Geomechanics: elastic to stress and fracture

From elastic to mechanical properties

The elastic cubes from Part 7 carry all the information needed to compute the mechanical properties geomechanics uses. The key identities:

• Poisson’s ratio $\nu = \dfrac{V_p^2 - 2 V_s^2}{2(V_p^2 - V_s^2)}$ — controls how much a rock expands laterally when compressed axially. Equivalently $\nu = \dfrac{(V_p/V_s)^2 - 2}{2(V_p/V_s)^2 - 2}$: Vp/Vs is the primary observable.
• Shear modulus $\mu = \rho V_s^2$. Rock rigidity. Derived directly from Is = ρVs.
• Bulk modulus $K = \rho(V_p^2 - \tfrac{4}{3}V_s^2)$. Compressibility stiffness.
• Young’s modulus $E = 2\mu(1+\nu)$. The "stiffness" number most engineers prefer.
• Brittleness index (an empirical combination) — typically (E_norm + (1 − ν_norm))/2 or weighted forms. HIGH brittleness means the rock fails in a brittle mode (useful for frac stimulation), LOW brittleness means ductile (plastic, hard to stimulate).

Every voxel in the QI cube can be converted to E, ν, K, μ, and brittleness using these identities. The result is a 3D cube of MECHANICAL properties that geomechanical modelers treat as their input — the seamless bridge from Part 7 to Part 8.

Exercise — walk through the scenarios

• Open the widget in Intact rock (in-situ) mode. The Mohr’s circle sits COMFORTABLY below the red Mohr-Coulomb failure envelope. Both endpoints (σ₃′ and σ₁′) of the circle are on the σ axis; the apex reaches up to τ = radius. The safety factor reads > 1.3, in the SAFE zone. This is the starting condition of every reservoir before we touch it.
• Drag the σv slider up and down. The circle expands (σ₁′ grows) or contracts. High σv means overburden is heavy — deep reservoirs, pushing the circle to the right. Drag σH: low σH lowers σ₃′, widening the circle. The MORE DIFFERENTIAL the stress, the larger the radius, and the closer the apex can get to the envelope.
• Switch to Depleted reservoir mode. Pore pressure drops by 10 MPa (typical for a mature oilfield). Effective stress rises: the circle shifts RIGHT. A ghost BASELINE circle shows where the circle was before depletion — you can see the shift directly. If the production run is deep enough, the depleted circle approaches or crosses the envelope (safety factor below 1) — that’s how compaction, subsidence, and casing-shear events happen in depleted fields.
• Switch to Injection mode. Pore pressure rises 8 MPa (water flood, gas injection, CO2 storage). Effective stress falls: the circle shifts LEFT. If σ₃′ approaches zero, the rock is on the verge of TENSILE (hydraulic) fracture — this is exactly how frac jobs work, but uncontrolled injection can fracture the CAP ROCK, compromising reservoir integrity.
• Switch to Critically stressed mode. The widget auto-tunes Pp so the Mohr’s circle JUST TOUCHES the failure envelope (safety factor = 1.00). This is the knife-edge state where ANY perturbation — a small pressure change, a minor earthquake, a temperature shock — can trigger failure. Real subsurface systems routinely operate close to this state; it’s why induced seismicity is a risk in every injection project.
• In any mode, drag Pp directly to watch the circle translate left or right in real time. The baseline/depleted/injected modes are just preset Pp shifts — the underlying physics is a single degree of freedom.

The stress regimes

The subsurface has THREE principal stresses: vertical σv (overburden weight), maximum horizontal σH, and minimum horizontal σh. Their RELATIVE MAGNITUDES determine the stress REGIME and the failure mode:

• Normal fault regime: σv > σH > σh. Dominant in passive-margin basins (Gulf of Mexico, North Sea). Fault motion is DIP-SLIP down; extension is the driver.
• Strike-slip regime: σH > σv > σh. Transform-margin regimes (San Andreas, Anatolia). Faults slip horizontally.
• Reverse fault regime: σH > σh > σv. Compressional regimes (Rockies, Andes). Faults slip up-dip; thrusts are the geometric expression.

For QI purposes, the regime matters because the FRAC AZIMUTH follows σH: hydraulic fractures open perpendicular to σh (the minimum stress direction). Knowing the regime tells you where to land horizontal wells, how to orient completions, and where induced fractures will propagate.

Effective stress and the pore-pressure lever

The single most important geomechanical idea: ROCK MATRIX carries EFFECTIVE stress, not total stress. The effective stress law (Terzaghi-Biot):

$\sigma' = \sigma - \alpha P_p$

where σ′ is effective stress, σ is total stress, Pp is pore pressure, and α is the Biot coefficient (typically 0.7-1.0 for reservoir rocks). This means:

• If you REDUCE Pp (production), σ′ goes UP — the rock matrix experiences MORE compressive stress. Eventually it may fail in compression (compaction, crushing).
• If you INCREASE Pp (injection), σ′ goes DOWN — the rock matrix experiences LESS stress. If σ₃′ drops to zero, the rock opens as a fracture (hydraulic fracturing).

The Biot coefficient α is a material property (not 1.0 in general). For a dense, tight rock α ≈ 0.7; for a loose high-porosity sand α ≈ 1.0. Getting α wrong by 20% gives an effective stress error of ∼3-5 MPa, which is enough to change the safety factor significantly.

The Mohr-Coulomb failure criterion

Rocks fail when shear stress τ on some plane exceeds a threshold that depends on the normal stress σ_n on that plane:

$\tau = \mu_f \sigma_n + C$

where μ_f = tan(φ) is the internal friction coefficient (φ is the friction angle, typically 25–40° for rocks) and C is the cohesion (a few MPa for intact rock, less for pre-existing faults). In (σ, τ) space this is a straight line — the FAILURE ENVELOPE. Mohr’s circle encodes all the (σ_n, τ) combinations that act on planes of every orientation through a given point in the rock; if the circle touches the envelope, some plane is at failure.

Three practical consequences:

• HIGH differential stress (large circle radius) → apex is high in τ → more likely to touch envelope. This is why highly deviatoric stress states fail easily.
• LOW effective stress (small σ_n) → circle sits far left → envelope line passes through LOWER τ values there → low-friction rocks fail at lower shear stresses.
• Pre-existing faults have LOWER C (cohesion) than intact rock. This is why ACTIVE faults are critically stressed at much lower differential stresses than a nearby intact rock mass.

Workflow — from QI cube to geomechanical deliverable

• Input: QI cubes (Ip, Is, ρ) from §7.3 + a pore-pressure cube (from basin modeling, mud weights, or 4D-derived pressure estimates).
• Compute: mechanical-property cubes (E, ν, K, μ, brittleness) voxel-by-voxel using the elastic identities above.
• Stress model: estimate the three principal stresses (σv from integrated density log; σH from well-log breakout analysis + regional stress maps; σh from leak-off tests + regional models). Output: σv, σH, σh cubes.
• Check: for every voxel, compute the Mohr’s circle in effective-stress space and the safety factor vs the Mohr-Coulomb envelope. Output: safety factor cube, failure-mode cube (shear vs tensile), breakout-azimuth cube.
• Forward-model scenarios: depletion (reservoir withdraws fluid) and injection (stimulation, disposal). For each, recompute the safety factor cube to find where failure risk emerges.
• Deliverable: 3D safety-factor cubes, predicted-breakout maps, recommended mud-weight windows, optimal-trajectory maps, frac-azimuth maps.

These deliverables flow into drilling engineering (trajectory + mud weight), completions engineering (frac design), and reservoir management (allowable pressure drawdown). In mature basins this is increasingly done through 4D geomechanical modeling, where pressure and stress are tracked through time — the bridge into §8.2.