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4.2 HOMOGENEOUS RESERVOIR MODEL- SLIGHTLY COMPRESSIBLE LIQUIDS

The basic of well-test analysis techniques for homogeneous-acting reservoir is the line-source (Ei-

funtion) solution to the diffusivity equation.

The relationship between bottomhole flowing pressure (BHFP), Pwf and the formation and well

characteristics for a well producing a slightly compressible liquid at a constant rate is

4.1

If we change from natural logarithms to base 10 logarithms and simplify, we can rewrite Eq. 4.1 in

a more familiar form,

..(4.2)

Where the skin factor,s, is used to quantify either formation or stimulation. Skin affects are

discussed later.

4.2.1 ANALYSIS OF CONSTANT-RATE FLOW TESTS

Eq. 4.2 describes the variation of the wellbore pressure with time when a well is produced at

constant rate. Production at a constant rate can be considered a pressure-drawdown or-flow test.

Comparing Eq. 4.2 whit the equation of a straight line, y=mx+b, suggests an analysis technique in

which the following term are analogous:

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(4.3)

These analogies indicate that a plot of Pwf Vs. log t will exhibit a straight line from whitch the

slope, m, allows us to estimate k and s. fig. 4.1 is an example semilog graph of constant-rate flow

test date. The slope of the line, m, is the difference between the pressure, Pwf and Pwf,, one log

cycle apart, or m=Pwf,-Pwf,.

For single-phase flow, the formation permeability in the drainage area of the well is computed

from

Where the absolute value of m is used rearranging Eq.4.2 and combining with Eq. 4.3 gives. For

convenience, we set the time, t, equal to 1hour, and use the symbol P1hr for the BHFP AT THIS

TIME. NOTE THAT p1HR necessarily lies on the semilog straight line substituting these into Eq. 4.4

yields

In summary, from the straight line predicted by theory for a plot of constant-rate flow test data on

semilog graph paper, we can estimate k and s.

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4.2.2 ANALYSIS OF PRESSURE-BUILDUP TESTS

AN EQUATION MODELING A PRESSURE-BUILDUP TEST CAN BE developed by use of superposition

in time. In term of the line sourcesolution given by Eq. 4.2, the bottomhole pressure (BHP) for the

rate history shown in fig. 4.2 is

Where

= shut-in BHP,

= DURATION OF THE CONSTANT-RATE PRODUCTION PERIOD BEFORE SHUT-IN, AND

= DURATION OF THE SHUT-IN PERIOD

IF WE COMBINE TERM AND SIMPLIFY, Eq 4.6 can be rewrite as

Comparing eq. 4.7 to the equation of straight line, y=mx+b, gives

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This suggests that a plot of shut-it BHP, Pws from a buildup test as a function of the log of the

horner time ratio function,(tp+t)/ t, will exhibit a straight line with slope m.

The slope is the difference between two values of pressure, Pws+1 and Pws, 2 one log cycle apart.

To calculate permeability, we use the absolute value of the slope, or

From the semilog, the original reservoir pressure, Pi, is estimated by extrapolating the straight line

to infinite shut-in time where fig.4.3 illustrates calculation of the slope and

original reservoir pressure.

We also can solve for the skin factor, s, from a pressure-buildup test. At the instant a well is shut-

in, the BHFP IS

COMBINING Eqs. 4.7.4., and 4.9, we can derive an expression for the skin factor

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where m=slope of the semilog straight line. Setting t= I hour, introducing the symbol P1hr for

Pws at t=1 hour on the semilog line, and neglecting the term log gives

Where at the instant of shut-in.in summary, using information obtained from a plot

of Pws vs log we can estimate k,pPb and s

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