A New Insight into Mathematical modeling for sand production prediction due to high-speed fluid flow in loosely Consolidated Sandstones

Technical Summary: A New Insight into Mathematical Modeling for Sand Production Prediction

Problem Statement
Sand production in poorly consolidated oil and gas reservoirs (unconfined compressive strength < 1000 psi) presents a critical challenge to the petroleum industry. It leads to wellbore instability, equipment erosion, production downtime, and significant environmental and economic costs. While sand control methods exist (mechanical, chemical, and restrictive production rates), effective control relies heavily on accurate prediction. Existing modeling approaches generally fall into strain-based, erosion-based, or particle-based categories. However, there is a need for a comprehensive model that integrates the physics of particle detachment, transport, and re-adhesion under high-velocity fluid flow to predict volumetric sand production and the resulting changes in reservoir properties.

Methodology
The paper proposes a combined erosion-based and particle-based numerical model to predict sand production in both one-dimensional (linear) and axisymmetric (radial) flow geometries. The methodology is structured as follows:

1. Physics of Particle Detachment: The study develops a novel sanding criterion based on a moment balance of forces acting on a particle attached to a pore wall. The forces considered include drag, lift, buoyancy, and DLVO (Derjagin–Landau–Verwey–Overbeek) forces. The analysis concludes that for typical sand sizes, DLVO forces are negligible compared to hydrodynamic forces. A critical velocity ($U_{cr}$) is derived; sand production occurs only when the fluid velocity exceeds this threshold, disrupting the torque balance and detaching the particle.
2. Mathematical Formulation:
   • Linear Flow (1D): Governing equations are established for mass balance involving fluidized, attached, and re-adhered particles. The model introduces three key tuning parameters: a filtration coefficient ($\lambda_1$), a first-erosion coefficient ($\lambda_2$) driven by suspended particle flux, and a second-erosion coefficient ($\lambda_3$) driven by fluid flow velocity. Porosity changes are linked to erosion via a modified Vardoulakis equation, and permeability is updated using the Carman-Kozeny equation.
   • Radial Flow (Axisymmetric): The linear equations are adapted for radial coordinates to simulate a production well, accounting for the variation in fluid velocity with distance from the wellbore.
3. Numerical Solution: An explicit finite difference scheme is employed to solve the governing equations for particle concentration and porosity over time. The model assumes incompressible flow, uniform particle size, and equal velocities for fluid and suspended particles.
4. Calibration and Validation: The model parameters ($\lambda_1, \lambda_2, \lambda_3$) were calibrated by matching the numerical results against experimental data from Papamichos et al. [40] using a least-squares method in MATLAB.
5. Sensitivity Analysis: The behavior of the model was assessed by varying the volumetric flow rate and the three erosion/filtration coefficients to observe their impact on cumulative sand production, particle concentration, porosity, permeability, and pressure drop.

Key Contributions

• Novel Sanding Criterion: Introduction of a critical velocity formula derived from a torque balance of hydrodynamic and inter-particle forces, providing a physical threshold for particle detachment.
• Integrated Numerical Model: Development of a unified model that accounts for particle detachment, transport, and re-adhesion, coupled with dynamic updates to porosity and permeability.
• Dual-Geometry Approach: Presentation of both 1D linear and axisymmetric radial solutions, allowing for the translation of laboratory data to field-scale well predictions.
• Parameter Identification: Successful tuning of model coefficients to match experimental data, demonstrating the model's capability to replicate observed sand production kinetics.

Results

• Model Accuracy: The numerical model showed high agreement with experimental data, achieving $R^2$ values of 0.9735 and 0.9961 for two different test cases.
• Flow Rate Impact: Sensitivity analysis revealed that while increasing the production rate accelerates the onset and rate of sand production, the total cumulative sand production is limited by the formation's capacity. Once the erodible material is depleted or the velocity drops below critical levels due to porosity changes, production stabilizes.
• Near-Wellbore Effects: In the radial model, sand erosion was found to be most severe near the wellbore where fluid velocity is highest. The eroded zone was limited to a few meters from the wellbore.
• Property Changes: Significant increases in porosity and permeability were observed in the near-wellbore region due to sand erosion.
• Pressure Drop: The study found that sand production can reduce the pressure drop in the near-wellbore region by approximately 36% for typical model coefficients. This reduction is attributed to the increase in permeability, which outweighs the density increase caused by suspended particles.
• Mitigation: The results indicate that sand production can be significantly mitigated by reducing production rates to keep fluid velocities below the critical detachment threshold.

Significance and Claims
The paper claims that its primary significance lies in providing a practical mathematical tool for predicting sand production that bridges the gap between laboratory experiments and field applications. By matching experimental data, the model's tuning parameters can be determined and subsequently used to predict field-scale sand production in radial systems. The authors assert that the model effectively captures the kinetics of particle detachment and the consequent evolution of reservoir properties (porosity and permeability).

The study maintains a modest scope, noting that the model does not account for complex physics such as wormhole development. Furthermore, the authors suggest that while the current explicit numerical solution provides good agreement, future work could focus on developing implicit or analytical solutions to further improve accuracy. The paper concludes that reducing production rates is a viable strategy for mitigating sand production, supported by the model's demonstration of the relationship between fluid velocity and particle detachment.