Question: Carbon dioxide flooding experiment. One of the most promising methods for extracting crude oil employs a carbon dioxide (CO ₂ ) flooding technique. CO ₂ , when flooded into oil pockets, enhances oil recovery by displacing the crude oil. In a microscopic investigation of the CO ₂ flooding process, flow tubes were dipped into sample oil pockets containing a known amount of oil (Journal of Petroleum Technology, Aug. 1982). The oil pockets were flooded with CO ₂ and the percentage of oil displaced was recorded. The experiment was conducted at three different flow pressures and three different dipping angles. The displacement test data are recorded in the accompanying table.

CO2FLOOD

1. Write the complete second-order model relating percentage oil recovery y to pressure x ₁ and dipping angle x ₂ .
2. Plot the sample data on a scattergram, with percentage oil recovery y on the vertical axis and pressure x ₁ on the horizontal axis. Connect the points corresponding to the same value of dipping angle x ₂ .Based on the scattergram, do you believe a complete second-order model is appropriate?
3. Fit the interaction model \(y={{\beta }_{0}}+{{\beta }_{1}}{{x}_{1}}+{{\beta }_{2}}{{x}_{2}}+{{\beta }_{3}}{{x}_{1}}{{x}_{2}}+\varepsilon \) to the data. Give the least-squares prediction equation.
4. Construct a plot similar to the scattergram of part b, but use the predicted values from the interaction model on the vertical axis. Compare the two plots. Do you believe the interaction model will provide an adequate fit?
5. Check model adequacy using a statistical test with a = .05.

Is there evidence of interaction between pressure x ₁ and dipping angle x ₂ ? Test using a = .05.

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