Carbon Storage (EAEE E4301)

Fall 2024

Homework #1 (Due Monday, October 7th, 11:59 pm)

Homework Guidelines:

Your solutions to homework assignments will be submitted and graded through Gradescope (see the Gradescope tab on your Courseworks dashboard).

You will have two options for submitting your work in Gradescope, either: 1) upload individual scanned images of   your handwritten pages (e.g., using your phone), one or more per question; or 2) upload a single PDF that you create which contains the whole submission (e.g., merge files on your computer or phone with a software of your choice).

Please use the naming convention Lastname_HWxx.pdf when submitting your homework assignment. You may choose to type up your calculations, in which case show all your steps and highlight your solution. Note: During the upload, Gradescope will ask you to mark which page/s each problem is on (see example here). It is important that you follow that step for grading purposes.

It is acceptable to discuss problems with your colleagues, and questions are encouraged during office hours, but all work must be done independently. Make sure to clearly show all work on each problem and that your solutions are  presented in an orderly fashion. It is your responsibility to make your solutions easy to grade.

Topics/Chapters covered:

● Class notes: Modules 1-4

● Rackley, Chapter 2, 11-14 and other resources mentioned in class notes (e.g., Smit book)

Problem #1 (The Carbon Cycle) 15 pts

In class we reviewed the box model for Earth’s carbon cycle, also shown below. The percentages in white text boxes represent the percentage of emitted anthropogenic carbon accumulated in the  planet’s major reservoirs.

(a) Briefly explain why the surface layer of the ocean shows an increase in concentration of CO2, but the deep ocean does not.

(b) Why is there no percentage shown in the sediments and crust reservoir? How can we change this?

(c) Of the 9 Gt of anthropogenic CO2 emissions, how many Gt must we offset with CCUS technologies to offset rising temperatures (e.g. the greenhouse effect)? Don’t overthink - this is a quick calculation.

(d) Estimate the maximum storage capacity of 10 geologic formations with the following average properties: lateral area of 10 km x 5 km, height of 30m, porosity = 0.2, irreducible water saturation (Sw_irr) = 0.15, and in situ scCO2 density = 700 kg/m^3. Hint: Back of the envelope calculations are fine.

Comment on the magnitude of this storage capacity.

Problem #2 (Geology / porous media) 25 pts

The porosity of a sandstone (or soil for that matter) is heavily dependent on the grain size and packing arrangement, among other factors. See the below figure.

(a) Mathematically prove that the porosity associated with the simple cubic packing of equally sized spheres, shown in the above figure is ~0.48. Show your work for credit.

Does this value of porosity change if the grain size changes from a radius of 0.1 mm to a radius of 1.0 mm? How about permeability?

(b) Using the same logic, derive the porosity of either an orthorhombic or rhombohedral packing of equal spheres (your choice!), where the grains are shifted and porosity reduced. Show your work for credit.

(c) Name 2 other factors, physical and/or chemical, that can degrade porosity in a rock formation. Briefly (in a few words) explain why.

(d) Would you expect a slight increase or decrease in porosity in an over-pressured formation? Why?

Microscale grain packing arrangements: Top: Cubic packing of equal spheres. Middle: orthorhombic packing of equal spheres. Bottom: Rhombohedral packing of equal spheres.

Problem #3 (Geomechanics) 10 pts

The below left plot is a generic Mohr–Coulomb plot with failure envelope.

If you want to review geomechanics more, see Rackley Chapter 12.

Imagine that this Mohr–Coulomb plot represents the key rock type in a candidate storage formation for hydrostatic pressure conditions. Your team tells you that the rock has the pore pressure properties as shown in the Pressure vs Depth plot on the right.

(a) At a depth of 2000m, describe or draw how the Mohr–Coulomb plot above would change. (b) At a depth of 3500m, describe or draw how the Mohr–Coulomb plot above would change.

(c) What possible scenarios may create the overpressure seen at the lower depths?

Problem #4 (Fluid properties, fluid-rock interactions) 15pts

A formation has similar pressures, temperatures and brine properties as the Sleipner-Utsira formation/CO2 storage pilot. Assume a completely hydrophilic caprock (e.g. completely water - wetting) of average pore size r = 100 nm. What is the maximum CO2 column height beyond which CO2 will enter the caprock through capillary forces?

Now perform the same calculations for an average pore size of average pore size r = 1 micron. Comment on their differences.

Problem #5 (Fluid-rock interactions) 10 pts

The below figure shows the water saturation with depth (or height above  100% water saturation line or “Free Water Line”) for several geologic layers. You can assume that the other fluid in the pore space is carbon dioxide. Assume normal hydrostatic conditions (no overpressure).

(a) Which layer is at residual water saturation? What does this mean for the flow of water and CO2 in this layer?

(b) Rank the layers from likely highest to lowest permeability based on the character of their saturation curves. Which layers might you recommend as caprocks?

Problem #6 (Geochemical interactions) 25 pts

The rate r of calcite dissolution in acid, aqueous solutions in moles/m2/s can be estimated via

where k =

in which k is the rate constant (a function of T & pH), ar is the reactive surface area of calcite in the rock,Q is the activity product for ions in the solution (in this case Q = 0.25*[Ca2+]*[H2CO3] at low pH), KS  is the equilibrium constant (use 10-5  in seawater at pH 3.5), n and m in the first expression are 1, kH  is a pre-exponential constant, EH  is the activation energy, R is the gas constant, T is the temperature of interest in Kelvin, T0  is the standard state temperature in Kelvin (25°C, 298.15 K), aH  is the activity of hydrogen ions in solution (which we will take to be equal  to [H+]) and nH  is an empirical constant.

For your reference, for calcite dissolving in acid water: kH  ≈ 10-0.3  moles/m2/s, EH  ≈ 14,400 J/mole, and nH  ≈ 1.

(a) What is the rate of calcite dissolution in water containing 500 ppmw Ca2+  and 150 ppmw H2CO3, at 50°C with a pH of 3.5, in moles/m2/s?

(b) When Q = KS, what is the rate of calcite dissolution? If Q > KS, what should happen?

(c) If the calcite dissolution rate were 2x10-4  mol/m2/s and the calcite in the rock matrix has a specific surface area of 15/mm and a density of 2710 kg/m3, what is the rate of calcite dissolution in moles/gram/s?

Hint: do not overthink this problem and let the formulas and units be your guide!