, we consider the case where the permeability varies transverse to the flow direction, exemplified by a V-shaped Hele-Shaw cell. Finally, we analyse a model where both the permeability and the porosity vary transverse to the flow direction. In each case, a self-similar solution for the shape of these gravity currents is found and a power-law behaviour in time is derived for the mass remaining in the system. Laboratory experiments are conducted in homogeneous and V-shaped Hele-Shaw cells, and the measured profile shapes and the mass remaining in the cells agree well with our model predictions. Our study provides new insights into drainage processes such as may occur in a variety of natural and industrial activities, including the geological storage of carbon dioxide.

- Al-Housseiny, Talal, Peichun Tsai, and Howard Stone, 2012:
**Control of interfacial instabilities using flow of geometry**. *Nature Physics*, **8**, doi:10.1038/nphys2396 747-750

[ Abstract ]The displacement of one fluid by another is one of the most common processes involving interfacial instabilities. It is universally accepted that, in a uniform medium, flow displacement is unstable when a low-viscosity fluid invades a fluid of higher viscosity: the classical viscous fingering instability. Consequently, once fluid properties are specified, opportunities for control become very limited. However, real systems where displacement instabilities occur, such as porous structures, lung airways and printing devices, are rarely uniform. We find that the simplest heterogeneity—a gradient in the flow passage—can lead to fundamentally different displacement behaviours. We use this finding to either inhibit or trigger an instability and, hence, to devise a strategy to manipulate instabilities in fluid–fluid systems. The control setting we identify has a wide spectrum of applications ranging from small-scale technologies such as microfluidics to large-scale operations such as enhanced oil recovery.

- Al-Housseiny, Talal, Peichun Tsai, Zhong Zheng, and Howard Stone, 2011:
**The effect of permeability gradients on immiscible displacement in Hele-Shaw flows**. *American Physical Society*,

[ Abstract ]In heterogeneous media, it is well known that when a fluid of high viscosity displaces a less viscous fluid, the interface can still be unstable and exhibit finger-like patterns due to capillary fingering. Motivated by porous media flows in natural geological formations, we consider homogeneous displacement in a Hele-Shaw cell subjected to a permeability gradient. The permeability gradient is introduced by linearly varying the Hele-Shaw cell depth. We study how capillary forces can affect interfacial stability in the presence of the gradient via linear stability analysis. Depending on the system, we find that surface tension can either have a stabilizing or a destabilizing role. We report the emergence of an important dimensionless parameter--the ratio of the permeability gradient to the capillary number--that determines the stability of the interface along with the well-studied viscosity ratio. Experiments testing the theoretical findings will also be presented.

Direct link to page: http://cmi.princeton.edu/bibliography/results.php?author=4720