INTRODUCTION TO PRESSURE INTRODUCTION TO PRESSURE TRANSIENT ANALYSISTRANSIENT ANALYSIS

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What is Pressure-Transient What is Pressure-Transient AnalysisAnalysis

• The analysis of pressure changes over time, especially those associated with small variations in the volume of fluid. It involves allowing a limited amount of fluid to flow from the formation while monitoring the pressure over time.

• The well is then shut-in and the pressure monitored while the fluid in the reservoir stabilizes.

• Analysis of these pressure changes provides information on the size and shape of the formation as well as its producibility.

Origin of Log-Log Type CurvesOrigin of Log-Log Type Curves

• The log-log analysis is a global approach as opposed to straight-line methods that use only one fraction of the data, corresponding to a specific flow regime.

• Stallman (1952) published log-log type curves for both the no-flow and the constant pressure linear boundaries. His curves are applicable for the analysis of single well tests and also for interference tests. These curves may be used to find the distance of the linear boundary and its orientation.

• Davis and Larkin (19631, Standing (1964), Witherspoon, et al. (1967) and Kruseman and De Ridder (1970) extended the log-log method for a single linear boundary. They introduced the semilog method for determining the distance to a linear boundary.

• Loucks and Guerrero (1961) and Bixel and van Poolen (1967) presented type curves for a well centered in a two region radial flow system. Ramey (1970) presented approximate solutions for unsteady liquid flow for a well centered in a radially concentric composite system.

• The present work concentrates on internal circular boundaries, yet, the same mathematical methods apply also to linear boundary configurations.

LOG-LOG SCALELOG-LOG SCALE

• For a given period of the test, the change in pressure is plotted on log-log scales versus the elapsed time. A test period is defined as a period of constant flowing conditions (constant flow rate for a drawdown and shut-in period for a build-up test).

• By comparing the log-log data plot to a set of theoretical curves, the model that best describes the pressure response is defined. Theoretical curves are expressed in dimensionless terms because the pressure responses become independent of the physical parameters magnitude (such as flow rate, fluid or rock properties).

• On log-log scales, the shape of the response curve is characteristic.

• The shape of the global log-log data plot is used for the diagnosis of the interpretation model(s).

• The dimensionless pressure pD and time tD are linear functions of Ap and At, the coefficients A and B being dependent upon different parameters such as the permeability k.– log pD =log A + log Ap– log tD =log B + log At

EquationsEquations

• Dimensionless Pressure

• Dimensionless Time

• Dimensionless wellbore storage coefficient

• Gringarten et al. (1979) dimensionless time group

EquationsEquations

• Pure Wellbore

• Infinite-Acting Radial Flow

Reference CurvesReference Curves

Wellbore StorageWellbore Storage

Liquid Re-injectionLiquid Re-injection

Types of FlowTypes of Flow

ChannellingChannelling

Infinite Acting Radial FlowInfinite Acting Radial Flow

Flow Regimes Flow Regimes For many engineering purposes, the actual flow geometry may be

represented by one of the following flow geometries:

• Radial flow

• Bilinear flow

• Linear flow

• Spherical and hemispherical flow

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• Radial flow

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P vs log t gives a straight line i.e. semilog straight line

• Linear flow

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Linear flow occurs in some reservoirs with long, highly conductive verticalfractures.

Straight line given with p vs √t with slope of 1/2

log–log graph of Δp vs t yields a straight line with ½ slope

• Bilinear flow

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It is a new type of flow behavior called bilinear flow because two linear flows occur simultaneously.

Straight line of p vs t1/4

Can be identified from a log–log plot of Δp versus t which will exhibit a straight line with a ¼ slope

• Spherical flow

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Straight line of p vs 1/√t

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Inflow Performance RelationshipProductivity Index, PI

This is a measure of the ability of a well to produce. It is defined by the symbol J, and is the ratio of the total liquid flow rate to the pressure drawdown.

PI changes in time, cumulative production, increased drawdown

• Inflow Performance Relationship (IPR)

Inflow performance represents ability of well to give up fluids

Plot production rate vs. flowing bottom hole pressure called Inflow Performance Relationship (IPR)

• IPR and PI not equivalent

IPR is relationship between flowing pressure and rate

PI represents the special case when Pwf is greater than the bubble point

• Absolute Open Flow

Maximum rate of flow qmax, corresponding if the bottom hole pressure opposite the producing face were reduced to zero psia

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• Rate pressure relationships

For under-saturated oil wells

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Straight-line IPR

When Pwf = PR, q=0 and no flow enters the wellbore

qmax , AOF corresponds to Pwf =0

Slope = 1/J (PI)

Straight-line IPR

When Pwf = PR, q=0 and no flow enters the wellbore

qmax , AOF corresponds to Pwf =0

Slope = 1/J (PI)

• Saturated Oil wells & Gas wells

• PI curve not normally linear for a solution gas drive field because: Increased free gas saturation, lowering Kro

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IPR curvature, indicating gas and/ or two-phase flow

J decreases with increasing drawdown

n; 0.5-1.0

Log-log plot of q vs Δp2 is a straight line with slope 1/n

IPR curvature, indicating gas and/ or two-phase flow

J decreases with increasing drawdown

n; 0.5-1.0

Log-log plot of q vs Δp2 is a straight line with slope 1/n

Vogel (1968) – Saturated Oil wells

Reservoir pressure above the bubble point but wellbore flowing Reservoir pressure above the bubble point but wellbore flowing pressure below the bubble point pressure below the bubble point

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IPR of an under-saturated oil well producing at flowing pressure

below the bubble point

For flowing pressures below the bubble point :

Evolution of PTA Evolution of PTA methodologiesmethodologies

Log-log type curves Bourdet Derivative

PC-based PTA software

Specialized plots(MDH Semi-log & Horner plots)

1950’s 1970’

s1983

1985 & onwards

Specialized plotsSpecialized plots These plots were focused on using a specific flow

regime (IARF), to determine well productivity and the main reservoir properties:

• Effective permeability (keff)• Skin factor (S)• Conductivity (kh)• Pressure drop due to skin (Δps)• Drainage area/OOIP• Time for well bore storage effects to cease, or

IARF to start.• Wellbore storage coefficient.

Specialized plotsSpecialized plots

MDH Pressure Drawdown

3000

3100

3200

3300

3400

3500

3600

3700

3800

3900

4000

4100

4200

4300

4400

0.01 0.1 1 10 100 1000

Flowing Time (hrs)

Pw

f (p

sia)

Specialized plotsSpecialized plots

MDH Pressure Buildup

10

100

1000

0.01 0.1 1 10 100 1000

Flowing Time (hrs)

(pi-p

wf)

(psi

a)

1 ½ cycles

Specialized plotsSpecialized plotsOil Well Horner Plot

4200

4250

4300

4350

4400

4450

4500

4550

1.0 10.0 100.0 1000.0 10000.0

(tp + delt)/delt

Pw

s (p

sia)

Log-log type curvesLog-log type curves• Developed to compliment straight line techniques.• A log-log plot of the pressure response vs. time on

tracing paper is placed over a set of predefined curves.

Results obtained from the specialized plots is used to help position data on the type curves.

• The choice and relative position of the data on the type curve, called the match point, were used to calculate physical results.

• This method was of poor resolution until Bourdet derivative was introduced.

Type Curve matching Type Curve matching techniquetechnique

Drawdown Type Curves Manual Drawdown Type Curve Matching

Bourdet DerivativeBourdet Derivative Was introduced to address the many shortcomings

of the type curve matching technique, and was at the origin of what is called modern PTA methodology.

It is defined as the slope of the superposition semi-log plot displayed on the log-log plot.

Considered the single most important breakthrough in the history of PTA.

Bourdet DerivativeBourdet Derivative

Bourdet derivative: semi-log and log-log

PC-based PTA softwareCategory I

• Relies heavily on graphics.• User inputs well test data into the

computer after which the computer graphically displays the data, derivative of the data, and derivative type curve on the screen.

• The user can then move the WT data on the screen until a match is achieved bet. the data and the type curve.

• The user then enters the match; as well as required reservoir and production characteristics.

• The program will then calculate and output k, S & C.

Category II

• Relies on numerical techniques to achieve a fit.

• The type curve is num. rep. in the program.

• The user enters the WT data, and reservoir and production parameters.

• The WT data is then smoothed using num methods and the derivative curve calculated.

• The program compares the type curve to the WT data and its derivative.

• When a match is achieved, the program outputs the reservoir parameters.

Bourdet Derivative and well/wellbore Bourdet Derivative and well/wellbore effectseffects

• Pure wellbore storage effects are only observed at early time when the well pressure behaviour is dominated by well fluid decompression.

• For pure wellbore storage:

• The derivative is:

• This implies that at early time, when wellbore storage is present, pressure and the Bourdet derivative curves will merge on a unit slope line on the log-log plot.

tCp

ptCtd

tdCtp

'

Bourdet Derivative and IARFBourdet Derivative and IARF• When IARF occurs: ∆p=m’sup(∆t), where m’ is the slope of the semilog str. line.

• Derivative is:

• This implies that the derivative will have zero slope.

')sup(

' mtd

pdp

Bourdet Derivative & PSSBourdet Derivative & PSS• After long stabilized production, PSS is reached, and the

pressure response is: ∆p=A∆t+B.• The superposition time can again be approximated by

sup(∆t)≈ln(∆t).

• The derivative is:

• At very large time, Δp = AΔt + B ≈ AΔt.• So, when PSS is reached, the pressure response on the log-

log plot will tend to a unit slope, while the derivative will reach the unit slope much earlier.

• In a BU the pressure stabilizes and the derivative plunges towards zero.

tAtd

BtAdtp

)(

'

References References

• http://www.glossary.oilfield.slb.com/Display.cfm?Term=pressure-transient%20analysis

• http://earthsci.stanford.edu/ERE/research/geoth/publications/techreports/SGP-TR-065.pdf

• Bourdet D. - Handbook of Petroleum Exploration and Production 3, Well Test Analysis, The Use Of Advanced Interpretation Models

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