# ⚛  L1 Principle — Well Productivity Index Inversion

**ID:** `L1-465` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Petroleum Engineering — *Well performance*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** reservoir property vector
> **📡 Carrier:** acoustic · **🌫 Noise:** measurement gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555280

---

## 🧠 1. Introduction

**Well Productivity Index Inversion** is a **parameter-estimation problem** whose unknown lives in **reservoir property vector** space, within the **Well performance** sub-domain of **Petroleum Engineering**.

Measurements consist of acoustic pressure waves recorded by transducers via a **pressure transient analysis** sensing mechanism.

The forward operator applies, in order: gradient / divergence with respect to position; S · pta · derivative analysis operator; O · semi log · straight line operator.

Observations are corrupted by measurement gaussian. Existence of the recovered reservoir_property_vector is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 25); wellbore_storage_distortion dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Measurement gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Spatial derivative → S · pta · derivative analysis → O · semi log · straight line → **y** (detector).

```
y = `O.semi_log.straight_line` `S.pta.derivative_analysis` ∇ x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.space` | Gradient / divergence with respect to position |
| `S.pta.derivative_analysis` | S · pta · derivative analysis operator |
| `O.semi_log.straight_line` | O · semi log · straight line operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Petroleum Engineering |
| Sub domain | Well performance |
| Carrier | acoustic |
| Problem class | parameter_estimation |
| Solution space | reservoir_property_vector |
| Noise model | measurement_gaussian |
| Integration axis | radial_wellbore |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered reservoir_property_vector is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 25); wellbore_storage_distortion dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Measurement gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | permeability_estimation_error_percent |
| Secondary | skin_factor_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `pressure_transient_analysis`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| K md | — | 50 |
| Skin factor | — | 5 |
| N pressure points | — | 500 |
| Wellbore storage c bbl psi | — | 0.01 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| K md | — | 0.1 – 10000 |
| Skin factor | — | -5 – 50 |
| N pressure points | — | 50 – 5000 |
| Wellbore storage c bbl psi | — | 0.001 – 1.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 15 permeability_estimation_error_percent

| Metric | Range |
|---|---|
| Permeability estimation error percent | 2 – 50 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **permeability_estimation_error_percent**, with κ = `500` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x8a1ccb9a2ebaacf2e139043aa5ad760ae46b7d6e4c022b3242e85b99d5d470a7`
- **Chain tx hash:** `0xbf8e44adcebeb838bc2c456575091cfb34c09e4034b0c7d52174b07343673dec`
- **Chain block:** `41555280`

---

## File Mapping

This bundle consists of: `L1-465.md`, `L1-465.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-465.md` | Source of truth — edit this | Human or LLM |
| `L1-465.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 `L1-465`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._