Bilinear Flow: Fracture Conductivity Analysis and PTA Diagnostics for Hydraulically Fractured Wells

Key Takeaways

• Bilinear flow mathematical derivation and the one-quarter slope: The bilinear flow solution to the diffusivity equation for a finite-conductivity vertical fracture in a homogeneous reservoir was developed analytically by Cinco-Ley, Samaniego, and Dominguez (1978, SPE 6014) — one of the landmark papers in petroleum engineering — and the derivation produces a closed-form expression for pressure change during bilinear flow: delta P = (4.06q / h) × (μ / kf w) ^(1/2) × (μ / k phi ct)^(1/4) × t^(1/4) where q is flow rate, h is net pay, μ is viscosity, kf w is fracture conductivity, k is matrix permeability, φ is porosity, ct is total compressibility, and t is elapsed time. The t^(1/4) dependence is the mathematical source of the one-quarter slope on the log-log diagnostic plot: log(delta P) = log(C) + (1/4) × log(t), where C is the collection of terms preceding t^(1/4). The slope of 0.25 on the log-log plot is diagnostic: any deviation from exactly 0.25 during the bilinear flow period indicates either wellbore storage domination (initial slope approaching 1.0), a transition to matrix linear flow (slope changing from 0.25 toward 0.50), or boundary effects. The dimensionless fracture conductivity (FcD = kf w / (k × Lf), where Lf is the fracture half-length) determines how long bilinear flow persists: for FcD above 100 (high-conductivity fractures), bilinear flow is brief and transitions quickly to fracture linear flow (1/2 slope); for FcD below 30 (finite-conductivity fractures common in Montney completions), bilinear flow is the dominant long-duration transient regime observed in well test data. Most WCSB Montney and Duvernay completions have FcD in the range 1-50, ensuring that bilinear flow is observed and analyzable in pressure transient tests of practical duration (hours to days of shut-in).
• Bourdet pressure derivative and bilinear flow identification: The Bourdet pressure derivative (d(delta P)/d(ln t) = t × d(delta P)/dt) is the standard tool for identifying bilinear flow in field PTA data because it amplifies the subtle diagnostic differences between flow regimes. On a log-log plot, the derivative of a pressure buildup or drawdown shows a slope of 0.25 during bilinear flow (coincident with the pressure change slope), a slope of 0.50 during fracture linear flow, a flat derivative (slope = 0) during pseudo-radial flow, and an ascending derivative during boundary effects. The simultaneous plot of delta P and its derivative (the Bourdet plot) allows the analyst to distinguish bilinear flow from wellbore storage-dominated early time (where both delta P and derivative show slope = 1.0) and from matrix linear flow (where delta P shows slope = 0.50 and derivative also shows 0.50). For a WCSB Montney pressure buildup test, the typical diagnostic sequence is: wellbore storage (slope = 1.0 for the first 0.5-4 hours of shut-in); transition; bilinear flow (slope = 0.25 from approximately 4-20 hours shut-in); possible transition to fracture linear flow (slope = 0.50 if the fracture half-length is short or the buildup is long enough); and pseudo-radial or interference flow at very long shut-in times (72-240+ hours). AER Directive 040 well test reporting requires that the PTA diagnostic plot be included in the Well Completion Report, and bilinear flow identification on the Bourdet plot is the basis for the fracture conductivity value reported to the AER for WCSB unconventional well completions.
• Fracture conductivity calculation from the bilinear flow period: Fracture conductivity (kf w, mD-m) is extracted from the bilinear flow period on a specialized bilinear flow plot: a Cartesian plot of bottom-hole pressure versus t^(1/4) (the fourth root of elapsed time) should produce a straight line during bilinear flow with slope mBLF. The fracture conductivity is then: kf w = (4.064 × q × B × μ)^2 / (mBLF × h)^2 × (k × φ × μ × ct)^(1/2) (in Darcy field units with appropriate conversion factors). This calculation requires an independent estimate of matrix permeability (k) — obtained from the pseudo-radial flow period if the test is long enough, from core analysis, or from analog wells — because fracture conductivity and matrix permeability are coupled in the bilinear flow solution. For a WCSB Montney well producing 2.5 MMcf/d from a 6-stage frac in 8 m net pay, with matrix permeability k = 0.001 mD (from core) and a measured bilinear flow slope mBLF = 1.82 MPa/hr^(1/4): kf w = (4.064 × 2.5 × 10^6 scf/d × 1.38 RCF/SCF × 0.025 cP)^2 / (1.82 × 8 m)^2 × (0.001 × 0.06 × 0.025 × 8×10^-5)^(1/2) = calculated fracture conductivity of approximately 28 mD-m. This value, combined with the fracture half-length (estimated from the fracture linear flow period if observed), gives the complete characterization of the hydraulic fracture's performance from field pressure transient data — information that is otherwise only accessible through expensive reservoir simulation or indirect inferences from production data matching.
• Bilinear flow in multistage horizontal Montney wells: In WCSB multistage horizontal frac wells (20-45 stages, 4-6 perforation clusters per stage, 150-500 m fracture half-lengths in the Montney), the bilinear flow regime observed in well test data is a composite response from all fracture stages simultaneously, not a single isolated fracture. The composite bilinear flow solution assumes all fractures are identical (same conductivity, same half-length, same matrix permeability around each fracture) and that there is no interference between fractures — approximations that are increasingly inaccurate as fracture density increases and adjacent fractures begin to share the same drainage volume. Despite these simplifications, the composite bilinear flow analysis gives an effective average fracture conductivity for the well that is diagnostically useful for comparing stimulation quality across wells in the same program. In practice, WCSB Montney PTA analysts observe bilinear flow slopes on Bourdet plots from horizontal multistage frac wells for shut-in periods of 4-50 hours, followed by a compound linear flow regime (slope = 0.50 on the Bourdet derivative) at longer shut-in times. The transition from bilinear to linear flow indicates that matrix linear flow from the reservoir volume between fractures begins to dominate the transient response — a progression that represents the fracture-matrix drainage system advancing from the fracture face outward into the formation. Identifying this transition time gives a lower bound on the effective fracture half-length (via the formula: Lf ≥ 0.159 × SQRT(k × t_transition / (φ × μ × ct)).
• Distinguishing bilinear flow from other flow regimes in Montney PTA: The one-quarter slope is relatively rare among diagnostic flow regimes (most common slopes in PTA are 0, 0.5, and 1.0), but misidentification can occur when the one-quarter slope falls near the transition between wellbore storage and linear flow, creating an apparent 0.25 slope that is actually a transition curve rather than a true bilinear flow signature. Reliable identification requires: (1) The 1/4 slope must persist for at least 0.5 log cycles on the Bourdet plot (10^0.5 = 3.16-fold increase in elapsed time with the derivative tracking 0.25 slope throughout); (2) The derivative must coincide with the pressure change on the log-log plot for the bilinear period (when the pressure change also shows 0.25 slope); (3) The bilinear plot (BHP vs t^(1/4)) must show a linear trend with R2 > 0.98 during the identified bilinear flow period; (4) The fracture conductivity calculated from the slope must be physically plausible given the formation and stimulation design (for Montney 30/50 sand proppant at 1,500 kg/m2 areal density, expected kf values are 80-200 D, giving kf w of 40-120 mD-m for a 0.5-0.6 mm average fracture width). Anomalously high or low calculated kf w relative to these design expectations triggers a diagnostic review: low kf w may indicate proppant crushing, gel damage, or embedment; high kf w may indicate unpropped natural fractures contributing to the signal.