CO2 Sequestration
Original Darcy’s Law in Algebraic Form:
Darcy found that the volumetric fluid flow rate through the saturated soil column was directly proportional to the drop in hydraulic head between the inlet and outlet; and inversely proportional to the length of the soil column.
Extended Darcy’s Law in Differential Form:
In order that Darcy’s observations may help formulate the differential equation of flow in a porous medium, it is customary, in the interpretation of Darcy’s experiment, to let the length of the soil column to tend in the limit to zero.
When this is done, Darcy’s observations lead to the inference that the flow rate through the soil column remains directly proportional to the hydraulic gradient.
Can we comfortably apply the differential form of extended Darcy’s law for spatially and temporally evolving boundaries of deep saline aquifers associated with CO2 sequestration in the early phase and CO2 leakage at later stages?
In CO2 sequestration fluid flow application, can we assume creeping flow and equilibrium of forces, where, the impelling pressure forces on CO2-brine fluid remain exactly balanced within the aquifer by the sum of frictional forces generated by the flowing Newtonian fluid and the body forces due to the weight of the pore-fluid itself?
Suresh Kumar Govindarajan
Professor (HAG) IIT-Madras
https://home.iitm.ac.in/gskumar/
https://iitm.irins.org/profile/61643
22-July-2024
Suresh Kumar Govindarajan
Applying the differential form of extended Darcy’s law to spatially and temporally evolving boundaries of deep saline aquifers can be challenging but is feasible under certain conditions. Here are some key considerations and conditions under which it can be comfortably applied:
1. Understanding Extended Darcy’s Law: The extended Darcy’s law incorporates additional factors such as non-linear flow regimes, multiphase flow, and the presence of chemical reactions. This makes it more complex than the classical Darcy’s law but also more accurate for complex systems like deep saline aquifers.
2. Spatial and Temporal Variability: For deep saline aquifers with evolving boundaries, both spatial and temporal variations must be accounted for. This involves recognizing that permeability, porosity, and other hydrological properties can change over time due to factors like pressure changes, temperature variations, and chemical interactions.
3. Mathematical and Computational Models: Implementing the differential form of extended Darcy’s law requires robust mathematical models that can handle partial differential equations (PDEs) with variable coefficients. Numerical methods such as finite element or finite difference methods are typically employed to solve these PDEs.
4. Data Availability: Accurate application relies heavily on the availability of detailed geological and hydrological data. High-resolution data on the aquifer’s properties and boundary conditions are crucial for modeling spatial and temporal changes accurately.
5. Boundary Conditions: Defining appropriate boundary conditions is essential. These include initial conditions for the state variables (pressure, saturation, concentration, etc.) and boundary conditions that reflect the physical and chemical processes at the boundaries of the aquifer.
6. Model Validation: Validation of the model with field data is necessary to ensure reliability. This involves comparing the model’s predictions with observed data and adjusting the model parameters to improve accuracy.
7. Computational Resources: The complexity of solving the extended Darcy’s law for evolving boundaries demands significant computational resources. High-performance computing may be required to handle the large datasets and complex calculations involved.
8. Simplifications and Assumptions: In practice, some simplifications and assumptions may be made to make the problem tractable. For instance, assuming quasi-steady-state conditions for certain periods or simplifying the geometry of the boundaries can reduce computational complexity.
In summary, while applying the differential form of extended Darcy’s law to deep saline aquifers with evolving boundaries is complex, it is achievable with robust mathematical modeling, high-quality data, computational resources, and careful consideration of the underlying physical and chemical processes
Hi Suresh and all.
I will assume that your “extended Darcy’s law” applied to saline aquifers, refers to the Forchheimer (1901) quadratic law, which holds for fluid flow in porous media at high velocities. In this case, inertial forces (that is, proportional to the fluid density) have magnitudes similar to the viscous forces. Except for the “low velocities” required by the Darcy’s original equation, in your case, the remaining requirements are: incompressible newtonian fluid, porous media with uniform porosity and isotropic (that is, its permeability is a scalar quantity or flow direction independent). Mind you, in this case, fluid convective accelerations will be present, even in a steady flow regime, so as to account for the increased pressure gradient (compared to Darcy’s prediction).
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