Multi-Rate Drawdown — Step-Rate Test

Step-rate drawdown test with three rate periods. Demonstrates PO's built-in superposition via the prod_data array parameter — a single PO.PTA.Pw.VW call handles the full rate history. Uses rate-normalized pressure ΔP/q for interpretation, which removes rate dependence and allows permeability and skin extraction from any single rate period.

prod_data array parameter. PO.PTA.Pw.VW accepts either a single flow rate or a 2-column array of [time, rate] pairs. When an array is passed, the function internally applies superposition to compute the pressure response for the complete rate history. This is the recommended approach for DSTs, extended flow tests, and any multi-rate scenario.

Rate normalization. Dividing ΔP by the current rate qn removes the rate dependence from the semilog analysis. The rate-normalized semilog slope m_rn = 162.6 × Bo × μ / (k × h) is rate-independent. In a real multi-rate test, plotting ΔP/q vs log₁₀(t) for each period would yield the same IARF slope, confirming radial flow across all rate levels.

Why Bourdet is period-1 only. The Bourdet derivative uses a log-time smoothing window. When this window spans a rate change, it averages pre- and post-change slopes, producing incorrect values. Computing within period 1 (where rate is constant) gives clean derivatives for interpretation. In practice, engineers either compute derivatives per period or use superposition time to collapse all periods.

For rigorous multi-rate analysis, use superposition time (Odeh-Jones method) instead of log₁₀(t). Superposition time collapses all rate periods onto a single semilog line, accounting for rate history effects. This blueprint uses the simpler rate-normalization approach with first-period interpretation.

Reference: Odeh, A.S. and Jones, L.G. (1965). "Pressure Drawdown Analysis, Variable-Rate Case." JPT.

• Interpretation Inputs (rows 3–10): Fluid and reservoir properties — Bo, μ, φ, ct, rw, h, Pi, and Bourdet smoothing L. No single flow rate q since it changes.
• Forward Model (rows 13–15): k, S, C for synthetic data generation.
• Rate Schedule (rows 17–20): Three periods defined as [time, rate] pairs. Period 1: 250 STB/D (0–24h). Period 2: 500 STB/D (24–48h). Period 3: 125 STB/D (48–72h). This array is passed directly as prod_data to PO.PTA.Pw.VW.
• Data Table (rows 22–35): Column B = Pwf from forward model using the full rate schedule. Column C = ΔP = Pi − Pwf. Column D = current rate qn at each time. Column E = rate-normalized ΔP/qn. Columns G–H = raw and rate-normalized Bourdet derivatives, computed for period 1 only (rows 23–29). Periods 2–3 omit Bourdet because computing derivatives across rate changes produces artifacts.
• Interpretation (rows 37–42): Uses first-period IARF. The rate-normalized slope m_rn = 162.6 × Bo × μ / (k × h) — the rate q drops out because ΔP is already divided by q.