Instant vs. Delayed 4D Response in Gas Reservoirs

A Batzle–Wang gas EOS coupled to Gassmann fluid substitution and a stress-sensitive rock frame, run forward through a depletion path — dry gas by default, no injection — with an optional water-breakthrough sequence (front arrival, breakthrough at the well, late life with residual gas) layered on top. It traces what pressure decline and then water influx do to fluid density, matrix velocity, full-stack impedance and a synthetic 4D seismic section, so the early-softening / late-hardening crossover and the later water hardening fall out of the physics rather than being asserted.

4D Seismic Gas Depletion Rock Physics
PVT & fluids
Sg + Sw + Sc is renormalized to 1 automatically. These are the base-survey (t = 0) saturations. They stay fixed through time unless the water-breakthrough option below is on, in which case Sw ramps up and Sg falls toward the residual-gas value.
Rock frame (virgin / base survey)
Stress & depletion path
Frame stiffening response
Plain elastic stress-stiffening (power-law) is monotonic and, under representative parameters, does not reproduce a softening→hardening crossover — one term simply dominates throughout. Reproducing the field's early-soft / late-hard pattern needs a frame response with a delay: little stiffening until cumulative depletion passes an onset (compaction/cementation threshold), then a faster ramp. That is a testable geomechanical claim, not a default assumption — the numbers here are illustrative, tuned to a ~3–5 year crossover, not measured.
Water breakthrough
Sw rises smoothly from the base value at "front enters" to 1 − Sgr − Sc at "late life"; the breakthrough time marks when water reaches the well and is a stage boundary. Aquifer support pulls pressure back toward its initial value in proportion to the flooded fraction. Frame compaction in threshold mode is treated as irreversible, so pressure support does not un-stiffen the rock. Times are illustrative — set them from your own water-cut and 4D history.
Synthetic seismic model
Overburden Vs is used only by the angle-stack sections (shear contrast sets the gradient).
Two-interface wedge (shale–gas sand–shale, symmetric encasing shale). Time axis is relative to the top-reservoir pick — absolute TWT isn't modeled. The base event's position moves with Vp(t): it pushes down while Vp softens, pulls up once the frame-hardening phase takes over.
Crossover age
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Peak softening (ΔAI)
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ΔAI at marked age
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ΔAI at end of projection
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Stage at marked age
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ΔAI vs. time since first production
Net (blue = hardening) Net (red = softening) Fluid term only Frame term only
Synthetic 2D section vs. production time
3.0×
softening0hardening
A two-interface wedge (shale / gas sand / shale) convolved with a zero-phase Ricker wavelet. The top-reservoir event is the near-stack-style response; the base-reservoir event inherits the same amplitude evolution plus a time shift, since the sand's own two-way time changes as Vp(t) changes — pushing down while Vp softens, pulling back up once the frame-hardening phase dominates. The 4D-difference mode (monitor minus the t = 0 base survey) is the direct analog of the dFar/dNear-style difference products used on the real data.
Angle-stack sections vs. production time
Uses the section mode and colour intensity set in the full-stack panel above. Dashed lines: water stages 2–4; yellow line: marked age.
Near 0–10°—
Mid 1 10–20°—
Mid 2 20–30°—
Far 30–40°—
Ultra-Far 40–50°—
softening0hardening
Top-event amplitude change by stack (monitor − base)
Near Mid 1 Mid 2 Far Ultra-Far Ultra-Far − Near (D)
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Each stack is a flat-gather, zero-phase Ricker synthetic of the same two-interface wedge, but with angle-dependent (Aki–Richards) reflectivity, averaged over five angles inside the bin. Angles are the incidence angle at the reservoir top, no NMO stretch, attenuation or angle-dependent wavelet. Reading it: Near senses impedance (A), Mid adds the Vs term (gradient), Far / Ultra-Far sense mostly Vp. Use the Fluid term view for the instant response (pressure and saturation, frame frozen), the Frame term view for the delayed response (fluid frozen), and Net with the water stages on to follow water movement. The amplitude chart plots the trace value at the top-reservoir pick, so it includes tuning; the box under it classifies the marked age by the signs of the near-stack change (A) and D = Ultra-Far − Near. With the default stiff sand, water mixed uniformly (Reuss) raises density long before Vp, so near hardens much more than far (D negative); switching the water-mixing mode to patchy (Brie) lets Vp respond early and the hardening grows with angle (D positive). That is the main thing to calibrate against a real field.
Fluid vs. bulk density vs. time
Pore-fluid density Bulk (rock+fluid) density
Pore-fluid density falls continuously with pressure while gas is the dominant phase — no delay, no threshold, it tracks P in real time — and climbs again once water replaces gas. Bulk density follows it down (small effect: gas only fills φ·Sg of the rock volume) and stays essentially flat afterward — the matrix (mineral) grains themselves never change density; what moves is the fluid they're mixed with, plus any compaction, which this model does not yet include as a separate porosity-loss term.
Vp and Vs vs. time
Vp Vs
Vp carries both terms (K and ρ), so it dips early and recovers late. Vs is fluid-blind by construction — it only moves once the frame term (μ) engages, so its rise lags Vp's dip and marks the shift into the hardening regime.
Vp/Vs vs. time — the discriminator
Vp/Vs
Fluid-term softening suppresses Vp while leaving Vs (shear, fluid-blind) essentially untouched, so Vp/Vs drops early. Frame stiffening raises both K and μ together, so Vp/Vs recovers as the response transitions toward hardening.
Water-breakthrough stages
Sw (%) Sg (%)
StageWindow (yr)Sw (%)ΔAI net (%)Vp/VsΔ interval TWT (ms)
Stages are set by the three times above. Hardening from water is strongly non-linear under uniform (Reuss) mixing — fluid bulk modulus stays gas-like until little gas remains, so most of the impedance gain arrives late in the Sw ramp — while patchy mixing brings it in earlier and more linearly. Interval TWT is the reservoir two-way time at the thickness set in the seismic fieldset: it lengthens while Vp softens and shortens (speed-up) as water and frame hardening raise Vp. Water production at a well and 4D hardening need not coincide: the seismic sees the front before it reaches the well, but the first 20–30 % of gas replacement barely moves Vp, so detection can lag real water arrival.

Model: Batzle & Wang (1992) real-gas density and adiabatic bulk modulus for the free-gas phase; Reuss (isostress) mixing for gas + water + condensate; Gassmann fluid substitution for K_sat (μ unaffected by fluid, by construction). K_dry₀ and μ₀ are inverted from the virgin Vp₀/Vs₀ inputs, so the model is internally consistent with the base-survey rock at t = 0. The frame term uses either a plain power-law stress response or a delayed/threshold one — switch modes above to see why the plain elastic case alone does not reproduce a sign reversal. Saturations are held fixed unless water breakthrough is enabled, in which case Sw ramps (smoothstep) between front arrival and late life toward 1 − Sgr − Sc with condensate held constant and no gas re-dissolution; gas–water mixing is Reuss (uniform) or Brie (patchy), aquifer support partly restores pressure, and threshold-mode frame compaction is irreversible. There is no other injection, and the dry-gas preset removes condensate entirely. The synthetic section is a simple 1D convolutional model (no compaction/porosity-loss term, no real absolute TWT, symmetric encasing shale) — a qualitative illustration of the mechanism's seismic expression, not a calibrated forward model. Defaults are representative values, not Sapphire-specific measurements — replace every field with your own PVT and log values to make this quantitative for a given well.

Theory

Batzle-Wang Gas EOS

Real-gas equation of state for calculating gas density and bulk modulus from pressure, temperature, and gravity. Accounts for non-ideal behavior at reservoir conditions through z-factor and compressibility calculations.

Gassmann Fluid Substitution

Relates saturated bulk modulus to dry-frame modulus, mineral modulus, fluid modulus, and porosity. Separates the instant fluid response from the delayed frame response, allowing decomposition of 4D signatures.

Stress-Sensitive Frame

Two models: power-law (elastic) or threshold (delayed compaction). The threshold model reproduces the early-softening / late-hardening crossover observed in real 4D data by introducing a delay before frame stiffening engages.

Water Breakthrough Stages

Four stages: pre-water, water approaching, breakthrough at well, and late life. Sw ramps smoothly while aquifer support partially restores pressure. Mixing mode (Reuss uniform vs. Brie patchy) controls the AVO gradient response.