Dake L.P. Fundamentals of reservoir engineering

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REAL GAS FLOW: GAS WELL TESTING 271

inadequate to define the B and F factors in the well inflow equation (8.44). This is

because if the permeability is small then the flowing time required to reach semi-steady

state conditions, which depends on t

= 0.000264kt/

(

A, can become extremely

large. Then, the use of a semi-steady state inflow equation to analyse the test is

inadmissible because the pressure response at the wellbore is strongly time

dependent. To cater for this time dependence, two methods have been presented in

the literature for analysing tests under the assumption that the wellbore pressure

response could be matched by superposing transient constant terminal rate solutions

of the radial diffusivity equation. The first of these was the Odeh-Jones technique

(1965), in which the test analysis equation was formulated using p-squared terms. This

was followed in 1971 by the method of Essis and Thomas

, in which the Odeh-Jones

test equation was modified by the use of real gas pseudo pressure to give

()

nnj1

iwf jDDDnn

mp mp Qm t t QS

1422T

−

′

−=∆−+

which is the general form of the superposed constant terminal rate solution,

equ. (8.39). Both Odeh and Jones and Essis and Thomas applied their analysis

techniques strictly for transient flow conditions, for which the superposed m

functions

in equ. (8.39) can each be expressed as

()

mt ln

(8.32)

rather than using the general expression, equ. (8.40), which assumes a knowledge of

the area drained and geometry. If transient m

functions can be used then the analysis

is simple and should yield values of k and S which in turn can be used to calculate a

value of the Darcy flow coefficient B, equ. (8.44), for any values of the area drained and

shape factor. The analysis also directly determines F (or D), the second coefficient in

the inflow equation. The technique will be fully illustrated in exercise 8.2.

The statements made in Chapter 7, sec. 8, about the possibility of incorrectly

interpreting multi-rate test data through making an a priori judgement concerning the

prevailing flow conditions are equally, if not more, valid in gas well test analysis. This is

because the

c product for gas is several times smaller than for oil which implies that,

for the same permeability, porosity and area drained, the boundary effects will be felt

much earlier in a gas well test. To apply transient analysis techniques it is insufficient to

assume that each separate flow period should be short enough so that transient

conditions prevail. Instead, the entire test duration must be so brief that the maximum

value of the m

function in equ. (8.39), which is

()

max

DD D

m t m total dimensionless testing time

′

can still be evaluated using the transient expression, equ. (8.32). The possible error

that can be made by making the flow periods too long will be illustrated in the following

exercise.

REAL GAS FLOW: GAS WELL TESTING 272

EXERCISE 8.2 MULTI-RATE GAS WELL TEST ANALYSED ASSUMING

UNSTABILIZED FLOW CONDITIONS

A gas well is tested by producing at four different rates for periods of one hour, followed

by a four hour pressure buildup. The rates, times and observed pressures during the

flow test are listed in table 8.5.

Rate (Q)

Mscf/d

Cum. Flowing Time (t)

hrs

psia

m (p

)

psia

/cp

10 × 10

1 4160 1023.35 × 10

20 × " 2 3997 967.00 "

30 × " 3 3802 899.59 "

40 × " 4 3577 821.81 "

TABLE 8.5

The reservoir temperature and gas properties are again the same as detailed in

table 8.1 and therefore, the pressure-pseudo pressure relationship for p > 2800 psia is

given by equ. (8.48). The reservoir and well data are listed below.

= 4290 psia (µc)

=3.6 × 10

-6

cp/psi

A = 200 acres

=.15

h = 50 ft

Geometry

=.3 ft

This exercise requires the evaluation of m

D(MBH)

) for values of t

< 0.01, which is

the lower limit of the dimensionless time scale in each of the MBH charts, figs. 7.11-15.

For t

< 0.01, values of m

D(MBH)

,for the 4:1 geometry, can be obtained from fig. 8.11,

which has been drawn from the tabulated values of m

D(MBH)

versus t

presented by

Earlougher

. (N.B. There appears to be a typographical error in the latter reference in

that the data listed for the 4:1 geometries

and

in table 2, p. 203, should be interchanged).

The pressure buildup is first analysed from which it is determined that the permeability

is 50 mD. (This technique will be fully described in sec. 8.11 and illustrated in

exercise 8.3).

Analyse this multi-rate test to determine values of k, S and F using

1) m

evaluated for transient conditions, equ. (8.32)

2) m

evaluated using the general expression, equ. (8.40)

REAL GAS FLOW: GAS WELL TESTING 273

EXERCISE 8.2 SOLUTION

1) The Essis-Thomas combination of equations (8.39) and (8.32) can be expressed in

more practical terms as

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

D (MBH)

0.005 t

0.010

Fig. 8.11 MBH chart for the indicated 4:1 rectangular geometry for t

< .01 (After

Earlougher, et.al

)

()

iwfn jnj1n

1637T k

m p m p FQ Q log t t Q log 3.23 .87S

φµ

−

æö

−−= ∆ −+ −+

ç÷

èø

()

iwfn

nj1

mp mp FQ

m log t t m log 3.23 .87S

QQ cr

φµ

−

−−

æö

∆

=−+−+

ç÷

èø

(8.49)

Thus a plot of

iwfnn nj1

(m(p ) m(p ) FQ ) / Q versus log

(t t )

−

∆

−−

−

should be linear with

slope m = 1637 T/kh and intercept =

()

m(log 3.23 .87S)

φµ

−+ from which

values of k and S can be calculated. In this type of analysis the value of F must be

obtained by trial and error until a straight line is achieved. The analysis is shown in

table 8.6 and the plot as fig. 8.12.

REAL GAS FLOW: GAS WELL TESTING 274

iwfn

m(p ) m(p ) FQ

−−

Mscf/d

hrs

nj1

log(t t )

−

psia

m(p

)

psia

/cp F = 0 .04 .05 .06

10×10

1 0 4160 1023.35×10

4494 4094 3994 3894

20× " 2 .1505 3997 967.00 " 5065 4265 4065 3865

30× " 3 .2594 3802 899.59 " 5623 4423 4123 3823

40× " 4 .3450 3577 821.81 " 6162 4562 4162 3762

TABLE 8.6

F = 0.0

F = 0.04

F = 0.05

F = 0.06

6000

5500

5000

4500

4000

3500

log (t t )

−

∆

−

m(p) FQ

∆−

(psia / cp / Mscf / d)

Fig. 8.12 Essis-Thomas analysis of a multi-rate gas well test under assumed transient

flow conditions. Data; table 8.6

Initially, the plot is made ignoring the effects of non-Darcy flow (F = 0) and this exhibits

a marked upward curvature for large values of the abscissa, corresponding to the fact

REAL GAS FLOW: GAS WELL TESTING 275

that the high flow rates occur towards the end of the test. If the well had been tested

with a decreasing rate sequence, the degree of curvature would have been greater for

smaller values of the abscissa

. The calculation of the left hand side of equ. (8.49) has

been repeated in table 8.6 for different values of F in an attempt to linearize the plot,

and from fig. 8.12 it can be seen that a value of F = .05 accomplishes this. For this

value, the plot has a slope of m = 491.5 and intercept 3993, from which the

permeability and skin factor can be calculated as

1637 T 1637 660

k44mD

mh 491.5 50

== =

()

intercept k

and S 1.151 log 3.23 2.7

mcr

φµ

æö

=−+=

ç÷

èø

2) The Essis-Thomas analysis, which assumes transient flow conditions for the evaluation

of m

, equ. (8.32), will now be compared with the more general case in which m

calculated using equ. (8.40) for the 4:1 rectangular geometry. The general well test

equation (8.41) can be re-arranged as

()

nj1

iwfn

DD D

m(p ) m(p ) FQ

1422T 1422T

mt t S

QkhQ kh

−

−−

∆

=−+

(8.50)

in which m

is evaluated as

11 1

22 2

D D DA DA D(MBH) DA

m(t) 2 t ln t ln m (t )

′′ ′ ′

=+ + −

(8.40)

DD D(MBH)DA

m(t) m (t )

′′

=−

(8.51)

Alternatively, adhering to the Essis-Thomas analysis expressed in terms of

dimensionless parameters

111

222

DD DA

m (t ) ln ln t ln

′′

==+

(8.32)

A plot of the left hand side of equ. (8.50) versus

nj1

DD D

m(t t )

−

∆

−

should be linear

with slope m = 1422 T/kh, and intercept = 1422 TS/kh, which will yield values of k and

S, respectively.

The first part of the analysis is to evaluate the m

functions, equs. (8.51) and (8.32), for

all values of the dimensionless time argument

nj1

(t t )

−

required in the analysis.

These functions are listed in table 8.7, for k = 50 mD, the permeability value obtained

from the pressure buildup analysis, for which

.000264kt

t .0028 t

(c)A

φµ

REAL GAS FLOW: GAS WELL TESTING 276

t (hrs) t

equ. (8.51) ½ m

D(MBH)

equ. (8.51)

equ. (8.32)

1 .0028 6.6772 .0174 6.6598 6.6590

2 .0056 7.0413 .0266 7.0147 7.0061

3 .0084 7.2617 .0230 7.2387 7.2080

4 .0112 7.4231 .0121 7.4110 7.3527

TABLE 8.7

Assuming the value of F = .05 determined from the Essis-Thomas analysis, equ. (8.50)

can be evaluated for both m

functions as listed in table 8.8.

nj1

DD D

m(t t )

−

∆

−

Mscf/d

hrs

m(p) FQ

∆−

(equ. (8.51) ) m

(equ. (8.32) )

10×10

1 3994 6.6598 6.6596

20 " 2 4065 6.8373 6.8329

30 " 3 4123 6.9711 6.9582

40 " 4 4162 7.0811 7.0568

TABLE 8.8

The plots of the data contained in table 8.8 are shown as fig. 8.13 (a) and it can be

seen that for the total test duration of four hours the difference between them is very

slight. The values of k and S calculated from the two plots are presented in table 8.9.

(equ. (8.51) ) m

(equ. (8.32) )

Slope, m 402.8 426.5

Intercept 1312 1153

k = 1422 T/mh 46.6 mD 44.0 mD

S = kh × intercept/1422 T 3.3 2.7

TABLE 8.9

As the duration of each flow period is increased, the difference between the transient

and the correct analysis becomes much more pronounced. Fig. 8.13 (b) shows the

difference for 4 × 2 hour flow periods while fig. 8.13 (c) is for a test of 4 × 4 hour flow

REAL GAS FLOW: GAS WELL TESTING 277

(m(p)FQ)/Qn

(psia / cp /Mscf / d)

∆−

(m(p)FQ)/Qn

(psia / cp / Mscf / d)

∆−

(m(p)FQ)/Qn

(psia / cp /Mscf / d)

∆−

nj1

DD D

m(t t )

−

∆

−

nj1

DD D

m(t t )

−

∆

−

nj1

DD D

m(t t )

−

∆

−

4100

4000

3900

6.5 6.75 7.0 7.25

4300

4200

4100

7.0

7.57.25

4400

4300

4200

8.07.5 7.75

4500

CORRECT ANALYSIS; m EVALUATED USING EQU. (8.40)

TRANSIENT

; m

(

8.32

)

“

““

“

Fig. 8.13 The effect of the length of the individual flow periods on the analysis of a

multi-rate gas well test; (a) 4×

××

×1 hr periods, (b) 4×

××

×2 hrs, (c) 4×

××

×4 hrs

REAL GAS FLOW: GAS WELL TESTING 278

periods. In each case the solid line represents the correct analysis technique, in which

is evaluated using equ. (8.51),while the dashed line is derived for the same

observed pressure data but with the m

function evaluated under the assumption that

transient flow conditions prevail throughout the test. Using the correct analysis

technique always gives the same values of k and S, but the results obtained from the

transient analysis are open to severe misinterpretation. One may either treat the

dashed lines in figs. 8.13 (b) and (c) as being approximately linear, which would result

in the calculated values of both k and S being too small. Alternatively, since both the

dashed lines have a distinct upward curvature one may be tempted to linearize them

either by increasing the non-Darcy flow coefficient (even though the correct value of

F = .05 has been used in the analysis to produce figs. 8.13 (b) and (c) ), or by reducing

the initial pressure as demonstrated in exercise 8.1.

The majority of gas well test analyses described in the literature, for non-stabilized flow

conditions, are for wells which have very low permeabilities; a few millidarcies or less.

Under these circumstances, the transient analysis technique of Essis and Thomas,

which is usually applied in one form or another, is probably quite justified. The

foregoing exercise illustrates what can go wrong in applying transient analysis

techniques in a moderately high permeability reservoir and, of course, at the time of

planning the test the permeability is unknown. It is therefore very difficult to estimate in

advance just how long the total test duration should be for the application of transient

analysis to be still valid.

As a safeguard, it is recommended that all sequential flow tests be followed by a

pressure buildup, which under normal circumstances should provide a more reliable

value of the permeability and one which can be determined independently of the

boundary condition or flow condition (refer Chapter 7, sec. 7). If the permeability

derived from the multi-rate test, assuming the transient flow condition, is less than that

from the buildup it is likely that the multi-rate test analysis is incorrect and should be re-

analysed, attempting to make allowance for the boundary condition and discarding the

assumption of transient flow. The latter is easier said than done, however, for although

the value of k, obtained from the buildup, facilitates the determination of t

, required in

the analysis, there always remains an ambiguity in the estimation of the correct shape

of the drainage area which, as shown in exercise 7.8, cannot be resolved by

conventional analysis techniques.

Odeh et al.

have also described an analysis technique for multi-rate flow tests which

are followed by a buildup. The obvious drawback to this method of testing is that it

negates one of the main aims of multi-rate testing which is to avoid well closure.

8.11 PRESSURE BUILDUP TESTING OF GAS WELLS

Just as in the case of oilwell testing, pressure buildup tests in gas wells, if analysed

correctly using the Horner buildup plot, can provide the most reliable values of the

permeability and skin factor. The only difference is that a buildup in a gas well must be

accompanied by two separate flow periods, one before and one after the buildup, as

shown in fig. 8.14. This measure is necessary in order to determine both components

of the skin factor, S and DQ.

REAL GAS FLOW: GAS WELL TESTING 279

Rate

∆t

max.

t ’

(a)

Time

(b)

∆t

max.

t ’

Bottom

hole

Pressure

Fig. 8.14 (a) Rate-time schedule, and (b) corresponding wellbore pressure response

during a pressure buildup test in a gas well

The theoretical buildup equation for the rates and times shown in fig. 8.14 is just a

special case of the general test equation (8.39).

iwsDDDDD

(m(p ) m(p )) m (t t ) m ( t )

1422Q T

−=+∆−∆

(8.52)

This is identical in form to equ. (7.32), the theoretical buildup equation for an oilwell

test. In deriving equ. (8.52) from equ. (8.39), for superposed constant terminal rate

solutions with rate changes Q

and (0 – Q

), both the mechanical and rate dependent

skin factors disappear, a fact which has been investigated by Ramey and

Wattenbarger

In analogy with the buildup theory described in Chapter 7, sec. 7, for small values of ∆t,

equ. (8.52) can be expressed as a linear relationship between m(p

) and log (t

+ ∆t)/

∆t. The equation of this straight line for any value of ∆t is

iws(LIN) DD

ttkh

(m(p ) m(p )) 1.151 log m (t ) ln

1422Q T t

+∆

−= +−

∆

(8.53)

in which m(p

ws(LIN)

) is the hypothetical pseudo pressure on the extrapolated linear trend,

and

m(t ) and

4t /

, which are both evaluated for the dimensionless, effective

flowing time before the buildup, are constants. For large values of ∆t the real pseudo

pressure m(p

), equ. (8.52), will deviate from m(p

ws(LIN)

) as demonstrated for the similar

liquid flow equations in exercise 7.7. Therefore,. the Horner plot of

REAL GAS FLOW: GAS WELL TESTING 280

m(p ) versus log

+∆

∆

for the recorded pressure data will be linear for small ∆t and the extrapolated trend can

theoretically be matched by equ. (8.53). The attractive feature of the Horner buildup is

that the analysis to determine k and S does not involve the specific evaluation of

(

t ) in equ. (8.53) but merely requires that the early linear buildup trend be

identified. The slope of this line is

1637Q T

(8.54)

from which kh and k can be calculated, and the total skin factor, corresponding to rate

, can be determined as

ws(LIN)1 hr wf

(m(p ) m(p ))

S S DQ 1.151 log 3.23

m(c)r

φµ

−

æö

′

=+ = − +

ç÷

èø

(8.55)

in which m(p

ws(LIN)1-hr

) is the pseudo pressure read from the extrapolated straight line at

∆t = 1 hour. The derivation of equ. (8.55) follows the same argument leading to

equ. (7.52) and therefore the calculated value of

′

is independent of the value of

(

t ).

The early, transient pressure response of both flow periods can be analysed to

determine values of k,

′

and

′

(= S + DQ

). The equation describing the transient

pseudo pressure drop at the wellbore at any time t during the first flow period is

iwf 1

(m(p ) m(p )) ln S

1422Q T

′

−= +

which can be expressed as

iwf 1

1637Q T

m(p ) m(p ) log t log 3.23 0.87S

kh ( c) r

φµ

æö

′

−= + −+

ç÷

èø

(8.56)

Thus a plot of m(p

) versus log t will be linear during the transient flow period with

slope

1637Q T

(8.54)

again giving the value of k, while the skin factor can be calculated by evaluating

equ. (8.56), for the specific value of m(p

) at t = 1 hr, as

iwf1hr

(m(p ) m(p ) )

S S DQ 1.151 log 3.23

m(c)r

φµ

−

æö

−

′

=+ = − +

ç÷

èø

(8.57)

Only the values of m(p

) which plot as a linear function of log t are used, which

ensures that the application of transient analysis is valid.