Semilog Plot

Semi-Log Plot - Drawdown

p mlogtb

Even though this is the analogous MDH for a DD, the principles are equally applicable to BU analysis

‘MDH’ (Miller-Dyes-Hutchinson) plot, for a drawdown,

• Considering the ideal case, of putting on production a well with no wellbore storage and noskin, the blue curve is obtained. The straight line representing radial flow is established almost instantaneously, and from the slope of the line the permeability-thickness product, kh, is obtained.

• With wellbore storage but no skin the red curve is obtained. • With skin but no storage, the green curve shows radial flow

immediately, parallel to but offset from the ideal blue line.• A typical test will reveal both wellbore storage and skin,

corresponding to the black curve transitioning on to the green curve. The storage causes the delay, the skin the offset, and once again the final straight line slope is unchanged, as permeability is a reservoir property and is unaffected by near-wellbore effects.

S = 1.151 m – log Φµ CtrW

k2

+ 3.23

m = 162.6qµBkh

Semi-Log Plot – Drawdown

∆p1hr

Pressure vs Delta t Plot

∆p1hr = logΦµCtrW

mk

2– 3.23 + 0.87 S

m = 162.6qµBkh

Semi-Log Plot – Drawdown - MDH

Delta Pressure vs Delta t Plot

It is strictly valid only for the first ,ever drawdown on a well but can in exceptional circumstances be used for analysis of a later drawdown or even a build-up.

Semilog – Example

The data summarized at Semilog Example.xls were recorded during apressure draw down test from an oil well. Estimate the effectivepermeability to oil and the skin factor using the semilog graphicalanalysis technique for a constant-rate drawdown flow test.

Q = 250 STB/DPi = 4,412 psiaH= 46 ftPoro = 12 %Rw= 0.365 ftB=1.136 RB/STBCt= 17 E-6 psi-1Visc= 0.8 cp

Semilog Example.xls

S = 1.151 m – log ΦµCtrW

Semilog – Example

qBµmh

k 162.6∆p1hr k

2+ 3.23

Pressure Derivative

• Pressure derivative analysis is based on the observation the pressure variation that occurs during a well test is more significant than the pressure itself.

• The use of pressure derivatives makes the well test interpretation easier in a number of ways:

• Derivative type curves increase the possibility of converging to a unique model (i.e., solution).

• Derivative analysis makes it easier to identify a semilog straight line.

• Derivative analysis makes it easier to identify the type of reservoir heterogeneity, because the signatures or patterns it provides are more definitive.

Pressure Derivative

⇒

Pressure DerivativeThe basic idea of the derivative is to calculate the slope at each point of the pressure curve on the semi-log (superposition) plot, and to display it on the log-log plot.

Points 1 and 2 fall on the wellbore storage unitslope in early time and, during the transition to IARF, the derivative peaks at point 4. The transition is complete at point 6, as the derivative flattens to a value equivalent to m.

Drawdown Test and Pressure Derivative AnalysisTransient flow period:

2332

.rc

kLogt Logmppp

wtpwfi

06162

m and kh

Bq.m sc

p

p

pp

p

pp t Lnd

dt

dt

pd.

t Logd

dt

dt

pd

t Logd

pd

30262

m

dt

pdt.

t Logd

pd

pp

p

30262

In radial flow geometry, the pressure drop during the transient period is expressed by:

where

From the above equation,

or

The above development implies that a log-log plot of [d(p)/d(Log tp)] versus tp should yield a

horizontal line with an intercept equal to m (Figure 1 ).

Figure 1

Procedure for Derivative Analysis

t

ptPDrivative

.

11

11)(

ii

iiiiDrivative tt

pptP

To calculate the pressure derivative curve we need to use the formula of derivative which is:

ti-1 Pi-1

ti Pi

ti+1 Pi+1

Illustration of Pressure Derivative Method

LOG

∆p

& LOG

∆p' ∆

t

Δt0

Wellbore Storage and Skin

Wellbore Storage

0

DERIVATIVE TYPE CURVE FOR DRAWDOWN ANALYSIS IN DIMENSIONLESS TERMS:

D

DD c

tp

1

D

D

D

ctd

dp

During early times when wellbore storage effects dominate,

where c D is the wellbore storage constant. By taking the logarithm of both sides, we obtain

Log pD = Log tD -Log cD

The above equation shows that log p D versus log tD is a unit slope line when wellbore

storage effects dominate. Now we can examine the behavior of the derivative

Therefore, when wellbore storage effects dominate, the derivative of the pressure curve with

respect to tD/cD also has a unit slope (Figure 2 ,

DRAWDOWN ANALYSIS IN DIMENSIONLESS TERMS:

During the radial flow period, the dimensionless form of the drawdown equation is:

s

DD

DD ecLn.c

tLnp 2809702

1Then, 2

1

D

D

D

ctd

dp

The above equation implies that the derivative plot during radial flow will generate a horizontal line with a value of 0.5 (Figure 2).

Properties of the Derivative

Wellbore Storage & Skin ResponseDimensionless Groups:

A “rule of thumb, ” developed from the fundamental solutions of the diffusivity equation including wellbore storage and skin effect (Agarwal et al., 1970), suggests that the “transition” period lasts 1.5 log cycles from the cessation of predominant wellbore storage effects (unit slope line). Points beyond that time fall on a semi-log straight line.

Log-Log Example.xls

Log-log – Example

The data summarized at Log-log Example.xls were recorded during apressure draw down test from an oil well.Your task is to change thevalues of k, S and C until they visually fit the test data in the Log-logplot.

k(md) 8.7

S 5

C(STB/d) 0.03

Log-log Example answer

The data summarized at Log-log Example.xls were recorded during apressure draw down test from an oil well.Your task is to change thevalues of k, S and C until they visually fit the test data in the Log-logplot.

Log-Log Example.xls

Build-up Tests

Build-up testA well already flowing (ideally at constant rate) is shut in andpressure is measured

Practical advantage: constant (zero) rate more easily achievedAnalysis often require slight modification of the techniques fordrawdown

Build-up

Build-up Analysis

q

tp

0

t

pi

RATE

p

tp

t = t - tp

Buildup Test

• Drawdown data quality is subject to many operational problems; slugging, turbulence, rate variation, inaccurate rate measurements, instability, unsteady flow, plugging, interruptions, equipment adjustments, etc…

• Buildup is measurement of pressure and time when well is shut-in.• In high permeability reservoirs the pressure will buildup to a stabilized

value quickly, but in tight formations the pressure may continue to buildup for month before stabilization attained.

• Buildup must be preceded by flow period.• Simplified Analysis assumes constant flow rate for a duration t hours.• Shut-in time, Δt, measured from end flow.• Buildup Analysis treated as superposition of flow and injection.• Analysis of buildup data may yield the values of K, S, and the average

reservoir pressure.

Methods of analysis:

•Horner plot (1951): Infinite acting reservoir

•Matthews-Brons-Hazebroek (MBH,1954): Extension of Horner plot to finite reservoir.

•Miller-Dyes-Hutchinson (MDH plot, 1950): Analysis of P.S.S. flow conditions.

Buildup is always preceded by a drawdown and the buildup data are directly affected by this drawdown.

Behavior of Static Sandface Pressure Upon Shut-in of a Well

Reflects “kh”

Reflects the wellbore storage (afterflow)

Reflects the effects of boundaries.

S

rc

kt

kh

qBpp

wt

o

wfi87.023.3loglog

6.1622

•Flowing sandface pressure during drawdown

•Shut-in wellbore pressure: The static sandface pressure is given by the sum of the continuing effect of the drawdown rate, qsc, and the superposed effect of the change in rate(0-qsc)

Src

kt

kh

Bq

Src

ktt

kh

qBpp

wt

o

wt

o

wsi

87.023.3loglog06.162

87.023.3loglog6.162

2

2

t

tt

kh

qBtpp o

wsilog

6.162 Horner plot relationship- Infinite acting reservoir

t

tt

kh

qBtpp o

wsilog

6.162

Horner plot relationship

kh

qBm o

6.162

t

tttimeHorner

Slope of semilog straight line same as drawdown – used to calculate permeability.

Buildup test does NOT allow for skin calculation. Skin is obtained from FLOWING pressure before shut-in.

t

tt

kh

qBS

rc

kt

kh

qBtpttp po

wt

p

o

pwfpwslog

6.16287.023.3loglog

6.1622

Src

k

tt

tt

kh

qBtpttp

wtp

po

pwfpws87.023.3loglog

6.1622

hrt 1

23.3

1log151.1

2

1

wtp

pwfhr

rct

tk

m

ppS

23.3

1log151.1

2

1

wtp

pwfhr

rct

tk

m

ppS

Detecting Faults from Buildup

41

42

Pw(∆t)=pi- mlogTp+∆t

∆t

∆t

late time

m

early time

Build-Up Analysis

Log tp + ∆t

∆ t

m =162.6qBµ

kh

Infinite acting flow period

Horner time

tp+ ∆t

∆p1hr – log

m = 162.6qµBkh

S = 1.151 mk

θµCtrW2

+ log tp+ 1 + 3.23tp

Horner Plot

Horner Plot – Example

The data summarized at Horner Example.xls were recorded during apressure build up test from an oil well. Estimate the effectivepermeability to oil and the skin factor using the Horner plot graphicalanalysis technique.

tp= 25 hrQ = 542 STB/DH= 87 ftPoro = 0.07 %Rw= 0.25 ftB= 1.56 RB/STBCt= 6E-6 psi-1Visc= 0.75 cp

Horner Example.xls

Equivalent Time

Agarwal proposed;

Valid for infinite acting radial flow

Build-up Type-curve will match Drawdown type-curve

(∆t)eq =∆t

1+∆ttp

Principle of Superposition

Superposition Concept

Superposition consists of making linear combinations ofsolutions of simple problems to form the solution of acomplex one.

Superposition in time: With simple drawdownsolutions, the solution of a test with complexproduction history is constructed

Boundary effects: The response of several wells(or virtual wells) is used to construct the solutionof test that sense boundaries

Main Superposition Principles

For a diffusion problem involving the diffusivity equation,initial conditions and boundary conditions that are alllinear:

A linear combination of solutions honoring thediffusion equation also honor this diffusionequation

At any well or boundary, the flux resulting fromthe linear combination of solutions will be thesame linear combination of the correspondingfluxes.

Main Superposition Principles

If a solution composed of a linear combination ofsolutions to the diffusivity equation is found andhonors all boundary and initial conditions, it is thesolution to the problem. Elementary solutions mayor not be physical as long as they honor thediffusivity equation.

6

Superposition Rules

Derived from the superposition principles

If one considers the pressure change due to aunit rate production, the pressure change due tothe production of the same system at rate q willbe q times the unit rate solution

Superposition Rules

To simulate the sequence of a constant rate q1

from time zero to time t1, followed by theproduction q2 from time t1 to infinity, you cansuperpose the production at rate q1 from time zeroto infinity and production of rate (q2 – q1) fromtime t1 to infinity.

Principle of Superposition

∆pBU ~ ∆pDD fortp >> ∆t

Superposition in Time - Buildup