Doublet Streamline Model Physics Knowledge

This skill provides physics knowledge for the 2-D doublet (injection-production well pair) model using streamline methods. The model solves advection-reaction-dispersion problems along circular streamlines in a potential flow field.

When This Skill Applies

• Modifying streamline geometry calculations (bipolar coordinates)
• Changing transport solvers (hydrodynamic, chemical, thermal)
• Implementing or modifying breakthrough curve integration
• Working with Green's function kernels for thermal transport
• Debugging unexpected breakthrough behavior
• Planning changes to physics-related code
• Explaining why the model behaves a certain way

Core Physics Reference

The model tracks transport in a 2-D potential flow field between an injector (+a, 0) and producer (-a, 0):

Key state variables:

For detailed equations, see [EQUATIONS.md](EQUATIONS.md). For symbol definitions and units, see [SYMBOLS.md](SYMBOLS.md). For derivation logic, see [DERIVATIONS.md](DERIVATIONS.md). For edge cases and sanity checks, see [SANITYCHECKS.md](SANITYCHECKS.md).

Workflow A: Physics Validation

Use this workflow when reviewing or planning code changes.

Step 1: Identify affected transport type

Determine which transport physics is affected:

• Hydrodynamic: Streamline geometry, time-of-flight, velocity field
• Chemical: Capacity-limited reaction, logistic traveling wave
• Thermal: Fluid-matrix exchange, Bessel function kernel

Step 2: Check coordinate system consistency

The model uses multiple coordinate systems:

• Cartesian (x, y) for plotting
• Bipolar (ξ, β) for streamline parameterization
• Polar about streamline center (R, φ) for local calculations

Verify transformations are applied correctly.

Step 3: Verify breakthrough integration

Breakthrough curves integrate over all streamlines:

C_prod(t) = (1/π) ∫_0^π C(β, t) dβ

Check that:

• Integration weights are correct (trapezoidal or custom)
• β limits span [0, π] (or arrived streamlines only)
• Singularities at β → 0, π are handled

Step 4: Check Green's function stability (thermal)

For thermal transport, the Bessel kernel G(τ, t) can overflow:

• Verify exponent recentering is applied
• Check that i1e (scaled Bessel I1) is used instead of i1
• Verify prefix integrals use cumulative_trapezoid

Step 5: Report findings

Summarize:

• Which transport type is affected
• Any coordinate transformation issues
• Any integration boundary issues
• Any numerical stability concerns
• Recommendations for correction

Workflow B: Physics Reasoning

Use this workflow when explaining behavior, debugging, or proposing solutions.

For explaining physics concepts:

1. Identify the relevant equations and parameters
2. State the physical interpretation
3. Explain cause-and-effect relationships
4. Use the derivation steps from [DERIVATIONS.md](DERIVATIONS.md) to show connections

For debugging simulation issues:

1. Identify which transport type shows unexpected behavior
2. Check if the issue relates to:

- Breakthrough timing (τ calculation) - Front retardation (reaction capacity) - Thermal lag (fluid-matrix exchange rates)

1. Trace through the analytical solution
2. Check edge cases against [SANITYCHECKS.md](SANITYCHECKS.md)

For proposing physics-consistent changes:

1. Identify which equations are affected
2. Show how new terms integrate into the transport equations
3. Demonstrate that mass/energy balance is preserved
4. Identify any new sanity checks needed

Quick Reference: Module Structure

Key Physical Constraints

These must always hold:

1. Streamline geometry: Streamlines are circular arcs through (±a, 0)
2. Travel time bounds: T(β=0) = 4πna²/(3Q) is minimum arrival time
3. Velocity singularity: v → ∞ at wells (handled by coordinate transformation)
4. Conservation: Integrated flux across all streamlines equals Q
5. Retardation: Chemical front travels at speed v·Cinj/(Cinj + S_0)
6. Thermal equilibrium: As t → ∞, Tf → Tm → T_inj (behind the front)

Transport Type Classification

Dimensionless Groups