semi-log analysis

7/22/2019 Semi-Log Analysis

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Well Test Interpretation Methodology

Gamma Experts Petroleum EngineeringYves Chauvel

PRACTICAL RESERVOIR MONITORING

September 2002

Semi-Log Analysis


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September 2002

Drawdown Semi-Log Analysis: the MDH Plot (Miller-Dyes-Hutchinson)

In drawdown analysis, the log approximation to the Exponential Integral gives:

which can be written as:

On the MDH plot, one can solve for m and b by reading the coordinates of two

points:

t = 0, pDd = pi, and

t = 1 hr, pDd = p1hr.

( )

+

+ S

rCkt

khq

pwt

Dd 86859.02275.3

loglog6.162

( ) btmpDd += log


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September 2002

Drawdown Pressure Profile: the MDH Plot

Because the pressure change is proportional to the logarithm of elapsed time when

IARF is reached, a graph ofP vs Log t will yield a straight line of slope m.

The effects ofwellbore storage and skin are superimposed onto the ideal response as

shown below.


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September 2002

Drawdown Semi-Log Analysis (contd)

The solution is then:

and

+

= 2275.3

log1513.1

1

wt

hri

rCk

mpp

S

m

qkh

6.162=


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September 2002

Build-up Analysis

In practice, it is not often possible to conduct drawdown analysis. This is because

drawdown analysis applies to a constant flow rate, a condition which is difficult tomaintain during well tests.

To remedy this shortcoming, it is more practical to analyze build-up periods by

resorting to the the principle ofsuperposition of states.

Modern well testing now offers multiple possibilities to analyze drawdown (flow)

periods by measuring the flow rates downhole during testing. For the interpretation,

the principle of superposition is generalized into a technique called thepressure-flow

convolution.


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September 2002

The Principle of Superposition of States

Because of the linearity of the pressure response equation, the response during a buid-

up period is equal to the sum of the responses of two drawdown periods:

- Flow rate q from time t = 0, and

- Flow rate -q from time t = tp (drawdown production time).


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September 2002

Single Flow Period Superposition for Build-up Analysis

Considering a single flow period of duration tp:

( ) ( )ttptpppp pDdDdwfiBu ++=


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September 2002

Build-up Semi-Log Analysis: the Horner Plot

For a single flow period, the superposition

function is the Horner time:

On a semi-log plot, the extrapolated pressureis the static reservoir pressure, provided that

- The reservoir has not entered

depletion regime during the drawdown.

-No late-time effects will affect the

buildup after the end of the buildup (this

is impossible to ascertain without

testing longer).

tttp

+


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September 2002

Build-up Semi-Log Analysis: the Horner Plot (contd)

On the Horner plot, the solution is again:

and

m

qkh

6.162=

+

= 2275.3rC

klogm

pp1513.1S

wt

wfhr1


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September 2002

Generalized Superposition for Build-up Analysis

When the well has been submitted to a series of flow periodsprior to build-up, one

must consider a generalized superposition function as follows:

( ) ( )( )

( )( )

=

=

=

tNi

1ii

tN

1iittln

qqq

tSn


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September 2002

Multi-Rate Build-up Analysis

When the pressures are plotted versus

Sn(t), the solution is identical to thecase of a single flow period (Horner

plot).

On a semi-log plot, the extrapolated

pressure is the static reservoir pressure,

with the same restrictions as apply to

the Horner plot.


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September 2002

Pressure Derivative

Log-Log Analysis


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September 2002

The Pressure Derivative

Modern well testing advances (1983) have culminated with the introduction of the

Pressure Derivative PD as an indispensable complement to plotting pressures versustime. By definition:

The Pressure Derivative is the slope of the semi-log plot as shown below.

( ) tdpd

ttdLn

pd'p

=

=


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September 2002

Dimensionless Variables

In order to solve the diffusivity equation in typical situations applicable to all possible

values of the physical parameters, one uses dimensionless variables defined asfollows:

Dimensionless distance: in which rw is the wellbore radius.

Dimensionless pressure: in which pi is the initial

pressure.

Dimensionless time: in which t is the elapsed time.

wD

rrr =

)(2 ppqkhp iD =

trC

ktwt

D =


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September 2002

Homogeneous Reservoir with Wellbore Storage and Skin

Because the skin just adds to the pressure drop in the wellbore, the dimensionless skin

S just adds to the PD function in the solution of the diffusivity equation for IARF:

In physical terms:

( )[ ]StLnp DD 280907.021 ++=

( )

+

+ SrC

kt

kh

qptp

wt

i 86859.02275.3

log)log(6.162


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September 2002

The IARF solution for a well with wellbore storage and skin has been expressed as:

In log-log analysis, it is preferrable to re-write the pressure response as:

in which CD is the dimensionless wellbore storage constant:

( )[ ]StLnp DD 280907.021 ++=

eLnCCtLnp

S

DD

DD

2

80907.021 ++=

hrCCC

wtD

2=

Homogeneous Reservoir with Wellbore Storage and Skin (contd)


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September 2002

Type Curves

By plotting the theoretical pressure

response PD versus tD/CD, (instead ofvs tD), one obtains a way of

characterising in a unique way the

IARF solution (for a well with

wellbore storage and skin forexample).

One thus defines an array of type

curves, each curve corresponding to a

value of the sensitivity parameterCDe**2S.

The inclusion of the pressure derivative

on this plot was a major breakthrough

in well test interpretation.


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September 2002

Attributes of the Log-Log Plot: Early Time Behaviour

At early times, the pressure response is dominated by the wellbore effect. The

solution of the diffusivity equation is:

This plots as a unit slope on a graph of pD vs tD/CD.

Then

and the derivative matches the pressure response on a unit slope.

This particularity of early time behaviour is one of the most conspicuous features of a

log-log plot in well test interpretation.

D

DD

Ctp =

( )D

D

D

D

DD

D

D

DD p

Ct

dtdp

t

Ct

dLn

dpp ===='


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September 2002

Attributes of the Log-Log Plot: IARF

The solution of the diffusivity equation for IARF is:

Then

When IARF is reached, the pressure derivative levels off to a plateau on the log-log

plot. The corresponding value of PD is 0.5. Again, this characteristic leveling off of

PD upon reaching IARF is one of the most conspicuous features of the log-log plot in

well test interpretation.

( )[ ]StLnp DD 280907.021 ++=

( ) 21' == DD

DtdLn

dpp


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September 2002

Type Curve Matching


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September 2002

Data Set and Type Curve Array

The data collected during a well test are in the form of couples (pressure-time). These

are initially presented as a log-log plot of pressure variations vs elapsed time, with thecomputation of the pressure derivative.

Type-curve matching has for objective the superposition of the data set over the array

of type curves corresponding to the model chosen, and the extraction of the test target

parameters.

This will be done by

- shifting the data horizontally (time match).

- shifting the data vertically (pressure match).

- finding the matching type curve (and its derivative) with its characteristic CDe**2S.


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September 2002

Data Set and Array of Type-Curves


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September 2002

Matched Data Set


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September 2002

Pressure Match: Extracting kh

From the expression of dimensionless pressure

one defines the pressure match Mp

Mp is read as the value of pD matching a specific value ofp. Then

pq

khpD =2.141

qkh

pp

MD

p2.141

=

=

pMqkh 2.141=


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September 2002

Time Match: Extracting C

From the expressions of dimensionless time and wellbore storage constant:

one defines the time match Mt

Mt is read as the value of tD/CD matching a specific value oft. Then

tM

khC 000295.0=

tC

khCtD

D =

000295.0

Ckh

tCt

M DD

t000295.0=

=


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September 2002

Skin Match: Extracting S

One reads the value of Ms on the matching type curve:

Then

with CD calculated from its dimensionless expression:

eCMS

DS2=

D

S

CMLnS

21=

hrCCC

wtD

8936.0

=


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September 2002

Type-Curve Match Example: Data Set

TCMATCH.WTD (Field Data)

1

10

100

1000

10000

0.001 0.01 0.1 1 10 100 1000

Pressure

change,p

si

Equivalent time, hrs


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September 2002

Type-Curve Match Example: Unmatched Overlay

TCMATCH.WTD (Drawdown type curve, Radial equivalent time)

Radial flow, Single porosity, Infinite-acting: Varying CDe2s

0.001

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10 100 1000 10000 100000

D

imensionlesspress

ure

Dimensionless time

0.001 0.01 0.1 1 10 100 1000

1

10

100

1000

Equivalent time, hr

P

ressurechange,psi


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September 2002

Type-Curve Match Example: Matched in Pressures



0.001

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10 100 1000 10000 100000

D

imensionlesspressure

Dimensionless time

0.001 0.01 0.1 1 10 100 1000

1

10

100

1000

Equivalent time, hr

Pressurechange,psi


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September 2002

Type-Curve Match Example: Matched in Both Times and Pressures



0.001

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10 100 1000 10000 100000

D

imensionlesspressure

Dimensionless time

0.001 0.01 0.1 1 10 100 10001

10

100

1000

Equivalent time, hr

Pressurech

ange,psi


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September 2002

Type-Curve Match Example: Extraction of Time, Pressure and Skin Match



0.001

0.01

0.1

1

10

100

0.001 0.01 0.1 1 10 100 1000 10000 100000

Dimensionlesspressure

Dimensionless time

0.001 0.01 0.1 1 10 100 10001

10

100

1000

Equivalent time, hr

Pressurecha

nge,psi

tD/CD=1

teq=0.0546 hr

p=262 psipD=10 CDe2s=7x109

semi-log analysis

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