# ⚛  L1 Principle — Buckley-Leverett — two-phase displacement saturation

**ID:** `L1-192` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Fluid Dynamics — *Buckley-Leverett in petroleum reservoirs*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** saturation shock
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41553978

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## 🧠 1. Introduction

**Buckley-Leverett — two-phase displacement saturation** is a **linear inverse problem** whose unknown lives in **saturation shock** space, within the **Buckley-Leverett in petroleum reservoirs** sub-domain of **Fluid Dynamics**.

Measurements consist of mechanical vibrations or strain signals via a **well production** sensing mechanism.

The forward operator applies, in order: L · flux function operator; L · shock admissible operator; L · fractional flow operator; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · flux function → L · shock admissible → L · fractional flow → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.fractional_flow` `L.shock_admissible` `L.flux_function` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.flux_function` | L · flux function operator |
| `L.shock_admissible` | L · shock admissible operator |
| `L.fractional_flow` | L · fractional flow operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Fluid Dynamics |
| Sub domain | Buckley-Leverett in petroleum reservoirs |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | saturation_shock |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `bl` · **Forward operator:** `bl_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 30 |
| N cells | — | 100 |
| N times | — | 1000 |
| Relperm error | — | 0 |
| Capillary error | — | 0 |
| Mobility ratio error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 0 – 40 |
| N cells | — | 10 – 100000 |
| N times | — | 100 – 100000 |
| Relperm error | — | 0.0 – 0.3 |
| Capillary error | — | 0.0 – 0.3 |
| Mobility ratio error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 24.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `300` and δ = `3`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0xb5d96e0f7bd9783b2718a08a9ec3928333aea9fca4152e3c8fbabdacaad67e69`
- **Chain tx hash:** `0x0af50dc66db6aeccdff0295e67dcff4107b9391b886e495aefa182b796ae97a1`
- **Chain block:** `41553978`

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## File Mapping

This bundle consists of: `L1-192.md`, `L1-192.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-192.md` | Source of truth — edit this | Human or LLM |
| `L1-192.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-192`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._