Pressure Transient Analysis COPYRIGHT. Introduction to Pressure Transient Analysis. This section will cover the following learning objectives:

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1 Pressure Transient Analysis Core Introduction to Pressure Transient Analysis This section will cover the following learning objectives: Describe pressure transient analysis (PTA) and explain its objectives Identify the data that is acquired through PTA and what it analyzes Identify the types of tests used in PTA and their limitations 1

2 In This Section Pressure Transient Analysis What is it? What data is acquired? What does it analyze? What types of tests are used? What are its objectives? 2

3 What Is Pressure Transient Analysis (PTA)? Input An input rate q(t) is put into a well, either as production or injection It is a function of time Output Pressure P(t) is recorded at the same well or a nearby well P(t) is recorded during: Producing/injection period (drawdown/injectivity test) Shut-in period (buildup/falloff test) and some time before shut-in P(t) is then analyzed What PTA is Not Not a Definite Answer PTA is not a definitive answer to your questions about the inter-well space It is an inverse problem, not a direct problem PTA requires corroboration with: Geology Geophysics 3

4 Data Gathering Valve Closed Tubing Head Pressure Gas Accumulates At Top of Oil Gas Bubbles Rise Flow rate Bottom Hole Pressure Fluid Flows Out WELL TESTING The Diffusivity Equation Record as a function of time: Tubing head pressure (TBH) Bottom hole pressure (BHP) Flow rate Quality Checks: Flow tubing head pressure (FTHP) Flow stable (BSW, GOR, etc. are constant) Flow rate as constant as possible The response of the P(t) from the input of the q(t) follows the Diffusivity Equation, which describes: Flow of a slightly compressible liquid through a porous medium Flow of heat in solids under the influence of a temperature gradient Diffusion of a solute in a solvent under the influence of a concentration gradient Continuity Equation A statement of the law of conservation of matter Darcy s Law Describes the movement of fluid in response to an imposed pressure gradient Equation of State Describes how volume of a slightly compressible liquid changes with pressure 4

5 Hydraulic Diffusivity Equation exp c p u r x b p b k dp dr t u r 2 p 2 r φμct k Hydraulic Diffusivity Equation p t Mass Balance In - Out = Gain bexp c p pb Continuity Equation 2 p 2 r Darcy Equation 1 p. r r φμct k pressure : radius : time p. t Equation of State Where: k effective permeability total porosity μ flowing fluid viscosity c t total compressibility c o S o c g S g c w S w c f Detailed Derivations: OGCI pg. 16-4, Dake (1978) Chapter 5 5

6 This section will cover the following learning objectives: Describe pressure transient analysis (PTA) and explain its objectives Identify the data that is acquired through PTA and what it analyzes Identify the types of tests used in PTA and their limitations 6

7 Learning Objectives Pressure Transient Analysis Core Drawdown Tests, the Concept of Skin, and Drainage Radius This section will cover the following learning objectives: Define drainage radius, radius of investigation, net pay, and skin as used in the diffusivity equation Describe the concept of skin, initially as having no thickness and its relationship to wellbore damage and stimulation Describe the effects of skin, including positive and negative skin Understand that skin has a thickness in order to stimulate through it, if damaged 7

8 In This Section Terminology and Basic Concepts Drainage radius Radius of Investigation Net pay Concept of Skin What is skin? Wellbore damage Skin factor equation Positive skin Negative skin h Damaged Wellbore 8

9 Drawdown Test Terminology p p i, p e p p - p wf p wf What is h? h h r w r r e p i -p wf Units: ft or m Typical values: ft ( m) h 9

10 Radius of Investigation, r inv What part of the reservoir do you see and how fast does a transient move in a reservoir? Field Units Pseudo SI Radius of Investigation q/2 q r inv r inv kt μc 6 t kt μc Parameter Field Units Practical SI Units k md md t hours hours fraction fraction c t psi -1 Pa -1 cp Pa s r ft m t 2q AWTI

11 The Concept of Skin Skin(s) is any phenomenon at the wellbore which results in flowing pressures that are different from radial flow theory. p r w Wellbore Damage Expected flowing pressure Actual flowing pressure r Damaged Wellbore Damage Model p reservoir p skin k a k r a 11

12 Additional Pressure Drop due to Skin Factor p a < 0 p s = 0 p a > 0 k a > k k a < k Distance from axis of wellbore of wellbore Where: k permeability of undisturbed formation, md ka permeability of damaged or stimulated zone near the wellbore, md ra radius of damaged or stimulated altered zone, ft or m pa additional pressure drop due to the presence of the altered zone, psi Kilopascals KP Wellbore Damage (Field Units) Pressure drop across damaged zone Δp d p d r w k a 141.2qB r a k r Δp u ln a 141.2qB ra pu ln kah rw kh rw Additional pressure drop due to damage 141.2qB k r p a s pd pu 1 ln kh k a rw k Pressure drop if no damage r w k r a k Nomenclature (p d ) : pressure drop between (r a ) and (r w ) for a permeability (k a ) (p u ) : pressure drop between (r a ) and (r w ) for a permeability (k) (p s ) : additional pressure drop due to damage 12

13 Skin Factor k r s 1ln a ka rw Where: s skin factor, dimensionless k ka formation permeability, md permeability of altered zone, md ra radius of altered zone, ft rw wellbore radius, ft 13

14 Learning Objectives This section will cover the following learning objectives: Define drainage radius, radius of investigation, net pay, and skin as used in the diffusivity equation Describe the concept of skin, initially as having no thickness, and its relationship to wellbore damage and stimulation Describe the effects of skin, including positive and negative skin Understand that skin has a thickness in order to stimulate through it, if damaged 14

15 Learning Objectives Pressure Transient Analysis Core Buildup Tests and Time Period Analysis This section will cover the following learning objectives: Understand the time periods analyzed in PTA Describe the effects occurring during Early Time Period Analysis and Middle Time Period Analysis Describe the objectives of PTA in each time period analysis Explain how to solve a buildup test problem using the Horner Method 15

16 In This Section Time Periods What they are What is happening in each time period Objectives of each analysis period Buildup Problem Equations needed for example problem Step-by-step instructions Review solution Bottomhole Pressure, psi Late Time Oil Reservoir Well Buildup Horner TIme Ratio Early Time 16

17 Time Period Analysis Let s look at what is important to in each period of a pressure transient test Early Time (ET) Middle Time (MT) Late Time (LT) P ws Wellbore and near wellbore effects Transient or Infinite Acting Period Reservoir parameters log What s Happening in Each Time Period 1 Early Time (ET) Reservoir boundaries Pressure response is controlled by wellbore and near-wellbore phenomena. 2 Middle Time (MT) Pressure response represents infinite-acting radial flow, unaffected by wellbore or reservoir boundaries. 3 Late Time (LT) Pressure response is controlled by reservoir boundaries. 17

18 Objectives of PTA by Analysis Period 1 Early Time (ET) Near-wellbore condition Quantify wellbore damage Evaluate stimulation treatment effectiveness 3 Early Time (ET) Results q 2 Late Time (LT) Reservoir Boundaries Detect presence of reservoir boundaries Identify nature of boundaries Estimate distances to boundaries Estimate drainage area Estimate average drainage area pressure Early Time Middle Time q surface (ET) (MT) Wellbore Theoretical Storage Wellbore Effects Wellbore storage Afterflow Middle Time (MT) Bulk Reservoir Characteristics Estimate in-situ permeability thickness product (kh) Late Time (LT) Theoretical ` Afterflow after Surface Shut In t t q sand face Pressure Draw Down (DD) Pressure Buildup (BU) 18

19 Early Time (ET) Results Pressure Draw Down (DD) q q surface Theoretical t q sand face Wellbore Storage Theoretical ` Pressure Buildup (BU) t Afterflow after Surface Shut In 19

20 Buildup Tests q res 0 p p i, p e p p - p wf p ws Buildup Tests For a Buildup, P ws is recorded at the bottom of the well where P wf was recorded for a drawdown. Buildup requires superposition to the Hydraulic Diffusivity Equation qb t qb = r w r r e p i -p wf + -qb p 0 = + t p t p + t t t p t p +t t p t p +t Horner Result Positive Rate Component Negative Rate Component = + 20

21 Buildup Tests - Equations Transient, S.I. units p p 4k qb ln( t) ln 2 4kh 1.781ctrw i wf 2 s Positive rate component Negative rate component Add: p p i ws qb t ln 4 k h p p p Buildup Tests - Parameters Permeability from slope (k in md): 162.6q o o B m o kh o P skin qb ln( t 4kh 4k t) ln s ct rw i wf p 2 p p 4k qb ln( t) ln 2 4kh 1.781ct rw wf ws 2 t t P 1Hr P wf s m Horner Time Ratio (HTR) Skin: t p t t s k log o 2 o c t r w 3.23 Flow Efficiency P Skin 0.87 ms FlowEfficiency P P P r wf Skin P r P wf 21

22 Buildup Semi-log Graphs Intercept p i 1780 p wsi p 1hr HTR 1hr = (t p +1)/ Horner Time Ratio Permeability and Skin Factor - Liquids SI Units Field Units qb k hm. p1hr pwf k s log m ctrw qb k hm. p1 hr pwf k s log m ctrw HTR 1hr = (t p +1)/1 10 Slope m m in Pa/(log 10 cycle) m in psi/(log 10 cycle) 1 Skin Factor p skin = 0.87ms 22

23 Permeability and Skin Factor Gas m(p) SI Units Field Units k 128.8qT mh m( p1 hr) m( pwf ) k s log m ctrw k 1637qT mh m( p1hr) m( pwf ) k s log m ctrw Radius of Investigation Pressure t = 0 t 1 t 2 t 3 m in Pa/sec / (log 10 cycle) q in m 3 /sec T in degrees Kelvin m in psi 2 /cp / (log 10 cycle) q in Mscfd T in degrees Rankine t 4 Distance from axis of of wellbore r i t 5 t 6 kt 948c t 23

24 Learning Objectives This section has covered the following learning objectives: Understand the time periods analyzed in PTA Describe the effects occurring during Early Time Period Analysis and Middle Time Period Analysis Describe the objectives of PTA in each time period analysis Explain how to solve a buildup test problem using the Horner Method 24

25 Learning Objectives Pressure Transient Analysis Core Late Time Analysis This section will cover the following learning objectives: Understand pressure behavior occurring in late time, from a Horner Time analysis Understand the difference between pseudosteady state and steady state, and their implications to late time Understand what is happening in the late transient period Understand what bounded reservoir behavior is, and that it is equivalent to: Volumetric flow behavior Pseudosteady state Semi-steady state 25

26 In This Section What Happens in Late Time Examples of Horner Plots Pressures Recorded in a Flowing Well Open and Closed Systems 26

27 Open System (full recharge qb) qb r w r w Closed System qb Increasing Time Radius Transient Steady State Increasing Time Transient Steady State Radius qb Pseudo-Steady State (Semi-Steady State) r e No influx r e 27

28 Pressure at the well When do Boundaries Occur? Boundaries None felt Some felt All felt Transient Late Transient Semi-Steady State Late Time Horner Changes Ideal Positive Skin Wellbore Storage Time +t +t +t +t Negative Skin Regular Boundary +t +t +t +t Phase Separation in Tubing ( Gas Hump Fault or Nearby Boundary Naturally Fractured Stratified System 28

29 How Long Does Middle Time Flow Last? Defined by rock and fluid properties for a specific reservoir ct r t 948 k 2 e (field units) Marks the approximate end of transient flow and the beginning of pseudosteady state flow for a well in the center of a circular reservoir Irregular drainage shapes and well locations shorten the time to the end of the transient flow and lengthen the time to the start of pseudosteady state flow 29

30 Learning Objectives This section has covered the following learning objectives: Understand pressure behavior occurring in late time, from a Horner Time analysis Understand the difference between pseudo-steady state and steady state, and their implications to late time Understand what is happening in the late transient period Understand what bounded reservoir behavior is, and that it is equivalent to: Volumetric flow behavior Pseudo-steady state Semi-steady state 30

31 Learning Objectives Pressure Transient Analysis Core Log-Log Analysis This section will cover the following learning objectives: Describe log-log analysis Explain why log-log analysis may be used to improve over semi-log analysis Describe dimensionless variables, explain why they are used, and explain their importance Discuss how flow regimes are used to identify models, and how they are implicit in model selection 31

32 In This Section Introduction to Log-Log Analysis Problems with semi-log analysis PTA as an inverse problem Dimensionless variables Bounded reservoir behavior Flow Regimes/Models Derivative type curves Example of a log-log plot PTA models that work with bounded reservoirs PTA variations 32

33 Problems with Semi-Log Analysis Semi-log straight line analysis is often difficult Radial flow straight line doesn t exist or cannot be correctly identified, due to: Wellbore effects Reservoir boundaries A flow regime other than radial PTA as an Inverse Problem q Log-Log techniques focus on comparing actual data to solutions for KNOWN theoretical models (the inverse problem) Two Log-Log Techniques Type curve matching by hand or by software Analytical history matching using software Both techniques require the use of dimensionless variables and its derivative, worked simultaneously t Known Input A1, A2, {a 1, a 2, a N } U1, U2, {u 1, u 2, u M }? Unknown Model System p Input: 2, 3 Output: 5, 6 Possible Models: Addition, Multiplication Known Output t 33

34 Dimensionless Variables Field Units Dimensionless variables permit theoretical solutions to be readily converted into dimensional values (time, pressure, rate) for any given set of reservoir conditions. Dimensionless Pressure (P D ) Dimensionless Time (t D ) based on wellbore radius Dimensionless Rate (q D ) Liquid khp 141.2q B kt c 2 t r w 141.2qB khp Bounded Reservoir Behavior Character of bounded reservoir response Pseudopressure khm( p) 1424qT 1424qT khm( p) Recognizing response is more important than analyzing it Many reservoir models produce similar responses Selected model must be consistent with best available geological and geophysical information Boundaries control pressure response after the end of the MT region Pressure Squared 2 khp 1424qzT 1424qzT 2 khp Shape of drawdown pressure response can be understood in terms of flow regimes exhibited 34

35 Derivative Type Curves It is sometimes difficult to identify unique match on standard log-log type curves Derivative functions are included with the pressure change curves p ' D d dp D dp D lnt D dtd Matching two curves simultaneously reduces interpretation ambiguity Is This the Right Model? t D 35

36 Flow Regime Slopes Derivative Curves Give Unique Flow Regime-Identifying Slope The values of the pressure derivative have very specific values for identified flow regimes Therefore, inspection of the log-log derivative type curve allows identification of flow regimes and distance from well The flow regime slope identifies the model If the correct model has been identified the reservoir parameters can be calculated Flow Regime Selection Drives Model Selection Flow Standard Specialized Straight Line Regime/Model Diagnostic Plot Diagnostic Plot Plot Radial 0 t(dp/dt) vs. t p vs. t Linear ½ dp/dt ½ vs. t p vs. t ½ Volumetric 1 dp/dt vs. t p vs. t Spherical ½ dp/dt ½ vs. t p vs. t ½ Bilinear ¼ dp/dt ¼ vs. t p vs. t ¼ Flow Regime/Model Data Needed Slope Gives Intercept Gives Radial h k s Linear k x or w; k w or k x ; L f s g ; s f,ch Volumetric c t C; V p, N J Spherical B, μ, c t, ø k s Bilinear k w f k f s f,ch 36

37 Vertical Well, Infinite-Acting Radial Flow t(dp/dt) vs. t Flow Regimes in Horizontal Wells p w D 1E+03 1E+02 1E+01 1E+00 Early Radial Flow (ERF) Early Hemiradial Flow (EHRF) Early Linear Flow (ELF) Pseudo Radial Flow (PRF) Late Linear Flow (LLF) Pseudo Steady State Flow (PSSF) 1E-01 1E-02 1E-03 1E-06 1E-05 1E-04 1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1E+03 1E+04 t Lh D 37

38 Learning Objectives This section has covered the following learning objectives: Describe log-log analysis Explain why log-log analysis may be used to improve over semi-log analysis Describe dimensionless variables, explain why they are used, and explain their importance Discuss how flow regimes are used to identify models, and how they are implicit in model selection Key Learning Points 1 PTA involves a q(t) impacting a reservoir and a P(t) analyzed 2 PTA uses the hydraulic diffusivity equation, combining: 3 The continuity equation (mass balance) Darcy s law The equation of state The amount of the reservoir seen is not a function of the rate of production/injection q 4 5 PTA is commonly an inverse problem, without a unique solution, and requires corroboration with geology and geophysics If you know the model, only then can you calculate the parameters, as you did with the buildup problem 38

39 Key Learning Points 6 PTA utility in bounded reservoir management decisions is greater knowing the correct model as opposed to its parameters PTA is analyzed in three distinct connected time periods, having carryover issues to adjoining periods Each PTA time period has different objectives, but these have challenges that impact the analysis in adjoining periods: A. Early Time: Impacted by wellbore storage and near-well reservoir effects B. Middle Time: Impacted by flow regime slope chosen C. Late Time: Impacted by boundaries and well interference Complex well trajectories and production systems, i.e. fractured reservoirs, encounter many more possible flow regimes making them more difficult to analyze PetroAcademy TM Applied Reservoir Engineering Skill Modules Properties Analysis Management This is Reservoir Engineering Core Reservoir Rock Properties Core Reservoir Rock Properties Fundamentals Reservoir Fluid Core Reservoir Fluid Fundamentals Reservoir Flow Properties Core Reservoir Flow Properties Fundamentals Reservoir Fluid Displacement Core Reservoir Fluid Displacement Fundamentals Reservoir Material Balance Core Reservoir Material Balance Fundamentals Decline Curve Analysis and Empirical Approaches Core Decline Curve Analysis and Empirical Approaches Fundamentals Pressure Transient Analysis Core Rate Transient Analysis Core Enhanced Oil Recovery Core Enhanced Oil Recovery Fundamentals Reservoir Simulation Core Reserves and Resources Core Reservoir Surveillance Core Reservoir Surveillance Fundamentals Reservoir Management Core Reservoir Management Fundamentals 39