Hydroshearing Stability Map: Radius vs Fracture Orientation (2D)

Injection Controls

Wellhead pressure ΔP 0.0 MPa

Differential stress σ₁−σ₃ 20.0 MPa

Friction coefficient μ 0.60

Stimulation Status

GREEN — safe to inject

Pressure front R(t)

0 m

Fractures slipped

0 / 0

Cumulative seismic moment ΣM₀

0 N·m

Largest event

Mw —

How to read this map

This is a 2D companion to the 3D hydroshearing simulator: instead of a rendered rock volume, every fracture is plotted directly in the two coordinates that actually govern whether it slips — its radial distance r from the injection well (horizontal axis) and its orientation θ to the maximum principal stress σ₁ (vertical axis). Those two numbers are exactly what the physics needs; the third spatial dimension the 3D scene carries is discarded here because it plays no role in the failure criterion.

The background is a live-computed stability field: for every (r, θ) cell it evaluates the same Mohr-Coulomb test the 3D sim runs per-fracture,

σn(θ) = (σ1+σ3)/2 + (σ1-σ3)/2 · cos(2θ)
τ(θ)  = (σ1-σ3)/2 · sin(2θ)
P(r,t) = ΔP · max(0, 1 − r / R(t))     (inside the diffusion front)
slip when τ(θ) ≥ μ · (σn(θ) − P(r,t))

so the amber/red band you see sweeping outward as R(t) grows is the failure zone, not a decorative gradient. Each dot is one fracture at its own (r, θ); once a dot's cell crosses the line it turns red and stays red — a real earthquake, sized with the same circular-crack source model as the 3D sim:

M0 = (16/7) · Δτ · a³
Mw = (2/3) · log10(M0) − 6.07

The thin curve above the map is the pressure-diffusion profile P(r,t) itself, sharing the same r-axis, so you can read off directly why fractures close to the well go critical first.

Wellhead pressure — raises the pressure-profile curve and pushes the failure band up the θ-axis at small r.
Differential stress — widens the Mohr circle, so more of the θ range sits close to failure even before pumping starts.
Friction coefficient μ — Byerlee's law puts crystalline basement at μ ≈ 0.6–0.85; lower μ shifts the failure band down toward θ=0.
Traffic light — green Mw<1.0, amber 1.0–2.0, red >2.0 (stop), same illustrative thresholds as the 3D sim.

Stable fracture Near critical Slipped (seismic event) Injection well (r=0)