Constant-Rate Drawdown Analysis

Constant-rate drawdown test interpretation using MDH (Miller-Dyes-Hutchinson) semilog analysis with Bourdet derivative for IARF identification. The most fundamental well test — a single rate from initial conditions. Includes forward model for synthetic data generation and round-trip validation.

No SORT required for Bourdet. Unlike the Horner blueprint where log₁₀(HT) is naturally descending, the drawdown log₁₀(t) axis is ascending — matching the Bourdet function requirement directly.

IARF flag column (F) uses 1/0 numeric values, not TRUE/FALSE. Avoids AVERAGEIF #DIV/0! from text-interpreted booleans.

For the default synthetic case (k=50 mD, S=5, C=0.001 bbl/psi), the unit-slope WBS ends at ~0.17h and IARF begins at approximately t ≥ 4h.

Why L = 0.2. PO.DCA.Diag.Bourdet.SemiLog takes the raw abscissa and returns dP/d(ln x), so the *LN(10) converts it to the per-log-cycle slope m needs. That is why time itself is passed now rather than its logarithm: L is a window measured in natural-log cycles of whatever x it receives, so on a pre-logged abscissa it would be taken on ln(log₁₀ x) — a second logarithm, and one undefined wherever log₁₀ x ≤ 0. A neighbour must sit at least L away in ln(x) to be used, so the window engages only where the data is sampled more finely than that. Measured on synthetic gauge data at 20 points per decade with 2 psi of noise, L = 0.2 cuts the RMS error in k from 9.9% to 8.5%; on clean data it costs under 0.01 points of accuracy. That trade is what Bourdet smoothing is for.

MDH vs Horner. This blueprint analyzes drawdown (flowing) data. For buildup (shut-in) data, see po.pta.horner.pstar. Both methods yield the same k and S for the same reservoir — they just use different test configurations.

For real data use. Replace column C (Pwf) with measured gauge pressures. Clear or ignore the forward model block.

LAMBDA functions defined here (_PO.PTA.*) represent common PTA calculations not yet available as native PO functions.

Reference: Miller, C.C., Dyes, A.B., and Hutchinson, C.A. (1950). "The Estimation of Permeability and Reservoir Pressure from Bottom-Hole Pressure Build-Up Characteristics." JPT.

• Interpretation Inputs (rows 3–11): Reservoir and fluid properties — q, Bo, μ, φ, ct, rw, h, Pi, and the Bourdet smoothing window L, which is 0.2 here — see Why L = 0.2. Initial pressure Pi is assumed known from pre-test surveys.
• Forward Model (rows 14–16): Optional validation block. Enter known k, S, C to generate synthetic Pwf via PO.PTA.Pw.VW at constant rate q.
• Data Table (rows 18–29): Column A = time t. Column B = Pwf from forward model. Column C = measured Pwf (defaults to =B for self-validation; replace with gauge data). Column D = pressure drop ΔP = Pi − Pwf. Column E = log₁₀(t). Column F = IARF flag (1/0). Column G = Bourdet derivative dΔP/dlog₁₀(t).
• Flow Regime Identification: On a log-log plot of ΔP and Bourdet derivative vs time: (1) unit-slope line during WBS — ΔP increases linearly with t; (2) transition hump; (3) flat derivative = IARF. Set F = 1 where derivative is approximately constant.
• Slope, m (row 31): AVERAGEIF of Bourdet values in IARF. This is the semilog slope dΔP/dlog₁₀(t) in psi/cycle — positive for drawdown.
• ΔP at t=1hr (row 32): Intercept of IARF line at log₁₀(t) = 0 (t = 1hr). Computed from mean ΔP and mean log₁₀(t) of IARF-flagged points.
• Permeability (row 33): Computed via _PO.PTA.PermFromSlope — k = 162.6 × q × Bo × μ / (|m| × h).
• Skin (row 34): Computed via _PO.PTA.SkinFromDp1hr — S = 1.151 × [ΔP₁ₕᵣ/|m| − log₁₀(k/(φμctrw²)) + 3.23].
• End of WBS (row 37): Computed via _PO.PTA.EndWBS. Useful for planning test duration — data before tWBS is dominated by wellbore storage and cannot be interpreted.
• Validation (rows 40–41): Compares forward model k, S against interpretation results. Expected: k < 1%, S < 2% on clean synthetic data.