Pore-scale network models

The Celia Group also has funding from the National Science Foundation to support mathematical developments related to CO2 modeling on very small scales. One aspect they are studying is hysteresis and phase trapping, with a focus on trapping models for nonwetting phases, including CO2. To understand how trapped nonwetting fluids evolve as a function of phase saturation, they are using pore-scale network models to simulate two-phase displacement and extracting a wide range of information about trapped nonwetting fluids. This information is being used to compare and test existing theories as well as new theories that include explicit representation of trapped phases.

In a project funded by the Department of Energy, with Catherine Peters as the lead PI, Celia and colleagues are using pore-scale network models to simulate reactive transport including dissolution and precipitation associated with carbonated brines injected into carbonate formations (Figure 8). These models provide insights into conditions under which dissolution, precipitation, or a complex combination of the two will occur, with associated calculations of changes in porosity and permeability. Like the two-phase hysteretic relationships, the response of porosity and permeability to geochemical reactions shows non-uniqueness, implying a complex relationship that depends on additional subscale factors.

Figure 8. Results from single‐phase reactive pore‐scale network models showing (a) pore network, (b) evolution of permeability and porosity for different invasion conditions, and (c) evolution of permeability as a function of injected pore volumes. Figure from Nogues et al. (2012).


Bench-scale investigations of multiphase flow

Carbon sequestration raises new questions regarding multiphase flows in porous media, which the Stone Group is investigating with a combination of bench-scale laboratory experiments and theory. The group is investigating four distinct scenarios: (i) two-phase injection flows with permeability gradients in the flow direction, (ii) two-phase invasion flows in a parameter space expected for CO2 injection, including the influence of vertical gradients of permeability, (iii) leakage from a porous medium, and (iv) buoyancy-driven flows driven by dissolution of one phase in a second phase, with the parameter space comparable to that expected for carbon sequestration. The group’s work on micron-scale processes is providing new insight into the controls on CO2 transport and dissolution, which can be used to inform large-scale models for predicting the fate of CO2 in injection reservoirs.

Impacts of permeability gradients on flow stability

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. For two-phase injection flows with permeability gradients in the flow direction, the Stone Group has shown that surface tension triggers a new type of hydrodynamic instability (analogous to the traditional form of “viscous fingering”) relevant to flows in model porous materials. Using a Hele-Shaw configuration of two parallel plates, the researchers showed that, for given parameters, there is a critical speed (or capillary number) for which the flow is either stable or unstable. The researchers found that a dimensionless parameter, Ca (the ratio of the permeability gradient to a traditional capillary number) determines the stability of the interface (Figure 9).

Figure 9. a) Schematic of the Hele‐Shaw experimental apparatus and b) the results from the two‐phase injection experiment with a permeability gradient in the flow direction (Cac from theory is shown). Height of apparatus is 1.4 cm at right, length is approximately 40 cm, and width (not shown) is 5.1 cm.

Propagation of gravity currents in porous media with vertical permeability gradients

In related work on invasion flows, the Stone Group has shown that the influence of a permeability gradient produces qualitative differences in the solution, and that the rate of spreading is influenced by transverse gradients of permeability. Zhong Zheng, co-advised by Stone and Robert Socolow, simulated the influence of a vertical permeability gradient on the horizontal propagation of gravity currents in porous media. A self-similar solution was found for the flow, which indicated a power-law behavior for the front propagation (Figure 10). To verify the modeling work, he designed fluid displacement experiments in Hele-Shaw cells using liquids that are expected to have similar dynamical responses as the case for CO2 injection, and measured the profile shape and the front propagation dynamics. Experimental results fit well with model predictions.

Figure 10. Xn, the position of the leading edge of the current, versus time (t). Blue “V” shape at lower right shows cross‐sectional shape.

Dissolution-driven flows

For flows driven by dissolution, as expected for brine solutions, the influence of inclination of the bounding walls has been explored with systematic experiments (Figure 11). Stone and colleagues found that the tilting angle of the inclined boundary profoundly affects the dynamics of the density-driven plumes, and that the permeability of the porous medium strongly changes the convective rate. These findings have key implications for geological CO2 storage in a saline aquifer when the dissolved CO2 into brine produces a heavier mixture with an enhancement of the mass transfer by convection. In such a scenario, inclined boundaries tend to increase the influence of convection in bounded systems, which should increase the dissolution rate of CO2.

Figure 11. Density‐driven plumes in a Hele‐Shaw cell inclined 10° to the horizontal.

Future directions by the Stone Group will include experiments with model porous materials in the form of packed beds of particles, with particular focus on gradients in the permeability. Where possible, analytical solutions are being developed to give quantitative descriptions to the experimental results.