reservoir · material balance
Aquifer Influx (Fetkovich Pseudosteady-State)
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Inputs
m³/d/kPa
psi
bbl
Paste one row per line — separate columns with a comma or tab: Time (years), Reservoir Pressure (psi)
⚠ Needs at least 2 valid rows to calculate — 0 so far.
Description
Computes cumulative water influx from an aquifer into a reservoir using Fetkovich's (1971) pseudosteady-state productivity-index method — the algebraic, non-tabulated alternative to van Everdingen-Hurst that this platform's formula-driven calc() pattern can actually implement. Unlike every other array-input calc on this platform, this is an ORDERED, STATEFUL recursion: each interval's influx depends on the aquifer pressure left over from every prior interval, not on that row's own values in isolation. Given a time-ordered reservoir pressure history plus the aquifer's productivity index (Jw) and initial encroachable water (Wei), it steps through the history one interval at a time, computing each interval's influx rate from the aquifer's currently-remaining shut-in pressure, then updating that pressure via the aquifer material balance before moving to the next interval.
Variables
| Symbol | Unit | Description |
|---|---|---|
| We | bbl | Cumulative Water Influx |
| p̄ | psi | Aquifer Pressure (Final) |
| Jw | m³/d/kPa | The aquifer's pseudosteady-state productivity index — the same PI-curve concept as an oil well's or a gas well's backpressure coefficient, applied to the aquifer itself (Fetkovich 1971, Eq 15). Usually history-matched against production-derived water-influx estimates rather than taken purely from idealized geometry; kept in this platform's own compound m³/d/kPa unit (not separately toggled imperial/metric) the same way the Inflow Performance discipline's own Productivity Index output already is. |
| pi | psi | The aquifer's own shut-in pressure before any water influx has occurred — typically equal to the reservoir's own initial pressure at discovery, since reservoir and aquifer are normally in pressure equilibrium prior to production. |
| Wei | bbl | The aquifer's total encroachable water volume at initial pressure (Wei=ct·Wi·pi in general form; for radial geometry, Wei=(π/5.61)(ra²−rr²)θh·ct·pi per Fetkovich 1971, Eq 17). Not the same as the aquifer's total pore volume — the aquifer remains 100% water-saturated even after full pressure depletion (p̄=0); Wei is the volume term in the linear pressure-decline relationship, not a physical storage capacity. |
| t, pwf | — | One row per historical (or forecast) reservoir-pressure reading at the aquifer boundary, in increasing time order starting at t=0. Row 1 is a baseline only — it fixes the starting time and reservoir pressure but by itself produces no influx; the recursion's first COMPUTED interval runs from row 1 to row 2, seeded with the aquifer held at the Initial Aquifer Pressure (pi) above (Fetkovich 1971's own p̄(0)=pi convention). Each additional row adds one more interval, and each interval's incremental influx updates the aquifer pressure that feeds directly into the next interval — unlike every other array-input calc on this platform, row order and every prior row's value genuinely matter, not just each row's own contents. |
Assumptions
- Pseudosteady-state (or steady-state) flow is already established in the aquifer — the early transient period is neglected, per Fetkovich's own derivation and stated limitations
- Jw and Wei are already-known or already-history-matched constants for this specific aquifer — this calc does not derive them from raw permeability/thickness/radius/porosity/compressibility geometry
- No interference from other reservoirs sharing the same aquifer, and no water injection into or production from the aquifer itself — this calc implements Fetkovich's simpler single-reservoir form (Eq 2/13), not the general interference/injection form (Eq 9)
- The pasted reservoir-pressure history is dense enough that each interval's two-point average meaningfully represents the true (unknown) pressure path within that interval
Limitations
- Row 1 of the pasted history is a baseline only, not a computed interval — minRows is 2, and the first computed interval always starts from the Initial Aquifer Pressure (pi) input, not from any aquifer pressure implied by row 1's own reservoir-pressure value
- Implements the step-function (constant-rate-per-interval) approximation, not the exact closed-form solution — coarser time steps (e.g. yearly, as in the paper's own worked example) carry a small, known, quantifiable discretization gap versus the exact analytical answer that shrinks as the interval is subdivided further (derivation step 5)
- No golden vector could be sourced from Fetkovich's own worked multi-year example (the paper's Table 5) — a genuine worked example exists in the primary source, but 1971-era scan quality defeated reliable digit-level transcription even for the two years (1 and 20) the paper itself presents in full iteration detail (reservoir-aquifer-influx-phase2a); this calc's golden vector is independently derived from the paper's own Eq5≡Eq6 closed-form identity instead, not from that table
- Assumes a single non-interfering aquifer feeding a single reservoir with constant Jw and Wei over the full pasted history — a materially changing effective drainage radius (e.g. from a second reservoir beginning to draw on the same aquifer) requires Fetkovich's more general interference form (Eqs 9, 20-22), not implemented here
- If the computed aquifer pressure reaches zero or below, the aquifer's Wei has been physically exhausted at pi — results beyond that point are not physically meaningful (flagged at runtime, not silently clamped)
Use Cases
- → Water-drive performance forecasting: Given an already history-matched Jw/Wei and a forecast reservoir-pressure schedule, project cumulative water influx and the aquifer's own pressure decline over the remaining life of the field — the paper's own primary intended application, demonstrated there against a 20-year gas-reservoir forecast.
- → Jw/Wei history matching: Run this calc against several candidate Jw/Wei combinations and compare the computed We against production-derived water-influx estimates (Fetkovich 1971, Eqs 27-28) to find the best-fitting pair — the same iterative-substitution process the paper's own worked example performs by hand.
- → Completing the material-balance water-drive picture: mbe_havlena_odeh's own derivation explicitly sets We=0 and defers the active-water-drive case to "the more general form F−We plotted against Et+m·Efg" (its own step 2 explanation) — this calc supplies the We term that extension needs.
Related Calculations
Region Notes
Permian Basin
mbe_havlena_odeh's own regionNotes already observe that Wolfcamp/Spraberry unconventional reservoirs are almost always closed, depletion-drive systems (We≈0) — this calc is for the comparatively rare Permian reservoir with genuine measurable edge or bottom water, where mbe_havlena_odeh's own We=0 form would systematically bias N.
Gulf Coast / Gulf of Mexico
Miocene and Frio sandstone reservoirs with active edge-water or bottom-water drive are classic Fetkovich-method candidates in the published literature — history-match Jw/Wei against early production data before relying on this calc for long-range forecasting.
North Sea
Several Brent Group and Forties Sandstone fields have documented aquifer support; Fetkovich's method is a commonly cited practical alternative to van Everdingen-Hurst in North Sea reservoir-engineering practice specifically because it avoids needing tabulated dimensionless-pressure data.
Global
Jw and Wei are best treated as history-matched fitting parameters (see Jw/Wei History Matching above), not purely geometric calculations — real aquifers routinely depart from the idealized radial or linear geometries Fetkovich's own PI formulas (Eqs 15, 17) assume.
References
Primary source
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