Adjusted Flow Time: Definition, Well Testing, and Horner Plot

Adjusted flow time is the approximated equivalent producing time used in pressure transient analysis (PTA) when a well's flow rate has varied before or during the test period. Rather than using the actual elapsed producing time, which may span months or years of irregular production, the engineer substitutes a single equivalent value that correctly represents the cumulative pressure drawdown history of the reservoir. The calculation is straightforward: divide the well's cumulative production since its last extended shut-in period (N_p, in stock-tank barrels or thousand cubic metres) by the stabilized flow rate measured immediately before the well is shut in for a buildup test (q_last, in barrels per day or m³/d). The result, denoted t_p* or simply t_p, is then used in the Horner time function and related semi-log analysis plots to derive reservoir permeability-thickness product (kh), skin factor (S), and extrapolated static reservoir pressure (p*).

Key Takeaways

• Adjusted flow time equals cumulative production since the last extended shut-in divided by the final flow rate before shut-in: t_p* = N_p / q_last.
• It is also called equivalent producing time or effective flowing time and is essential for constructing a valid Horner plot from a pressure buildup (BU) test.
• Using actual clock time instead of adjusted flow time when a well has had variable rates leads to an incorrect Horner straight line, causing errors in permeability, skin, and p* calculations.
• Modern pressure transient analysis software (Kappa Saphir, IHS Harmony) applies superposition time or Agarwal equivalent time, which generalise adjusted flow time to fully variable-rate histories without requiring the single-rate approximation.
• The concept is foundational to SPE 2172 (Ramey and Cobb, 1971) and is covered in all major PTA references including Bourdet (2002) and Lee, Rollins, and Spivey (2003).

How Adjusted Flow Time Works in Pressure Transient Analysis

When a reservoir engineer designs a pressure buildup test, the goal is to observe how shut-in bottomhole pressure (BHP) recovers over time and then use that recovery curve to quantify formation properties. The theoretical basis of buildup analysis rests on the principle of superposition in time: the shut-in response is mathematically equivalent to the original drawdown response plus a mirror-image injection that begins at shut-in. The Horner method (Horner, 1951) exploits this by plotting BHP against the dimensionless Horner time ratio (t_p + Delta-t) / Delta-t on a semi-log scale, where Delta-t is the elapsed shut-in time. The resulting straight line has a slope m related to permeability by the classical equation: kh = 162.6 q B mu / m, where q is flow rate in STB/d, B is formation volume factor in res bbl/STB, and mu is viscosity in centipoise. Extrapolating the straight line to a Horner time ratio of 1 gives p*, the extrapolated pressure that approximates average reservoir pressure in a new or lightly depleted reservoir.

The critical problem is that this equation was derived for a well that produced at a single constant rate q for time t_p before shut-in. Real production history is never that simple. A well may have been on production for three years, been shut in twice for workovers, and had its choke adjusted dozens of times. If the engineer simply uses the total elapsed time since first production, the Horner time function will be wrong, the straight line will be distorted, and all derived parameters will be in error. The adjusted flow time corrects for this by computing what constant-rate producing time would have produced the same cumulative volume at the final rate. Formally: t_p* = N_p / q_last. This is also called the Equivalent Horner Time or Equivalent Producing Time in many textbooks. For example, if a well has produced 45,000 STB total since its last extended shut-in and the stabilized rate before the current shut-in was 500 STB/d, the adjusted flow time is 45,000 / 500 = 90 days, regardless of whether the elapsed calendar time was 120 days or 180 days.

It is important to understand what the adjusted flow time does and does not fix. It correctly accounts for variable-rate history when the early-time rates are large relative to the final rate, or when the well has had temporary shut-ins that reduced cumulative production below what continuous production at q_last would have produced. It does not fully correct for severe rate variation where early transients are still propagating through the reservoir at the time of the buildup test. In those cases, the engineer must use full multi-rate superposition, implemented analytically as a sum of logarithmic terms for each rate change, or numerically in PTA software. The adjusted flow time is best viewed as a practical, field-ready approximation that works well for most conventional buildup tests where the test duration is much shorter than the producing time.

The Horner Plot and the Role of Producing Time

The Horner plot remains the most widely used diagnostic tool in pressure transient analysis despite being introduced more than 70 years ago. It is a semi-log plot of shut-in BHP (y-axis, linear) against the Horner time ratio (t_p + Delta-t) / Delta-t (x-axis, logarithmic, decreasing to the right). The middle-time region (MTR), where the pressure transient is fully within the reservoir away from both the wellbore and any boundaries, appears as a straight line. The slope of this line yields kh and subsequently permeability k if net pay h is known from a wireline log or core analysis. The skin factor S quantifies near-wellbore damage or stimulation and is computed from the pressure difference between the extrapolated MTR line at one hour of shut-in and the actual BHP at one hour.

If t_p* is too large (i.e., actual calendar time is used when it greatly exceeds the adjusted value), the Horner time ratio at early shut-in times will be only slightly greater than 1, compressing the x-axis and making the MTR appear flatter than it truly is. This leads to underestimation of the slope m, overestimation of kh, and a skin factor that may be incorrectly positive or negative. Conversely, if t_p* is too small, the x-axis is stretched, the slope appears steeper, and kh is underestimated. In wells with a long, complex production history, even the adjusted flow time may not be sufficient: the engineer should compare the adjusted flow time approach against the full superposition solution to confirm that the two methods give consistent straight-line slopes before trusting the result.

A related concept is the Agarwal equivalent time (Agarwal, 1980), which reformulates the superposition principle as a single equivalent drawdown time. Rather than using the Horner time ratio, Agarwal's equivalent time Delta-t_e is defined as (t_p * Delta-t) / (t_p + Delta-t). Plotting BHP against Delta-t_e on a log-log or semi-log scale transforms the buildup into a pseudo-drawdown curve, which can then be analysed using the same type curves and derivative methods as a drawdown test. This is particularly powerful for identifying flow regimes in fractured or heterogeneous reservoirs. The type curves of Bourdet et al. (1983) for pressure derivative analysis are most conveniently applied using Agarwal time.

Multi-Rate Superposition and Variable-Rate Wells

Many wells, particularly those on artificial lift, gas-lifted wells, and horizontal wells with varying drawdown, cannot be adequately represented by a single adjusted flow time. The rigorous treatment requires multi-rate superposition, also known as the principle of superposition in time. For a well that has produced at rates q₁, q₂, ..., q_n during time intervals t₁, t₂, ..., t_n before shut-in, the superposition time function at shut-in time Delta-t is:

t_sup(Delta-t) = sum from j=1 to n of [(q_j - q_j-1) / q_n] * log(t_n+1 - t_j-1 + Delta-t)

where q₀ = 0 and t₀ = 0 by convention. Plotting BHP against the superposition time function yields a straight line whose slope is directly proportional to kh, exactly as in the Horner method but without any approximation for rate variation. The adjusted flow time method produces results identical to superposition only in the limiting case where one rate dominates the history or the flow time is much shorter than any boundary effects.

In modern reservoir characterisation workflows, the full rate history is loaded into PTA software and the superposition time is computed automatically. The engineer's role is to QC the rate history data, identify any extended shut-ins that reset the pressure transient, and select the correct reference point for N_p. An extended shut-in is one long enough for the reservoir pressure to re-equalise to near-static conditions, effectively resetting the cumulative production counter. In practice, if a well was shut in for more than a few weeks in a low-permeability reservoir, the engineer may choose to count cumulative production only from the end of that shut-in rather than from first production.