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# Estimating average reservoir pressure from diagnostic plots

Diagnostic plots are a log-log plot of the pressure change and pressure derivative (vertical axis) from a pressure transient test vs. elapsed time (horizontal axis). They are typically divided into three time regions: early, middle, and late.

Two different method types, one using data from the middle-time region and the second using data from the late-time region (LTR), are commonly applied in estimating average reservoir pressure. The middle-time region methods are the Matthews-Brons-Hazebroek (MBH) method^{[1]} and the Ramey-Cobb method. ^{[2]} The LTR methods are the modified Muskat method^{[3]} and the Arps-Smith method. ^{[4]}

## Middle-time region (MTR) methods

The MTR methods are based on extrapolation of the middle-time region and the correction of the extrapolated pressure. The advantage of these methods is that they use pressure data only from the middle-time region, which means they require relatively short tests. The disadvantages are the need for accurate fluid property estimates, a known drainage area shape and size, and the location of the well within the drainage area.

### Drainage area shapes

The MTR methods depend on the shape of the drainage area. Matthews-Brons-Hazebroek^{[1]} developed a series of curves that model buildup tests in many shapes. As a matter of interest, these graphs were generated using image wells to simulate boundaries.

**Figs. 1 through 3** illustrate representative dimensionless pressures as calculated by the MBH method. **Fig. 1** is a plot of dimensionless pressure as defined by the MBH method plotted against dimensionless producing time calculated using the drainage area. Dimensionless pressure is defined as

and dimensionless time is

In **Eq. 1**, *p** = the extrapolated pressure at a HTR of unity, = the current average drainage area pressure, and *m* = the slope of the MTR straight line on a Horner plot. In **Eq. 2**, *t*_{p} = the producing time before shut-in, and *A* = the well’s drainage area expressed in square feet.

The four different curves in **Fig. 1** represent four different locations of a well within a square drainage area. On this plot of dimensionless pressure on a linear scale vs. dimensionless time on a logarithmic scale, these curves eventually become straight lines. For example, for a well centered in a square drainage area, the line becomes straight at a dimensionless time of approximately 0.2. The time at which the line becomes straight is an indication that a well has reached pseudosteady-state flow at that dimensionless time.

**Fig. 2** shows the Matthews-Brons-Hazebroek correlations for a well in the various positions in a 2×1 rectangle. The wells eventually reach pseudosteady state and the lines become straight, but, in general, the time to reach pseudosteady state is longer for the 2×1 rectangle than it was for the square rectangle. Furthermore, the farther the well is off center within the drainage area, the longer the time required to reach pseudosteady state. The difference is on the order of one full log cycle between the case in which the well is centered in the drainage area and that for a well most off-centered in the drainage area, which is the lowest curve on this plot. **Fig. 3** shows the MBH pressures for wells in various positions in a 4×1 rectangle. Matthews-Brons-Hazebroek generated many similar graphs for other drainage-area shapes.

### Example of the Matthews-Brons-Hazebroek method

This method will be applied to a well in a reservoir with the following properties: t p = 482 hours, *ϕ* = 0.15, *μ* = 0.25 cp, *c*_{t} = 1.615 × 10^{–5}, and *A* = 1,500 × 3,000 ft (a 2 × 1 reservoir, well centered).

First, plot well shut-in pressure against the HTR on semilog coordinates. In **Fig. 4**, which is an ordinary HTR plot, the wellbore storage affects the data at large values of HTR, followed by the straight-line middle-time region, in turn followed by a deviation of the curve as it begins moving toward a fully built-up pressure.

The MTR straight line on this Horner graph is extrapolated to a HTR of 1 to determine *p**. In this case, *p** = 2,689.4 psi. From the slope of the semilog straight line, 26.7 psi/cycle, we calculate *k* = 7.5 md.

Next, calculate the dimensionless producing time, *t*_{pAD}, with **Eq. 2**.

To calculate dimensionless production time, use the same producing time used in preparing the Horner graph. If the actual producing time is quite long, replace it with the time required to reach pseudosteady state, but remember to use the same producing time in the HTR and in calculating the dimensionless time for the MBH function. The time to reach pseudosteady state is determined by observing the appropriate MBH graph and finding when the dimensionless pressure vs. time becomes a straight line.

The next step is to select the appropriate MBH chart for the drainage area shape and well location being evaluated. Because the example well is centered in a 2×1 rectangle, choose **Fig. 2**. On this chart, enter the graph at a dimensionless producing time of 0.35, as illustrated in **Fig. 5**, and read across to find the dimensionless pressure, *p*_{MBHD}, which has a value of 2.05.

The next step is to calculate the average reservoir pressure, . From rearrangement of **Eq. 1**,

In this case, the extrapolated *p** = 2,689.4 psi, the slope of the MTR = 26.7, and the dimensionless pressure= 2.05. Thus,

### Ramey-Cobb method

The Ramey-Cobb method^{[2]} also uses information from a Horner plot of buildup test data. After determining permeability from the Horner plot, dimensionless producing time, *t*_{pAD}, can be calculated.

The third step differs from the MBH method in that the Dietz shape factors, *C*_{A}, from **Table 1** for the drainage-area shape and well location that best describes the tested well are used. (For the physical significance of the shape factor, see Ramey and Cobb. ^{[2]}) For the example well, the drainage area is a 2 × 1 rectangle, and the shape factor is 21.8369. Ramey and Cobb found a relationship between shape factor and the HTR at which the pressure on the MTR is current average drainage area pressure, . The relationship is

In the example test, the dimensionless producing time is 0.35, so the HTR that corresponds to the average reservoir pressure is 7.63.

Enter the Horner plot at a HTR of 7.63, read up to the extrapolated MTR straight line, then read across to the vertical axis. The resulting average reservoir pressure is 2,665.8 (**Fig. 6**). The result, for practical purposes, is identical to the result obtained using the MBH method.

The MBH and Ramey-Cobb methods use only data in the MTR. Once enough data is available to identify the MTR, the test can be terminated, which reduces test costs. The disadvantages of these methods are the need to know the drainage area size, shape, location of the well within that drainage area, and an accurate measurement of fluid properties. In the MBH method, the well can be in transient flow at the time of shut-in, but in the Ramey-Cobb method, the well must have reached pseudosteady state before shut-in. Results for the two methods should be identical, because they are based on the same theory. When it is applicable (pseudosteady state before shut-in), the Ramey-Cobb method is preferred because it is easier to apply.

## Late-time region methods

Methods using LTR data are based on extrapolation of the post-middle-time region data trend. The advantages of these methods are that they require neither accurate fluid property estimates nor the drainage area size and shape. They do require that the well be reasonably centered within its drainage area. The disadvantage is that they require the post-middle-time region transient data. Thus, they require longer and more expensive shut-in tests to provide the data required for analysis.

### Modified Muskat method

The modified Muskat method is based on the theoretical observation first published by Larsen^{[3]} that, for late-time data (after boundary effects have appeared), the difference between current average reservoir pressure, , and shut-in BHP, *p*_{ws}, declines exponentially. In equation form,

**Eq. 6** leads to a procedure for estimating average drainage-area pressure, . This method requires a trial-and-error approach. To select data suitable for analysis with this method, use the diagnostic plot to determine the start of boundary effects. Then assume a value for , and plot log vs. time. If the curve is concave downward, the assumed pressure is too low; if the curve is concave upward, the assumed pressure is too high. Try different values for until the graph is a straight line, as predicted by theory.

On the example (**Fig. 7**), once the data begin to fall on a straight line, they tend to remain on that straight line. Shown are curves for assumed values of = 5,600; 5,575; and 5,560. On the first curve, for = 5,600, the final data points are trending above the straight line. For the lower curve, with = 5,560, the last few data points are trending below the straight line. For the assumed value = 5,575, all of the data points fall on a straight line making this assumption the right estimate of . The advantage to this method is that it is very easy to apply. It works best with a well reasonably centered within a drainage area.

The weaknesses of this method are that it is more sensitive to estimates that are too low rather than to estimates that are too high and that it is not easily automated and, therefore, not as widely incorporated into well-test analysis software as some other methods.

### Arps-Smith method

This is an alternative method for analyzing LTR data. ^{[4]} The theoretical basis for this originally empirical method is also **Eq. 5**. Differentiating **Eq. 5** with respect to time,

To apply this method, plot the change in BHP with time, *dp*_{ws}/*dt* vs. *p*_{ws}, on Cartesian coordinates. On such a plot, data for the LTR should fall on a straight line, and extrapolation of that line to *dp*_{ws}/*dt* = 0 provides an estimate of the average drainage area pressure, .

In **Fig. 8**, the final points from an example test fall on a straight line. Extrapolating the straight line to the horizontal axis gives the average pressure at the intercept. For this example, which is the same test illustrated with the modified Muskat method, the average pressure is 5,575 psi, which is the same value found with the Muskat method.

The advantages of the Arps-Smith method are that it is simple to apply and easily automated (which means that it is easily implemented into well-test analysis software or into spreadsheets). The disadvantages are that it requires data in the LTR, which means that it requires longer, more expensive tests. It assumes that shut-in pressure approaches average pressure exponentially, which is most nearly true for wells centered in the drainage area, and it requires numerical differentiation of pressure with respect to time, which tends to magnify any noise that may be present in the data.

The modified Muskat and Arps-Smith methods actually apply for shut-in times in the range,

In **Fig. 9**, the data points with darker dots are on the type curve for the derivative. These are the data in the range for which the modified Muskat and Arps-Smith methods work.

**Fig. 9** illustrates one of the disadvantages of these two methods. Many other reservoir models will exhibit similar diagnostic plots, but data like that shown with the dark dots in this figure will not extrapolate to the correct average reservoir drainage area pressure. Examples of these other cases include dual-porosity reservoirs during the early transition from fracture flow to total system flow, layered reservoirs, and composite reservoirs with an inner zone mobility much lower than the outer zone mobility.

## Nomenclature

## References

- ↑
^{1.0}^{1.1}Matthews, C.S., Brons, F., and Hazebroek, P. 1954. A Method for Determination of Average Pressure in a Bounded Reservoir. Trans., AIME 201, 182–191. - ↑
^{2.0}^{2.1}^{2.2}H.J. Ramey, J. and Cobb, W.M. 1971. A General Pressure Buildup Theory for a Well in a Closed Drainage Area (includes associated paper 6563). J Pet Technol 23 (12): 1493-1505. SPE-3012-PA. http://dx.doi.org/10.2118/3012-PA - ↑
^{3.0}^{3.1}Larson, V.C. 1963. Understanding the Muskat Method of Analysing Pressure Build-up Curves. J Can Pet Technol 2 (3): 136-141. http://dx.doi.org/10.2118/63-03-05. - ↑
^{4.0}^{4.1}Arps, J.J. and Smith, A.E. 1949. Practical Use of Bottom Hole Pressure Build-Up Curves. Reprint Paper No. 851-23-I, Tulsa Meeting, API, March.

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## See also

Fluid flow through permeable media

Boundary effects in diagnostic plots

PEH:Fluid_Flow_Through_Permeable_Media