## 6 Expansion Rate of the Dam
Moreover, our data suggest a subtle yet consistent increase in peak expansion rates over time. This trend is not prominently discussed in the current literature, which often assumes stable soil properties over time. Our findings challenge this assumption, hinting at potential long-term changes in soil behavior, possibly due to climate change or gradual alterations in soil mineralogy. To quantify this trend, we calculated the average peak expansion rate (using July and August data) for three periods: 2010-2013: 16.94 mm/month, 2014-2016: 17.47 mm/month, and 2017-2020: 18.88 mm/month. This steady increase (0.53 mm/month from periods 1 to 2 and 0.41 mm/month from periods 2 to 3) supports our hypothesis of evolving soil properties. While the changes appear small, they accumulate over time and could significantly impact long-term dam safety and maintenance strategies. Figure 16: Expansion rate results on twelve soil samples from the downstream area of the Tiga Dam in Kano State, Nigeria. The performance metrics for the linear regression model with noise, which predicts the expansion rate of downstream soil in a dam, demonstrate remarkably high accuracy and robustness. The model's performance was evaluated using Mean Squared Error (MSE) and R-squared (R2) metrics on both training and testing datasets, with an additional adjusted R-squared (R2a) to account for model complexity. The MSE values of 0.231 for training and 0.271 for testing indicate that, on average, the model's predictions deviate from the actual expansion rates by approximately 0.48 mm/month (\(\surd\)0.231) and 0.52 mm/month (\(\surd\)0.271) in the training and testing sets, respectively. Given that the result's expansion rates range from nearly 4 to about 19.45 mm/month, these errors are minimal, suggesting high precision in the model's predictions. The R2 values are even more impressive, with 0.990 for training and 0.989 for testing. These scores indicate that the linear regression model explains about 99% of the variance in soil expansion rates. Such high R2 values are rare in geotechnical engineering, where soil behavior is often complex and influenced by numerous factors. Moreover, the adjusted R2 (R2a) values of 0.989 for training and 0.978 for testing are nearly as high as the standard R2 values. R2a penalizes the addition of unnecessary predictors, so these high scores suggest that the model's performance is not artificially inflated by overfitting. Instead, it implies that the chosen variables (likely time-based features like month and year) genuinely predict soil expansion rates. [PERSON] et al. used Support Vector Regression (SVR) to predict soil deformation in the Three Gorges Dam's downstream area, achieving R2 values of 0.942 and 0.937 on training and testing sets, respectively. While impressive, these fall short of our model's R2 scores by about five percentage points [67]. Similarly, [PERSON] et al. applied Random Forest regression to forecast soil displacement at the Xiaowan Dam, obtaining R2 values of 0.983 (training) and 0.972 (testing). Although close to our results, their model's performance on data (testing R2 = 0.972) is notably lower than our testing R2 of 0.989, indicating superior generalization by our linear regression model [66]. In terms of MSE, our model also outperforms our peers. [PERSON] et al. used Artificial Neural Networks (ANNs) to predict soil expansion in the Longtan Dam's downstream, reporting MSEs of 0.412 (training) and 0.453 (testing) [67]. Our MSEs (0.231 and 0.271) are nearly half these values, signifying substantially more accurate predictions. The novelty of our approach lies in its simplicity and effectiveness. Most recent studies in this domain lean towards complex machine learning models--SVR, Random Forest, ANN--presuming that soil behavior is too intricate for simpler models. For example, [PERSON] et al. argue that \"nonlinear models are essential for capturing the complex interactions in soil dynamics\", supporting their claim with an LSTM model that achieves R2 = 0.976 [66].