Fractional-Order Modeling of Sediment Transport and Coastal Erosion Mitigation in Shorelines Under Extreme Climate Conditions: A Case Study in Iraq
Abstract
:1. Introduction
2. Model Settings
Modified Gamma Function
3. Three-Dimensional Fractional Calculus
- The term indicates an exponential decay, modifying the memory effect.
- The power generalizes the weighting of past values.
- The 3D gamma function normalizes the operator.
- Temporal fading memory is controlled by the exponential term (stronger decay for α).
- Scaling qualities can be adjusted by the power term (via β).
- The exponent becomes nonlinear due to the parameter κ.
- Classical Fractional Calculus: Setting and reduces these operators to classical Riemann–Liouville and Caputo operators. Moreover, when we have the fractional operators in [15].
- Tempered Fractional Calculus: Choosing recovers the tempered fractional derivative.
- If we replace with the Mittag–Leffler function , we obtain a generalized Prabhakar derivative.
- A discrete analog of the integral operator is
3.1. Applications
3.2. Algebraic Properties
- (i)
- (ii)
- (iii)
4. Ocean Engineering: Modeling Wave Attenuation in Porous Media
4.1. Problem Setup
4.1.1. Physical Interpretation
- describes anomalous attenuation of pore pressure due to the viscoelastic response of the porous medium;
- captures the spatial damping effect;
- k is the wave number, which dictates the phase oscillation of pressure.
4.1.2. Numerical Example:
- (initial pressure amplitude);
- (fractional order for memory effects)
- (wave dissipation);
- (modified parameters of the 3D gamma function);
- (wave number for a 10-meter wavelength);
- and
4.2. Extending the Model: Incorporating Erosion and Cave Dynamics
4.2.1. Seabed Erosion Effect
4.2.2. Cave Formation and Seabed Instability
4.2.3. Numerical Simulation
- Data 1, with
- Compare cases with low erosion (δ = 0.1 vs. high erosion (δ = 0.5);
- Introduce a cavity at χ = 10 m, where locally increases pressure retention.
4.2.4. Engineering Insights
- Data 1, with
- Varying Cave Dimensions and Sites: We will compare smaller caves, which are located at χ = 10 m, with larger caverns, which are dispersed over m. Stronger pressure amplification and wave trapping may occur in larger caverns.
- Higher k waves that are stronger allow us to observe the intensification of pressure fluctuations and increases in wave frequency k (shorter waves). Longer waves with a lower k result in deeper penetration.
- Changing Seabed Materials (higher or lower . When modeling sandy seabeds, a higher α (more damping) causes pressure to dissipate more quickly. With reduced α (reduced damping), clayey seabeds are represented by deeper pressure penetration. To mimic non-uniform seafloor features, we change κ.
5. Real-World Seabed Simulation
5.1. Features of the Seabed
- Geological Formation: the melting of polar ice sheets caused the Arabian Gulf to form between 3000 and 6000 years ago [16].
- Sediment Composition: Although there are patches of rocky and coral reefs, the bottom is mostly made up of silty and sandy sediments. These sediments affect the permeability of the seabed and, in turn, the pressure penetration caused by waves.
- Archaeological Significance: Archaeological remains discovered in marine areas in northwest Qatar date back 8000 years, suggesting that the seafloor was open during the Last Glacial Maximum [17].
5.1.1. Dynamics of Waves
- Wave Heights: The significant wave heights in the Arabian Gulf vary depending on region and water depth. Wave heights at offshore locations vary from 5.0 to 7.0 m, while water levels range from 30 to 60 m [18].
- Seasonal fluctuation: Wind seas, swells, and the resulting wave characteristics exhibit notable seasonal and regional fluctuation throughout the area. The design and functionality of marine structures are impacted by this variability [19].
5.1.2. Implications for Ocean Engineering Structural Design
- When constructing foundations for offshore platforms, pipelines, and other marine structures, engineers need to take into consideration the sediment composition of the seabed as well as wave dynamics.
- Erosion and Sediment Transport: Predicting erosion patterns and sediment transport is made easier with an understanding of the seabed’s properties. These processes are essential to preserving the stability of offshore and coastal facilities.
- Environmental Considerations: Careful planning is required to reduce the environmental impact during construction and operation due to the presence of coral reefs and archaeological sites [20].
5.1.3. Erosion and Sediment Transport in Arabian Gulf
- Aeolian Dust Deposition: Every year, a significant amount of aeolian (wind-blown) desert dust is dumped into the Northern Arabian Gulf. According to studies, more than 90% of the sediment in this area comes from aeolian sources, with the remaining portion coming from rivers. Sediment Composition: although there are patches of rocky and coral reefs, the bottom is mostly made up of silty and sandy sediments [21].
- Coastal Development: The dynamics of the coast have been profoundly changed by projects like Dubai’s Palm Islands. Wave patterns, coastal erosion, and alongshore sediment transport have all changed as a result of the building. Such developments have disturbed sediments, which have reduced sunlight penetration and impacted marine life. Dams and Water Retention Structures: Water retention dams catch about 48% of the sediment supply from wadis (dry riverbeds) in areas like Oman’s Al Batinah coast. Because there is less material available to restore beaches, this decrease in the availability of sediment adds to coastal erosion [22].
- Significance for Coastal Management: It is essential to comprehend these elements in order to manage the Arabian Gulf’s coastline effectively. To reduce erosion and protect marine ecosystems, strategies should take into account both the effects of human activity and the natural processes of sediment transport [23].
5.2. Modified Fractional Equation for Erosion and Sediment Transport
5.2.1. Engineering Insights
- Coastal Protection: while dam-induced sediment entrapment lessens resilience, aeolian processes can aid in seabed stabilization.
- In order to avoid foundation instability, offshore structure design must take varying sediment transport rates into consideration.
- Erosion Control: to prevent seabed deterioration, coastal planners should strike a balance between human activities and natural sediment inflow.
- Dredging solutions should avoid excessive sediment loss in ports and harbors because high erosion erodes foundations.
- Pipeline Stability: To prevent exposure, submerged pipes in high-erosion regions require reinforcing.
- Sediment barriers and artificial reefs can assist prevent excessive erosion while preserving the natural equilibrium of the seabed.
5.2.2. Long-Term Sediment Erosion Trends (50-Year Simulation)
5.2.3. Optimizing Engineering Solutions for Dynamic Seabed Conditions
- In areas with significant erosion and frequent storms, pipelines ought to be buried deeper.
- Layers of Scour Protection: to lessen seafloor erosion, we should use artificial reefs, riprap, or geotextiles, whereThe goal is to minimize seabed erosion by choosing the optimal material size and placement.
- Sediment Nourishment and Beach Replenishment: To combat seafloor loss brought on by climate change, we should place sand in degraded regions. The ideal sediment volume equation is
5.2.4. Complete Engineering Suggestions
- Offshore Pipelines: in high-erosion zones, we should use scour protection and deep burial.
- Port and Harbor Stability: we should implement floating structures and maximize dredging cycles.
- Wave Energy Reduction: we should protect coastal areas by appropriately placing artificial reefs.
- Climate Adaptation: we should track trends in sediment movement and strengthen vulnerable areas.
5.2.5. Case Study: Seabed Erosion and Sediment Transport in Iraq’s Coastal Waters (Shatt al-Arab and the Northern Arabian Gulf)
- Storm surge and seasonal wind-driven sediment movement;
- Oil export terminals and pipeline vulnerability owing to seabed changes;
- Shatt al-Arab sediment discharge oscillations impacting seabed stability.
5.2.6. Erosion Model:
- A reduction in sediment supply from upstream river management causes coastal retreat in the vicinity of Basra.
- Over a 20-year period, the seabed in offshore oil terminal zones will drop by 1.5 to 3 m.
- The risk of pipeline exposure is rising as a result of sediment loss close to important infrastructure.
5.2.7. Engineering Solutions
5.3. Sediment Transport and Stabilization Equation with Fractional Derivatives
- Absent stability (dashed red line) (Figure 12a): Storms, erosion, and human activity all contribute to the steady drop in sediment volume. Sediment loss surges are noticeable during extreme storm events, which occur every five years. Coastal stability is impacted by the substantial long-term loss of sediment.
- Using stabilizing methods (blue line): Sediment loss is decreased via artificial reefs, geotextiles, and nutrition. Stabilization over time enhances sediment retention and avoids severe erosion. In comparison to the non-stabilized instance, about 30% more silt is retained after ten years.
- Not stabilized (dashed red line):There is ongoing loss of sediment as a result of storms, dredging, and sea level rise. Sediment lowers sharply after extreme storms that occur roughly every five years. Nearly 60% of the sediment will have been lost by 2050, causing significant coastal erosion.
- Using the blue line for stabilization:
- Erosion is slowed by artificial reefs, geotextiles, and sediment nutrition. Approximately 40% more sediment is left after 50 years than in the non-stabilized scenario. Although it happens considerably more slowly and controllably, sediment loss still happens.
- With normal weather and no stabilization (dashed red line), there is severe erosion over time. Almost 60% of the sediment will be gone by 2050.
- With the combined strategy and a standard climate (green line), the best-case situation involves artificial stabilization and biotechnology. Sediment retention is around 50% higher than in the absence of stabilization.
- With a combined strategy and an extreme climate (purple dotted line), sediment loss is accelerated by frequent major storms (every three years). Coastal erosion is made worse by a faster rise in sea level (0.2 m by 2075).
- Sediment levels drop more quickly even with combined stabilization than they would otherwise. Although there is definitely some protection, erosion is still very bad.
6. Conclusions and Discussions
6.1. Key Findings and Numerical Comparisons
- Fractional Models Improve Sediment Transport Predictions
- The fractional order (time derivative) provided a 30% better fit to real seabed erosion data compared to classical models.
- Fractional spatial transport () successfully modeled slow sediment diffusion, which is often underestimated in standard approaches (see Table 2).
- 2.
- Cavity Effects Increase Pressure Retention and Erosion Risk
- Higher liquefaction risk in sediments near offshore infrastructure.
- Reduced dissipation rates of wave-induced pressure, prolonging instability.
- 3.
- Storm Impact Simulations Show Stronger Sediment Transport in Fractional Models
- Traditional integer-order models underestimate erosion by 25% in high-energy environments.
- Fractional models withbetter capture delayed pore pressure dissipation and long-term seabed evolution.
- Severe storm cases () showed a 60% increase in sediment displacement, proving that non-local models better account for storm-driven changes.
- 4.
- Long-Term Sediment Trends Are More Accurately Predicted with Fractional Operators
- Baseline (no storm impact): Traditional and fractional models gave similar results.
- Moderate erosion (seasonal waves and human activities): Fractional models predicted 15% higher sediment displacement over time.
- Severe Erosion (storms and human impact): The fractional model estimated 40% more sediment loss than classical methods.
6.2. Practical Implications and Future Work
- Offshore engineering: there can be more accurate erosion risk assessments for pipelines, platforms, and cables.
- Coastal protection: there may be improved modeling of wave-driven sediment transport for breakwaters and artificial reefs.
- Dredging optimization: there can be better long-term sediment distribution predictions in order to reduce over-dredging risks.
- By enhancing sediment transport models, fractional operators made it possible to more accurately depict erosion memory effects.
- Although artificial stabilization offered instant advantages, performance deteriorated as a result.
- Bioengineering solutions promoted ecological resilience and sediment retention, providing long-term sustainability.
- Hybrid strategies that combine artificial stability and bioengineering may maximize sediment retention and storm resilience.
- Optimize hybrid protection measures that combine bioengineering and designed structures;
- Incorporate real-world field data from Iraq’s coastal zones for validation.
- Expand the model to evaluate the effects of climate change after 2100.
- Conduct cost–benefit evaluations for the extensive use of sediment stabilization methods.
6.3. Standardized Accuracy Assessment of the Sediment Transport Model
6.4. Summary of Study Limitations
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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0.5 | 0.5 | 0.5 | 1.772 | 2.119 | 3.217 | 5.806 | 12.00 |
0.5 | 0.5 | 1.0 | 2.061 | 2.021 | 2.306 | 2.919 | 4.00 |
0.5 | 0.5 | 1.5 | 2.047 | 1.865 | 1.926 | 2.161 | 2.576 |
1.0 | 1.0 | 1.0 | 1.772 | 0.942 | 0.919 | 1.222 | 2.000 |
2.0 | 1.5 | 1.5 | 3.537 | 0.623 | 0.393 | 0.377 | 0.466 |
0.5 | 1.0 | 1.0 | 2.5066 | 2.0539 | 3.0913 | 6.3402 | 16.000 |
0.5 | 1.0 | 1.5 | 2.5757 | 1.8622 | 2.2282 | 3.3852 | 6.000 |
0.5 | 1.5 | 0.5 | 3.5449 | 2.4445 | 16.5701 | 297.4688 | 10,080.00 |
0.5 | 1.5 | 1.0 | 4.3116 | 2.098 | 4.6214 | 17.71389 | 6.0000 |
0.5 | 1.5 | 1.5 | 4.4565 | 1.8672 | 2.8022 | 6.396213 | 18.7778 |
Parameter | Value | Description |
---|---|---|
0.8 | Memory decay factor (erosion history effect) | |
1.5 | Controls sediment diffusion speed | |
1.2 | Controls long-term sediment retention | |
0.3 | Fractional order for sediment deposition | |
1.5 | Fractional order for seabed erosion transport | |
Varies | Input from Shatt al-Arab and Euphrates | |
High | Dredging and port expansion | |
Medium | Using geotextiles, reefs, and nourishment |
Evaluation Criteria | Method Used | Accuracy Level: | Remarks and Limitations |
---|---|---|---|
√√√ = High, | |||
√√ = Moderate, | |||
√ = Low | |||
Model Validation (compared with real-world data) | Empirical fitting to past sediment trends | √√ | Model accuracy depends on the dataset used for calibration. There is uncertainty in long-term predictions. |
Fractional Parameters Calibration () | Empirical selection and fitting | √ | No fundamental physical derivation; parameters mainly adjusted for best fit. |
Boundary Condition Sensitivity | Sensitivity analysis on seabed boundary types | √√√ | Robust response under different conditions, but requires additional real-world validation. |
Wave-Induced Pore Pressure | Modeled based on field data and equations | √√ | Some assumptions on pore pressure distribution may not generalize to all regions. |
Sediment Retention Accuracy (with stabilization) | Computational model + limited empirical validation | √√ | Needs more observational data to confirm the estimated 40–50% retention improvement. |
Uncertainty Quantification | Monte Carlo and ensemble testing | √ | Uncertainty not fully incorporated; deterministic values presented as fixed outcomes. |
Comparison with Classical Models | Compared with Van Rijn and Soulsby–Whitehouse models | √√ | Fractional model offers advantages in complex terrain, but validation is still limited. |
Long-Term Predictions (50-year trends) | Extrapolated based on current trends | √ | Climate variability and extreme events introduce high uncertainty. |
Storm Impact Robustness | Tested with different storm intensities | √√√ | Model accounts for increased sediment transport during storms, but lacks uncertainty bounds. |
Applicability to the Arabian Gulf (Iraq case study) | Local environmental data integrated | √√ | General trends match observed Gulf dynamics, but local erosion rates require validation. |
Limitation & Description |
---|
Limited Field Data Availability & Lack of high-resolution, site-specific data (e.g., wave patterns, pore pressure, erosion rates) hinders model validation and calibration accuracy. |
Empirical Calibration of Fractional Parameters & Parameters such as are chosen through fitting, lacking theoretical derivation, which affects generalizability and confidence. |
Uncertainty Representation & The model provides deterministic outcomes without incorporating stochastic variability due to storms, sea level rise, or human activities. |
Oversimplified Sediment Properties & Assumes homogeneous seabed sediment, while natural beds are heterogeneous with varied composition, affecting erosion and retention dynamics. |
Static Long-Term Assumptions & No dynamic feedback mechanisms are included (e.g., vegetation growth, structural degradation), reducing realism in 50-year projections. |
Regional Specificity & Model is customized for the Arabian Gulf (Iraq coast); results may not be transferrable to other geographical areas with different hydrodynamic regimes. |
High Computational Load & Use of modified gamma-based fractional operators increases numerical complexity, challenging scalability and real-time simulation feasibility. |
Simplified Bioengineering Modeling & Stabilization techniques are abstracted into retention coefficients without simulating mechanical interactions. |
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Aldawish, I.; Ibrahim, R.W. Fractional-Order Modeling of Sediment Transport and Coastal Erosion Mitigation in Shorelines Under Extreme Climate Conditions: A Case Study in Iraq. Computation 2025, 13, 104. https://doi.org/10.3390/computation13050104
Aldawish I, Ibrahim RW. Fractional-Order Modeling of Sediment Transport and Coastal Erosion Mitigation in Shorelines Under Extreme Climate Conditions: A Case Study in Iraq. Computation. 2025; 13(5):104. https://doi.org/10.3390/computation13050104
Chicago/Turabian StyleAldawish, Ibtisam, and Rabha W. Ibrahim. 2025. "Fractional-Order Modeling of Sediment Transport and Coastal Erosion Mitigation in Shorelines Under Extreme Climate Conditions: A Case Study in Iraq" Computation 13, no. 5: 104. https://doi.org/10.3390/computation13050104
APA StyleAldawish, I., & Ibrahim, R. W. (2025). Fractional-Order Modeling of Sediment Transport and Coastal Erosion Mitigation in Shorelines Under Extreme Climate Conditions: A Case Study in Iraq. Computation, 13(5), 104. https://doi.org/10.3390/computation13050104