# CVEEN 7920: Carbon Capture and Storage Wednesday, 20 October 2010 - PowerPoint PPT Presentation

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CVEEN 7920: Carbon Capture and Storage Wednesday, 20 October 2010 Topic: Solving PDE’s (solutions to mathematical models). Models of CCS = Models of Fluid Flow & Other Processes. Reprise: the continuity equation and groundwater (fluid) flow equation.

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CVEEN 7920: Carbon Capture and Storage Wednesday, 20 October 2010

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#### Presentation Transcript

CVEEN 7920: Carbon Capture and Storage

Wednesday, 20 October 2010

Topic: Solving PDE’s (solutions to mathematical models)

Models of CCS = Models of Fluid Flow & Other Processes

• Reprise: the continuity equation and groundwater (fluid) flow equation.

• What do you need to solve the groundwater flow equation?

• Taylor Series - the backbone of the FDM

• Building an FDM Approximation

Groundwater Flow Equation

Left side of continuity equation has q, this is driven by

Let’s try to relate the left side to h

Assume the axes of are parallel to x,y,z

Then

If K axes not aligned with x, y, z, then we must use

Groundwater Flow Equation

Possible combinations of (in)homogeneity and (an)isotropicity

(Freeze and Cherry, 1979)

Groundwater Flow Equation

heterogeneous, anisotropic medium

heterogeneous, anisotropic media

heterogeneous, anisotropic media

heterogeneous, anisotropic media

Groundwater Flow Equation

All transient conditions:

heterogeneous, anisotropic medium

heterogeneous, isotropic media

homogeneous, anisotropic media

homogeneous, isotropic media

Groundwater Flow Equation

Models of CCS = Models of Fluid Flow & Other Processes

• Reprise: the continuity equation and groundwater (fluid) flow equation.

• What do you need to solve the groundwater flow equation?

• Taylor Series - the backbone of the FDM

• Building an FDM Approximation

Solving the Groundwater Flow Equation

Three primary approaches:

(1) Analytical (applied more often than you might think)

(2) Graphical (flow nets)

(3) Numerical (most common)

Solving the Groundwater Flow Equation

The ‘Boundary Value Problem’ (BVP):

Governing equation

2)Region of flow (e.g., geometry, dimensions, etc.)

Material properties, or parameterization

Boundary conditions

Initial conditions (transient problems)

Method of solution

Models of CCS = Models of Fluid Flow & Other Processes

• Reprise: the continuity equation and groundwater (fluid) flow equation.

• What do you need to solve the groundwater flow equation?

• Taylor Series - the backbone of the FDM

• Building an FDM Approximation

Backbone of FDM: Taylor Series Expansion

Backbone of FDM: Taylor Series Expansion

Backbone of FDM: Taylor Series Expansion

Backbone of FDM: Taylor Series Expansion

Backbone of FDM: Taylor Series Expansion

Models of CCS = Models of Fluid Flow & Other Processes

• Reprise: the continuity equation and groundwater (fluid) flow equation.

• What do you need to solve the groundwater flow equation?

• Taylor Series - the backbone of the FDM

• Building an FDM Approximation

The FDM

The FDM

Construction of an FDM Approximation

Construction of an FDM Approximation