Dake L.P. Fundamentals of reservoir engineering

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OILWELL TESTING 151

Equation (5.20) can be expressed with respect to this new variable as

1d dp s s cdp s

r ds ds r r k ds t

φµ

æö

∂∂ ∂

ç÷

∂∂ ∂

èø

and using equs. (7.2) and (7.3), this becomes

1 cr d cr dp cr dp

r 2k t ds 2k t ds 2k t ds

φµ φµ φµ

æö

=−

ç÷

èø

which can be simplified as

ddp dp

ds ds ds

æö

=−

ç÷

èø

dp d dp dp

ds ds ds ds

æö

+=−

ç÷

èø

This is an ordinary differential equation which can be solved by letting

′

Then

()

ps sp

′

′′

+=−

′

=−

′

(7.4)

Integrating equ. (7.4) gives

ln p ln s s C

′

=− − +

−

′

(7.5)

where C

and C

are constants of integration and C

can be evaluated using the line

source boundary condition

pq dps dp

lim r r 2s

r2kh dsr ds

→

∂∂

== =

∂∂

therefore,

OILWELL TESTING 152

4kh

−

which yields, as r (and therefore s) tends to zero

4kh

Equation (7.5) can now be integrated between the limits t = 0 (s → ∞ ) and the current

value of t, for which s = x; and p

(initial pressure) and the current pressure p.

i.e.

dp ds

4kh s

−

∞

òò

which gives

r,t i

4kt

pp ds

4kh s

φµ

−

∞

=−

(7.6)

Equation (7.6) is the line source solution of the diffusivity equation giving the pressure

r,t

as a function of position and time.

The integral

4kt

ds ds

φµ

−−

∞∞

òò

(7.7)

is a standard integral, called the exponential integral, and is denoted by ei(x).

Qualitatively, the nature of this integral can be understood by considering the

component parts, fig. 7.2.

The integral of curve (c) between x and

∞ will have the shape shown in fig. 7.2 (d).

Thus ei (x) is large for small values of x, since the ei-function is the area under the

graph from the particular value of x out to infinity (i.e. the shaded area in curve (c) of

fig. 7.2) and, conversely, small for large values of x. The ei-function is normally plotted

on a log-log scale and such a version is included as fig. 7.3. From this curve it can be

seen that if x < 0.01, ei (x) can be approximated as

ei(x) ln x 0.5772≈− − (7.8)

OILWELL TESTING 153

-s

(a)

(b)

s = x

(c)

(d)

−

ei (x) ds

−

∞

Fig. 7.2 The exponential integral function ei(x)

where the number 0.5772 is Euler's constant, the exponential of which is denoted by

0.5772

e1.781

and therefore equ. (7.8) can be expressed as

()

ei(x) ln x for x 0.01

≈− < (7.9)

The separate plots of ei(x) and −In(

x ), in Fig. 7.3, demonstrate the range of validity of

equ. (7.9). The significance of this approximation; is that reservoir engineers are

frequently concerned with the analysis of pressures measured in the wellbore, at r = r

Since in this case

xcr/4kt

φµ

= , it is usually found that for measurements in the

wellbore, x will be less than 0.01 even for small values of t. Equation (7.6) can then be

approximated as

rt wf i

q4kt

ppp ln

4kh

γφµ

==−

Or, if the van Everdingen mechanical skin factor is included as a time independent

perturbation (ref. Chapter 4, sec. 7), then

wf i

q4kt

pp ln 2S

4kh cr

πγφµ

æö

=− +

ç÷

èø

(7.10)

As expected for this transient solution there is no dependence at all upon the area

drained or well position with respect to the boundary since for the short time when

equ. (7.10) is applicable the reservoir appears to be infinite in extent.

OILWELL TESTING 154

-1

-2

-3

-2

-1

255

ei (x)

x→0

ei (x) ~ (-In γ x) - × -0.5772

ei (x)

-In x

∝

ei (x)

-In γ x

Fig. 7.3 Graph of the ei-function for 0.001 ≤ ×

××

× ≤ 5.0

Sometimes pressure tests are conducted in order to determine the degree of

communication between wells, for example, pulse testing. In such cases pressure

transients caused in one well are recorded in a distant well and, under these

circumstances, r is large and the approximation of equ. (7.9) is no longer valid.

Equation (7.6) must then be used in its full form, i.e.

r,t i

qcr

pp ei

4kh 4kt

µφµ

æö

=−

ç÷

èø

(7.11)

in which the values of the exponential integral can be obtained from fig. 7.3.

EXERCISE 7.1 ei-FUNCTION: LOGARITHMIC APPROXIMATION

A well is initially producing at a rate of 400 stb/d from a reservoir which has the

following rock and fluid properties

k = 50 mD

h = 30 ft

= 6 inches

=0.3

=3 cp

c = 10 × 10

-6

/ psi

= 1.25 rb/stb

1) After what value of the flowing time is the approximation

ei(x) = − ln(γx ) valid for this system?

OILWELL TESTING 155

2) What will be the pressure drop at the well after flowing at the steady rate of 400 stb/d

for 3 hours, assuming transient conditions still prevail.

EXERCISE 7.1 SOLUTION

Converting the reservoir and fluid properties to Darcy units.

k = 0.05 D

h = 30 ft × 30.48 = 914.4 cm

= 0.25 sq.ft. = 0.25 × (30.48)

= 232.3 cm

c = 10 × 10

-6

/psi = 10 × 10

-6

× 14.7/atm

= 14.7 × 10

-5

/atm

q = 400stb/d = 400× 1.25× 1.84 r.cc/sec = 920 r.cc/sec.

The approximation ei(x) ≈ − In(γx) applies for x < 0.01 i.e.

0.01

4kt

0.04k

.3314.710 232.3

.04 .05

for t 15.4 seconds

φµ

−

×× × ×

Now in a practical sense, nobody is concerned with what happens in the well during the

first 15 seconds, after which the pressure decline can be calculated using the

logarithmic approximation for ei(x) i.e.

iwf

pp ei

4kh 4kt

q4kt

4kh cr

φµ

πγφµ

æö

−=

ç÷

èø

æö

ç÷

èø

After producing for 3 hours at a steady rate of 400 stb/d the pressure drop at the

wellbore is

iwf

9203 4.053360010

pp ln

4 .05 914.4 1.781 .3 3 14.7 232.3

50.8atm, or 747psi

×××××

−=

×× ××× ×

OILWELL TESTING 156

The constant terminal rate solution of the radial diffusivity equation during the late

transient flow period is too complicated to include at this stage. A simplified method of

obtaining this solution will be described in sec. 7.6.

Once semi-steady state conditions prevail the solution can be determined by adding

the simple material balance equation for the bounded drainage volume

()

cAh p p qt

−= (7.12)

to the semi-steady state inflow equation

q4A

pp ½ln S

2kh Cr

πγ

æö

−= +

ç÷

èø

(6.22)

to give

wf i

q4Akt

pp ½ln 2 S

2kh cACr

πφµγ

æö

=− + +

ç÷

èø

(7.13)

In this equation

p is the current average pressure within the drainage boundary and C

is the Dietz shape factor introduced in Chapter 6, sec.5. The magnitude of C

depends

on the shape of the area being drained and also upon the position of the well with

respect to the boundary.

Theoretically, for the constant terminal rate solution, the rate q in equs. (7.12) and

(6.22) is the same. In practice, it is sometimes difficult to maintain the production rate of

a well constant over a long period of time and therefore, the current rate in equ. (6.22)

may differ from the average rate which is implicitly used in material balance, equ.

(7.12). In this case the rate in equ. (7.12) is set equal to the current, or final flow rate,

and the flowing time is expressed as an effective flowing time, where

Cumulative Production

t Effective flowing time

Final flow rate

(7.14)

Use of the effective flowing time is therefore simply a method for equalising the rates

and preserving the material balance and is frequently used in pressure analysis, as will

be described later.

Even though no equation for describing the pressure decline during the late transient

flow period has yet been developed, equs. (7.10) and (7.13), which are appropriate for

transient and semi-steady state flow, can be usefully employed by themselves in well

test analysis.

Well testing involves producing a well at a constant rate or series of rates, some of

which may be zero (well closed in), while simultaneously taking a continuous recording

of the changing pressure in the wellbore using some form of pressure recording device.

The retrieved record of wellbore pressure as a function of time can be analysed in

conjunction with the known rate sequence to determine some or all of the following

reservoir parameters:

OILWELL TESTING 157

− initial pressure (p

)

− average pressure within the drainage boundary (

p )

− permeability thickness product (kh), and permeability (k)

− mechanical skin factor (S)

− area drained (A)

− Dietz shape factor (C

)

In the following example of a pressure drawdown test, a well is produced at a single

constant rate from a known initial equilibrium pressure p

and p

analysed as a function

of the flowing time t. Equation (7.10) is used to determine k and S while equ. (7.13) is

used for large values of t to determine A and C

. This latter part of the test is

sometimes referred to as reservoir limit testing and the analysis technique used to

determine the shape factor follows that presented by Earlougher

EXERCISE 7.2 PRESSURE DRAWDOWN TESTING

A well is tested by producing it at a constant rate of 1500 stb/d for a period of

100 hours. It is suspected, from seismic and geological evidence, that the well is

draining an isolated reservoir block which has approximately a 2:1 rectangular

geometrical shape and the extended drawdown test is intended to confirm this. The

reservoir data and flowing bottom hole pressures recorded during the test are detailed

below and in table 7.1

h = 20 ft c = 15 × 10

-6

/psi

= .33 ft

=1 cp

=.18 B

= 1.20 rb/stb

Flowing time

(hours)

(psia)

Flowing time

(hours)

(psia)

0 3500 (p

) 20 2762

1 2917 30 2703

2 2900 40 2650

3 2888 50 2597

4 2879 60 2545

5 2869 70 2495

7.5 2848 80 2443

10 2830 90 2392

15 2794 100 2341

TABLE 7.1

1) Calculate the effective permeability and skin factor of the well.

2) Make an estimate of the area being drained by the well and the Dietz shape factor.

OILWELL TESTING 158

EXERCISE 7.2 SOLUTION

1) The permeability and skin factor can be obtained by the analysis of the transient flow

period for which

wf i

q4kt

pp ln 2S

4kh cr

πγφµ

æö

=− +

ç÷

èø

or in field units

wf i

162.6q B

p p log t log 3.23 0.87S

kh cr

φµ

æö

=− + − +

ç÷

èø

Thus for the initial period, when transient flow conditions prevail, a plot of p

versus

log t should be linear with slope m = 162.6 q

/kh, from which kh and k can be

determined. Furthermore, using the value of p

wf(1hr)

taken from the linear trend for a

flowing time of one hour and solving explicitly for S gives

iwf(1hr)

(p p )

S 1.151 log 3.23

mcr

φµ

−

æö

=−+

ç÷

èø

The plot of p

versus log t for the first few recorded pressures, fig. 7.4, indicates that

transient flow conditions last for about four hours and that the values of m and p

wf(1 hr)

are 61 psi/log cycle and 2917 psia, respectively.

OILWELL TESTING 159

(psi)

2900

2800

2700

0.51.01.5

SLOPE m = 61 psi / log cycle

p = 2917 psi

wf (1hr)

log t

020406080100

t HOURS

(psi)

2900

2800

2700

2600

2500

2400

2300

= 5.08 psi / hr

p = 2848 psi

Fig. 7.4 Single rate drawdown test; (a) wellbore flowing pressure decline during the

early transient flow period, (b) during the subsequent semi-steady state

decline (Exercise 7.2)

OILWELL TESTING 160

Therefore,

162.6q B

162.6 1500 1 1.2

kh 4798mD.ft

m61

×××

== =

and k = 240 mD. The skin factor can be evaluated

()

3500 2917

240 10

S 1.151 log 3.23 4.5

61 .18 1 15 .109

æö

−

=−+=

ç÷

×× ×

èø

2) The area being drained and the shape factor can be determined from the later part of

the flow test when semi-steady state flow conditions prevail. Under these

circumstances dp/dt ≈ constant, and as shown in the plot of p

versus t, fig. 7.4(b), this

occurs after a flowing time of approximately 50 hours, after which dp/dt = −5.08 psi/hr.

Therefore, equ. (5.9) can be used to determine the area, since

dp q

(atm / sec)

dt cAh

=−

or, in field units

0.2339qB

5.08 (psi / hr)

dt cAh

=− = −

and hence

.2339 1500 1.2

A35acres

15 10 20 .18 5.08 43560

−

××

×××××

The equation of the linear pressure decline under semi-steady state conditions is

wf i

q4Akt

pp ln 2 S

2kh cACr

πφµγ

æö

=− + +

ç÷

èø

(7.13)

The linear extrapolation of this line to small values of t gives the specific value of

= 2848 psia when t = 0 and inserting this condition in equ. (7.13) gives

q4A

pp ln 2S

4kh Cr

πγ

æö

−= +

ç÷

èø

or in field units

io A

p p m log logC 0.87S

æö

−= − +

ç÷

èø