Adjusted Flow Time

Key Takeaways

• The formula t_p* = N_p / q_last computes adjusted flow time from cumulative production since the last extended shut-in and the final stabilised rate before shut-in. The key subtlety is the reference point for N_p: it must be reset at the most recent extended shut-in (one long enough for reservoir pressure to approach static conditions, typically a few weeks or more depending on permeability), not at first production. Using total cumulative production over a multi-year history without resetting at an intervening extended shut-in inflates t_p* to months or years, compresses the Horner time ratio axis, reduces the apparent slope of the MTR straight line, and overstates permeability by the ratio of the inflated t_p* to the correct value. A well that was shut in for 3 weeks for a workover 6 weeks before the current buildup test has N_p counted only from the end of that workover shut-in, not from first production 2 years ago.
• The Horner time ratio (t_p* + Δt)/Δt is plotted on the x-axis (logarithmic, decreasing to the right) against shut-in bottomhole pressure p_ws on the y-axis (linear, increasing upward). The middle-time region (MTR) appears as a straight line with slope m = 162.6 × q × B × μ / (k × h) in field units, where q is the rate in STB/d, B is the formation volume factor in res bbl/STB, μ is viscosity in centipoise, k is permeability in millidarcies, and h is net pay in feet. This slope gives kh, and if h is known from logs or core, permeability k directly. Extrapolating the MTR straight line to the Horner time ratio of 1.0 (equivalent to infinite shut-in time) gives p*, the extrapolated pressure that approximates average reservoir pressure in a new or lightly depleted reservoir. The skin factor S is calculated from the pressure difference between the extrapolated straight line at Δt = 1 hour and the actual flowing wellbore pressure at shut-in: S = 1.1513[(p_1hr - p_wf)/m - log(k/(φ μ c_t r_w²)) + 3.2275] in field units. Every one of these results depends on a correct slope m, which in turn depends on the correct adjusted flow time.
• The Agarwal equivalent time Δt_e = (t_p* × Δt)/(t_p* + Δt) is a more rigorous time transformation that converts buildup data into a pseudo-drawdown dataset suitable for type-curve analysis. Rather than using the Horner semi-log plot, the engineer plots p_ws versus Δt_e on a log-log scale and applies Bourdet et al. pressure derivative type curves. The derivative plot d(p_ws)/d(ln Δt_e) against Δt_e reveals flow regimes with characteristic shapes: unit slope for wellbore storage, half slope for linear flow in a hydraulic fracture, flat plateau for infinite-acting radial flow (the MTR), and slope doubling for a sealing fault boundary. Agarwal time is more accurate than the Horner approach when t_p* is comparable to the shut-in duration Δt, because the Horner straight-line derivation is strictly valid only when t_p* >> Δt. In tight oil and shale wells where t_p* (computed from cumulative production to date divided by last rate) may be days or weeks while the desired buildup duration is also days or weeks, Agarwal time is preferred.
• The MTR identification on the Horner or Agarwal-time plot requires recognising three successive regions. At early shut-in times (large Horner time ratio), wellbore storage afterflow masks the reservoir signal: the unit slope on the log-log derivative plot indicates that fluid stored in the wellbore (rather than from the formation) is driving the pressure recovery. The MTR begins when wellbore storage effects die out, typically after 1 to 1.5 log cycles after the peak of the pressure derivative. At late shut-in times (small Horner time ratio), boundary effects emerge: doubling of the semi-log slope indicates a single sealing fault or closed reservoir boundary; a flat Horner plot (zero slope) indicates a constant-pressure boundary such as an aquifer. The MTR window must be selected between these two effects to obtain the correct slope m. Using data contaminated by wellbore storage (early time) gives too shallow a slope and overestimates k; using data affected by boundary effects (late time) gives too steep a slope and underestimates k.
• Full multi-rate superposition is required instead of adjusted flow time when the production rate history shows large variations (rate ratio greater than approximately 3:1 between early and final rates), when the well has had multiple shut-ins of varying lengths, or when the reservoir permeability is so low that early-time transients are still propagating at the time of the buildup. The superposition time function at each shut-in increment Δt is: t_sup = sum from j=1 to n of [(q_j - q_{j-1})/q_n] × log(t_{n+1} - t_{j-1} + Δt), where the q_j are the successive rates and t_j are the cumulative times at each rate change. Plotting p_ws versus t_sup gives a straight line of the same slope m as the correct Horner MTR, without any approximation for rate variation. In the Western Canada Sedimentary Basin, wells on plunger lift, rod pump, or intermittent gas lift have highly variable daily rates that are poorly represented by adjusted flow time; PTA software is essentially mandatory to compute t_sup correctly from monthly production allocation records and to avoid the systematic errors that arise from using a simple t_p* approximation on these variable-rate producers.