Bilinear Flow: Definition, Fractured Wells, and Well Test Analysis

Bilinear flow is a transient flow regime that occurs in hydraulically fractured wells when fluid moves simultaneously in two perpendicular linear directions: from the formation matrix into the fracture plane, and from within the fracture along its length toward the wellbore. The two flows are coupled and occur at the same time, giving the regime its name. Bilinear flow is identified on a log-log diagnostic plot by a characteristic one-quarter slope in the Bourdet pressure derivative curve, and analysis of the bilinear flow period allows engineers to calculate fracture conductivity, one of the most critical parameters controlling the productivity of a hydraulically fractured well.

What Is Bilinear Flow?

When a hydraulically fractured well is placed on production or subjected to a pressure transient test such as a drillstem test, the pressure disturbance propagates outward from the wellbore and through the fracture system into the surrounding reservoir. The manner in which this propagation occurs depends on the hydraulic properties of the fracture relative to the formation. In an infinite-conductivity fracture, one where fracture permeability is so high relative to formation permeability that pressure is essentially uniform along the entire fracture length, fluid flows linearly from the matrix into the fracture and then essentially instantaneously to the wellbore. This is pure linear flow, characterized by a one-half slope on the log-log diagnostic plot.

A finite-conductivity fracture, by contrast, is one in which the fracture permeability is limited, so there is a measurable pressure gradient along the fracture from its tip toward the wellbore. In this case, fluid entering the fracture from the matrix near the fracture tip must still travel a significant distance along the fracture to reach the wellbore. Two linear flows are therefore occurring simultaneously and at right angles to each other: linear flow from the matrix into the fracture plane (perpendicular to the fracture face), and linear flow from within the fracture along its length toward the wellbore (parallel to the fracture). These two coupled linear flows together define the bilinear flow regime.

The concept was formally developed and published by Cinco-Ley and Samaniego in 1981, in a landmark paper that remains a foundational reference in well test analysis. Their work identified the one-quarter slope diagnostic signature and derived the analytical equations relating wellbore pressure response to fracture conductivity during the bilinear flow period. The bilinear flow model applies to both vertical wells with a single transverse hydraulic fracture and to horizontal wells with multiple transverse fractures, the latter being the dominant completion design for unconventional tight gas, shale gas, and tight oil wells.

How Bilinear Flow Works: The Physics and the Math

During bilinear flow, the pressure drawdown at the wellbore increases proportionally to the fourth root of elapsed time. This relationship arises because pressure propagation is occurring simultaneously in two perpendicular directions, each individually exhibiting linear diffusivity, but coupled such that the combined system evolves as t to the one-quarter power. The governing equation for wellbore pressure drawdown during bilinear flow in a drawdown test is:

Delta P = m_BL x t^1/4

Where Delta P is the pressure drawdown in psi (or kPa), t is elapsed time in hours, and m_BL is the bilinear flow slope on a Cartesian plot of Delta P versus t to the one-quarter power. The bilinear flow slope is related to fracture conductivity by:

m_BL = (444.75 x q x B x mu) / (h x (Fc x k x h)^1/2)

In field units: q is the flow rate in reservoir barrels per day (res bbl/d), B is the formation volume factor in res bbl/STB, mu is fluid viscosity in centipoise (cp), h is the net pay thickness in feet (ft), Fc is fracture conductivity in millidarcy-feet (md-ft), and k is formation permeability in millidarcies (md). In SI units, the constant changes and the units become cubic metres per day, metres, Pascal-seconds, and millidarcy-metres. The critical point is that m_BL, derived from the slope of the bilinear Cartesian plot, directly yields Fc when all other parameters are known. Fracture conductivity is then decomposed into fracture permeability (kf) and fracture width (wf), though only their product is directly measurable from the pressure transient.

Fracture conductivity Fc (in field units, md-ft, or SI units, md-m) determines how efficiently the fracture connects the reservoir to the wellbore. A fracture with high conductivity delivers fluid to the wellbore with low resistance even at high flow rates. A fracture with low conductivity becomes a bottleneck: the fracture may penetrate deep into the reservoir but cannot transport the produced fluids efficiently, resulting in lower-than-expected productivity. The dimensionless fracture conductivity, Fcd, defined as Fc divided by the product of formation permeability and fracture half-length (k x xf), is the key scaling parameter. When Fcd exceeds approximately 300, the fracture behaves as infinite conductivity and bilinear flow is bypassed. When Fcd is below approximately 10, bilinear flow is the dominant early-time regime, and when Fcd falls below 1 the fracture is severely capacity-limited regardless of its length.

Flow Regime Sequence in a Hydraulically Fractured Well

A hydraulically fractured well passes through several distinct flow regimes as a pressure transient propagates outward from the wellbore over time. Understanding this sequence is fundamental to interpreting pressure transient tests, type curve matching, and rate transient analysis on tight-gas, shale gas, and tight oil wells. The standard progression, from earliest to latest, is as follows.

Wellbore storage is the earliest identifiable flow period. Immediately after a well is shut in or opened, the compressible fluid in the wellbore itself is expanding or compressing, masking the reservoir signal. On the log-log diagnostic plot, wellbore storage appears as a unit slope (slope = 1) on both the pressure change and pressure derivative. The duration of wellbore storage depends on wellbore volume and fluid compressibility, and in wells tested with downhole shut-in tools, it can be minimized to a few minutes.

Bilinear flow follows wellbore storage for finite-conductivity fractures. The log-log diagnostic plot shows a one-quarter slope. This is the period from which fracture conductivity is calculated, as described above. In tight gas and shale wells, bilinear flow may persist for hundreds of hours or even years, because the matrix permeability, which governs the rate at which the linear component of flow from the matrix into the fracture evolves, is so low that the fracture linear flow component is sustained indefinitely on practical timescales. Analyzing bilinear flow in these wells requires long shut-in times during pressure buildup tests, and rate transient analysis methods applied to production data are often more practical than conventional pressure transient tests.

Linear flow develops after bilinear flow when the pressure disturbance has propagated sufficiently far into the matrix that the fracture length is the controlling dimension of the flow geometry. In pure linear flow, fluid moves essentially perpendicular to the fracture plane, with the fracture acting as a line sink. The log-log diagnostic shows a one-half slope. Analysis of the linear flow period yields the product of formation permeability and fracture half-length squared (k x xf squared), which together with Fc from bilinear flow allows independent determination of both k and xf in some cases.

Pseudoradial flow is the latest flow regime, developing when the pressure disturbance has propagated far enough from the fracture that the fracture appears as a point source and radial symmetry is approximately established. The Bourdet pressure derivative on the log-log diagnostic plot flattens to a zero slope (horizontal line), identical to the middle-time radial flow signature seen in unfractured wells. Pseudoradial flow yields formation permeability from the conventional radial flow equation. In tight gas wells with long hydraulic fractures and very low permeability, reaching pseudoradial flow may require weeks to months of shut-in, making it impractical in many field settings.