The Hall plot analyzes steady-state flow at an injection well. In general, the slope of a Hall plot is interpreted as an indicator of the average well injectivity. At normal conditions, the plot is a straight line. Kinks on the plot indicate changes to injection conditions.
Hall (in 1963) presented this technique to interpret routinely collected injection well data to draw conclusions regarding near-wellbore skin effects and average injectivity performance. The data required for a Hall plot analysis includes the following:
- monthly sandface injection pressures (monthly average)
- average reservoir pressure
- monthly water injection volumes
- injection days for the month
The Hall method assumes steady-state injection such that the injection rate can be expressed as:
(Equation
1)where:
k = permeability,
h = reservoir thickness
pwi = flowing wellhead pressure
pavg = average reservoir pressure
μ = fluid viscosity
re = reservoir effective radius
rw = wellbore radius
S = skin
Equation 1 is based on the following assumptions:
- the fluid is homogenous and incompressible
- the reservoir is vertically confined and uniform, both with respect to permeability and thickness
- the reservoir is horizontal and gravity does not affect flow (consequently, the flow is radial)
- the flow is steady-state
- the mobility ratio is equal to 1
- during the time of observation, the pressure at the distance equal to re is constant, and this distance itself is constant as well
At this point, it is assumed that k, h, μ, re, rw, and S are constant. Therefore, Equation 1 reduces to:
(Equation
2)
where:
(Equation
3)
Rearranging Equation 2 yields the following:
(Equation
4)
Integrating both sides of Equation 4 with respect to time gives:
(Equation
5)
The integral on the right- side of Equation 5 is cumulative water injected, so Equation 5 can be represented as:
(Equation
6)
where:
Wi = cumulative volume of water injected at time t, bbls
Closer inspection of Equation 6 indicates that a coordinate graph of its left-side of versus the right-side should form a straight line with a slope of 1/C. This type of graph is called the Hall plot. If h, μ, re, rw, and S are constant, then from Equation 3, the value of C is constant and the slope is constant. However, if the parameters change, C changes, and thus the slope of the Hall plot changes, which is where the diagnostic value of the plot lies.
Changes in injection conditions may be noted from the Hall plot. For example, if wellbore plugging, or other restrictions to injection are gradually occurring, the net effect is a gradual increase in the skin factor, S. As S increases, C decreases; thus, the slope of the Hall plot increases. Conversely, if S decreases (as would be the case if injecting pressure exceeds fracture pressure, causing fracture growth), then C increases and the slope of the Hall plot decreases. See Figure 1 for various injection well conditions and their Hall plot signatures.
Figure 1. Hall plot characteristic signatures
The most challenging part of developing the Hall plot is calculating the pressure-integral function of the y-axis. Fortunately, the integral can easily be solved. Consider Figure 2, which shows a graph of monthly sandface injection pressure, pwi, and periodic estimates of average reservoir pressure, pavg.
Figure 2. Sandface (bottomhole) injection and reservoir pressure vs. time
If it can be assumed that pwi and pavg are average for the month, then
(Equation
7)
where:
Δp = pwi - pavg
Δt = number of injection days for the month
Changes in the slope of the Hall plot typically occur gradually, so several months (six months or more) of injection history may be needed to reach reliable conclusions about injection behavior, as is the case for production decline curve analysis.
It is important to note that changes in the slope of the Hall plot can be the result of other factors. Early in the life of an injection well (before gas fillup), the radius of the water and oil zones increase with cumulative injection and cause the value of C to increase, resulting in a concave upward trend in the Hall plot. Recall that the Hall plot technique assumes a mobility ratio of 1.0. If the mobility ratio is greater than 1, then the Hall plot gradually trends concave downwards after gas fillup (as shown in curve D in Figure 1); if the mobility ratio is less than 1.0, it gradually trends concave upwards (see curve C). Also, as the average water saturation in the reservoir increases with time, kw may increase, which can also affect the slope of the plot.
If after gas fillup it can be assumed that pavg does not change significantly, then calculating the y-axis on the Hall plot is greatly simplified by dropping pavg. This is because if pavg is constant and ignored, the Hall plot is only shifted on the y-axis without changing the slope, or its diagnostic interpretive value. Under this condition, the sandface injection pressure (pwi) is simply the wellhead injection pressure, plus a hydrostatic gradient, minus a frictional loss term. Since these two terms can usually be assumed to be constant and neglected, the left-side of Equation 6 can simply reduce to the integral of the wellhead injection pressure, a dataset that is more readily available.
To determine whether average reservoir pressure is changing, it is necessary to conduct regular pressure buildup / falloff tests, and to monitor monthly voidage replacement ratio (VRR) plots. The objective of the Hall plot is to detect changes in the injection well's skin factor. It is not a perfect tool, but can, under certain conditions, provide reasonable insight on skin changes. The best tool for quantifying injection wellbore skin damage is a properly designed, well executed, and fully analyzed pressure fallout test.
References: Waterflooding Course, William M. Cobb & James T. Smith.