Chapter 10

Applicationof PressureDerivative in OilWell Test Analysis

10.1 Introduction

The pressure derivative application in oil well test analysis involves thecombined use of existing type curves in both the conventional dimensionlesspressure form (pD) and the new dimensionless pressure derivative grouping(PD X IDICD). Thus, this new approach has combined the most powerfulaspects of the two previously distinct methods into a single-stage interpretiveplot. Use of the pressure derivative with pressure behavior type curvesreduces the uniqueness problem in type curve matching and gives greaterconfidence in the results. Features that are hardly visible on the Horner plotor that are hard to distinguish because of similarities between a reservoirsystem and another are easier to recognize on the pressure derivative plot.

10.2 Pressure Derivative Applicationsin Well Test Analysis

Figure 10-1 illustrates the application of pressure derivative to homo-geneous reservoirs, naturally fractured reservoirs, and vertically fracturedreservoirs.

10.3 Pressure Derivative Analysis Methods

Bourdet etal.3 developed a new set of type curves (see Figure 10-2) basedon the pressure and pressure derivative. In Figure 10-2, at early time, thecurves follow a unit-slope log-log straight line. When infinite-acting radial

Pressure derivativeapplications

Homogeneousreservoirs

Type curves presented in Figure 10-2 can be used to analyze awell test with wellbore storage and skin effects. Unique matchpoints are obtained by matching the two straight lines (unit-slope and the late-time horizontal lines)

Pressure match gives khTime match gives C and CD

Curve match gives s

Pseudo-Steady-State Interporosity Flow

This type of reservoir shows pseudo-steady-state interporosityflow. Figure 10-4 is used for diagnosing dual-porosity behavior.The derivative curve displays the double-porosity behavior andlimits of the three characteristic regimes such as:1. Well on production

Initial flow regime - homogeneous behavior2. Production continuous

Transient flow regime3. When pressure equilibrates between two media

Homogeneous behavior characterizes the total system

Naturallyfracturedreservoirs

Pressure, time and data match points canbe used to calculate reservoir parameters

such as s, UJ, X and (CD)f+m usingEqs. 10-1 through 10-4

Transient Interporosity FlowFigure 10-5 can be applied to analyze transient interporosity flow.Transient regime is described by a family of fi curves that areidentical to homogeneous C^e2* curves except that pressure and timeare divided by 2.

At early time - fissured flow -> /3' and (CD e2s)f+m

Pressure curvesLate time - homogeneous flow

Pressure, time, curve and transient curve match points can beused to determine reservoir parameters kf, C, s and X

Vertical fracturedreservoirs

Figures 10-4 and 10-5 can be used to analyze transienttests in both the pseudo-steady-state interporosity and transientinterporosity flows for fractured oil reservoir systems

Figure 10-1. Well with wellbore storage and skin in a homogeneous reservoir(© SPE, [J. Pet. Techno/. Oct. 1988]).1

flow is reached at late time, the curves become horizontal at a value ofP'D^DICD) = 0.5. Between these two asymptotes, at intermediate times, eachCD Q2S curve produces a specific shape and is different for varying values of

Dimensionless groups are:

Dim

ensi

onle

ss p

ress

ure

grou

ps

dim

ensi

onle

ss p

ress

ure

grou

ps

damaged well

non-damaged well

acidized well

Figure 10-2. Derivative and pressure type curves for a well with wellbore storageand skin in infinite-acting homogeneous reservoir.3

CD e2*. Thus, it is easy and simple to identify the correct C^e2* curvecorresponding to the data. By matching the two straight lines (unit-slopeand the late-time horizontal lines) a unique match point is provided. Thedouble plot provides as a check, as the two data set must match on theirrespective curves. The matching procedures for pressure buildup and draw-down tests are as follows.

Pressure Buildup Test Data Matching Procedure

• Plot Ap versus Ar on 3 x 5-cycle log-log paper.• Calculate the pressure derivative of the field data.• Plot Af(Ap') versus At on same 3 x 5-cycle log-log paper.• Select the best match by sliding the actual test data plot both horizon-

tally and vertically.• Note the values of the match points:

• Determine the formation permeability, k, from pressure match points:

• Estimate the wellbore storage, C, from time match point

(10-1)

(10-2)

• Calculate the dimensionless wellbore storage, CD, from

(10-3)

• Estimate the skin factor, s, from curve match point:

(10-4)

Pressure Drawdown Test Data Matching Procedure

• Plot Ap = (pi — pwf) versus t on 3 x 5-cycle log-log paper.• Calculate the pressure derivative of the field data.• Plot t'(Ap') versus t on same 3 x 5-cycle log-log paper.• Select the best match by sliding the actual test data plot both horizon-

tally and vertically.• Note the values of the match points:

• Determine the formation permeability, k, from pressure match points:

(10-5)

• Estimate the wellbore storage, C, from time match point:

(10-6)

• Calculate the dimensionless wellbore storage, CD, from

(10-7)

• Estimate the skin factor, s, from curve match point:

(10-8)

Example 10-1 Analyzing Single-Rate Buildup Test Using Pressure Deriva-tive Curves

A single-rate pressure buildup test was run in an oil well. Table 10-1shows the pressure-time data. The reservoir and well data are: oil rate,q0 = 550stb/s; h = 100 ft; /x = 0.95 cP; (3O = 1.05 rb/day; 0 = 0.16; ct =1.95 x 10"5PSi"1; rw — 0.29 ft. Find the reservoir permeability and skinfactor. Table 10-1 shows single-rate pressure buildup data.

Solution The points are plotted on tracing paper superimposed on the"homogeneous reservoir" set of curves. Figure 10-3 shows the matchingcurve. The match points are:

Table 10-1

At (hr) Ap (psi)

0.19 700.28 980.58 1450.82 1711.12 1882.20 2194.01 2486.75 277

10.71 29014.92 30120.70 31029.82 31540.45 32560.00 330

Dimensionless groups are:

Dim

ensi

onle

ss p

ress

ure

grou

ps

Figure 10-3. Buildup data plotted and matched to pressure derivative type curveof Figure 10-2.

For the same point, the following are read on the type curve set:

and the designation of the superimposed type curve is noted as

From the pressure match points, calculate k using Eq. 10-5:

From the time match points, find the wellbore storage constant, C, usingEq. 10-6:

Find the dimensionless wellbore constant, C/>, using Eq. 10-7:

Find the skin factor, s, from Eq. 10-8:

The above values indicate that the well has been improved.

10.4 Fractured Reservoir Systems

New type curves suitable for practical applications, based on the modelby Warren and Root, were introduced by Bourdet et al.2'3 These curves areprimarily used for diagnosing dual-porosity behavior and for ensuring thatan optimum, conclusive test is obtained. The idea behind these curves is thatthe log-log plot consists of three typical flow regimes as follows:

1. The first flow regime represents radial flow in a homogeneous reservoirwith wellbore storage, skin, permeability, k/, and reservoir storage, St.

2. The second flow represents a transient period.3. The third flow represents radial flow in a homogeneous system with

wellbore storage, skin, permeability, kf, and reservoir storage(Sf + Sm).

Pseudo-State Interporosity Flow

Bourdet etal. type curves, as shown in Figure 10-4, can be used for theanalysis of fractured reservoirs with pseudo-steady-state interporosity flow.Only two parameters (a; and A) characterize the reservoir heterogeneity.The parameter UJ is defined as the ratio of fracture storage to total storage.

Dimensionless pressure, pD, and pressure derivative group, (tD /CD) pD

Approximate endof unit slope

log—log straight line

Dimensionless time group, tjJCD

Figure 10-4. Type curve showing the behavior of both the pressure and itsderivative (after Bourdet et al., 1984).3 It can be used to analyze test data fromfractured reservoirs. The behavior of the pressure derivative is reasonablyresolutive for the identification of the transition period.

The interporosity flow parameter A is proportional to the ratio of matrixpermeability to fracture permeability. Thus,

(10-9)

where

Sf = fracture storage = <j>fCfhf

Sm = matrix storage = <j>mcmhm

and

(10-10)

where a is a shape factor, which is defined as

n = number of normal set of fracturesL — length of matrix, ft.In Figure 10-4, the dimensionless pressure (po versus tj)/Cj)) curves

show two families of component curves such as the Cp Q2S curves thatcorrespond to homogeneous behavior and the A e2s curves that show pres-sure behavior during transition. The pressure derivative curve responsefollows this sequence:

• Initially, due to wellbore storage effects, the derivative curve follows(CD zls)f = 1 type curve.

• When the infinite-acting radial flow occurs in the fissured system, thepressure derivative group will follow the 0.5 horizontal straight lines.

• During the transition period, when pressure stabilizes, the derivativewith respect to natural logarithm of time drops and follows theAQ)/a;(l — J) type curve until it reaches a minimum and then bouncesback up along the ACz>/(l — J)2 curve before returning to the 0.5straight line. The 0.5 horizontal lines correspond to the infinite-actingradial flow in the total system (CD Q2s)f+m.

To use the type curve in Figure 10-4, one has to match the early data withone of the type curves labeled CD e2s. The label of the matched curve is nowreferred to as (CD e25)/- The permeability, kf, is calculated from the pressurematch and C is calculated from the time match. The matching procedure isas follows:

• Plot pressure derivative versus A^ on log-log graph paper.• Plot Apws versus At on log-log paper.• Match the derivative curve with one of the derivative type curves of

Figure 10-4.• Choose any point and read its coordinates on both figures. Thus,

would become known. Also, read the matched derivative curve labeledXCD/(I - UJ)2; here CD is CDf+m.

• Now, with the match still maintained, change your focus from thederivative curve to the data curve. Read the values of the curves labeledCDQ2S, which match initial and final segments of the data curves,(CD Q2s)f and (CD e2s)f+m, respectively.

• Calculate the different parameters as follows:

(10-11)

(10-12)

(10-13)

(10-14)

Assuming that the total reservoir storage (S/ + Sm) is known from well logs,

(10-15)

and A can be calculated from the label of the matched derivative curveXCDfJ(l-uj)2:

(10-16)

Transient Interporosity Flow

Bourdet etal. type curves, as shown in Figure 10-5, can be used for theanalysis of fractured reservoir with transient interporosity flow.3 The tran-sient period is described by a family of /?' curves that are identical tohomogeneous CD e2s curves except that pressure and time are divided by 2.In the transient interporosity flow period, the double-porosity responses donot flatten out but tend to develop a semilog straight line, the slope of whichis half of the true radial flow slope. The dimensionless interporosity transientflow parameter /3f is defined by the following equation:

(10-17)

Dimensionless pressure, pD and pressure derivative group, \tD ICn) p'n

Approximate endof unit slope

log-log straight line

Dimensionless time group, tD ICn

Figure 10-5. Type curve matching the behavior of both the pressure and itsderivative (after Bourdet et al., 1984).18 Transient interporosity fluid flow behavior.

whereSf = matrix block shape factor

= 1.8914 for slab matrix block= 1.0508 for spherical matrix block.

(The choice of matrix geometry for interpretation has to be supported bygeological models.)

Figure 10-5 shows the following characteristics:

1. /?': At early time, the fissured flow (CDe2s)f is masked by wellborestorage, and the pressure response starts on the transition curve.

2. (CD Q2s)f+m: At late time, the homogeneous behaviors corresponding tothe total system parameters are reached.

The derivative of pressure curves is three-component curves.

1. /?': At early time, the response first follows an early-transition deriva-tive curve.

2. 7(C/))/+m/(l — oo)2\ A late-transition curved is reached.3. 0.5: At late time, the homogeneous behavior corresponding to

(CD z2s)f+m is, in general, reached on the 0.5 lines.

As presented in the previous sections, kf and C are calculated frompressure and time match, respectively. The skin factor, s, is obtained from

the curve match and Eq. 10-18. The parameter A is calculated from thetransition curve match and the following equation:

(10-18)

10.5 Pressure Derivative Trends for OtherCommon Flow Regimes

Figure 10-6 shows pressure derivative trends for common flow regimes.

Wellbore storagedual-porosity matrix to

fissure flowSemilog straight lines with slope= 1.151.

Parallel straight-line responses are characteristics of naturallyfractured reservoirs

Dual-porosity withpseudo-steady-stateinterporosity flow

Pressure change slope •* increasing, leveling off increasingPressure derivative slope = 0, valley = 0

Additional distinguishing characteristic is middle-time valley trendDuration is more than 1 log cycle

Dual-porosity withtransient interporosity

flow

Pressure change slope, A(p) -> steepeningPressure derivative slope, A(p')At = 0, upward trend = 0

Additional distinguishing characteristic -> middle-time slope doubles

Pseudo-steady state

Pressure change slope, A(p) -> 1 for drawdown and zero for buildupPressure derivative slope, A(p')At -> 1 for drawdown and steeply

descending for buildupAdditional distinguishing characteristic -^ late-time drawdown

pressure change and derivative are overlain; slope of 1 occurs muchearlier in the derivative

Constant-pressureboundary (steady state)

Pressure change slope -> 0Pressure derivative slope -> steeply descending

Additional distinguishing characteristic -> cannot be distinguishedfrom pseudo-steady state in pressure buildup test

Single sealing fault(pseudo-radial flow)

Pressure change slope -> steepeningPressure derivative slope -> 0, upward trend -> 0

Additional distinguishing characteristic -> late-time slope doubles

Elongated reservoirlinear flow

Pressure change slope -> 0.5Pressure derivative slope -^ 0.5

Additional distinguishing characteristic -> late-time pressure changeand derivative are offset by factor of 2; slope of 0.5 occurs much

earlier in the derivative

Figure 10-6. Illustration of pressure derivative trends for other common flow regimes.

10.6 Summary

• A new technique is presented to analyze data in the bilinear flow period.It is shown that, during this flow period, a graph of pw/ versus t1^4 yieldsa straight line when slope is inversely proportional to /*/(&/&/)1/2.

• New type curves are now available for pressure analysis of fractured oilwells, and the problem in the analysis is reduced considerably with theuse of these type curves.

• Prefracture information about the reservoir is necessary to estimatefracture parameters.

• The type curve analysis method must be used simultaneously with thespecific analysis methods pw/ versus /1/4, pw/ versus £1/2, and pw/ versuslog t to produce reliable results.

References

1. Economides, C. E., "Use of the Pressure Derivative for DiagnosingPressure-Transient Behavior," / . Pet. Technol. (Oct. 1988), 1280-1282.

2. Bourdet, D., Whittle, T. M., Douglas, A. A., and Pirard, Y. M., "A NewSet of Type Curves Simplifies Well Test Analysis," World Oil (May 1983).

3. Bourdet, D., Alagoa, A., Ayoub, J. A., and Pirard, Y. M., "New TypeCurves Aid Analysis of Fissured Zone Well Tests," World Oil (April1984).

Additional Reading

1. Bourdet, D., Ayoub, J. A., Whittle, T. M., Pirard, Y. M., and Kniazeff, Y.,"Interpreting Well Tests in Fractured Reservoirs," World Oil (Oct. 1983).