Finite-aquifer history replay and discrete Bayesian uncertainty

This workflow extends the frozen-EOS measured-history tank. Keep the original sample and independently laboratory-tuned EOS in CalibratedPvtModel. Reservoir matching changes volume/support assumptions; it does not retune that EOS.

Physical aquifer and units

FetkovichAquifer accepts initial water volume $W_i$ in reservoir m3, total water/rock compressibility $c_t$ in 1/bar, productivity $J$ in reservoir m3/(s bar), and initial average aquifer pressure $p_{ai}$ in bara. Its rate and storage equations are

\[C=c_tW_i,\qquad\frac{dW_e}{dt}=J(p_a-p_r),\qquad p_a=p_{ai}-\frac{W_e}{C}.\]

Positive $W_e$ means water entering the reservoir. No surface-volume conversion or extra multiplication by Bw is applied. The water-volume basis is fixed; the slightly-compressible, constant-property assumptions need engineering justification. The model follows the rate-plus-material-balance structure of Fetkovich (1971), SPE 2603. The OPM AQUFETP reference documents corresponding physical parameters. OPM deck water volumes/productivity use a surface-volume basis with PVTW; convert that basis before reusing them here.

initialState() returns an immutable zero-influx checkpoint. advance(state, durationSeconds, pressureStartBara, pressureEndBara) integrates the linear pressure path exactly. For $x=J\Delta t/C$, $L=1-e^{-x}$ and $R=1-L/x$, the finite-storage interval increment is

\[\Delta W_e=C[(p_{a0}-p_{r0})L-(p_{r1}-p_{r0})R].\]

The implementation uses series expansions and expm1 near zero exposure. Zero productivity gives no exchange. Zero storage gives no volumetric support. Explicitly infinite water volume with positive compressibility gives fixed aquifer pressure. Signed exchange permits recharge; finite-water exhaustion, including an interior pressure-ramp extremum, is rejected. This lumped storage law is distinct from radial transient diffusion in VanEverdingenHurstAquifer; only common storage limits are compared, not claimed transient equivalence.

Coupling to the existing EOS tank

CompositionalReservoirHistory.replay(initialVolumeM3, aquifer) solves the reservoir end pressure implicitly with the aquifer interval response. The existing SimpleReservoir TV flash closes the tracked EOS fluid volume to

\[V_{\mathrm{tracked}}=V_{\mathrm{pore}}-W_e.\]

Aquifer water stays segregated and immiscible. It occupies pore space, but does not enter the tracked component inventory. The measured component schedule therefore cannot produce this segregated water. Initial EOS water and measured component injection remain part of the tracked fluid; they are separate from aquifer support. Water mixing, dissolution, salinity, water chemistry, rock expansion and thermal balances are not added by this approximation. Use a different qualified model when those effects matter.

For opt-in phase-selective withdrawals, the aquifer is coupled on each explicit extraction substep. Only tracked EOS aqueous fluid can be selected for production; segregated support water remains separate.

Each accepted result retains initial/end reservoir pressures, average aquifer pressures, cumulative aquifer volume, times and tracked component amounts. Reject a nonpositive remaining fluid volume, unavailable components, negative cumulative segregated support, failed TV-flash volume closure or a failed coupled root. Root probes outside physical storage bounds never become predictions. Minimum/maximum interval influx are checked as well as endpoints, so recharge cannot conceal temporarily negative segregated water or exhausted pore space. SimpleReservoir.advanceComponentAmounts has an overload accepting the complete cumulative external water volume; later ordinary component steps retain the last accepted external volume. Rejected steps leave state, time and calculation identity unchanged.

fitVolume(minimumVolumeM3, maximumVolumeM3, aquifer) holds aquifer parameters fixed while estimating initial pore volume against the supplied pressure sigmas. Report bound-limited solutions and rejected trials. Refine the measurement/flow intervals: exact integration of a prescribed linear pressure path does not make the unknown coupled nonlinear pressure trajectory exact.

Joint Bayesian scenarios

ReservoirHistoryEnsemble.Candidate combines one frozen history, pore volume, optional physical aquifer, positive prior probability mass and provenance. Use initial gas/oil/aqueous fractions in each history to encode geological gas-cap assumptions. Different independently accepted laboratory models and measured component-amount realizations can represent PVT and production-history uncertainty. These uncertainties are propagated only when explicitly supplied; the utility does not infer missing distributions or repair a miscalibrated EOS.

All candidates must share named component order, initial temperature/pressure, observation/forecast times, pressure observations, sigmas and holdout selection. For independent Gaussian pressure errors, evaluation uses

\[w_i=\frac{\pi_i\exp(-\chi_i^2/2)}{\sum_j\pi_j\exp(-\chi_j^2/2)}.\]

Here $\pi_i$ is a complete joint prior mass, not a density at a grid point. A nonuniform grid needs cell-integrated masses. Logarithmic normalization prevents likelihood underflow. Correlated pressure errors require another likelihood; production realizations and laboratory uncertainty should not count the same measurement uncertainty twice.

Returned evidence	Interpretation
Normalized prior/posterior masses	Discrete joint hypotheses, including EOS/history assumptions
Best likelihood index	Minimum calibration chi-square, separately from prior preference
Rejected candidates and prior mass fraction	Infeasible hypotheses retain explicit diagnostic weight
Posterior means and parameter correlations	Volume, initial gas/aqueous fractions, aquifer capacity and productivity; constant/non-finite dimensions report NaN
Prior standardized shifts	Posterior-minus-prior mean in prior standard deviations; a conflict diagnostic, not a significance test
Effective scenario count	Inverse sum of squared weights; concentration of this ensemble, not MCMC sample size
Log evidence without Gaussian constant	Relative normalizer only; not absolute data-density evidence
pressureQuantiles(q)	Lower-tail model-pressure quantiles over all history/forecast times

Held-out pressures never enter the posterior likelihood. Prediction-only intervals use NaN pressure/sigma pairs and remain available for forecast quantiles. Quantile 0.1 is the lower tenth percentile, not the petroleum exceedance convention. Quantiles exclude future measurement noise. Only the product $C=c_tW_i$ enters pressure/storage response; pressure history cannot distinguish $c_t$ from $W_i$ separately. Include observations around the response timescale $C/J$ and independent geological bounds. Refine the joint hypothesis grid and check sensitivity to prior/support assumptions before interpreting intervals.

The tests check independent Bayes odds, heldout exclusion, sigma weighting, underflow-resistant normalization, rejected prior mass, volume/aquifer confounding, production-realization uncertainty, serialization and quantile ordering. Aquifer tests compare analytical storage limits and an independent RK4 integration; coupled tests check separate inventories, pore-volume closure, known-volume recovery and interval refinement. These are analytical/synthetic software qualifications. They are not a matched public laboratory-to-field validation or recoverable-reserves estimate.

Remaining qualification

Issue #4740 also requires independent laboratory/field qualification, broader phase-path coverage and independently predicted produced-phase validation. Multi-tank transmissibility follows qualified single-tank matching. ES-MDA/external ensemble integration and the 2025 rescaled-CVD research method are separate gated extensions. None is implied by this finite scenario integrator.