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Diagnostic plots
The diagnostic plot is a log-log plot of the pressure change and pressure derivative (vertical axis) from a pressure transient test vs. elapsed time (horizontal axis).
Fig. 1 shows an example of a diagnostic plot. The diagnostic plot can be divided into three time regions: early, middle, and late. At the earliest times on a plot (the early-time region), wellbore and near-wellbore effects dominate. These effects include wellbore storage, formation damage, partial penetration, phase redistribution, and stimulation (hydraulic fractures or acidization). At intermediate times (the middle-time region), a reservoir will ordinarily be infinite acting. For a homogeneous reservoir, the pressure derivative will be horizontal during this time region. Data in this region lead to the most accurate estimates of formation permeability. At the latest times in a test (the late-time region), boundary effects dominate curve shapes. The types of boundaries that may affect the pressure response include sealing faults, closed reservoirs, and gas/water, gas/oil, and oil/water contacts. Several common flow regimes and the diagnostic plots associated with these flow regimes are discussed here.
Fig. 1 – Diagnostic plot showing time regions.
Volumetric behavior
Volumetric behavior is defined as that pressure response time dominated by the wellbore, reservoir, or part of the reservoir acting like a uniform-pressure "tank" with fluid entering or leaving the tank. The most common example of volumetric behavior is wellbore storage, which dominates during the early-time region. The "tank" is the wellbore, in which the pressure is uniform. Fluid either leaves this tank (earliest times in a flow test, before the reservoir begins to respond) or enters the tank (earliest times in a buildup test). Another example is pseudosteady-state (boundary-dominated) flow in a closed reservoir during constant-rate production. In this case, the reservoir is the tank; pressure is changing at the same rate throughout (although it is not the same at all points), and fluid is leaving the reservoir through the producing well. As a final example, in a test the reservoir may behave like a tank with recharge (fluid influx) entering from a secondary source of pressure support, such as a large supply of hydrocarbons in a lower-permeability medium in pressure communication with the reservoir being tested.
The equation modeling wellbore storage (derived from a mass balance on the wellbore) is
....................(1)
The equation modeling pseudosteady-state flow in a cylindrical drainage area is
....................(2)
The general form is
....................(3)
The derivative of the general form is
....................(4)
The implication is that the derivative plot will have unit slope (up one log cycle as it moves over one log cycle) on log-log coordinates, and the pressure change plot will approach unity at long times when b v is not equal to zero (Fig. 2). In wellbore storage, bv is zero, and the derivative and pressure change plots will lie on top of one another. During pseudosteady-state flow or recharge, the pressure change and pressure derivative plots will not coincide.
Fig. 2 – Volumetric flow produces derivative with unit slope.
Radial flow
Infinite-acting radial flow is common in reservoirs, and data in the radial flow regime can be used to estimate formation permeability and skin factor. Common situations in which radial flow occurs include flow into vertical wells after wellbore storage distortion has ceased and before boundary effects, hydraulically fractured wells after the transient has moved well beyond the tips of the fracture, horizontal wells before the transient has reached the top and bottom of the productive interval, and horizontal wells after the transient has moved beyond the ends of the wellbore.
The equation used to model radial flow for a well producing at constant rate is the familiar logarithmic approximation to the line-source solution,
....................(5)
Equations modeling radial flow have the general form
....................(6)
with derivative
....................(7)
On the diagnostic plot (Fig. 3), radial flow is indicated by a horizontal derivative.
Fig. 3 – Radial flow appears as a horizontal derivative on the diagnostic plot.
Linear flow
Linear flow is also common and occurs in channel reservoirs, hydraulically fractured wells, and horizontal wells. Data from linear flow regimes can be used to estimate channel width or fracture half-length if an estimate of permeability is available. In horizontal wells, an estimate of permeability perpendicular to the well can be made if the productive well length open to flow is known.
An equation that models linear flow in a channel reservoir of width w is
....................(8)
For a hydraulically fractured well with fracture half-length Lf,
....................(9)
The general form is
....................(10)
The derivative is
....................(11)
Linear flow on the diagnostic plot is indicated when a derivative follows a half-slope line—that is, a line that moves up vertically by one log cycle for each two cycles of horizontal movement (Fig. 4). The pressure change may or may not also follow a half-slope line. In a hydraulically fractured well, the pressure change will follow a half-slope line unless the fracture is damaged. In a channel reservoir, a hydraulically fractured well with damage, or a horizontal well, the pressure change will approach the half-slope line from above.
Fig. 4 – Linear flow derivative follows a half-slope line on a diagnostic plot.
Bilinear flow
Bilinear flow occurs primarily in wells with low-conductivity hydraulic fractures. Flow is linear within the fracture to the well, and also linear (normal to fracture flow) from the formation into the fracture. Estimates of fracture conductivity, wfkf, can be made with data from this flow regime when estimates of formation permeability are available.
For a hydraulically fractured well, an equation that models bilinear flow is
....................(12)
The general form is
....................(13)
The derivative is
....................(14)
Bilinear flow derivatives plot as a quarter-slope line on the diagnostic plot (Fig. 5). The quarter-slope line moves up one log cycle as it moves over four log cycles. The pressure change does not necessarily follow a quarter-slope line. In a damaged, hydraulically fractured well, the pressure change curve will approach the quarter-slope line from above; in an undamaged hydraulically fractured well (Δps = 0), the pressure change will typically follow the quarter-slope line when the effects of wellbore storage have ended.
Fig. 5 – Bilinear flow derivative follows a quarter-slope line on the diagnostic plot.
Spherical flow
The flow pattern is spherical when the pressure transient can propagate freely in three dimensions and converge into a "point." This can occur for wells that penetrate only a short distance into the formation (actually hemispherical flow), wells that have only a limited number of perforations open to flow, horizontal wells with inflow over only short intervals, and during wireline formation tests. Data in the spherical-flow regime can be used to estimate the mean permeability,
....................(15)
An equation that models spherical flow is
....................(16)
where 
....................(17)
and rsp is the radius of the sphere into which flow converges. The general form is
....................(18)
and the derivative is
....................(19)
Spherical flow on the diagnostic plot produces a derivative line with a slope of −1/2. The pressure change during spherical flow approaches a horizontal line from below, and never exhibits a straight line with the same slope as the derivative (Fig. 6). Spherical flow can occur during either buildup or drawdown tests.
Fig. 6 – Spherical flow derivative has slope of –1/2.
Flow regimes on the diagnostic plot
A major application of the diagnostic plot is the potential that it provides in identifying the flow regimes that appear in a logical sequence during a buildup or flow test. For example, consider Fig. 7. At early times, the unit slope line on both derivative and pressure change, indicating wellbore storage. Later, a derivative with a slope of −1/2, indicating possible spherical flow, followed by a horizontal derivative, indicating infinite-acting radial flow. Boundary effects, including a unit-slope line, follow, indicating possible recharge of the reservoir pressure.
Fig. 7 – Well test diagnostic plot indicating several flow regimes.
Nomenclature
| b | = |  (gas flow equation)
|
| bB | = | intercept of Cartesian plot of bilinear flow data, psi |
| bL | = | intercept of Cartesian plot of linear flow data, psi |
| bV | = | intercept of Cartesian plot of data during volumetric behavior, psi |
| B | = | formation volume factor, res vol/surface vol |
| ct | = | Soco + Swcw + Sgcg + cf = total compressibility, psi–1 |
| C | = | performance coefficient in gas-well deliverability equation, or wellbore storage coefficient, bbl/psi |
| CD | = | 0.8936 C/ϕcthrw2 , dimensionless wellbore storage coefficient |
| h | = | net formation thickness, ft |
| k | = | matrix permeability, md |
| kf | = | permeability of the proppant in the fracture, md |
| kh | = | horizontal permeability, md |
| kr | = | permeability in horizontal radial direction, md |
| ks | = | permeability of altered zone, md |
| kz | = | permeability in z-direction, md |
| Lf | = | fracture half length, ft |
| m | = | 162.2 qBμ/kh = slope of middle-time line, psi/cycle |
| ms | = |  , slope of spherical flow plot, psi-hr1/2
|
| mB | = | slope of bilinear flow graph, psi/hr1/4 |
| mL | = | slope of linear flow graph, psi/hr1/2 |
| mV | = | slope of volumetric flow graph, psi/hr |
| p | = | pressure, psi |
| pi | = | original reservoir pressure, psi |
| pp | = | pseudopressure, psia2/cp |
| ps | = | stabilized shut-in BHP measured just before start of a deliverability test, psia |
| psc | = | standard-condition pressure, psia |
| pwf | = | flowing BHP, psi |
| pws | = | shut-in BHP, psi |
| p0 | = | arbitrary reference or base pressure, psi |
| q | = | flow rate at surface, STB/D |
| r | = | distance from the center of wellbore, ft |
| re | = | external drainage radius, ft |
| rs | = | outer radius of the altered zone, ft |
| Rs | = | dissolved GOR, scf/STB |
| s | = | skin factor, dimensionless |
| s′ | = | s + Dq = apparent skin factor, dimensionless |
| t | = | elapsed time, hours |
| z | = | gas-law deviation factor, dimensionless |
| Δp | = | pressure change since start of transient test, psi |
| Δt | = | time elapsed since start of test, hours |
| β | = | turbulence factor |
| λ | = | interporosity flow coefficient |
| λt | = |  , total mobility, md/cp
|
| μ | = | viscosity, cp |
| ϕ | = | porosity, dimensionless |
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See also
Fluid flow through permeable media
Boundary effects in diagnostic plots
Estimating average reservoir pressure from diagnostic plots
Flow equations for gas and multiphase flow
PEH:Fluid_Flow_Through_Permeable_Media