ENM200 Well Testing Notes

PgDip/Msc Energy Programme/ Subsurface Well Testing

© Robert Gordon University Page 1 of 25

Well Testing

Summary

This topic reviews:

Purpose of Well Testing

Appraisal, Exploration and Development Well Testing

Transient Well Test Analysis

o Essential Equations

o Pressure Drawdown and Pressure Buildup Tests

o Type Curve Analysis

o Gas Well Testing

o Deliverability Testing

o Drill Stem Testing

Table of ContentsSummary .................................................................................................. 1

Table of Contents ....................................................................................... 1

1. Introduction ........................................................................................ 2

2. Purpose of Well Testing......................................................................... 2

2.1 Appraisal/Exploration Well Testing.................................................... 2

2.2 Development Well Testing ............................................................... 5

3. Transient Well Test Analysis - Fundamentals............................................ 5

3.1 Radial Flow Equation........................................................................... 5

3.2 Pressure Response ............................................................................. 7

3.3 Pressure Drawdown Analysis................................................................ 9

3.4 Pressure Buildup Analysis .................................................................. 12

3.5 Type Curve Analysis ......................................................................... 16

3.6 Gas Well Testing .............................................................................. 19

3.7 Deliverability Testing ........................................................................ 21

3.8 Drill Stem Testing ............................................................................ 22

4. Example Calculations .......................................................................... 23

Horner Pressure Buildup Analysis ............................................................. 23

References .............................................................................................. 25



1. Introduction

There are a number of different sources and techniques available to help generate acomprehensive description of the reservoir in order to make important development decisions andreliable performance predictions.

Some of the most important are:

Geophysical and Geological Studies.

Well Drilling Data; Core analysis and Log interpretation.

Pressure Testing of Wells.

It is essential that information from these different areas is brought together to assess andcompare when generating a description of the reservoir. This will help eliminate inconsistenciesin the data, minimise assumptions and provide the most complete representation of the reservoir.

Pressure testing of wells involves inducing pressure changes in the reservoir by flowing (andclosing in) a production well and measuring the resultant pressure response. Pressure testingcan also be carried during injection into a well and is termed injectivity testing.

The pressures are measured on a gauge in the wellbore. The wellbore flowing pressure (pwf) and

the wellbore static (shut-in) pressure (pws) are recorded as a function of time.

These rate, time and wellbore pressure measurements are then interpreted to determinereservoir pressure, reservoir size and productivity.

2. Purpose of Well Testing

2.1 Appraisal/Exploration Well Testing

Most offshore exploration and appraisal well tests are designed to measure or determine thefollowing:

Initial Reservoir Pressure (pi psia)

The initial reservoir pressure can be determined from a pressure buildup plot,although it is more routinely obtained by running an MDT (Modular Dynamic Tester –proprietary name for Schlumberger’s tool).

Production Rate (Oil, qo stb/d)

The production rate is essential in determining the commercial viability of theaccumulation.

Collection of Fluid Samples

The importance of collecting reliable fluid samples cannot be overemphasized sincethe PVT properties are used in virtually all reservoir engineering calculations:determination of hydrocarbons initially in place, the behaviour of the reservoirduring production, the recovery efficiency and design of the surface facilities.



Skin Factor (S, dimensionless)

The skin factor, S, is a dimensionless number that represents the level of formationdamage caused while drilling and completing a well. This damage affects theefficiency with which a well can produce. In appraisal wells the damage could besubstantial and must be determined in order to evaluate the performance of futuredevelopment wells which will be designed and drilled to eliminate formation damage.This will help determine the number of wells required to develop the accumulation.

A wells performance is defined by its Productivity Index (PI) which is the ratio of awells production rate to it’s drawdown at that particular rate. The drawdown is thedifference between the flowing bottom hole pressure and the wells static pressure.Basically, the higher the PI, the more productive the well for a given drawdown.Ways to increase a well’s PI include skin removal, fraccing and horizontal drilling.

During a well test the observed productivity index of a well can be defined byEquation 2.1.

psi/d/stbp-p

q

DrawdownPressure

RateOilPI

wfi

o

EQUATION 2.1

Where:

qo is the oil rate (stb/d).

Pi is the initial reservoir pressure (psia).

Pwf is the wellbore flowing pressure (psia).

In Equation 2.1 the total drawdown includes ∆pskin, the additional pressure dropacross the damaged zone (reduced permeability zone). It’s effect on flowing bottomhole pressure is shown in Figure 2.1 and it is defined in terms of the dimensionlessskin factor, S, in Equation 2.2.

psiShk

Bμq141.2Δp oo

skin

EQUATION 2.2

Where:

qo is the oil rate (stb/d).

µ is oil viscosity (cp).

Bo is the oil formation volume factor (rb/stb).

k is the average effective permeability to oil in the presence of irreducible water (mD).

h is the effective formation thickness contributing to flow (ft).

S is the skin factor (dimensionless).



Figure 2.1 Effect of ∆pskin on flowing bottom hole pressure

Reservoir Characteristics

Permeability

The permeability of the formation determined from well testing is the ‘average,effective permeability to oil in the presence of irreducible water’. It can be used tomatch or predict well performance using analytical or numerical simulationtechniques.

Its value is normally smaller than the permeability value obtained from routine coreanalysis which is the ‘absolute permeability of the reservoir rock’. The absolutepermeability is defined as a measure of the ability of the permeable rock to transmit afluid when only one fluid is present in the rock.

Fractured and Layered Reservoirs (Dual Porosity Systems)

Naturally fractured reservoirs in which there is a connected network of fractures andsurrounding supporting system (the matrix) are often described as classic dualporosity reservoirs. Dual porosity systems consist of two porous regions whichexhibit markedly different properties. In the case of fractured reservoirs theconnected network of fractures feeds fluid to the well and the matrix supplies thefracture network.

Two layer reservoirs can also be described as dual porosity systems. In this case alow permeability or tight layer would be adjacent to a much higher permeability zone.Well test interpretation in these two cases is more complex than reservoirs whichexhibit homogeneous behaviour.

Reservoir Boundaries

Reservoir boundaries (faults) and pressure depletion can be determined frompressure buildup analysis.

Hydrocarbon contacts may also be inferred if there is a significant mobility contrastbetween the fluids.

pwf

Log r

∆ps

FormationDamaged

Zone

rw



2.2 Development Well Testing

Most development well tests are designed to monitor reservoir pressure trends and determinechanges to formation parameters for reservoir management purposes:

Reservoir Pressure

The average pressure ( p ) within the drainage area of an individual well can be

determined from pressure buildup tests. Average pressure decline trends forindividual wells and the reservoir system can then be monitored and history matchedusing material balance and numerical simulation techniques.

Skin Factor

Changes in the magnitude of the skin factor of a well can be monitored from regularpressure buildup tests. Large skin factors could then be reduced or eliminated byremedial workover treatments to improve the wells productivity.

Productivity Index

Changes in a wells flow efficiency can be monitored from regular pressure builduptests.

Effective Permeability

Changes to the effective permeability of development wells can be monitored fromregular pressure buildup tests. A reduction in permeability could be attributed toreservoir compaction in depletion drive developments. The effectiveness ofstimulation treatments could also be regularly monitored by assessing changes in theeffective permeability.

3. Transient Well Test Analysis - Fundamentals

3.1 Radial Flow Equation

Most pressure analysis techniques are based on solutions to the radial diffusivity equation shownin Equation 3.1 This equation models how pressure changes with location and time. It is given inradial form to represent the flow of fluids into a wellbore. (For full derivation of this equation seeRef 2, Chapter 5 or Ref 3, Chapter 2)



t

p

k

cμ

0.000264

1

r

p

r

1

r

p t

2

2

EQUATION 3.1

Where:

p = pressure, psi

r = radial distance, ft

ct = total compressibility, psi-1

µ = fluid viscosity, cp

k = effective permeability, mD

= porosity, fraction

t = time, hours

This linearised differential equation assumes horizontal fluid (liquid) flow, negligible gravity effectsand a homogeneous, isotropic formation. It assumes that the well is perforated across the entireformation thickness and the pore space is 100% saturated with any fluid. It also assumes that thefluid has a constant viscosity and a small and constant compressibility. It combines the basicphysical principles of mass conservation, Darcy’s law and isothermal compressibility.

Darcy’s law is defined as a measure of the ability of the permeable rock to transmit a fluid whenonly one fluid is present in the rock. It is a measure of the ease of flow of a fluid through a porousmedium and for one dimensional linear horizontal flow Darcy’s law (in Darcy units) is:

cmlengthL

cpviscosityfluidμ

atmpressurep

cmflowtoareasectionalcrossA

DDarcy,ty,permeabilik

cc/secrateflowq

ΔL

Δp

μ

kAq

2



And can be expressed in field units as;

rb/stbfactorvolumeformationoilo

B

cpviscosityfluidμ

psi/ftgradientpressureΔl

Δp

ftflowtoareasectionalcrossA

mDtypermeabilikstb/drateproductionq

Δl

Δp

oμB

kA1.127x10q

2

3

Field units of permeability are Darcy, D and milliDarcy, mD, (Darcy/1000).

Equations similar in form to Equation 3.1 are encountered in the theory of diffusional transportprocesses, for example in heat conduction and equations of this type are referred to as ‘diffusivity’equations.

The quantity:cμ

k

tis known as the hydraulic diffusivity constant and its magnitude determines

the depth of investigation into the reservoir as observed by the pressure response during thetest.

3.2 Pressure Response

Solutions to the radial diffusivity equation (Equation 3.1) are dependent on the initial andboundary conditions imposed in the well test. One important solution is known as the ConstantTerminal Rate (CTR) solution and it forms the basis of other more complicated solutions.

This solution describes the pressure response observed on a gauge located in the wellbore of awell that has been produced at a constant rate, q, from time t = 0. It is the equation of bottom

hole flowing pressure (pwf) versus time for constant rate production for any value of flowing time.

The CTR solution can then be described for two extreme outer boundary conditions; the BoundedReservoir System and the Steady State System (Open outer boundary) as discussed below.

Bounded Reservoir System

The flowing bottom hole pressure (BHP) response for a closed system for the constant terminalrate solution is shown in Figure 3.2.



Figure 3.2 Decline in bottom hole flowing pressure for a bounded system

In this case the pressure declines continuously and can be described by three different periods;transient, late transient and semi–steady State (sometimes called pseudo-steady state) eachcorresponding to a specific physical state of the reservoir.

In the transient flow period the wellbore pressure response is unaffected by any faults orboundaries in the reservoir. This period is also called the ‘Infinite Acting Radial Flow (IARF)period’.

The most common values calculated from the transient flow period are the effectivepermeability, the skin factor and the initial reservoir pressure. These values together withthe shape and size of the drainage area can be used to calculate or predict the well

production rate for different flowing bottom hole pressures, pwf .

In the late transient flow period the wellbore pressure response is affected by the reservoirboundaries. Both the shape of the drainage area and the location of the well with respect to theboundaries may influence the pressure response.

In the semi-steady state flow period the wellbore pressure response is affected by all the outerboundaries in the reservoir and for a constant production rate, the rate of change of pressure withrespect to time is constant i.e.

constanttd

pd wf

In an appraisal well test this pressure response may indicate the presence of a very smallhydrocarbon volume.

In a producing field, equilibrium between the production from all wells may be reached and foreach well pseudo-steady state flow will occur within the well drainage area. The analysis ofpressure behaviour in non-symmetrical drainage systems involves a shape dependent constant,CA, known as a Dietz shape factor.

Steady State Reservoir System

The flowing bottom hole pressure response for a steady state system for the constant terminalrate solution of the radial diffusivity equation is shown in Figure 3.3.

Time, t

Late Transient

Transient

Semi-Steady StatePressure, pwf



Figure 3.3 Steady-state wellbore pressure response

In this case, after the initial transient pressure drop, the pressure at the wellbore and throughoutthe whole system does not vary with time, i.e.

0td

pd

This pressure response can be observed for the following systems:

The reservoir is completely ‘recharged’ by a strong aquifer.

In high flow capacity reservoirs.

In reservoirs with a gas-cap.

When injection and production are balanced (total voidage replacement).

3.3 Pressure Drawdown Analysis

A pressure drawdown test involves producing a well at a constant rate from time t = 0, after it hasbeen shut in. Bottom hole flowing pressures are then continuously measured as a function oftime. This is known as a single-rate drawdown test and the rate schedule and pressure responsefor this test is shown in Figure 3.4.

Variable rate testing involves flowing the well at a series of different rates (increasing ordecreasing) for different periods of time. A rate schedule and resulting pressure response for anexample multi-rate test is shown in Figure 3.5.

Time, t

Transient

Steady-state

Pressure, pwf

0dt

pd



Figure 3.4 Rate and bottom hole pressure response for single–rate test

Figure 3.5 Rate and bottom hole pressure response for multi–rate test

In pressure drawdown analysis during the purely ‘transient’ pressure response, the constantterminal rate solution of the radial diffusivity equation can be approximated by the ‘line-source’solution, also known as the ‘exponential integral’ solution. A detailed review of this solution canbe found in References 1,2,3 and 4.

The solution results in the following familiar form of the pressure drawdown equation (Equation3.2) for the transient flow period:

BHP, pwf

Time, tt = 0

Rate, q

Time, t

q = 0Closed -In

Producing

t = 0

Rate, q

Time, t

BHP, pwf

Time, t



0.87S3.23-rc

klogtlog

hk

Bq162.6-pp

2

wt

iwf

EQUATION 3.2

Where:

p = pressure, psi

q = flow rate, stb/d

B = formation volume factor, rb/stb

ct = total compressibility, psi-1


h = formation thickness, ft

= porosity, fraction

µ = fluid viscosity, cp

r = radial distance, ft

S = skin factor, dimensionless

t = time, hours

Equation 3.2 can be rearranged to give a straight line relationship between pwf and log t as:

hr)(1wfwf ptLogmp

The semilog plot of pwf vs log t is shown in Figure 3.6. Any deviation from the straight line duringthe very early time period could be attributed to wellbore storage effects. Wellbore storageeffects make the very early transient pressure behave as though it were reflecting production onlyfrom the expansion of the fluid in the wellbore rather than the formation.

Deviation from the straight line during the late time period can be attributed to boundary effects asthey interrupt the infinite acting pressure behaviour and the system moves into semi-steady statebehaviour.

Figure 3.6 Semilog plot of flowing pressure performance

From this semilog straight line plot the slope, m is:

BHP, pwf

Log t

Infinite acting,

Transient period

Early time,Wellbore Storage

Late time,Boundary effects



cyclelog/psihk

Bq162.6-m

From which the permeability thickness product, kh can be obtained.

The skin factor can be obtained from the value of pwf taken from the straight line for a flowing timeof one hour and solving Equation 3.2 to give:

3.23rc

klog-

m

p-p1.151S

2

wt

ihr)(1wf

Multi-Rate Testing and the Principle of Superposition

The analysis presented above assumes a constant rate production history. In practicemaintaining a constant flow rate is difficult and analysing more complex variable rate histories ispossible using the Principle of Superposition. Basically this principle states that it ismathematically possible to generate the solution of a complex problem by combination of thesimpler linear solutions.

In well testing superposition in time is used to account for flow rate changes (including zero ratewhen the well is shut-in) and superposition in space is used to account for multiple wells anddifferent boundaries. Superposition is also known as Convolution.

Reservoir Limit Testing

The flow period of a drawdown test can be extended to increase the reservoir volumeinvestigated by the test. If produced for long enough the test can be used to estimate the initialoil in place. Once a semi-steady state pressure decline has been established the slope, m*, of

the straight line section of a plot of pwf versus t may be used to estimate the connected reservoirdrainage volume using the following equation:

*mc

Bq0.234-hA

t

o Where Ah is the net rock volume in cuft

and the STOIIP, N, can be estimated from:

STOIIP =*mBc

)S1(Bq0.0417-

oit

wco

3.4 Pressure Buildup Analysis

A pressure buildup test involves shutting in a producing well and recording the closed in pressureas a function of time. It is probably one of the most extensively used transient well testingprocedures. The rate schedule and pressure response for this test are shown in Figure 3.7.



Figure 3.7 Rate and pressure response for a buildup test

There are a number of ways of analysing the results of a buildup test, one of the most well knownbeing the Horner method. The buildup test rate schedule is the simplest form of a two rate testin which the second rate is zero and analysis involves the principle of superposition.

Horner Buildup Analysis

If the producing time, tp was long enough to establish infinite acting radial flow (IARF) then the‘Horner’ pressure buildup equation (for IARF flow) is given by Equation 3.3:

t

ttlog

hk

Bq162.6-pt)(p

p

iws

EQUATION 3.3

Where:

p = pressure, psi

q = flow rate, stb/d

B = formation volume factor, rb/stb


h = formation thickness, ft

t = time, hours

Equation 3.3 describes a straight line with intercept pi and slope –m. It is known as the HornerBuildup Plot or Semilog Plot:

(Wellbore Static Pressure)

Rate

Time

q

tp ∆t

Time

Pressure

pi

pwf

pws

tp ∆t



t

ttlogm-pt)(p

p

iws

The permeability-thickness product can be calculated from:

ftmDm

Bq162.6kh

Using the superposition principle means that the skin factor does not appear in the pressurebuildup equation but it can still be obtained from the buildup data and the flowing pressure prior tothe buildup as:

3.23rc

klog-

t

1tlog

m

p-p1.151S

2

wtp

p0)t(wfhr)(1ws

Where pws(1 hr) is obtained from the straight line portion of the pressure buildup curve 1 hour after

shut-in, m is the slope of the linear buildup (psi/log cycle) and pwf is the final flowing pressure.

This skin factor equation can be used if the producing time, tp is of the order of 1 hour (as in

drillstem testing), for much larger flowing times the term: log (( tp + 1) / tp), can be omitted.

The Horner Buildup plot is shown in Figure 3.8. The initial part of the plot is non-linear due to

wellbore storage effects. The extrapolated buildup pressure, p*

has no real physical meaningexcept in the infinite reservoir case where the pressure would continue to build up in infiniteacting radial flow to the initial reservoir pressure.

Of course no reservoir is infinite, but if the production volume prior to the buildup was negligible

compared to the oil in place then; p*

~ pi ~ p . Where p is the average drainage area pressure.

Figure 3.8 Horner Buildup Plot

t

ttlog

p

pws

Small ∆t

kh

Bq162.6m

p*

p(1 Hr)



Miller-Dynes-Hutchinson (MDH) Buildup Analysis

Another well known technique for analysing the results of a transient build up test is that of Miller,Dynes and Hutchinson (MDH).

The MDH buildup analysis involves plotting pws (or the pressure change, ∆p) versus log ∆t as

shown in Figure 3.9. This can be carried out when the producing time (tp) is significantly greaterthan the shut in time (∆t).

The slope of the straight line (m), gives the permeability-thickness product which is identical tothat obtained from the Horner plot:

ftmDm

Bq162.6kh

and the skin factor (which for the MDH method is independent of the producing time, tp), can becalculated as:

3.23rc

klog-

m

p-p1.151S

2

wt

0)t(wfhr)(1ws

Where pws(1 hr) is obtained from the straight line portion of the pressure buildup curve 1 hour after

shut-in, m is the slope of the linear buildup (psi/log cycle) and pwf is the final flowing pressure.

The initial part of the plot is non-linear due wellbore storage effects. The linear buildup trend can

be extrapolated to give the initial reservoir pressure, pi in an initial well test (infinite-actingreservoir).

Figure 3.9 MDH Buildup Plot

Radius of Investigation (Drainage)

The radius of investigation is defined as the distance seen into the reservoir when infinite acting(transient) flow conditions exist. It is defined as:

t

invc

tk0.03r

pws

Shut in time, log ∆t

kh

Bq162.6mSlope

End of wellborestorage

pi



Where rinv is the radius of investigation in feet and t is the total producing or closed in time inhours during infinite acting radial flow.

Fault Detection

Single faults or a system of faults (boundaries) can be detected from various types of well testand are determined using the method of images, (Ref 5). An example Horner buildup plot for asingle linear fault is shown in Figure 3.13.

In this plot the permeability and skin factor can be obtained in the usual way from the infiniteacting flow period (first straight line). The doubling of the slope then indicates the presence of asingle sealing fault. The distance to the fault can be estimated from:

t

x

c

tk0.01217d

Where d is the distance to the fault in feet and ∆tx (the closed in time in hours) is the point atwhich the two semilog straight lines intersect.

Figure 3.13 Horner plot for buildup data (Single fault case)

3.5 Type Curve Analysis

Many of the analysis techniques used in modern well testing software packages are based ontype curve interpretation.

Early type curve matching involved plotting ‘dimensionless’ pressure versus a ‘dimensionless’time group on a log-log scale. This could be carried out for either drawdown or buildup pressureresponses.

t

ttlog

p

pws

Slope = m

Slope = 2m



Dimensionless pressure is defined as:

p-pBq141.2

hkp iD

and the dimensionless time group as:

C

thk0.000295

C

t

D

D

Where CD is the dimensionless wellbore storage constant and C is the wellbore storageconstant which is a measure of the wells capacity to store fluid.

The technique of converting variables into their ‘dimensionless’ form is a well knowntechnique which simplifies the mathematics involved in solving complex equations. In thecase of well test interpretation the technique helps simplify solutions of the radial diffusivityequation.

The type curve analysis technique is possible because the actual pressure response has thesame shape as the dimensionless pressure response on a log-log scale. Matching the datainvolves plotting actual pressure differences as a function of time and overlying the plot on pre-defined dimensionless type curves. The matched data can then be used to determine thepermeability and skin factor from the infinite acting radial flow period. (Although the skin factor isnot defined in the dimensionless parameters given above, pre-defined type curves have been

generated for specific values of the grouped term,2S

DeC , from which the skin factor can be

obtained)

A set of pre-defined log-log type curves are shown in Figure 3.10 where the approximate start ofthe semi-log straight line refers to the beginning of infinite acting radial flow (IARF).

This early type curve analysis had various shortcomings including poor resolution of the log scaleand was improved significantly by the addition of the ‘derivative’ type curve which is the currenttechnique used for flow regime diagnostics.

Figure 3.10 Log-log type curves (from A.C. Gringarten, SPE Paper 8205, 1979)



Derivative Type Curve Analysis

The reservoir properties of permeability and skin factor have been determined from semilog(Horner, MDH) or log-log graphs of pressure versus time. It is also possible to determinereservoir characteristics from techniques which are based on the rate of change of pressure withtime. This type curve matching involves matching a ‘derivative’ type curve where the derivative isthe slope of the semilog plot presented on a log-log scale, (Figure 3.11).

On the derivative plot in Figure 3.11 the early time wellbore storage is seen as a unit slopestraight line, (m1). The infinite acting radial flow (IARF) period is shown as m2, a horizontal line,which makes identifying this flow period much easier than on the semi-log plot. Figure 3.11 alsoshows late time effects as a unit slope, m3.

The matched data can then be used to determine the permeability and skin factor from the infiniteacting radial flow period and also help define flow regimes. Figure 3.12 depicts various flowregimes during early, middle and late time pressure responses.

Figure 3.11 Semilog and derivative type curve.

Figure 3.12 Derivative type curves showing various flow regimes

Derivative type curves are routinely used in well test analysis and are considered to beone of the key techniques for flow regime diagnostics. All modern software packagesavailable for pressure transient analysis are based on identifying flow regimes in order togenerate the most appropriate model which can be used to help predict future wellperformance.

t

Early Time

Near Wellbore

Middle Time

Reservoir

Late Time

Boundaries

Constant-pressure

Outer Boundary

Dual Porosity

(Infinite acting)

Radial flow

Well

boreSto

rage

Linear Flow

(Fracture, Horizontal Well)

Pressure

Derivative

∆p

m1

m3

m2IARF

t t

Pressure Derivative Plot

m3

m2

IARF

m1

Semilog Plot

Semilog straight line



Another interpretation technique routinely used is that of ‘Deconvolution’. This is the reverse ofsuperposition. Basically, deconvolution is the conversion of a variable rate pressure profile intoan equivalent constant rate production sequence. It is a way of reconstructing the transientpressure behaviour often hidden in well test data due to flow rate variation. The technique isextremely sensitive to the quality of the actual pressure and rate data available and is mainlyused to complement other analysis techniques.

3.6 Gas Well Testing

The analysis techniques used for gas well testing are similar to those available for oil well testing.In gas well testing, interpretation techniques also account for the change in gas properties withpressure and ‘Non-Darcy’ flow. Non-Darcy or turbulent flow near the wellbore occurs due to anincrease in gas velocity and results in an additional pressure drop that is treated as skin. This‘skin’ due to turbulence is rate dependent and multi-rate testing is required to quantify its value togive the apparent total skin as:

Apparent total skin = Damage skin factor (S) + Rate dependent skin (DQ)

Where D is the Non-Darcy flow coefficient, (1/(Mscf/d)) and Q is the gas flow rate, (Mscf/d)

To account for the effect of changing gas properties the real gas pseudopressure, m(p) in

psia2/cp is used. (A full review of the real gas pseudopressure is given in Ref 2, Chapter 8).

Various multi-rate testing schedules are used in gas well testing to determine the permeabilityand total skin factor, one of the simplest being a flow period followed by a buildup which isfollowed by a second flow period as shown in Figure 3.14. (From Ref 2, Chapter 8)

Figure 3.14 Gas well test: Rate schedule and bottom hole pressure response.

Rate, QQ1

t ∆t

t1

t’

∆tmax

Q2

Bottomhole

pressure

t1 ∆tmax

t t’

pwf

pws



Analysis of Buildup Period

The Horner buildup plot of m(pws) versus log ((t + ∆t) / ∆t ) for the shut-in period gives the slope ofthe straight line section as:

cyclelog/cp/psiahk

TQ1637mSlope, 21

Where Q1 is the first gas flow rate in Mscf/d and T is the temperature in Deg R.

The total skin factor can be determined (from the buildup data) as:

3.23r)c(

klog-

m

)m(p-)m(p1.151DQSS

2

wit

0)t(wfhr)(1ws

11'

The linear trend can be extrapolated to give m(pi) from which the initial reservoir pressure can becalculated.

Analysis of Flow Periods

The early transient pressure responses from each flow period can be analysed to estimate thepermeability and total skin factors from which the damage and rate dependent skin factors can be

determined. Semilog plots of m(pwf) versus log t for each flow period give the slope of thestraight line section as:

cyclelog/cp/psiahk

TQ1637mSlope, 2

from which the permeability can be calculated and compared to the value obtained from thebuildup analysis.

The total skin factors for each flow period can be calculated from the following equations:

3.23

r)c(

klog-

m

pm-)m(p1.151DQSS

2

wit

hr1wfi11

'

3.23

r)c(

klog-

m

pm-)m(p1.151DQSS

2

wit

hr1wfhr1ws'

22'

Where m(pwf) is determined for t = 1 hour and m(p’ws) is determined from extrapolating the lineartrend on the buildup plot to one hour after the shut-in period has ended.

From these total skin factor equations the damage skin factor and the rate dependent skin factorcan then be determined.



3.7 Deliverability Testing

Deliverability testing of oil and gas wells is carried out to provide an estimate of well performanceagainst specific bottom hole flowing pressures including atmospheric pressure. This type of testevaluates the ability of a well to deliver and not the characteristics of the reservoir. The estimatescan be used in inflow performance relationship (IPR) calculations for oil wells and as the absoluteopen hole flow (AOF) potential of a gas well; the maximum gas rate when the back pressure atthe sandface is zero.

One type of deliverability test for an oil well is a ‘flow after flow’ test. This is where the well isproduced at a certain rate until a stable flowing bottom hole pressure is reached. The rate is thenincreased or decreased and again kept constant until the pressure stabilises. Data from this testcan provide the inflow performance relationship (IPR) for the oil well as shown in Figure 3.15.

Above the bubble point pressure a straight line relationship exists between the bottom holeflowing pressure and the flow rate which gives a constant Productivity Index (PI). Below thebubble point pressure the wells deliverability can be predicted by the Vogel IPR.

Figure 3.15 Example IPR for Oil Well

Above the Bubble Point Pressure:

constantp-p

qPI

wf

Where p is the average reservoir pressure.

Below the Bubble Point Pressure (Vogel IPR):

p

p0.8-

p

p0.2-1

q

q2

wfwf

max

In order to evaluate the AOF potential of a gas well, multi-rate testing is carried out to evaluatenon-darcy flow. One such multi-rate test is the modified isochronal flow test where the well isproduced then shut-in for equal times for a number of sequences followed by a final flow period toa stabilised pressure.

q Max. Flow Rate

pwf

pb

PI = constant

Vogel IPR

p



3.8 Drill Stem Testing

Drill stem tests (DST) are typically carried out in exploration wells to determine the possibility ofcommercial production. The DST provides a temporary completion of the test interval.

A common test sequences involves:

A short flow period of five or ten minutes

A buildup period of one hour

A flow period to establish stable flow

A final shut-in period (approximately one and a half times as long as the final flow period).

The permeability and skin factor can be obtained by analysing the transient pressure data by thevarious techniques described in the previous sections.

Drill stem testing is reviewed fully in the DST documentation and Well Test Manual provided byEXPRO and will not be covered further in this section.



4. Example Calculations

Horner Pressure Buildup Analysis

An appraisal well is produced for 1 hour then shut-in for a pressure buildup. Production, pressureand fluid data for the undersaturated reservoir are detailed below:

Oil Rate q = 1000 stb/d

Reservoir Thickness, h = 130 ft

Porosity, = 35 %

Oil Viscosity, = 2.2 cp

Oil Formation Volume Factor, Boi = 2.1 rb/stb

Total compressibility, ct = (coSo + cwSwc + cf) = 36 x 10-6

psi-1

Shut-In time

∆t (Hours)

Wellbore Pressure

Pws (psia) t

ttp

t

ttlog

p

0.0 1300 (Final Pwf)

0.5 3375 3.00 0.477

1.0 3709 2.00 0.301

1.5 3809 1.67 0.222

2.0 3847 1.50 0.176

2.5 3865 1.40 0.146

3.0 3877 1.33 0.125

3.5 3887 1.29 0.109

4.0 3896 1.25 0.097

4.5 3901 1.22 0.087

5.0 3907 1.20 0.079

5.5 3912 1.18 0.073

6.0 3916 1.17 0.067

From these data the initial reservoir pressure, effective permeability and skin factor are

determined from the Horner buildup plot of pws versus

t

ttlog

p shown below:



Effective permeability:

Slope of linear section (IARF) of semilog plot, m = 630.5 psi/log cycle.

mDhm

Bq162.6k

mD9k

mD130x630.5

2.2x2.1x1000x162.6k

Skin Factor:

3.23rc

klog-

t

1tlog

m

p-p1.151S

2

wtp

p0)t(wfhr)(1ws

3.23(0.3)x10x36x2.2x35.0

9log-

1

11log

630.5

1300-37671.151S

26-

1.0S

3.236.557-0.3013.9131.151S

Initial Reservoir Pressure:

Extrapolation of the linear section (IARF),

t

ttlog

p to zero gives the initial reservoir pressure

as 3957 psia.

3200

3300

3400

3500

3600

3700

3800

3900

4000

0.00.10.20.30.40.50.6

Pws (1 hr) = 3767 psia

Pi = 3957 psia

t

ttlog

p

pws



References

1. The Practice of Reservoir Engineering (Revised Ed.), L.P. Dake, Elsevier, 2001

2. Fundamentals of Reservoir Engineering, L.P. Dake, Elsevier, 1978

3. Pressure Buildup and Flow Tests in Wells, C.S. Matthews and D.G. Russell, SPEMonograph, 1967

4. Advances in Well Test Analysis, R.C. Earlougher, SPE Monograph, 1977

Documents

ENM200 Well Testing Notes