Optimal Well Placement (OWP)

Information-Theoretic Framework for Spatial Sampling Design

Configuration

Field Type:

Field Size:

Sampling Method:

Number of Samples: 20

Pattern Radius: 3

Training Images: 5

Reconstruction:

Ready. Generate a field to begin.

Run sampling and reconstruction to see metrics.

Given a 2D binary random field (e.g., sand/shale reservoir), select K measurement positions (wells) that maximize information about the entire field.

Binary entropy: H(p) = -p log2(p) - (1-p) log2(1-p)

Measures uncertainty at each position. Maximum at p=0.5 (1 bit).

f* = argmax H(X_f)

Select positions that maximize the joint entropy of measurements, equivalently minimizing posterior uncertainty.

Adaptive Sequential Empirical Maximum Entropy Sampling:

Estimate conditional entropy at all unsampled positions using pattern matching against the Training Image.
Select the position with maximum entropy.
Reveal the true value (drill the well).
Repeat K times.

Greedy with (1 - 1/e) approximation guarantee (submodular optimization).

C_k = (H_0 - H_k) / H_0

Fraction of total uncertainty resolved after k measurements. Ranges from 0 to 1.

Random Uniform: Baseline, no spatial structure.
Stratified Grid: Deterministic uniform coverage.
Random Stratified: Randomized within strata.
Multiscale: Hierarchical multi-resolution.
Oracle Entropy: Uses true field (upper bound).
Adaptive Entropy: AdSEMES (the main algorithm).
Penalized Adaptive: AdSEMES with spatial penalty to avoid locality clustering.
Hybrid Stratified + Adaptive: Stratified seeding + penalized adaptive refinement.
Multiscale Adaptive: Multi-resolution entropy-guided sampling.

Shannon (1948); Cover & Thomas (2006); Silva et al., Fondecyt 1140840; Nemhauser, Wolsey & Fisher (1978)