Rate Transient Analysis

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Modern Production Data Analysis Day 1 - Theory

1. Introduction to Well Performance Analysis 2. Arps – Theory

a) Exponential b) Hyperbolic c) Harmonic

3. Analytical Solutions

a) Transient versus Boundary Dominated Flow

b) Boundary Dominated Flow

i. Material Balance Equation ii. Pseudo Steady-State Concept iii. Rate Equations

c) Transient Flow

i. Radius of Investigation Concept ii. Transient Equation (Radial Flow)

4. Theory of Type Curves

a) Dimensionless variables b) The log-log plot

c) Type Curve matching 5. Principle of Superposition

a) Superposition b) Desuperposition

c) Material Balance Time 6. Gas Corrections

a) Pseudo-Pressure b) Pseudo-Time

Modern Production Data Analysis Day 2 - Practice

7. Arps – Practical Considerations a) Guidelines

b) Advantages c) Limitations

8. Analysis Using Type Curves a) Fetkovich

b) Blasingame (Integrals) c) AG and NPI (Derivatives) d) Transient

e) Wattenbarger

9. Flowing Material Balance 10. Specialized

11. Modeling and History Matching 12. A Systematic and Comprehensive

Approach 13. Practical Diagnostics a) Data validation b) Pressure support c) Interference d) Liquid loading e) Accumulating skin damage

f) Transient flow regimes 14. Tutorials

Introduction to Well Performance Analysis

Traditional

- Production rate only

- Using historical trends to predict future - Empirical (curve fitting)

- Based on analogy - Deliverables:

- Production forecast

Modern

- Rates AND Flowing Pressures - Based on physics, not empirical

- Reservoir signal extraction and characterization - Deliverables:

- OGIP / OOIP and Reserves - Permeability and skin

- Drainage area and shape

- Production optimization screening - Infill potential

Recommended Approach

- Use BOTH Traditional and Modern together - Production Data Analysis should include a comparison of multiple methods

- No single method always works

- Production data is varied in frequency, quality and duration

Welltest Analysis - High resolution early-time characterization - High resolution characterization of the near-wellbore -Point-in-time characterization of wellbore skin - Estimation of reserves when flowing pressure is unknown Empirical Decline Analysis - Flow regime characterization over life of well - Estimation of fluids-in-place - Performance based recovery factor - Able to analyze transient production data (early-time production, tight gas etc)

- Characterization of perm and skin -Estimation of contacted drainage area -Estimation of reservoir pressure - Projection of recovery constrained by historical operating conditions

Modern Production Analysis

Modern Production Analysis -Integration of Knowledge

Traditional Decline Curves – J.J. Arps

- Graphical – Curve fitting exercise - Empirical – No theoretical basis

The Exponential Decline Curve 2001 2002 2003 2004 2005 2006 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 Ga s R ate , MM sc fd Rate vs Time Unnam e d We ll 2001 2002 2003 2004 2005 2006 10-1 1.0 101 2 3 4 5 6 7 2 3 4 5 6 7 G a s R a te , M M s c fd Rate vs Time Unnam ed Well 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 Gas Cum . Prod., Bscf

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 G a s R a te , M M s c fd

Rate vs. Cumulative Prod.

Unnam ed Well t D ie i q q   log log 2.302 i i D t q  q  i i q  q D Q 2.302* i D  Slope Di  Slope i Slope D q 

The Hyperbolic Decline Curve

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 Gas Cum . Prod., Bscf

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 G a s R a te , M M s c fd

Rate vs. Cumulative Prod. Unnam ed Well b i i t bD q q _1/ ) 1 (   ( ) D  f t i _b b i D D q q 

Hyperbolic Exponent “b”

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 Gas Cum . Pr od., Bs cf

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 Ga s R ate , MM sc fd

Rate vs. Cumulative Prod.

Unnam e d We ll

Mild Hyperbolic – b ~ 0

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 Gas Cum ulative , Bs cf

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 G as R at e, M M sc fd

Rate vs. Cumulative Prod.

NBU 921-22G

Transient Flow

- Early-time OR Low Permeability

- Flow that occurs while a pressure “pulse” is

moving out into an infinite or semi-infinite acting reservoir

- Like the “fingerprint” of the reservoir

- Contains information about reservoir properties (permeability, drainage shape)

Boundary Dominated Flow

- Late-time flow behavior

- Typically dominates long-term production data - Reservoir is in a state of pseudo-equilibrium – physics reduces to a mass balance

- Contains information about reservoir pore volume (OOIP and OGIP)

Definition of Compressibility V p_i V dV p_i-dp p V V c     1

Compressibility Defines Material Balance of a Closed Oil Reservoir (above bubble point)

1 p i p i t i pss p N c N p p N p p c N p p m N      

Note: only valid if c is constant

V=N

DV = N_p

Single Phase Oil MB i p  p p N pss m slope  p pss i p m N p mx y   

Distance press ure r_w Constant Rate q 1 p 1 Illustration of Pseudo-Steady-State p_wf1 r_e 2 p p_wf2 2 3 p p_wf3 3 time

Flowing Material Balance p N pss m slope  b N m p p b mx y p pss wf i      wf i p p  b

Steady-State Inflow Equation Distance press ure r_w r_e p p_wf p_i

Inflow (Darcy) pressure drop- Constant-Productivity Index ) , , (kh s area f b qb p p pss pss wf   

Flowing Material Balance Variable Rate q Np pss m slope  pss p pss wf i b q N m q p p b mx y      q p pi  wf pss b

The Three Most Important Equations in Modern Production Analysis

i pss p p  p m N wf pss p  p  qb pss p pss wf i p m N qb p   

Constant Pressure = Production Constant Rate = Welltest q p_wf q p_wf

- Invert the PSS equation 1 1 ( ) 1 ( ) 1 pss p i wf pss pss pss pss pss i wf pss q m N p p t m t b b q q _b m p p t t b    _    _

Constant Rate Solution

Constant Flowing Pressure Solution

- Required: q(t), N_pmax and N for constant pwf

- Take derivative of both equations and solve for q

- Integrate to find Np(t), as t goes to infinity Np goes to N_pmax   max ( ) pss pss m t i wf _b pss i wf p i wf t pss p p q t e b p p N p p c N m       

Constant Flowing Pressure Solution

Relate Back to Arps Exponential, Determine N

max max ( ) ( ) i wf i pss pss i pss i p i t i wf t i wf i p i p p q b m D b q N D c p p c p p D N N q        

Plot Constant p and Constant q together 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1 0 5 10 15 20 25 30 35 40 45

Constant rate q/Dp (Harmonic)

Constant pressure q/Dp (Exponential)

1 ( ) 1 pss pss i wf pss q _b m p p t t b   _ ( ) 1 m_psspsst b i wf pss q t e p p b   

-4000 -3600 -3200 -2800 -2400 -2000 -1600 -1200 -800 -400 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 Radii, ft 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 P re s s u re , p s i

Cross Section Pressure Plot

Numerical Radial Model

10

Cross Section

Plan View

Transient and Boundary Dominated Flow

Boundary Dominated Well Performance = f(Volume, PI) Transient Well Performance = f(k, skin, time)

-4000 -3600 -3200 -2800 -2400 -2000 -1600 -1200 -800 -400 0 400 800 1200 1600 2000 2400 2800 3200 3600 4000 Radii, ft 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 P re s s u re , p s i

Cross Section Pressure Plot

Numerical Radial Model

10 Cross Section Plan View 948 948 inv inv kt r c kt A c     

Transient Equation 1 ( ) 141.2 1 0.0063 ln 0.4045 2 i wf t q kh p p B kt s c       _{ }    

q(t)’s compared 0 0. 2 0. 4 0. 6 0. 8 1 1. 2 1. 4 1. 6 0 5 10 15 20 25 30 35 40 45

Transient flow: compares to Arps “super hyperbolic” (b>1)

Blending of Transient into Boundary Dominated Flow

0 0. 5 1 1. 5 2 2. 5 3 0 5 10 15 20 25 30 35 40 45

Complete q(t) consists of: Transient q(t) from t=0 to tpss

Log-Log Plot: Adds a New Visual Dynamic

Comparison of qD with 1/pD

Cylindrical Reservoir with Vertical Well in Center

0.000001 0.00001 0.0001 0.001 0.01 0.1 1 10 100 1000

0.000001 0.0001 0.01 1 100 10000 1000000 100000000 1E+10 1E+12 1E+14

tD q D a n d 1 /p D 0.9

Constant Pressure Solution Exponential

Constant Rate Solution

Harmonic

Type Curve

- Dimensionless model for reservoir / well system - Log-log plot

- Assumes constant operating conditions

- Valuable tool for interpretation of production and pressure data

Type Curve Example - Fetkovich 10-1 2 3 4 5 6 7 8 9_1.0 2 3 4 5 6 7 8 ₁₀1 2 3 4 5 6 7 8 Tim e 10-2 10-1 1.0 2 3 4 5 6 7 9 2 3 4 5 6 7 R a te ,

Fetkovich Typecurve Analysis

Exponential Harmonic q_Dd t_Dd Dd t Dd e q   1 1 Dd Dd q t   t D t q t q q i Dd i Dd   ( ) Hyperbolic 1/ 1 (1 ) Dd b Dd q bt  

Plotting Fetkovich Type Curves-Example Well 1 (exponential) qi= 2.5 MMscfd D_i= 10 % per year Well 2 (exponential) q_i= 10 MMscfd D_i= 20 % per year

Raw Data Plot

0.00 2.00 4.00 6.00 8.00 10.00 12.00 0 5 10 15 Time (years) R a te (M M s c fd ) Well 1 Well 2 Dimensionless Plot 0.10 1.00 0.01 0.10 1.00 10.00 tDd q D d Well 1 Well 2 Time (years)

Well 1 Well 2 Well 1 Well 2 Well 1 Well 2

0 2.50 10.00 0.00 0.00 1.00 1.00 1 2.26 8.19 0.10 0.20 0.90 0.82 2 2.05 6.70 0.20 0.40 0.82 0.67 3 1.85 5.49 0.30 0.60 0.74 0.55 4 1.68 4.49 0.40 0.80 0.67 0.45 5 1.52 3.68 0.50 1.00 0.61 0.37 6 1.37 3.01 0.60 1.20 0.55 0.30 7 1.24 2.47 0.70 1.40 0.50 0.25 8 1.12 2.02 0.80 1.60 0.45 0.20 9 1.02 1.65 0.90 1.80 0.41 0.17 10 0.92 1.35 1.00 2.00 0.37 0.14 Rate (MMscfd) tDd qDd

Fetkovich Typecurve Matching

In most cases, we don’t know what “qi” and “Di” are ahead of time. Thus, qi and Di are calculated based on the typecurve match (ie. The typecurve is superimposed on the data set

t t D q t q q Dd i Dd i   ( )

Knowing qi and Di, EUR (expected ultimate recovery) can be calculated

1.0 101 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 Tim e 10-1 1.0 5 6 7 8 9 2 3 4 5 6 7 8 R a te ,

Fetkovich Typecurve Analysis

NBU 921-22G

qDd

t_Dd

q

Analytical Model Type Curve 10-4 2 3 4 5 6 7 9₁₀-3 2 3 4 5 6 78₁₀-2 2 3 4 5 6 7 8₁₀-1 2 3 4 5 6 78_1.0 2 3 4 5 6 78₁₀1 2 3 4 5 6 7 Tim e 10-2 10-1 1.0 101 2 3 4 6 9 2 3 4 6 9 2 3 4 6 9 2 3 4 6 R a te ,

Fetkovich Typecurve Analysis

Boundary Dominated Flow

Exponential

Transient Flow

r_e/r_wa= 10 r_e/r_wa= 100 r_e/r_wa= 10,000

q_Dd

Modeling Skin using Apparent Wellbore Radius rw re rwa (s) rwa(d) s w wa r e r   ΔP(s) ΔP(d)

Dimensionless Variable Definitions (Fetkovich) 2 2 141.2 1 ln ( ) 2 0.00634 1 1 ln 1 2 2 e Dd i wf wa wa Dd e e wa wa q B r q kh p p r kt ctr t r r r r    _{ }   _{  }  _  _{   }     _{ }  _{ }  _  _      _{ }  _{ }   _ _

Type Curve Matching (Fetkovich) 2 141.2 1 ln ( ) 2 0.00634 1 ln 1 1 ln 1 2 2 141.2 0.00634 2 ( ) e i wf wa Dd _match w wa t _{e e} Dd wa wa wa match e i wf t Dd _match Dd match B r q k h p p r q k t r r s c _{r r} t r r r B q t r h p p c q t         _{  } _  _{   }    _{  }   _{ }   _    _        _{ }  _{ }   _ _  

The Fetkovich analytical typecurves can be used to calculate three parameters: permeability, skin and reservoir radius

Type Curve Matching - Example 10-4 2 3 4 5 6 7 8₁₀-3 2 3 4 5 6 7 8₁₀-2 2 3 4 5 6 7 8₁₀-1 2 3 4 5 6 7 8_1.0 2 3 4 5 6 7 8₁₀1 2 3 4 5 6 7 8 Tim e 10-3 10-2 10-1 1.0 101 2 3 4 6 8 2 3 4 6 8 2 3 4 6 8 2 3 4 6 8 R at e,

Fetkovich Typecurve Analysis 10

Boundary Dominated Flow

Exponential Transient Flow t_Dd r_eD= 50 q_Dd q t k = f(q/q_Dd) s = f(q/q_Dd * t/t_Dd, r_eD) r_e = f(q/q_Dd* t/t_Dd)

What about Variable Rate / Variable Pressure Production? The Principle of Superposition

Superposition in Time:

1. Divide the production history into a series of constant rate periods

2. The observed pressure response is a result of the additive effect of each rate change in the history

Example: Two Rate History

q₁ q₂ Effect of (q₂-q₁) t₁ 1 ( ) ( 2 1) ( 1) i wf p  p  q f t  q q f t t q p_wf

The Principle of Superposition

1 ( ) ( 2 1) ( 1)

i wf

p  p  q f t  q  q f t t

Two Rate History

N - Rate History 1 1 1 ( ) ( ) N i wf j j j j p p q q  f t t    



Superposition versus Desuperposition

Simple

- Unit step response f(t) - Type Curve

- Superposition Time

Complex

- Real rate and pressure history - Modeling (history matching) Superposition Desuperposition q pwf q pwf

Superposition Time

Convert multiple rate history into an equivalent single rate history by re-plotting data points at their “superposed” times

1 1 1 ( ) ( ) N i wf j j j N _j N p p q q f t t q q     _  _



The Principle of Superposition – PSS Case 1 1 1 ( ) ( ) N i wf j j j N _j N p p q q f t t q q     _  _  141.2 3 ( ) ln 4 i wf e t wa p p t B r f t q c N kh r        _  _   1 1 1 1 ( ) 141.2 3 ( ) ln 4 1 141.2 3 ln 4 N i wf j j e j N t _j N wa i wf p e N t N wa p p q q B r t t q c N q kh r p p N B r q c N q kh r       _  _{ }  _       _{ }   _  _   

Definition of Material Balance Time (Blasingame et al)

Actual Rate Decline Equivalent Constant Rate

q Q actual time (t) Q = Q/q material balance time (t_c)

Features of Material Balance Time

-MBT is a superposition time function

- MBT converts VARIABLE RATE data into an EQUIVALENT CONSTANT RATE solution.

- MBT is RIGOROUS for the BOUNDARY DOMINATED flow regime

- MBT works very well for transient data also, but is only an approximation (errors can be up to 20% for linear flow)

Comparison of qD (Material Balance Time Corrected) with 1/pD

Cylindrical Reservoir with Vertical Well in Center

0.000001 0.00001 0.0001 0.001 0.01 0.1 1 10 100 1000

0.000001 0.0001 0.01 1 100 10000 1000000 100000000 1E+10 1E+12 1E+14

tD q D a n d 1 /p D 0 0.2 0.4 0.6 0.8 1 1.2 R a ti o 1 /p D t o q D

Beginning of "semi-log" radial flow (tD=25) Ratio (qD to 1/pD) ~ 97%

0.97 Very early time radial flow

Ratio (qD to 1/pD) ~ 90%

MBT Shifts Constant Pressure to Equivalent Constant Rate

Constant Pressure Solution q_D Corrected to Harmonic

Constant Rate Solution

1/p_D

Corrections Required for Gas Reservoirs

• Gas properties vary with pressure

– Formation Volume Factor – Compressibility

Corrections Required for Gas Reservoirs 141.2 3 ln 4 o e i wf o wa qt qB r p p c N kh r       _  _   Depletion Term Depends on compressibility Reservoir FlowTerm: Depends on “B” and Viscosity

Darcy’s Law Correction for Gas Reservoirs

Darcy’s Law states : D_p  _q



For Gas Flow, this is not true because

Depletion Correction for Gas Reservoirs

Gas properties (compressibility and viscosity) vary significantly with pressure

Gas Compressibility 0 0.002 0.004 0.006 0.008 0.01 0.012 0 1000 2000 3000 4000 5000 6000 Pressure (psi) C om pre s s ibi li ty ( 1 /ps i) p cg  1

Solution: Pseudo-Time   



Evaluated at average reservoir pressure

Not to be confused with welltest pseudo-time which evaluates properties at well flowing pressure

Depletion Correction for Gas Reservoirs: Pseudo-Time

Boundary Dominated Flow Equation for Gas

      _     D 4 3 ln * 6 417 . 1 ) ( 2 wa e a i i g i pwf pi p r r kh Tq e qt G Z c p p p p  Pseudo-pressure _Pseudo-time

Constant Rate Case

Variable Rate Case

pss i pa p b qG G q p _{ } D  Pseudo-Cumulative Production

Overall time function - Material Balance Pseudo-time         t g i g ta a ca t c c qdt q c qdt q t qdt q t 0 0 0 1 1  

    0 ca ) ( 1 ) ( t i f t i t dt p p c c t q q c t      

Improved Material Balance Pseudo-time

Overall material balance pseudo-time function (corrected for

variable fluid saturations, water encroachment, in-situ fluids & formation expansion and desorption):

Notes About Drive Mechanism and b Value (from Arps and Fetkovich)

b value Reservoir Drive Mechanism

0 Single phase liquid expansion (oil above bubble point) Single phase gas expansion at high pressure

Water or gas breakthrough in an oil well

0.1 - 0.4 Solution gas drive

0.4 - 0.5 Single phase gas expansion

0.5 Effective edge water drive

0.5 - 1.0 Layered reservoirs

Advantages of Traditional

- Easy and convenient

- No simplifying assumptions are required regarding the

physics of fluid flow. Thus, can be used to model very complex systems

Limitations of Traditional

- Implicitly assumes constant operating conditions

- Non-unique results, especially for tight gas (transient flow) - Provides limited information about the reservoir

Example 1: Decline Overpredicts Reserves

October Novem ber Decem ber January February March April

2001 2002 4 G a s Ra te , M M s c fd Rate vs Time Unnam ed Well 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50

Gas Cum . Prod., Bscf

0 1 2 3 4 G a s Ra te , M M s c fd

Rate vs. Cumulative Prod.

Unnam ed Well

Example 1 (cont’d)

Flowing Pressure and Rate vs Cumulative Production

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 4 5 6 7 8 9 10 Cumulative Production (bcf) R a te ( M M s c fd) 0 200 400 600 800 1000 1200 Fl ow ing P re s s ure ( ps ia )

True EUR does not exceed 4.5 bcf Rates

Pressures

Forecast is not valid here

Example 2: Decline Underpredicts Reserves

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 Gas Cum . Prod., Bscf

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 G a s R a te , M M s c fd

Rate vs. Cumulative Prod.

Unnam ed Well

Example 2 (cont’d)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Norm alized Cum ulative Production, Bscf

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070 0.075 0.080 0.085 N o rm a li z e d R a te , M M s c fd /( 1 0 6p s i 2/c P )

Flowing Material Balance

Unnam ed Well

Original Gas In Place

Legend

Decline FMB

Example 2 (cont’d) 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 620 640 660 680 700 720 Tim e, days 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 G a s , M M s c fd 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 P re s s u re , p s i Data Chart Unnam ed Well Legend Pressure Actual Gas Data

Operating conditions: Low drawdown Increasing back pressure

Arps Production Forecast

0.01 0.1 1 10

Dec-00 May-06 Nov-11 May-17 Oct-22 Apr-28 Oct-33

Time G a s R a te ( M M s c fd) Economic Limit = 0.05 MMscfd _{b = 0.25,} EUR = 2.0 bcf b = 0.50, EUR = 2.5 bcf b = 0.80, EUR = 3.6 bcf

Example 3 – Illustration of Non-Uniqueness

Blasingame Typecurve Analysis

Blasingame typecurves have identical format to those of Fetkovich. However, there are three important differences in presentation:

1. Models are based on constant RATE solution instead of constant pressure

2. Exponential and Hyperbolic stems are absent, only HARMONIC stem is plotted

3. Rate Integral and Rate Integral - Derivative typecurves are used (simultaneous typecurve match)

Data plotted on Blasingame typecurves makes use of MODERN DECLINE ANALYSIS methods:

- NORMALIZED RATE (q/Dp)

Blasingame Typecurve Analysis-Comparison to Fetkovich log(q_Dd) log(t_Dd) log(q/Dp) log(tca) log(q_Dd) log(t_Dd) log(q) log(t) Fetkovich _Blasingame

- Usage of q/Dp and tca allow boundary dominated flow to be represented by harmonic stem only, regardless of flowing conditions

- Blasingame harmonic stem offers an ANALYTICAL fluids-in-place solution - Transient stems (not shown) are similar to Fetkovich

Blasingame Typecurve Analysis-Definitions

Normalized Rate

Typecurves Data - Oil Data - Gas

      _       D  2 1 ln 2 . 141 wa e Dd r r P kh q q  P q D P_p q D  t dt q t q DA t Dd DA Ddi   0 1 _D        D c t c i dt P q t P q 0 1 _D          D ca t p ca i p dt P q t P q 0 1 DA Ddi DA Ddid dt dq t q  c i c id dt P q d t P q      D        D ca i p ca id p dt P q d t P q         D          D Rate Integral

Q rate integral = Q/t actual rate Q actual time

Concept of Rate Integral (Blasingame et al)

actual time

Rate Integral: Like a Cumulative Average

Effective way to remove noise

t₁

Average rate over time period “0 to t₁”

q

Average rate over time period “0 to t₂”



Typecurve Interpretation Aids: Integrals, Derivatives

Integral / Cumulative

Removing the scatter from noisy data sets

Dilutes the reservoir signal

Fetkovich,

Blasingame, NPI

Derivative

Amplifying the reservoir signal embedded in production data Amplifies noise - often unusable Agarwal-Gardner, PTA

Integral-Derivative Maximizing the strengths

of Integral and Derivative Can still be noisy Blasingame, NPI

Used in Analysis

Typecurve Most Useful For Drawback

Rate Integral and Rate Integral Derivative (Blasingame et al)

Rate Integral

Rate (Normalized)

Blasingame Typecurve Analysis-Transient Calculations

Oil:

k is obtained from rearranging the definition of

                      2 1 r r ln kh 2 . 141 p q q match wa e Dd  D                                 2 1 r r ln h 2 . 141 q p q k match wa e match Dd  D

Solve for rwa from the definition of

                               2 1 r r ln 1 r r r c 2 1 kt 006328 . 0 t match wa e 2 match wa e 2 wa t c Dd                                                 2 1 match wa r e r ln 1 2 match wa r e r 2 1 t c k 006328 . 0 match Dd t t wa r c         wa w r r ln s

Blasingame Typecurve

Analysis-Boundary Dominated Calculations-Oil

Oil-in-Place calculation is based on the harmonic stem of Fetkovich typecurves. In Blasingame typecurve analysis, q_Dd and t_Ddare defined as follows:

    Dd i c i Dd t Dt p q p q q  D D  and / /

Recall the Fetkovich definition for the harmonic typecurve and the PSS equation for oil in harmonic form: 1 1 1 and 1 1   D   c t Dd Dd t Nb c b p q t q

From the above equations:

Nb c D b p q t D p q p q t i i c i i 1 and , 1 ere wh 1           D  D  D          

PSS equation for oil in harmonic form, using material balance time

Definition of Harmonic typecurve

Blasingame Typecurve

Analysis-Boundary Dominated Calculations-Oil

Oil-in-Place (N) is calculated as follows: Rearranging the equation for Di:

b D c N i t 1 

Now, substitute the definitions of q_Ddand t_Ddback into the above equation:

          D            D      Dd Dd c t Dd c Dd t q p q t t c p q q t t c N 1 / / 1 Y-axis “match-point” from typecurve analysis

X-axis “match-point from typecurve analysis

Z c Gb p D b p q t D p q p q i i t i i i p c i i  2 and , 1 ere wh 1           D  D  D          

Blasingame Typecurve Analysis- Boundary Dominated Calculations- Gas

Gas-in-Place calculation is similar to that of oil, with the additional complications of pseudo-time and pseudo-pressure.

In Blasingame typecurve analysis, q_Dd and t_Ddare defined as follows:

    Dd i ca i p p Dd t Dt p q p q q  D D  and / /

Recall the Fetkovich definition for the harmonic typecurve and the PSS equation for gas in harmonic form:  2  1 1 and 1 1   D   ca i i t i p Dd Dd t b G c Z p b p q t q 

PSS equation for gas in harmonic form, using material balance pseudo-time

Definition of Harmonic typecurve

Gas-in-Place (G_i) is calculated as follows:

Rearranging the equation for Di:

Z c b D p G i t i i i  2 

Now, substitute the definitions of q_Ddand t_Dd back into the above equation:

Y-axis “match-point” from typecurve analysis

X-axis “match-point from typecurve analysis

Blasingame Typecurve

Analysis-Boundary Dominated Calculations- Gas

            D      D                  Dd p Dd ca i t i p Dd i t ca Dd i i q p q t t c Z p p q q c Z t t p G 2 / ) / ( 2  

Agarwal-Gardner Typecurve Analysis

Agarwal and Gardner have developed several different diagnostic methods, each based on modern decline analysis theory. The AG typecurves are all derived using the WELLTESTING definitions of dimensionless rate and time (as opposed to the Fetkovich

definitions). The models are all based on the constant RATE solution. The methods they present are as follows:

1. Rate vs. Time typecurves (tD and tDA format)

2. Cumulative Production vs. Time typecurves (tD and tDA format)

3. Rate vs. Cumulative Production typecurves (tDA format)

- linear format

Agarwal-Gardner - Rate vs. Time Typecurves

Agarwal and Gardner Rate vs. Time typecurves are the same as conventional drawdown typecurves, but are inverted and plotted in tDA (time based on area) format.

qD vs tDA

The AG derivative plot is not a rate derivative (as per Blasingame). Rather, it is an INVERSE PRESSURE DERIVATIVE.

p_D(der) = t(dp_D/dt) q_D(der) = t(dq_D/dt) 1/p_D(der) = ( t(dp_D/dt) ) -1

Agarwal-Gardner - Rate vs. Time Typecurves

Comparison to Blasingame Typecurves

Rate Integral-Derivative Inv. Pressure Integral-Derivative qDd and tDd plotting format qD and tDA plotting fomat

Agarwal-Gardner - Rate vs. Cumulative Typecurves

Agarwal and Gardner Rate vs. Cumulative typecurves are different from conventional typecurves because they are plotted on LINEAR

coordinates.

They are designed to analyze BOUNDARY DOMINATED data only. Thus, they do not yield estimates of permeability and skin, only fluid-in-place. Plot: qD (1/pD) vs QDA

Where (for oil):

   t p p t q kh B q wf i D   141.2 wf i i wf i t DA D DA p p p p p p N c Q t q Q        2 1 ely alternativ or ) ( 2 1 *

Where (for gas): Agarwal-Gardner - Rate vs. Cumulative Typecurves    t t q kh T e q wf i D     1.417 6*  t i i _{i wf i}i _wf ca DA D DA G Z c qt t q Q               2 1 ely alternativ or ) ( 2 2 1 *

Agarwal-Gardner - Rate vs. Cumulative Typecurves

qD vs QDA typecurves always converge to 1/2

NPI (Normalized Pressure Integral)

NPI analysis plots a normalized PRESSURE rather than a normalized RATE. The analysis consists of three sets of typecurves:

1. Normalized pressure vs. tc (material balance time) 2. Pressure integral vs. tc

3. Pressure integral - derivative vs. tc

- Pressure integral methodology was developed by Tom Blasingame; originally used to interpret drawdown data with a lot of noise. (ie. conventional pressure derivative contains far too much scatter)

- NPI utilizes a PRESSRE that is normalized using the current RATE. It also utilizes the concepts of material balance time and pseudo-time.

NPI (Normalized Pressure Integral): Definitions

Normalized Pressure

Typecurves Data - Oil Data - Gas

 q P kh P_D 2 . 141 D  q P D q P_p D  DA D Dd t d dP P ln   c d d t q P d q P ln       D        D    ca p i p t d q P d q P ln D        D  t dt P t P DA t p DA Di   0 1 D        D tc c i dt q P t q P 0 1  D        D tca p ca i p dt q P t q P 0 1 DA Di DA Did dt dP t P  c i c id dt q P d t q P      D        D ca i p ca id p dt q P d t q P _     D        D Conventional Pressure Derivative Pressure Integral Pressure Integral -Derivative

NPI (Normalized Pressure Integral): Diagnostics Transient Boundary Dominated Integral - Derivative Typecurve Normalized Pressure Typecruve

NPI (Normalized Pressure Integral): Calculation of Parameters- Oil

Oil - Radial  q P kh P_D 2 . 141 D  0.00634₂ e t c DA r C kt t   match D q P P h k             D 141.2 match DA c t e t t C k r _        00634 . 0 match wa e wq r re r r        _       wa w r r S ln match DA c match D t t t q P P S C N _                  D        1000 * 615 . 5 2 . 141 00634 . 0 ₀ (MBBIS)

Gas – Radial Tq P kh P_D p 6 417 . 1  D  0.00634 ₂ e ti i ca DA r C kt t   match p D q P P h T k             D  1.417 6 match DA ca ti i e t t C k r _        00634 . 0 match wa e e wa r r r r        _       wa w r r S ln    ₉ 10 * 6 417 . 1 00634 . 0 match p D match DA ca sc i ti i sc i g q P P t t P z c T P S G             D          (bcf)

NPI (Normalized Pressure Integral): Calculation of Parameters- Gas

Transient (tD format) Typecurves

Transient typecurves plot a normalized rate against material balance time (similar to other methods), but use a dimensionless time based on

WELLBORE RADIUS (welltest definition of dimensionless time), rather than AREA. The analysis consists of two sets of typecurves:

1. Normalized rate vs. tc (material balance time) 2. Inverse pressure integral - derivative vs. tc

- Transient typecurves are designed for analyzing EARLY-TIME data to estimate PERMEABILITY and SKIN. They should not be used (on their own) for estimating fluid-in-place

- Because of the tD format, the typecurves blend together in the early-time and diverge during boundary dominated flow (opposite of tDA and tDd format typecurves)

Transient versus Boundary Scaling Formats log(q_Dd) log(t_Dd) log(t_D) log(q_D)

Transient (tD format) Typecurves: Definitions

Normalized Rate

Typecurves Data - Oil Data - Gas

P kh q qD D 141.2  P q D Pp q D   1 0 1 / 1           P t dt t P DA t p DA Di 1 0 1          _D        D c t c i dt q P t q P Inv 1 0 1          D        D ca t p ca i p dt q P t q P Inv 1 / 1         DA Di DA Did dt dP t P 1                      D        D c i c id dt q P d t q P Inv 1                      D        D ca i p ca id p dt q P d t q P Inv Inverse Pressure Integral Inverse Presssure Integral - Derivative

Transient (tD format) Typecurves: Diagnostics (Radial Model)

Transient _{Transition to Boundary}

Dominated occurs at different points for different typecurves Inverse Integral -Derivative Typecurve Normalized Rate Typecurve

Transient (tD format) Typecurves: Finite Conductivity Fracture Model

Increasing Fracture Conductivity (FCD stems) Increasing Reservoir Size (xe/xf stems)

Transient (tD format) Typecurves: Calculations (Radial Model)

Oil Wells:

Using the definition of qD,

permeability is calculated as follows:

From the definition of tD,

rwa is calculated as follows:

Skin is calculated as follows:

/ 2 . 141 match D q p q h B k _      D   / 2 . 141 00634 . 0 match D c match D t wa t t q p q h B c r             D         ) ( 2 . 141 wf i D p p kh qB q    00634 . 0 2 wa t c D r c kt t   ln w       wa r r s Gas Wells:

For gas wells, qD is defined as follows:

The permeability is calculated from above, as follows:

From the definition of tD and k, rwa is calculated as follows

Skin is calculated as follows:

q 6 .417 1 p R D p kh T E q D  / 6 .417 1 match D p R q p q h T E k _      D  / 6 .417 1 00634 . 0 match D p match D ca R ti i wa q p q t t h T E c r _      D               ln w       wa r r s

Flowing p/z Method for Gas – Constant Rate p G Measured at well during flow

Pressure loss due to flow in reservoir (Darcy’s Law) is constant with time

i G i i z p wf wf z p

- Mattar L., McNeil, R., "The 'Flowing' Gas Material Balance", JCPT, Volume 37 #2, 1998

constant         wf z p z p

p G Measured at well during flow wf wf z p

Graphical Method Doesn’t Work!

Graphical Flowing p/z Method for Gas – Variable Rate

i G ? i i z p

p

G

Measured at well during flow

Pressure loss due to flow in reservoir is NOT constant i G i i z p wf wf z p pss wf qb z p z p _        Unknown

Flowing p/z Method for Gas – Variable Rate

Variable Rate p/z – Procedure (1)

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 Cum ulative Production, Bscf

0 50 100 150 200 250 300 350 400 450 500 550 F lo w in g P re s s u re , p s i

Flowing Material Balance Unnam ed Well

Original Gas In Place

Legend Static P/Z*

P/Z Line Flow ing Pressure

Step 1: Estimate OGIP and

plot a straight line from pi/zi to OGIP. Include flowing pressures (p/z)wf on plot

Variable Rate p/z – Procedure (2)

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 Cum ulative Production, Bscf

0.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00 4.40 P ro d u c ti v it y I n d e x , M M s c fd /( 1 0 6p s i 2/c P ) 0 50 100 150 200 250 300 350 400 450 500 550 F lo w in g P re s s u re , p s i

Flowing Material Balance Unnam ed Well

Original Gas In Place Legend Static P/Z*

P/Z Line Flow ing Pressure Productivity Index

Step 2: Calculate bpss for

each production point using the following formula:

Plot 1/bpss as a function of Gp line wf pss p p z z b q   _          

Variable Rate p/z – Procedure (3)

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 Cum ulative Production, Bscf

0.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00 4.40 P ro d u c ti v it y I n d e x , M M s c fd /( 1 0 6p s i 2/c P ) 0 50 100 150 200 250 300 350 400 450 500 550 F lo w in g P re s s u re , p s i

Flowing Material Balance Unnam ed Well

Original Gas In Place Legend

Static P/Z*

P/Z Line Flow ing Pressure Productivity Index

Step 3: 1/bpss should tend

towards a flat line. Iterate on OGIP estimates until this happens

Variable Rate p/z – Procedure (4)

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 Cum ulative Production, Bscf

0.00 0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.00 4.40 P ro d u c ti v it y I n d e x , M M s c fd /( 1 0 6p s i 2/c P ) 0 50 100 150 200 250 300 350 400 450 500 550 P /Z * , F lo w in g P re s s u re , p s i

Flowing Material Balance Unnam ed Well

Original Gas In Place Legend

Static P/Z*

P/Z Line Flow ing P/Z*

Flow ing Pressure Productivity Index

Step 4: Plot p/z points on the

p/z line using the following formula:

“Fine tune” the OGIP estimate

pss data wf p p qb z z   _   _         1/b_pss

Modeling and History Matching

Well / Reservoir Model

Well Pressure at Sandface Production Volumes Constraint (Input) Signal (Output)

Well / Reservoir Model

Production Volumes Well Pressure at Sandface Constraint (Input) Signal (Output)

1. Pressure Constrained System:

Modeling and History Matching

Models - Horizontal

Rectangular reservoir with a horizontal well located anywhere inside.

L

Models - Radial

Rectangular reservoir with a vertical well located anywhere inside.

Models - Fracture

A Systematic and Comprehensive Method for Analysis

Modern Production Analysis Methodology

Diagnostics _{Interpretation and} Analysis Modeling and History Matching Forecasting - Data Chart - Typecurves - Analytical Models - Numerical Models - Data Validation - Reservoir signal extraction - Identifying dominant flow regimes - Estimating reservoir characteristics - Identifying important system parameters - Qualifying uncertainty - Traditional - Fetkovich - Blasingame - AG / NPI - Flowing p/z - Transient - Validating interpretation - Optimizing solution - Enabling additional flexibility and complexity

- Reserves

• Qualitative investigation of data

– Pre-analysis, pre-modeling – Must be quick and simple

• A VITAL component of production data analysis (and reservoir engineering in general)

Illustration- Typical Dataset 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 520 540 Tim e, days 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 L iq u id R a te s , b b l/ d 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 G a s , M M c fd 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 P re s s u re , p s i Data Chart Unnam ed Well Legend Pressure Actual Gas Data

“Face Value” Analysis of Data