DST Sequence Analysis — Horner Interpretation

Standard four-period drill stem test analysis: Initial Flow (IF) → Initial Shut-In (ISI) → Final Flow (FF) → Final Shut-In (FSI). Forward model generates synthetic gauge data using PO.PTA.Pw.VW with built-in superposition across all rate changes. Horner analysis of the FSI buildup extracts permeability, skin, and extrapolated pressure P*. Cross-checks ISIP against Horner P* as an independent reservoir pressure estimate.

DST convention. A standard DST has four periods: IF flushes the tool and provides a preliminary flow test; ISI allows initial pressure recovery; FF is the main flow period for permeability determination; FSI is the main buildup for quantitative analysis. The IF is intentionally short (5–30 min) to minimize reservoir depletion before ISI.

Horner producing time. The Horner method uses tp = FF duration (4h), ignoring the IF period. This is standard DST practice — the ISI recovery effectively resets the pressure field if the IF was short relative to the ISI. For the default case (3 min IF, 57 min ISI), the IF transient has largely dissipated by the start of FF.

ISIP as pressure estimate. The Initial Shut-In Pressure is the best direct Pi estimate from a DST. Unlike the Horner P* (which is extrapolated), ISIP is measured at the end of a short-flow/long-shut-in sequence. In low-permeability formations where ISI doesn't reach IARF, ISIP still systematically underestimates Pi — Horner analysis of the FSI then provides a better extrapolated value.

Full superposition vs simplified Horner. PO.PTA.Pw.VW computes the exact pressure response accounting for all four rate changes via superposition. The Horner interpretation, however, uses the simplified tp-approximation that assumes a single drawdown at rate q for tp hours. For short IF with adequate ISI recovery, this approximation introduces negligible error. For two-cycle DSTs where IF is a significant fraction of FF, consider the rigorous superposition time approach (Odeh-Jones 1965).

Two-cycle Horner error. When the IF period is long relative to FF, or when ISI fails to reach radial flow, the single-cycle Horner analysis can overestimate permeability by 2× or more (Russell, PETSOC-90-06). This blueprint's default parameters (3 min IF, 4h FF) keep the error small, but the cross-check row quantifies any discrepancy.

Reference: Earlougher, R.C. Jr. (1977). "Advances in Well Test Analysis." SPE Monograph Vol. 5, Chapter 14. Horne, R.N. (1995). "Modern Well Test Analysis." Petroway Inc.

• Interpretation Inputs (rows 3–10): Reservoir and fluid properties for the test well — q (DST flow rate), Bo, μ, φ, ct, rw, h, and Bourdet smoothing L. Flow rate q is common to both IF and FF periods.
• Forward Model (rows 12–16): k, S, Pi, C for synthetic data generation. In real DST interpretation, these are unknown and the goal of analysis.
• DST Schedule (rows 18–22): Four periods defined as [time, rate] pairs. IF starts at t=0 (3 min flow), ISI at t=0.05h (57 min shut-in), FF at t=1.0h (4h flow), FSI at t=5.0h (6h shut-in). The array $B$19:$C$22 is passed as prod_data to PO.PTA.Pw.VW.
• Computed Values (rows 24–27): tp = FF duration (used as producing time for Horner analysis). ISIP = pressure at end of ISI (independent Pi estimate). Pwf = last flowing pressure at end of FF (needed for skin calculation).
• FSI Data Table (rows 29–40): Column A = shut-in time Δt during FSI. Column B = Pws from forward model with full 4-period superposition. Column C = pressure rise ΔPws = Pws − Pwf. Column D–E = Horner time coordinates. Column F = IARF flag. Column G = Bourdet derivative on the log₁₀(HT) axis, with SORT/SORTBY to handle descending x-values.
• IARF Selection: Examine column G for a stabilization plateau. For the default case (k=100 mD, S=3, C=0.0005 bbl/psi), IARF begins at approximately Δt ≥ 2h. Set column F = 1 for these points.
• Interpretation Results (rows 42–50): Same Horner analysis as po.pta.horner.pstar — slope m from AVERAGEIF on Bourdet, P* at HT=1, P1hr, k, S, rinv, rwa, and end-of-WBS estimate.
• DST Cross-Check (row 53): Compares ISIP (from ISI period) against P* (from FSI Horner extrapolation). In a good test, both should approximate Pi. The difference quantifies how well the ISI recovered from the IF disturbance.
• Validation (rows 55–58): Round-trip error check. Expected: k < 2%, P* < 0.01%. Skin error is typically 5–10% due to the Horner tp-approximation ignoring the IF period — this is an inherent limitation of simplified DST analysis, not a blueprint error.