Houston university pressure transient analysis

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Houston university pressure transient analysis

  1. 1. University of Houston Pressure Transient Analysis Petroleum Engineering Spring 2001 Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Description of a well test: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Types of tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Why we do transient testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Flow States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Development of Flow Equations for Flow in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Solutions to the Diffusivity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Skin Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Wellbore Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Wellbore Storage (WBS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Radius of Investigation (ROI). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Pseudo Steady-State. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Shape Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Principle of Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Horner’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Buildup Test Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Derivative Analysis (Drawdown case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Ideal vs. Actual PBU/DD Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Flow Regimes & Model Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Gas Well Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Gas Tests - Pseudo (Ψ(P)) Equation Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Pseudopressure or Real Gas Potential (Ψ(P)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Determination of Skin and D for Gas Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Multiple Rate Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Odeh-Jones Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Horizontal wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Pressure level in surrounding reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Drill Stem Tests (DST) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Conducting Well Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Wellbore Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 © 2000-2001 M. Peter Ferrero, IX
  2. 2. Pressure Transient Analysis Introduction Instructors: Jeff App email: app@wt.net B.S.: Civil Engineering, Rice University M.S.: Chemical Engineering, University o f H ou sto n Currently completing Ph.D. in Chemical Engineering, University of Houston Dr. Christine Ehlig-Economides email: cee@slb.com M.S.: Chemical Engineering, University of Kansas Ph.D.: Petroleum Engineering, Stanford University Grading: • 20% homework • 40% midterm • 40% final Textbook: • Well Testing by John Lee Introduction Page 2 © 2000-2001 M. Peter Ferrero, IX
  3. 3. Pressure Transient Analysis Description of a well test: Schematic: Gas Choke Separator Oil Water Packer Pressure gauge Perforations Fig. 1. Schematic of well test set-up Process: • flow well at single or multiple rates for time, tp. • shut well in for pressure buildup (PBU), ∆t. • measure P, T, and q’s (pressure, temperature, and flow rates, respectively). Information gained: • • • • reservoir fluids [BHS (bottom hole sample), separator samples for PVT analysis] reservoir temperature and pressure (from gauge) permeability and skin (completion efficiency) reservoir description, qualitative (faults, changes in permeability, oil/water contact) Description of a well test: Page 3 © 2000-2001 M. Peter Ferrero, IX
  4. 4. Pressure Transient Analysis Types of tests Drawdown test (DD) • difficult to maintain constant rate • this introduces scatter in measured FBHP (flowing bottom hole pressure) q P T im e Fig. 2. Drawdown test Pressure buildup test (PBU) • advantage: rate is known, i.e. q=0 • disadvantage: lost production q P T im e Fig. 3. Pressure buildup test Injection test • advantage: injection rates are easily controlled • disadvantage: analysis is complicated by multiphase effects and possible fracturing q P T im e Fig. 4. Injection test Types of tests Page 4 © 2000-2001 M. Peter Ferrero, IX
  5. 5. Pressure Transient Analysis Falloff test q P T im e Fig. 5. Falloff test Interference/pulse test • Tests connectivity of wells using a producers and observation wells kh • Used to estimate transmissibility ------ , and storativity φ hc t µ Drillstem test (DST) • • • • Way to go for exploration Utilize downhole shut-in which greatly reduces wellbore storage (WBS) Accurate production rate measurement on site production facilities Why we do transient testing When we make a rate change, the system goes through a transition state during which the steady-state solutions are not valid – this is known as transient flow. This is the period that is the basis for well testing or pressure transient analysis. • Steady-state equations do not yield “unique” values for k, h, & s: re 141.2 q µβ ∆ P = -------------------------  ln ---- + S  r  kh w • Log derived/core kh values are not always representative of system/reservoir kh. • Well testing yields macroscopic, average system kh. Why we do transient testing Page 5 © 2000-2001 M. Peter Ferrero, IX
  6. 6. Pressure Transient Analysis Flow States -----• Steady-state, ∂ P = 0 , pressures ∂t in reservoir/wellbore do not vary with time. For all time P rw Fig. 6. Steady-state flow regime • Pseudo steady state, ∂P ------ = constan t ∂t re , pressures in reservoir/wellbore are changing in a constant (linear) man- ner t1 t2 t3 P rw P re Linear Time Fig. 7. Pseudo steady-state flow regime Flow States Page 6 © 2000-2001 M. Peter Ferrero, IX
  7. 7. Pressure Transient Analysis -----• Transient, ∂ P = f ( x, y, z, t ), pres∂t sure in reservoir/wellbore are changing as a function of both time and location. t1 t2 t3 P rw re Fig. 8. Transient flow regime Development of Flow Equations for Flow in Porous Media Note: there is a good writeup in Appendix A of Lee. What’s needed to derive the diffusivity equation is: • A. Conservation of Mass (Continuity equation) • B. Darcy’s Law • C. Equation of State (EOS) A. Continuity equation, cylindrical coordinates (r, z, θ) ∂ ρv z r dr dθ + ( ρv z ) dz r dr dθ ∂z ρv r r dθ dz + ∂ ( ρrv r ) dθ dr dz ∂r ρv z r dr dθ + ∂ ( ρv z ) dz r dr dθ ∂z ρv r dθ dz r dz rd θ dθ ρv θ dr dz ρv z r dr dθ dr Fig. 9. Cylindrical coordinate system Development of Flow Equations for Flow in Porous Media © 2000-2001 M. Peter Ferrero, IX Page 7
  8. 8. Pressure Transient Analysis lbm ft lbm - ρ v = --------- × -- = ------------3 2 s ft ft ⋅ s mass flux, [Rate of mass accumulation] + [Rate of mass outflow] - [Rate of mass inflow] = 0 ∂ ∂ ( ρθr dθ dr dz ) = r dθ dr dz ( ρθ ) ∂t ∂t ρv r r dθ dz + ρv θ dr dz + ∂ ( ρrv r ) dθ dr dz – [ ρv r r dθ dz ] ....r direction ∂r ∂ ( ρv θ ) dθ dr dz – [ ρv θ dr dz ] ....θ direction ∂θ ρv z r dr dθ + ∂ ( ρv z ) dz r dr dθ – [ ρv z r dr dθ ] ....z direction ∂z ∂ ∂ ∂ ∂ r dθ dr dz ( ρθ ) + ( ρ rv r ) dθ dr dz + ( ρ v θ ) dθ dr dz + ( ρ v z ) r dθ dr dz = 0 ∂t ∂r ∂θ ∂z .... divide by r dθ dr dz ∂ ∂ 1∂ 1∂ ( ρθ ) + -- ( ρ rv r ) + -- ( ρ v θ ) + ( ρ v z ) = 0 r ∂r r ∂θ ∂t ∂z .... note that since there is no z or θ, the last two terms are 0 .... therefore, for a fully perforated interval, the RADIAL DIFFUSIVITY EQUATION is ∂ 1∂ ( ρθ ) + -- ( ρ rv r ) = 0 r ∂r ∂t B. Darcy’s Law k ν = – -- ∆ P µ k dP ν r = – ---r µ dr k θ dP ν θ = – ---µ dθ k z dP ν z = – ---µ dz Isotropic: k=kr=kθ=kz k dP ∂ 1∂ ∴ ( ρφ ) + --  – ρ r ---r  = 0 r ∂r ∂t µ dr  or ∂ 1 ∂  k r dP = ( ρφ ) -- ρ r --r ∂r µ dr  ∂t Assume single slightly compressible fluid - compressibility, c= constant 1 dρ 1 c ≡  – --------  d Vol → 1 ; Vol = -- Vol d P ρ ρdP By integration: Development of Flow Equations for Flow in Porous Media © 2000-2001 M. Peter Ferrero, IX Page 8
  9. 9. Pressure Transient Analysis ρ = ρ0e c ( P – P0 ) ρ P = ∫P c dP = 0 ∫ρ ∂ρ ----- ; ρ 0 ≡ base ρ ρ 0 c ( P – P0 ) ∂P ∂P ∂ρ = c ρ0 e = cρ ∂r ∂r ∂r c ( P – P 0 ) ∂P ∂P ∂ρ = c ρ0 e = cρ ∂r ∂r ∂r 1 ∂  k ∂P ∂ = ( ρφ ) -- ρ r -r ∂r µ ∂r  ∂t 2  ∂θ 1 k k k --  ∂P r -- ∂P + ρ -- ∂P + ρ r -- ∂ P = φ ∂ρ + ρ   ∂t  r ∂r µ∂r µ∂r µ ∂ r2  ∂t 2 ∂P 2 ∂P k 1 ∂ P ∂P -- --  cr ρ   + -+ ρr  = c φρ  ∂r  µr ∂r ∂r  ∂t Note: 1. Since φ doesn’t change wrt time, ρ ∂φ → 0 ∂t 2 2 2. Also, since the pressure gradient is small,  ∂P « 1 ; ∴cr ρ  ∂P → 0 ∂r  ∂r  2 ∂ P ∂P ρ k  ∂P -- --  + ρ r - = c φρ r µ∂r ∂r  ∂t k Canceling ρ’s, and dividing through by -- µ 2 c φµ ∂P ∂ P 1  ∂P --  + ρ r  = --------r ∂r k ∂t ∂r  .... therefore, for a fully perforated interval, the RADIAL DIFFUSIVITY EQUATION including Darcy’s law is k 1 ∂  ∂P 1 ∂P --  r  = -where η = --------r ∂r ∂r φµ c η∂t To solve this you need two boundary conditions and one initial condition. For a closed system: Initial condition: P = Pi @ t=0 Boundary condition 1: No flow - ∂P ∂r = 0 re Development of Flow Equations for Flow in Porous Media © 2000-2001 M. Peter Ferrero, IX Page 9
  10. 10. Pressure Transient Analysis Boundary condition 2: ∂P ∂r rw qµ = --------------2π hr w For an infinite reservoir, BC1 becomes P → P i as r → ∞ . Darcy’s law came from Darcy’s investigation of the sewers in Paris. He conducted his experiments on flow through gravel. Steady-state linear flow: P1 q P2 k dP velocity, u = – 0.001127 ⋅ ------ ⋅ µβ d l P e rm e a b i l i ty , k W a te r v i sc o si ty , µ w q kA dP q = – 0.001127 ⋅ ------ ⋅ µβ d l l Fig. 10. Steady-state linear flow Darcy velocity in Cylindrical coordinates k dP velocity, u = – 0.001127 ⋅ -- ⋅ µ dr 2π r w k dP q = – 0.001127 ⋅ --------------- ⋅ dr µ dr q ---r r2 rw 2π r w k = – 0.00708 ⋅ --------------- ⋅ dP µ rw h P2 Pw kh ( P 2 – P w ) q = – 0.00708 ⋅ ------ ⋅ -----------------------r2 µβ  ----  ln  r w A re a , A = 2 π r w h (a re a o f c y lin d e r) Fig. 11. Darcy velocity in cylindrical coordinates Examples of tests: • In transient flow, pressure will decrease wrt time at constant flow rate. • Separation of log-log and derivative plot indicates skin (larger separation=larger skin) 1. Derived diffusivity equation based on: Development of Flow Equations for Flow in Porous Media © 2000-2001 M. Peter Ferrero, IX Page 10
  11. 11. Pressure Transient Analysis • Continuity equation • Darcy’s law • EOS 2. Assumptions: a. Radial flow over entire net thickness b. Homogeneous and isotropic porous media (kr=kθ=kz) c. Uniform net thickness d. q and k are constant (independant of pressure) e. Fluid is of small and constant compressibility f. Constant µ 2 g. Small pressure gradients (  ∂P « 1 )  ∂r  h. Negligible gravity forces Solutions to the Diffusivity Equation 3. Develop solutions to diffusivity equation. • “Exact solution” - Van Everdingen & Hurst terminal rate solution (center, bounded, circular system!). (We won’t use this!) P wf 141.2 q µβ 2 tD 3 = Pi – ------------------------- ⋅ -------- + ln r eD – -- + 2 2 kh 4 r eD ∞ 2 –α tD 2 e J 1 ( α η r eD ) ∑ ---------------------------------------------------2 J1 η=1 2 αη 2 ( α η r eD – J 1 α η ) • Infinite reservoir, line source well - constant rate, q - unbounded (infinite acting) reservoir 1 ∂P 1 ∂  ∂P -- r = -r ∂r ∂r  η∂t a. Initial condition: P=Pi at t=o for all radius, r qµ b. Boundary condition (BC) #1:   r ∂P = ------------- f for t>0...constant rate condition   ∂ r  r w 2π kh c. BC #2: P → P i as r → ∞ for all t Replace BC#1 to obtain “line source” approximation lim r r→0 ∂P ∂r rw qµ = ------------2π kh for t>0 ∞ –µ   –r --------------Line source solution: P (r,t) = P i + 70.6 q µβ E i  -------- ; where  – Ei ( –x ) = ∫ e - dµ   2 kh 4η t  Solutions to the Diffusivity Equation x µ  Page 11 © 2000-2001 M. Peter Ferrero, IX
  12. 12. Pressure Transient Analysis • DRAWDOWN ONLY • Constant rate • Unbounded reservoir Limitations of the line source solution (Ei) 2 a. Check to insure that Ei solution is 100 r η 2 re 100 r w valid --------------- ≤ t ≤ -----η 4η 2 w - for t < --------------- , the assumption of zero wellbore radius limits the accuracy of the solution 2 - for re t > ------ , effects of boundaries are felt, E i solution no longer 4η 2 –r ---------valid. P (r,t) = P i + 70.6 q µβ Ei  --------    kh 4η t b. If Ei solution is valid, check applicability of ln approximation. E i ( x ) = ln ( 1.781 x ) , x ≤ 0.02 2 2 2 r 0.445 r –r E i  --------  = ln  ------------------- , -------- ≤ 0.02  4η t  η t  4η t For wellbore, Pw (if Ei is valid, then it’s always valid at the wellbore) 2 ln approximation rw -------- ≤ 0.02 4η t 2 but for Ei r w 0.01 -------- ≤ ---------4 4η t - If Ei function is valid at the wellbore, then ln approximation will always be valid at the wellbore! - Even if though the Ei function may be valid at radius, r (rw < r < re), the ln approximation won’t always be valid. Skin Development Skin, S, refers to a region near the wellbore of improved or reduced permeability compared to the bulk formation permeability. Impairment (+S): • Overbalanced drilling (filtrate loss) • Perforating damage • Unfiltered completion fluid • Fines migration after long term production • Non-darcy flow (predominantly gas well) • Condensate banking- acts like turbulence Stimulation (-S) • Frac pack (0 to -0.5) • Acidizing Skin Development Page 12 © 2000-2001 M. Peter Ferrero, IX
  13. 13. Pressure Transient Analysis • Hydraulic fracturing Generally S>5 is considered bad; S= -3.5 to -4 is excellent. Flow efficiency, FE, is the ratio of flow without skin to the flow with skin, Darcy w/o S 8 ------------------------------- , or FE ≈ ------------S+8 Darcy /w S ∆ P s = ∆ P ks – ∆ Pk Pressure rs rs q µβ q µβ ∆ P s = 141.2 ---------- ln  ----  – 141.2 --------- ln  ----   r w  r w ks h kh rs q µβ k ∆ P s = 141.2 ----------  ---- – 1 ln  ----    r w kh  k s rs k We define  ---- – 1 ln  ----  = S k   r w s ∆ Pk ∆Pks q µβ ∴∆ Ps = 141.2 --------- S kh ∆ Ps k including skin rw k of formation rs Radius Fig. 12. Skin pressure drop Combine with Darcy’s law: ∆ P total = ∆ PS = 0 + ∆ P S re re q µβ q µβ q µβ ∆ P total = 141.2 ---------- ln  ----  + 141.2 --------- S = 141.2 --------- ln  ----  + S  r w  r w kh kh kh S > 0 → Damaged ∴k s < k S < 0 → Stimulated ∴k s > k S = 0 → Undamaged ∴k s = k Skin Development Page 13 © 2000-2001 M. Peter Ferrero, IX
  14. 14. Pressure Transient Analysis SEM examples of various clays which can cause formation damage Fig. 13. Smectite (left) and kaolinite (right) coat grains and fill a pore. Note distinct differences in morphology of each clay ("honeycomb" smectite; vermicular booklets of kaolinite (x2000) (image courtesey of Westport Technology Center) Fig. 14. Delicate wisps of "hairy" illite project into a pore. Note that the fibers not only form a highly rugose surface within the pore, but the fibers could break and migrate under extreme fluid pressures (x2500) (image courtesey of Westport Technology Center) Fig. 15. Well-formed chlorite platelets form partial rosettes adjacent to, and coating quartz overgrowths (x2500) (image courtesey of Westport Technology Center) Fig. 16. Well-formed, but rather randomly oriented kaolinite booklets post-date quartz overgrowths (x700) (image courtesey of Westport Technology Center) Skin Development Page 14 © 2000-2001 M. Peter Ferrero, IX
  15. 15. Pressure Transient Analysis SEM examples of formation damage and stimulation Fig. 17. SEM image of perforation damage with percussion fines (x305) Fig. 18. SEM image of completion damage with polymer filament (x105) Fig. 19. SEM image of pre-acid treatment (x3100) Fig. 20. SEM image of post-acid treatment (x3100) Skin Development Page 15 © 2000-2001 M. Peter Ferrero, IX
  16. 16. Pressure Transient Analysis Wellbore Solutions 1. Ideal reservoir (no skin) 2 –4 q µβ  0.445 r w ×10 k P w (r,t) = Pi + 70.6 ---------- ln  -------------------- ; where η = 2.637 -------------------------------kh  η t  φµ c t 2 q µβ  1688φµ c t r w P w ( t ) = P i + 70.6 ---------- ln  ------------------------------  ; from Lee ktp kh   2. Solution at sandface (including skin) 2 q µβ  0.445 r w q µβ ∆ P wf = P i – Pwf = ∆ P k + ∆ P skin = – 70.6 --------- ln  -------------------- + 141.2 --------- S kh  η t  kh 2  q µβ  0.445 r w ∆ P wf = – 70.6 --------- ln  -------------------- – 2 S kh  η t  2 q µβ  0.445 r w P wf = P i + 70.6 ---------- ln  -------------------- – 2 S ; kh  ηt  –4 2.637 ×10 k η = -------------------------------φµ c t Wellbore Storage (WBS) • Unit slope on log-log plot of ∆P vs. time • Straight line on cartesian, b ≠ 0 Storage between the sandface and shut-in valve allow the formation to continue to flow when we affect a shut-in. This is due to fluid compressibility. We will consider two cases: 1. A well with a gas-liquid interface 2. A liquid filled well Wellbore Solutions Page 16 © 2000-2001 M. Peter Ferrero, IX
  17. 17. Pressure Transient Analysis General definitions Pt qβ (RB/D) β Vwb = volume of liquid in well (ft3) Awb = cross-sectional area of well (ft2) ρl = density of wellbore fluid (lbm/ft3) h = height of liquid column in wellbore (ft) Vwb dh Awb hliquid Gas-liquid interface • pumping wells, gas lift wells • injection wells (on vacuum) • an approximation for most naturally flowing oil wells (except highly undersaturated oils, P >Pb) qSFβ (RB/D) Pw + Pt + ρlgh 144 Fig. 21. Wellbore storage definitions Wellbore mass balance [Mass inflow] - [Mass outflow] = Accumulation of Mass 24 d ( q SF β – q β )ρ = -------------- ( ρ v WB ) 5.615 d t 3 ft 24  bbl • lbm ------- = --------------  lbm • ft 3 ------- --------- --------3  bbl   D 5.615  ft 3 ft Assume constant density, ρl dv dh 24 dv ∴( q SF – q )β = -------------- WB ; where v WB = A WB h ; WB = A WB dt dt 5.615 d t dh 24 ( q SF – q )β = -------------- A WB dt 5.615 Note: 144 ( Pw – P t ) h = --------------------------------ρg dP 144 dP w dh → t= = --------; assume ρg dt dt dt 24 144 A WB dP ∴( q SF – q ) )β = -------------- --------------------- w 5.615 ρ g d t Wellbore Storage (WBS) Page 17 © 2000-2001 M. Peter Ferrero, IX
  18. 18. Pressure Transient Analysis Definition: Wellbore storage coefficient for a gas-liquid interface 25.65 A BW bbl 144 ABW c s = --------------------- = -------------------------- ------psi ρl 5.615ρ l Example: 3.5” tubing, A WB = 0.041 ft 2 ρ o = 50 lbm/ft 3 vwb = 100 bbl depth = 17,000 ft A WB -------------------Solution cs = 25.65 ----------- = 25.65  0.041 = 0.02 bbl  50  psi ρl (note that for a gas-liquid interface the cs is independent of well depth!) Governing Equation (WBS) c s dP ( q SF – q ) = 24 ---- w β dt qsf = sandface flowrate, STB/D q = surface flowrate, STB/D cs = WBS coefficient, bbl/psi β = formation volume factor, RB/STB dP w dt = change in BHP wrt time BIG NOTE: Using downhole shut-in eliminates most WBS Pure Wellbore Storage B - Unit slope on log-log plot A - straight line on cartesian plot Why? A - 100% WBS, q=0 (PBU) c dP w β dt s • qSF = 24 ----- • Therefore, cs can be calculated from the slope of a straight line (intercept must be zero!) ∆P β q SF ------------ = m 24 c s ∆t B - Log-log plot, 100% WBS, q=0 @surface (PBU) Fig. 22. cs from cartesian plot Wellbore Storage (WBS) Page 18 © 2000-2001 M. Peter Ferrero, IX
  19. 19. Pressure Transient Analysis cs ∆ P qSF = 24 ---- ------- β ∆t β q SF ∆ P w = ------------ ∆ t 24 c s β q SF log ( ∆ P w ) = log ------------ ∆ t 24 c s β q SF log ( ∆ P w ) = m log ( ∆ t ) + log  ------------   24 c s  log ∆Pw m=1 β q SF -----------24 c s Estimate c s from any ( ∆ Pw ,∆ t ) pair on unit slope line log ∆t Fig. 23. cs from log-log plot. Estimate cs from any (∆Pw, ∆t) pair on unit slope ∆ line d (x) d ln t d d d d ( x ) = ln ( t ) ⋅ ( x) = t ( x) d ln ( t ) d ln ( t ) dt dt d (x) = t d (x) d ln ( t ) dt ∴∆ P W = β q SF d ( ∆ P W ) = t ⋅ -----------24 c S dt [ Take log of both sides ] β q SF d log ( ∆ P W ) = log ( t ) + log  ------------   24 c S dt β q SF ∴m = 1 intercept = ------------ for ∆P 24 c S Completely liquid filled wellbore Wellbore mass balance [Mass inflow] - [Mass outflow] = Accumulation of Mass Wellbore Storage (WBS) Page 19 © 2000-2001 M. Peter Ferrero, IX
  20. 20. Pressure Transient Analysis 24 d ( q SF β – q β )ρ = -------------- ( ρ v WB ) 5.615 d t [ Note → v WB = A WB h ] dρ 24 ( q SF – q )βρ = -------------- v WB   dt  5.615 dP dP  dρ   dP w 1 ∂ρ → by chain rule c ≡ -= = cρ w d P w  d t  dt ρ∂P dt dP 24 ( q SF – q )βρ = -------------- v WB c ρ w dt 5.615 dP 24 ( q SF – q )β = -------------- v WB c w dt 5.615 v WB c c s ≡ -------------5.615 bbl ------psi where c = average fluid compressibility Example: vWB = 100 bbl c = 1X10 -5 psi -1 –5 v WB c ) -------------------------------------Solution cs = -------------- = 100 ( 1 ×10 - = 0.0002 bbl 5.615 5.615 psi Note: for cs < 0.003 there is basically no WBS Determining the end of WBS c s dP qSF – q = 24 ---- w β dt Drawdown case (100% WBS) q SF = 0 initially as open to rate q c s dP q = 24 ---- w β dt Buildup case (100% WBS) q = 0 initially as the well is shut in qSF = fixed c s dP qSF = 24 ---- w β dt c s dP WBS is over when 24 ---- w ≤ 0.01 q β dt q = rate prior to a PBU Wellbore Storage (WBS) Page 20 © 2000-2001 M. Peter Ferrero, IX
  21. 21. Pressure Transient Analysis = production rate for a drawdown test PWF t q3 q2 q1 t Radius of Investigation (ROI) This is one of the basic concepts to well test analysis. From the error function: –4 Ri = 4η t 2.637 ×10 k ; η = -------------------------------φµ c t R feet t hou P i k mD P f frac t1 t2 t3 m cp c psi-r w r1 r2 r3 re Fig. 24. Illustration of ROI Radius of investigation is INDEPENDENT of q Radius of Investigation (ROI) Page 21 © 2000-2001 M. Peter Ferrero, IX
  22. 22. Pressure Transient Analysis Pseudo Steady-State Depletion of a closed system Pseudo steady-state occurs when the pressure transient has reached all boundaries in a closed system. The solution, based on the Van Everdingen & Hurst terminal exact solution of a bounded, cylindrical reservoir is 2 re re q µβ t PWF = P i – 141.2 - 2η- + ln  ----  – 0.75 for t ≥ ------------------------------ ------- r w 2 4η kh re ∴ ∂P WF 141.2 q µβ 2η - 2= – ------------------------- -----kh ∂t re –4 2.637 ×10 k ; η = -------------------------------φµ c t –4 ∂P WF – 0.0744 q β 141.2 q µβ 2 ( 2.637 ×10 ) k = – ------------------------- ---------------------------------------- = ---------------------------2 2 kh ∂t re φµ c t φ c t hr e 2 Note: V p = π r e φ h reservoir volume ∂P WF 0.234 q β ∆P = – --------------------- = ------ct Vp ∂t ∆t This is very difficult to apply! Shape Factors p. 9-10 of Lee text Principle of Superposition The diffusivity equation is a linear homogeneous equation (with homogeneous BC’s). 1 ∂  ∂P 1 ∂P -- r = -r ∂r ∂r  η ∂t Therefore, linear combinations of solutions are also solutions. The combined linear solution eliminates the following restrictions: • Single well • Reservoir boundaries • Constant rate Pseudo Steady-State Page 22 © 2000-2001 M. Peter Ferrero, IX
  23. 23. Pressure Transient Analysis Multi-well solution A qC q rAB qB rAC B qA C t Determine ∆ P A ∆ P TOTAL A = ∆ PA + ∆ PB + ∆ PC 2 qµβ 0.445r ∆P (r,t) = P i + 70.6 ---------- ln ------------------- – 2S kh ηt 2 0.445r qµβ ∆P = P i – P (r,t) = – 70.6 ---------- ln  ------------------- – 2S  ηt  kh 2 ∴∆ P TOTAL A 2 2 q B µβ  – r AB q A µβ q C µβ  – r AC 0.445r = – 70.6 ------------- ln  -------------------  – 2S A – 70.6 ------------- E i  ----------- – 70.6 ------------- E i  -----------   ηt  kh kh kh  4η t   4η t  2 r Check for ln ( 1.781 x ) if -------- < 0.02 4η t Principle of Superposition Page 23 © 2000-2001 M. Peter Ferrero, IX
  24. 24. Pressure Transient Analysis Boundaries Single fault Geologic model Mathematical Model L L L q actual q q image no flow boundary use image well) Fig. 25. Single fault geologic model Fig. 26. Single fault geologic model ∆ Ptotal = ∆ P actual + ∆ P imag e 2 2 –( 2 L ) q µβ qµβ  0.445rw = P i – PWF = – 70.6 --------- ln  -------------------- – 2S – 70.6 --------- E i  -----------------  4η t  kh kh  ηt  2 2 ------------------------------------For long time, E i  4 L  ≈ ln  0.445 ( 2 L ) -  4η t    ηt For not totally sealing faults use FOG FACTORS (for q of image well): • 1 = sealing • 0 = no fault • -1 = water drive (constant P) Principle of Superposition Page 24 © 2000-2001 M. Peter Ferrero, IX
  25. 25. Pressure Transient Analysis Intersecting faults (90 degree) Need three image wells Geologic model Mathematical Model q image q image L 2 L L L 2 L L q L L q actual q image no flow boundary (use e well) Fig. 27. 90 degree intersecting fault geologic model Fig. 28. 90 degree intersecting fault mathematical model Principle of Superposition Page 25 © 2000-2001 M. Peter Ferrero, IX
  26. 26. Pressure Transient Analysis Intersecting faults (45 degree) Need seven image wells Geologic model Mathematical Model q image q image q image q image q q image q image q actual q image (use image well) Fig. 29. 45 degree intersecting fault geologic model Fig. 30. 45 degree intersecting fault mathematical model Principle of Superposition Page 26 © 2000-2001 M. Peter Ferrero, IX
  27. 27. Pressure Transient Analysis Variable rate Single well producing at variable rates (ideal, infinite reservoir) q2 q1 ∆P = f(q,t) q3 t0 = t1 t2 q1 + -q1 + q2 + -q2 + q3 OR q1 + q2 -q 1 + q3-q 2 Principle of Superposition Page 27 © 2000-2001 M. Peter Ferrero, IX
  28. 28. Pressure Transient Analysis General Solution ∆ P = P i – P WF µβ = – 70.6 -----kh m ∑ i=1 2  0.445 r w  ( q i – q i – 1 ) ln  -------------------------  – 2 S  η ( t – ti – 1 ) Can incorporate dozens of rates Horner’s Approximation • Avoids the use of superposition to model variable rates • Can replace the need for multiple E i ( ln x ) function evaluation each representing a rate change, with a single function ( E i ) that contain a single rate and producing time. Procedure • Single rate used is most recent non-zero rate, qlast • Producing time is cumulative production (Np) divided by qlast ∑ Production from well NP t P = 24 ------------------------------------------------------------ = ---------q last Most recent rate 2  0.445 r w q last µβ ∆ P = Pi – P WF = – 70.6 ----------------- ln  -------------------- – 2 S kh  η tP  qlast qnext PBU q=0 Note: t last > 2 ⋅ tnext to last Buildup Test Solutions (Chapter 2 - Lee) Ideal pressure buildup test • Infinite acting reservoir (no boundaries have been felt by transient) • Formation and fluid properties are uniform (Ei and ln function apply) Horner’s Approximation Page 28 © 2000-2001 M. Peter Ferrero, IX
  29. 29. Pressure Transient Analysis • Use superposition to model variable rates q ∆t tP -q ∆ P = ∆ Pq t + ∆t p – ∆ Pq ∆t = DD – PBU = ( P i – P WF ) – ( P WS – P WF ) = P i – P WS 2 2 ( – q )µβ  0.445r w qµβ  0.445rw - ∆ P = P i – P WS = – 70.6 --------- ln  ------------------------  – 2S – 70.6 ------------------ ln  -------------------- – 2S η ( t p + ∆ t ) kh kh   η( ∆t )  2 2  0.445rw qµβ  0.445r w - P i – P WS = – 70.6 ---------- ln  ------------------------  – 2S – ln  -------------------- + 2S η ( t p + ∆ t ) kh   η( ∆t )  2 2  0.445r w qµβ  0.445r w - P WS = Pi + 70.6 ---------- ln  ------------------------  – ln  -------------------- η ( tp + ∆ t ) kh   η(∆t)  ∆t qµβ = Pi + 70.6 ---------- ln  --------------------   ( t p + ∆ t ) kh Note: ln x = 2.302 log x ( tp + ∆ t ) qµβ ∴P WS = Pi – 162.6 --------- log  --------------------   ∆t  kh q is the rate prior to PBU. Use Horner’s approximation with multiple rates Pi = P* (infinite shut-in) 162.6 q µβ m = ------------------------kh P2 – P1 ∆y m = ------ = -------------------------------------------------------------------------tP + ∆ t2 tP + ∆ t1 ∆x log  ------------------- – log  -------------------  ∆ t2   ∆ t1  PWS P2 – P 1 P2 – P1 = ------------------------------------------------ = ------------------ = P 1 – P 2 log ( 10 ) – log ( 100 ) 1–2 1000 100 10 tP + ∆ t ----------------∆t 1 t Note: lim t P + ∆- = 1 ---------------∆t → ∞ ∆t P* is always taken as the extrapolation from the MTR irregardless of whether boundaries or late time effects are seen. If late time effects are observed, P* may not correspond to Pi or P Buildup Test Solutions Page 29 © 2000-2001 M. Peter Ferrero, IX
  30. 30. Pressure Transient Analysis Derivative Analysis (Drawdown case) Bourdet derivative d 1 d = -------------- ⋅ 2.302 d log t d ln t By chain rule d( ) 1 d( ) d( ) d --------- = ln t ⋅ ---------- = -- ⋅ ---------d ln t t d ln t dt dt d( ) d( ) ---------- = t ⋅ --------dt d ln t Drawdown solution 2 q µβ  0.445 r w P WF = P i + 70.6 - ln  -------------------- – 2 S --------------------kh  ηt  2 2  0.445 r w 70.6 q µβ 70.6 q µβ  0.445 r w P i – PWF = – ---------------------- ln  -------------------- – 2 S = – ---------------------- – ln t + ln  -------------------- – 2 S η  kh kh   ηt  Take Bourdet Derivative 70.6 q µβ 1 70.6 q µβ d (P P ) – WF = t  – ---------------------   – --  = -------------------- kh   t  kh d ln t i ; 1 d t ln = – -t dt 70.6 q µβ d d ( P – P WF ) = ( ∆ P ) = m = ---------------------kh d ln t i d ln t PWF 162.6 q µβ m = ------------------------kh 1000 1 log t P 70.6 q µβ m = ----------------------kh 70.6 q µβ ∴kh = ---------------------------------------------------------d(∆P) --------------d ln t @ stabilization d(∆P) log --------------d ln t MTR 1000 1 log t Derivative Analysis (Drawdown case) Page 30 © 2000-2001 M. Peter Ferrero, IX
  31. 31. Pressure Transient Analysis Skin a. DD 2 2  0.445 r w 162.6 q µβ 70.6 q µβ  0.445 r w Pi – P WF = – ---------------------- ln  -------------------- – 2 S = – ------------------------- log  -------------------- – 0.87 S ηt  kh kh   ηt   ηt   η  Pi – P WF = m log  -------------------- + 0.87 S = m log t + log  -------------------- + 0.87 S 2  0.445 r w  0.445 r 2  w P i – P WF  η  ---------------------- = log t + log  -------------------- + 0.87 S m  0.445 r 2  w P i – P WF 2.25η ∴S = 1.151 ---------------------- – log -------------- – log t 2 m r w Take t = 1 hour P i – P WF 2.25η 1hr ∴S DD = 1.151 --------------------------- – log -------------2 m r w Semi-log MTR! PWF 1hr 162.6 q µβ m = ------------------------kh tp =1 log t P ∆ P = P i – P WF ∆P′ 70.6 q µβ kh = – -----------------------d(∆P) --------------d ln t t ps tP Derivative Analysis (Drawdown case) Page 31 © 2000-2001 M. Peter Ferrero, IX
  32. 32. Pressure Transient Analysis b. PBU The instant a well is shut-in, PWF : 2  0.445 rw q µβ P WF = P i + 162.6 - log  -------------------- – 0.87 S ------------------------kh  η tP   η tP  P WF = P i + m – log  -------------------- – 0.87 S  0.445 r2  w P WF = P i – m log tP + log 2.25η + 0.87 S ………………1, from Drawdown -------------2 rw Shut-in pressure (during PBU), tP + ∆ t P WS = P i – m log ---------------- ………………………………… 2 ∆t Subtract 1 from 2 tP + ∆ t  2.25η P WS – P WF = – m log  ---------------- + m log t P + m log  --------------  + 0.87 S  ∆t   r2  w tP + ∆ t P WS – P WF  k  ----------------------------- = – log  ---------------- + log  ----------------  – 3.23 + 0.87 S  tP ∆ t  m  φµ c r 2  t w ∴S PBU P WS – P WF tP + ∆ t  k  ∆t = 0 = 1.151 ------------------------------------------------- – log  ----------------  + 3.23 + log  ---------------- – log ∆ t 2  tP ∆ t  m Horner semi-log MTR  φµ c t r w 162.6 q µβ m = ------------------------kh PWS tP + ∆ t log ---------------∆t ∆ P = P WS – P WF PWSskin ∆P ∆t = 0 q ----------------------m ′ = 70.6 µβ kh d(∆P) ------------------------------tP + ∆ t  ---------------- d ln  ∆t  log ∆ t ∆t s Derivative Analysis (Drawdown case) Page 32 © 2000-2001 M. Peter Ferrero, IX
  33. 33. Pressure Transient Analysis Ideal vs. Actual PBU/DD Tests a. Drawdown case PWF 162.6 q µβ m = ------------------------kh Ideal (no WBS or LTR) log t P LTR ETR Actual PWF Transient reaches boundaries Reservoir heterogeneity WBS MTR kh, S Infinite acting R adial flow log t P b. Drawdown: log-log plot ∆P ∆P Ideal ∆P’ ∆P’ log t P ∆P ∆P Actual ∆P’ ∆P’ ETR MTR LTR log t P Ideal vs. Actual PBU/DD Tests Page 33 © 2000-2001 M. Peter Ferrero, IX
  34. 34. Pressure Transient Analysis Flow Regimes & Model Recognition Radial flow homogeneous, infinite acting system q µβ Pi – P WF = 162.6 ---------- log t + constant kh ∆P ∆P’ ETR WBS dominates MTR d( ∆P) q µβ --------------- = 70.6 --------d ln t kh 70.6 q µβ kh = ---------------------------∆ P ′ stabilized ∆t single fault Using superposition and image wells ∆ P total = ∆ P well + ∆ P imag e 2 2 0.445 ( 2 L ) q µβ qµβ  0.445r w = Pi – P WF = – 70.6 --------- ln  -------------------- – 2S – 70.6 ---------  ln -----------------------------  -  ηt kh kh  ηt  2 2 0.445 qµβ = – 70.6 --------- 2 ln  -------------- + ln r w + ln ( 2 L ) – 2 S  ηt  kh 2 2 0.445 qµβ PWF = P i + 162.6 ---------- 2 log  -------------- + log r w + log ( 2 L ) – 2 S  ηt  kh Note: 1 d ( ln t ) = – -t dt ) d( ∆P) d( ) d(∆P qµβ 1 --------------- = t --------- → --------------- = t 70.6 --------- – 2 -d ln t d ln t dt t kh ∴slope doubles 2 faults, slope x4 3 faults, slope x8, etc. LTR ∆P ETR MTR ∆P’ MTR ETR LTR 2m P WS ∆P m ∆P’ 2m m 1 log t P Flow Regimes & Model Recognition tP + ∆ t log ---------------∆t 1000 Page 34 © 2000-2001 M. Peter Ferrero, IX
  35. 35. Pressure Transient Analysis increase/decrease in kh or decrease in kh Concentric model: ETR ∆P MTR LTR ∆P ∆P’ inner (kh)inner (kh)outer outer ∆P’ log tP Radius for kinner: –4 ROI = increase decrease 4η t 2.637 ×10 k ; η = ---------------------------------i φµ c t t is where slope becomes negative [For ROI’s in outer zone, use k of outer zone! No matter if the k is higher or lower] contacts Same kh! ETR ∆P MTR 70.6 q µβ kh  -----  --------------------- µ o ( ∆ P o )′ --------------- = ---------------------- → same kh! 70.6 q µβ  kh -------------------------( ∆ P w )′  µ w LTR ∆P ∆P’ ∆Po’ ∆Pw’ µw ( ∆ P w )′ = ------ ( ∆ P o )′ µo log t P variable kh! kh  -----   µ o ( ∆ P w )′ = --------------- ( ∆ P o )′ kh  -----   µ w µw < µo µw > µo constant pressure boundary aquifer (strong) gas cap (high compressibility) water/gas support (pressure support) Flow Regimes & Model Recognition Page 35 © 2000-2001 M. Peter Ferrero, IX
  36. 36. Pressure Transient Analysis 2 2  0.445r w 0.445 ( 2 L ) µβ ∆ P = = – 70.6 ------ q ln  -------------------- – q ln  -----------------------------   ηt kh  ηt  2 rw 0.445 0.445 µβ = – 70.6 ------ q ln  -------------- – q ln  -------------- + q ln ------------ ηt   ηt  2 kh (2L) rw 2 q µβ ∆ P = – 70.6 --------- ln  ------ kh  2 L Spherical (Partial Penetration Completions) m=0 P WS g o e s to z e ro (in th e o ry ) 1 tP + ∆ t log ---------------∆t re a lity 1000 ∆t early radial late radial hp hT transition region between early radial and late radial early radial: khp, mechanical skin (usually masked by WBS) - can estimate kv/kh ratio spherical - t-0.5 m=0.5 late radial: khT, Sglobal=Smech+Spartial penetration Sglobal can be very large (maybe 400-500) ∆t Flow Regimes & Model Recognition Page 36 © 2000-2001 M. Peter Ferrero, IX
  37. 37. Pressure Transient Analysis Linear flow (Infinite conductivity fractures) Radial flow Linear flow ∆P m=0.5 log tP • • • • linear flow region (0.5 slope) represents stimulated well fracture conductivity > 10,000 mD-ft time transition between linear and radial flow corresponds to the frac. length (half length kh and skin are calculated from the radial flow region (need kh to estimate frac length). Therefore, to estimate the frac. length, for a large frac. into a “low” permeability zone, you may need a pre-frac. test. Bi-linear flow (finite conductivity fractures) Bi-linear flow Linear flow Radial flow ∆P m=0.25 m=0.5 log tP The bi-linear flow is very fast, need a very long fracture to distinguish! • fracture conductivity < 10,000 mD•ft • pressure drop in fracture is not negligible • almost never happens Flow Regimes & Model Recognition Page 37 © 2000-2001 M. Peter Ferrero, IX
  38. 38. Pressure Transient Analysis - if you can see the bi-linear region, it can be used to estimate frac-conductivity (if the matrix permeability is known) - linear region is used to estimate frac. half length - radial flow region is used to estimate kh, S Gas Well Testing Same analysis procedure as for oil well testing with the following exceptions: • Gas properties (transport), µ, z, cg vary as a function of pressure. Gas is considered a highly compressible fluid whereas oil is considered a slightly compressible fluid. • Non-darcy flow, or turbulence, can exist in gas wells which shows up as a skin due to extra pressure drop. Therefore, differentiation between true mechanical skin and skin due to non-darcy flow is important - non-darcy flow signifies that Darcy’s law does not properly predict the ∆P due to flow of gas in porous media ρν d - in porous media, non-darcy flow develops when Re > 50 ( R e = --------- ) µ - low µ and high velocities (close to the wellbore) are the contributing factors to nondarcy flow Gas tests - Diffusivity Equation Development MW P R - a. EOS for gas: =  ----------  --- → P = ρ z ---------- T  RT   z  MW For gases: µ and z may vary considerably as a function of pressure. Therefore, to account for this, the pseudo-pressure function was developed. ψ(P) = 2∫ P P ------ dP PB µ z Gas Tests - Pseudo (Ψ(P)) Equation Development a. Continuity equation ∂ ∂ ∂ ( ρ u x ) + ( ρ u y ) + ( ρ u z ) = – ∂ ( ρφ ) ∂x ∂y ∂z ∂t b. Darcy’s law k u = -- ∇P µ k x ∂P ; u x = ---µ ∂x k y ∂P ; u y = ---µ ∂y k z ∂P ; u z = ---µ ∂z c. EOS Gas Well Testing Page 38 © 2000-2001 M. Peter Ferrero, IX
  39. 39. Pressure Transient Analysis - oil and water: slightly compressible fluids ρ = ρo e c (ρ – ρo ) - For gases MW P ρ = ----------  --- - RT  z  (b) + (c) into (a) isotropic → k x = k y = k z 2 2 2 φµ c t ∂P 2 ∂P 2 ∂P 2 ∂P ∂ P ∂ P ∂ P c   +  +  + + + = ------------------------------- ∂x  ∂y  ∂z  2 2 2 –4 ∂ t 2.637 ×10 k ∂x ∂y ∂z ψ( P) = 2∫ P P ------ dP PB µ z Differentiating Ψ(P) wrt x, y, z, and t ∂ψ 2 P ∂P = -----∂x µz ∂x ; ∂P µ z ∂ψ = -----∂x 2P∂x ∂ψ 2 P ∂P = -----∂y µz ∂y ; ∂P µ z ∂ψ = -----∂y 2P∂y ∂ψ 2 P ∂P = -----∂z µz ∂z ; ∂P µ z ∂ψ = -----∂z 2P∂z ∂ψ 2 P ∂P = -----∂t µz ∂t ; ∂P µ z ∂ψ = -----∂t 2P∂t Input Darcy’s law into Continuity equation: ∂  k x ∂P ∂  k y ∂P ∂  k z ∂P ρ ---ρ ---ρ ---+ + ∂x µ ∂x ∂y µ ∂y ∂z µ ∂z Input EOS: = ∂ ( φρ ) ∂t MW P ρ = ----------  --- - RT  z  MW ∂  k x P ∂P MW ∂  k y P ∂P MW ∂  k z P ∂P MW ∂ P ----------  ---- ---  + ----------  ---- ---  + ----------  ---- ---  = ---------- φ  --- - - - RT ∂ x µ z ∂ x RT ∂ y µ z ∂ y RT ∂ z µ z ∂ z RT ∂ t  z  assume isotropic conditions k = k x = k y = k z Gas Tests - Pseudo (Ψ(P)) Equation Development Ψ © 2000-2001 M. Peter Ferrero, IX Page 39
  40. 40. Pressure Transient Analysis P ∂  P --- = ∂  --- Þ ∂P ∂t z  ∂P z  ∂t MW P ρ = ----------  --- - RT  z  RT z MW ∂ P 1 MW ∂ P 1 ∂ρ c g = -- = -- ----------  --- = ----------  --- ----------  --- MW  P RT ∂ P  z  ρ RT ∂ P  z  ρ ∂P z P P P c g =  --- ∂  --- ⇒ ∂  --- =  --- c g  P ∂ P  z   z ∂ P z  P ∂  P --- =  --- c g ∂P  z  ∂t ∂ t z  Substituting c g ≈ c t for gas reservoir ∂Ψ ∂Ψ ∂Ψ ∂  P ; ; ; --∂x ∂ y ∂ z ∂ t z  P φ c g  ---  z  ∂P P P P ∂   -----  µ z ∂Ψ ∂   -----  µ z ∂Ψ ∂   -----  µ z ∂Ψ - ------  +   - ------  +   - ------  = -----------------k ∂t ∂ x   µ z 2 P ∂ x ∂ y µz 2P ∂ y ∂z µz 2 P∂z P 2 2 2 φ c g  ---  z  µ z ∂Ψ 1 d Ψ d Ψ d Ψ = ------------------ ------ -+ + k 2P ∂t 2 d x2 d y2 d z 2 2 d Ψ dx 2 2 + d Ψ dy 2 2 + d Ψ dz 2 1 ∂Ψ = ----ηg ∂ t In radial coordinates: 1 ∂ ∂Ψ 1 ∂Ψ --  r  = ----r ∂r ∂r  ηg ∂ t –4 2.637 ×10 k where η g = -------------------------------φµ g c g Gas Tests - Pseudo (Ψ(P)) Equation Development Ψ © 2000-2001 M. Peter Ferrero, IX Page 40
  41. 41. Pressure Transient Analysis Pseudopressure or Real Gas Potential (Ψ(P)) Gas Li n ea r Liquid µz µz Liquid: slightly compressible system Constant P Ψ(P) 2 2 0 00 ρ = ρo e P c ( P – Po ) 3 0 00 P P Approximation to Ψ(P) 0 ≤ P ≤ 2000 P 2000 ≤ P ≤ 3000 Ψ(P) 3000 < P P 2 Note: is good for all pressures a. Ψ(P) (good for all pressures) Transient development Drawdown equation 2 P sc q g T  1688φµ c t r w Ψ ( P wf ) = Ψ ( P i ) + 50300 ------------------ 1.151 log  ------------------------------  – S + D q g ktp T sc kh   where Psc = atmospheric pressure (usually 14.7 psia) Tsc = 520 ° R T = ° R, reservoir temperature S = mechanical skin D = turbulence factor (non-Darcy flow) OR, 2  0.445 rw 1637 q g T Ψ ( P wf ) = Ψ ( P i ) + ----------------------- log  -------------------- – η ( S + D qg ) kh  ηtp  –4 2.637 ×10 k where η = -------------------------------φµ g c t P Note: no µ g ,β g because Ψ ( P ) = 2 ∫ ------ dP µz Buildup equation Pseudopressure or Real Gas Potential (Ψ(P)) Ψ © 2000-2001 M. Peter Ferrero, IX Page 41
  42. 42. Pressure Transient Analysis q tp ∆t time tp + ∆ t 1637 q g T Ψ ( P ws ) = Ψ ( P i ) + ----------------------- log  ---------------   ∆t  kh Pseudo-steady state equation (PSS): when transient reaches all boundaries of reservoir must be a closed system. re P sc q g T Ψ ( P wf ) = Ψ ( P i ) + 50300 ------------------ ln  ----  – 0.75 + ( S + D qg ) -  - rw T sc kh OR re qg T Ψ ( P wf ) = Ψ ( P i ) + 1422 --------- ln  ----  – 0.75 + ( S + D q g )  r w kh b. P2- valid for low pressures (P<2000psi) where µz is constant. Gas properties µ, z, Bg, etc. are evaluated at static pressure or initial pressure Pseudopressure or Real Gas Potential (Ψ(P)) Ψ © 2000-2001 M. Peter Ferrero, IX Page 42
  43. 43. Pressure Transient Analysis Ψ(P ) = 2∫ P P ------ dP PB µ z - µz is a constant P 2 P 2 Ψ ( P ) = ----- ∫ P dP = ----µz µ z PB 2 P - Drawdown transient equation: replaceΨ ( P ) with ----µz 2 2 2 P wf Pi 1637 q g T  0.445 r w ----------- = ------- + ----------------------- log  -------------------- – 0.87 ( S + D qg ) kh µz µz  η tp  for P 2000 0 → µz is constant 2  0.445 rw 2 2 1637 q g µ zT P wf = Pi + ------------------------------ log  -------------------- – 0.87 ( S + D q g ) kh  ηtp  - Buildup transient equation: tp + ∆ t 2 2 1637 q g µ zT P ws = Pwi + ------------------------------ log  ---------------   ∆t  kh PSS equation: re q g µ zT 2 2 P wf = P i + 1422 ---------------- ln  ----  – 0.75 + ( S + D q g ) -  - rw kh c. P- valid for high pressures (P>3000psi) where uz/P is constant. Gas properties evaluated a initial/static pressure. Can use P for tests where Pi and lowest Pware greater than 3000 psi. Pseudopressure or Real Gas Potential (Ψ(P)) Ψ © 2000-2001 M. Peter Ferrero, IX Page 43
  44. 44. Pressure Transient Analysis Pi P Assume ----- = constant = -------- → at initial reservoir pressure µi zi µz Ψ(P ) = 2∫ Pi P 2 ( Pi ) P P ------ dP = 2 -------- ∫ dP = ----------------µi zi PB µi zi PB µ z P Drawdown transient equation: 2 2 ( P i ) P 1637 q g T 2 ( P i ) P wf  0.445 r w ------------- -------- = ------------ -----i + ----------------------- log  -------------------- – 0.87 ( S + D q g ) - kh µi zi µ z µi zi µ z  ηg tp  2 P wf  0.445 rw 1637 q g µ i z i T = P i + -------------------------------- log  -------------------- – 0.87 ( S + D q g ) kh 2 P i  ηg tp  Consider real gas law:  PV =  PV ------------ zT  sc  zT  res βg i Scf 1000  ---------   Mcf 14.7 z i Tr Vr P sc z i T r ziTr RB - - = -------- = -------- --------- = ----------------------------- ---------- --------- = 5.035 --------- in  ---------   Mcf V sc T sc P i Pi Scf 520 P i  -------5.615  bbl  2 162.6 q g µβ g  0.445 r w P wf = P i + -------------------------------i log  -------------------- – 0.87 ( S + D qg ) kh  ηg tp  where µ is at end of drawdown Buildup equation in terms of P: 162.6 q g µβ g tp + ∆ t P ws = P i + -------------------------------i log  ---------------   ∆t  kh  k  ∆P S G = 1.151 ------- – log  ----------------- + 3.23 m  φµ c r 2  t w PSS equation: qg µi βg re P wf = P i + 141.2 -----------------i ln  ----  – 0.75 + ( S + D q g )  r w kh Summary 1. Buildup and drawdown analysis are conducted on gas wells in the same manner as for oil wells. 2. Choose Ψ(P), P2, or P depending upon the pressure range during test period • Ψ(P) - valid for all pressures ranges. Gas properties for diffusivity, η , are evaluated at the static or initial pressure. • P2 - valid for low pressures (below 2000 psi) where µz is constant. Gas properties µ, z, βg, etc. are evaluated at static or initial pressure. P • P - valid for high pressures (above 3000 psi) where ------ is constant. Gas properties are µz Pseudopressure or Real Gas Potential (Ψ(P)) Ψ © 2000-2001 M. Peter Ferrero, IX Page 44
  45. 45. Pressure Transient Analysis evaluated at static or initial pressure. Can use P for tests where Pi and lowest Pwf are greater than 3000 psi. 3. For critical systems or systems where large variation in gas properties occur across the range of test pressures, use Ψ(P). Determination of Skin and D for Gas Wells Global skin, Sg, calculated from gas well tests: S g = Sm + D q where Sm is mechanical skin and D is a turbulence factor. Well deliverability or potential is not linear with P, but is dependent upon rate if D ≠ 0 . For D ≠ 0 , Sg increases as a function of rate. P D = 0 D≠0 q PSS Equation: re q µβ P wf = P i – 141.2 - ln  ----  – 0.75 + [ S + D q ] ------------------------ r w kh Example of the effect of turbulence S =5 D=1x10-5(MCF/D) q=40,000MCF/D S =5 D=1x10-4(MCF/D) q=40,000MCF/D Sg= Sm + D|q| = 5 + (40000)(1x10-5) = 5.4 Sg= Sm + D|q| = 5 + (40000)(1x10-4) = 9 m m Multiple Rate Testing • Method for discriminating between Sm and non-Darcy skin. Determination of Skin and D for Gas Wells © 2000-2001 M. Peter Ferrero, IX Page 45
  46. 46. Pressure Transient Analysis • kh and Sg are evaluated in standard fashion through PBU’s. a. Theoretical method b. Empirical method Multi-rate test types: • • • • Flow after flow tests - usually with increasing flow rate Isochronal Modified isochronal - most popular Multi-flows followed by one PBU a. Theoretical method The flow equation can be written in the form (Deliverability equations): ψ ( P i ) – ψ ( P wf ) = aq + bq 2 2 P i – P wf = aq + bq P i – P wf = aq + bq 2 2 2 Consider P>3000 psi, ψ ( P ) → P 1. Transient flow equation (DD)  ηg tP  q µβ P i – PWF = 162.6 ---------- log  -------------------- + 0.87 ( S m + D q ) kh  0.445 r 2  w  ηg tP  q µβ µβ 2 P i – PWF = 162.6 ---------- log  -------------------- + 0.87 Sm + 141.2 ------ Dq 2 kh kh  0.445 r w P i – P WF  ηg tP  µβ µβ ---------------------- = 162.6 ------ log  -------------------- + 0.87 Sm + 141.2 ------ Dq 2 q kh kh  0.445 r w P i – P WF ---------------------- = a ( t ) + bq q Multi-rate test (say 4 points) - flow times must be equal Multiple Rate Testing Page 46 © 2000-2001 M. Peter Ferrero, IX
  47. 47. Pressure Transient Analysis l Pi – P wf ------------------q b (turbulence) a(t) q From intercept, mechanical skin, Sm:  ηg tP  µβ a ( t ) = 162.6 ------ log  -------------------- + 0.87 S m kh  0.445 r2  w Sm  ηg tP    µβ  a ( t ) – 162.6 ------ log  --------------------  kh  0.445 r2    w = -------------------------------------------------------------------------------------µβ 141.2 -----kh  ηg tP  a ( t ) kh S m = --------------------- – 1.151 log  -------------------- 141.2µβ  0.445 r 2  w From slope, turbulence coefficient, D: µβ b = 141.2 ------ D kh  MSCF ---------------- D  bkh D = --------------------141.2µβ –1 2 r 4η e 2. Pseudo-steady state flow attained ( t P > ------ for well centered in circular drainage area) re q µβ P i – P WF = 141.2 ---------- ln  ----  – 0.75 + ( S m + D q )  r w kh re P i – PWF µβ µβ ---------------------- = 141.2 ------ ln  ----  – 0.75 + Sm + 141.2 ------ Dq  r w q kh kh re µβ ∴a = 141.2 ------ ln  ----  – 0.75 + Sm  r w kh µβ b = 141.2 ------ D kh Multiple Rate Testing Page 47 © 2000-2001 M. Peter Ferrero, IX
  48. 48. Pressure Transient Analysis b P i – P wf ------------------q a q from intercept, a, calculate Sm from slope b, calculate turbulence coefficient D This yields the stabilized flow equation: P i – P WF = aq + bq2 . Use this to estimate flow rates as a function of ∆P . Therefore, given “a” and “b”, you can estimate a drawdown for a specified rate, or a rate for a specified drawdown. NOTE: This development is possible only if PSS is reached during all rates in the multirate test. Same methodology is used for P2 and Ψ(P) analysis: P2 : • Transient flow  η tP  2 2 zT zT 2 P i – P WF = 1637µ - q log  -------------------- + 0.87 S m + 1422µ - Dq --------------------------------------------kh kh  0.445 r2  w 2 2 P i – P WF ----------------------- = a ( t ) + bq q  η tP  zT a ( t ) = 1637µ - q log  -------------------- + 0.87 S m ----------------------kh  0.445 r2  w zT b = 1422µ - D ----------------------kh Multiple Rate Testing Page 48 © 2000-2001 M. Peter Ferrero, IX
  49. 49. Pressure Transient Analysis (Flow times must be equal) 2 Pi  η tP  a ( t )kh S m = 1.151 ----------------------- – log  -------------------- 1637µ zT  0.445 r 2  b 2 P wf – --------------------q w bkh D = ----------------------1422µ zT a(t) q • PSS (all rates need to reach PSS) re 2 2 zT 1422µ zT 2 P i – P WF = 1422µ - q ln  ----  – 0.75 + Sm + ----------------------- Dq ---------------------- r w kh kh 2 2 P i – P WF ----------------------- = a ( t ) + bq q re zT a ( t ) = 1422µ - q ln  ----  – 0.75 + Sm ---------------------- r w kh zT b = 1422µ - D ----------------------kh Deliverability equations: Now, say we want a deliverability equation of the form Pi – P WF = aq + bq2 , but cannot flow each rate to PSS. Alternative - flow 3 rates at transient conditions and final rate to PSS. PSS b P i – P wf ------------------q a Transient b a(t) q Multiple Rate Testing Page 49 © 2000-2001 M. Peter Ferrero, IX
  50. 50. Pressure Transient Analysis ----------Note that the slope, b ≈ µβ D , is the same irregardless of whether flow is transient or kh pseudo steady state. However, the intercept, “a”, is different as shown on preceding graph. The intercept from the stabilized or PSS flow is required for the deliverability equation P i – PWF = aq + bq2 (“a” in this equation IS NOT a function of time). c. Empirical method • AOF - absolute open (hole) flow - ( PWS ≈ 14.7 psia ) • based on historical observation that a log-log plot of P 2 – P 2 vs. q is approximately a i WF straight line. Empirical equation: 2 2 q = c ( P i – P WF ) n n 2 2 q ( Pi – PWF ) = -c 2 ( Pi – 2 P WF ) = q 1 1 --n  1 n - c 2 2 1 1 1 1 1 log ( P i – P WF ) = -- log q + -- log -- where -- log -- is constant n n c n c 2 n = 1: Darcy flow 2 ( P i – ( 14.7 ) ) n = 0.5: non-Darcy flow Therefore, 2 2 log ( P i – P wf ) slope = 1/n slope = 1: Darcy flow slope = 2: non-Darcy flow AOF log (q) q -Once slope is determined, 1 , estimate c from measured data: c = ------------------------------- . Then the n n 2 2 ( Pi – PWF ) deliverability equation becomes: q = c ( P 2 – P 2 ) i WF n • Flow after flow • Isochronal • Modified isochronal Multiple Rate Testing Page 50 © 2000-2001 M. Peter Ferrero, IX
  51. 51. Pressure Transient Analysis • Multi-flows followed by one PBU a. Flow after flow (discussed) Theoretical (equal flow times) P i – P wf = a ( t ) q + bq 2 P i – P wf = a ( t ) q + bq 2 - all rates in transient flow - stabilized deliverability equation (1 rate in PSS) Lee’s book refers to stabilization or PSS for each rate, i.e. each rate must reach PSS. Generally this is never feasible and not necessary. Usually never possible to have even one rate reach stabilization. b. Isochronal testing • Applicable for any permeability - required for lower permeabilities • Well is produced at four rates of equal time length • Well is shut-in for PBU between each flow period until pressure builds back up to initial or static pressure before proceeding to next rate • Flow time of last rate may be extended until stabilization (PSS). This is done only if feasible (need high permeability, small reservoir) • Isochronal tests performed on wells where time to reach PSS too long • Data recorded in isochronal tests is transient (except for last rate possibly) • kh is estimated from PBU’s Multiple Rate Testing Page 51 © 2000-2001 M. Peter Ferrero, IX
  52. 52. Pressure Transient Analysis q4 q3 q q2 q1 t P t Flow equation (transient period) P i – PWF µβg µβ g  ηtP  ---------------------- = 162.6 --------- log  -------------------- + 0.87 S m + 141.2 --------- D q 2 q kh kh  0.445 rw µβg  ηtP  a ( t ) = 162.6 --------- log  -------------------- + 0.87 S m kh  0.445 r2  w µβg b = 141.2 --------- D kh STANDARD: all 4 rates in transient flow RARE: 3 rates transient flow, last rate in PSS Comments: • Estimation of D is independent of flow regime (transient/PSS) • Calculation of intercept, “a”, is dependent upon flow regime which will impact deliverability equation. - If final rate reaches stabilization, deliverability equation will be more accurate - If all rates are in transient regime, extrapolated rates based on deliverability equation will be high (optimistic) c. Modified isochronal • Applicable to any permeability system Multiple Rate Testing Page 52 © 2000-2001 M. Peter Ferrero, IX
  53. 53. Pressure Transient Analysis • Reduced time required to conduct • Well produced at 4 rates, PBU following each rate. Flowing periods/PBU’s all same time duration. • Last pressure in each PBU is Pi for analysis of following flowing period (derivative, Odeh-Jones) • As with isochronal testing, last rate can be extended to stabilization (if practical) to provide more accurate deliverability equation. • Same analysis procedure as for isochronal testing • kh is estimated from PBU’s q4 q3 q q2 q1 t Pi 1 Pi 2 Pi 3 Pi 4 P t Analysis procedure: 1. Analyze each PBU for • kh, S • kh should be roughly the same from each PBU. If not, most likely error is in rate measurement 2. Estimate Sm and D • Plot Sg vs. q ( Sg = Sm + Dq ) - if Sg is constant then there is no turbulence - if Sg is linear with q then there no turbulence Multiple Rate Testing Page 53 © 2000-2001 M. Peter Ferrero, IX
  54. 54. Pressure Transient Analysis D SG SG = Sm + Dq Sm q 3. Develop deliverability equation: Pi – P WF = aq + bq2 • transient µβg  η tP  µβ P i – PWF = 162.6 ------ log  -------------------- + 0.87 S m + 141.2 --------- Dq 2 kh kh  0.445 r w - kh is calculated from PBU’s - Sm and D are calculated intercept and slope, respectively, from a plot of Sg vs. q - µ and Bg are from static (phase behavior - PVT) data - t is from test data • PSS re q µβ g µβ 2 P i – P WF = 141.2 ------------ ln  ----  – 0.75 + S m + 141.2 ------ Dq -  - rw kh kh - can be developed if accurate estimates for kh, Sm and D are made from multi-rate/ PBU testing. - need estimate of reservoir size, re. However, this is normally not very sensitive to the r rw e answer ( ln  ----- ≈ 7.5 )   d. Multi-flows followed by one PBU Multiple Rate Testing Page 54 © 2000-2001 M. Peter Ferrero, IX
  55. 55. Pressure Transient Analysis P t • Measure BHP vs. time • Analyze PBU based on multiple rates - superposition - Guess values for kh, Sg, Pi, Sm, and D until match all flowing pressures. - Perform non-linear regression on flowing data to estimate Sm and D. - Use Odeh-Jones analysis to estimate turbulence (pertains only to flowing pressures) The advantage of flow after flow followed by a PBU is that it saves time. It does not require multiple PBU’s. The disadvantage is that if a reliable kh value cannot be estimated from the final PBU, then the entire analysis can be in error. NOTE: SG = Sm + Dq IS NOT VALID FOR FLOW AFTER FLOW! SG = Sm + Dq only works for flow-PBU-flow-PBU... D SG SG = Sm + Dq Sm q Multiple Rate Testing Page 55 © 2000-2001 M. Peter Ferrero, IX
  56. 56. Pressure Transient Analysis Odeh-Jones Analysis Skin analysis (Sm) for gas wells based on flowing pressures. Extension of theoretical development presented earlier. Multi-rate Drawdown Test Analysis 2  0.445 r w µβ PWF = P i + 162.6 ------ log  -------------------- + 0.87 S G kh  ηt  Assume 2 rate test (both rates are non-zero) and apply superposition. q2 q1 t0 t1 t ( P i – P WF ) = ∆ P 2 2 2  0.445 r w  0.445 r w  0.445 rw µβ = – 162.6 ------ q 1 log  -------------------- – 0.87 Sq 1 – q 1 log  -------------------- + 0.87 Sq 1 + q 2 log  -------------------- – 0.87 Sq 2 ηt  η ( t – t 1 ) kh   η ( t – t 1 )  µβ let m ′ = 162.6 ------ kh ∴( P i – P WF ) 2 2 2  0.445 rw  0.445 r w  0.445 r w = – m ′ – q 1 log t + q 1 log ( t – t1 ) – q 2 log ( t – t 1 ) + q 1 log  -------------------- – q 1 log  -------------------- – m ′ q 2 log  -------------------- – 0.87 S  ηt   ηt   ηt  divide through by q2 ( P i – P WF )  η  m′ --------------------------- = ------ [ q 1 log t + ( q 2 – q 1 ) log ( t – t1 ) ] + m ′ log  -------------------- + 0.87 S q2 q2  0.445 r 2  w where q 1 log t + ( q 2 – q1 ) log ( t – t 1 ) is the superposition time function, STF ( P i – P WF ) ----------Plot --------------------------- vs. STF q2 q2 Odeh-Jones Analysis Page 56 © 2000-2001 M. Peter Ferrero, IX
  57. 57. Pressure Transient Analysis ----------slope = m' =162.6 µβ ; kh = 162.6 µβ P i – Pwf ------------------q2 kh m' intercept = b' STF/q2  η  b ′ = m ′ log  -------------------- + 0.87 S  0.445 r 2  w  η  b′ S G = 1.151 ------ – log  -------------------- m′  0.445 r 2  w Now, if non-Darcy flow effects are present, skin increases with increasing rate. Therefore, intercept values, b ′ , increases as skin increases. q2 P i – P wf ------------------q2 b2' q1 b2′  η  S 2 = 1.151 ------- – log  -------------------- m′  0.445 r 2  w b1′  η  S 1 = 1.151 ------- – log  -------------------- m′  0.445 r 2  w b1' S2 > S1 STF/q2 Odeh-Jones Analysis Page 57 © 2000-2001 M. Peter Ferrero, IX
  58. 58. Pressure Transient Analysis Based on relation SG = Sm + Dq (if flow tests are performed followed by PBU’s) D SG Sm q Flow Regimes 1. Radial flow - increase in separation of ∆P and ∆P' indicates increasing skin ∆P cs ∆P′ kh, S, Pi 2. Spherical flow (partial penetration completions) Flow regime sequence: - early radial (khp, Sm) - hp is the thickness of the perforated zone - spherical (kv/kh) - late radial (kht, SG, Pi) - ht is the total zone thickness early radial: khp, Sm (usually masked by WBS) kv/kh hp hT khT , Sg, Pi m=0.5 3. Linear flow (hydraulically fractured wells) - Infinite conductivity (no ∆P in the fracture) Flow Regimes Page 58 © 2000-2001 M. Peter Ferrero, IX
  59. 59. Pressure Transient Analysis Flow regime sequence: - linear (fracture half-length) - late radial (kh, S) log ∆P, ∆P' ∆P ∆P' Same rate q Pi fractured kh, S m = 0.5 (linear region) - characteristic of stimulated wells unfractured ∆t t 4. Bi-linear flow - Finite conductivity fracture (∆P in fracture accounted for) Flow regime sequence: - bilinear - flow through fractures (usually masked- rarely seen) - linear - flow from matrix to fractures - late radial - radial flow in matrix (basically pure radial) log ∆P, ∆P' ∆P ∆P ' late radial: kh, S m = 0.5 (linear, fracture length) m = 0.25 (fracture conductivity, R AR ELY seen) ∆t Flow Regimes Page 59 © 2000-2001 M. Peter Ferrero, IX
  60. 60. Pressure Transient Analysis Horizontal wells L h kv, k h kh h L L kvkh late radial transition early radial Early radial: L h Transition: L h Late radial: L h Flow regime sequence: k kh v • Early radial - L ----- , perforation skin (Sm, if have khh) k kh v kv and kh play role in early radial response. Estimate L ----- and, if khh is known, then you can estimate the perforation skin, Sm. • Transition region - estimate L (drainhole length) from beginning of transition. You need khh to estimate L. Horizontal wells Page 60 © 2000-2001 M. Peter Ferrero, IX
  61. 61. Pressure Transient Analysis • Late radial - khh, SG Need late radial to estimate the well’s productivity index (PI), Sm and drainhole length. k kh v For long drainholes with low ----- , it can take long times to reach late radial. When do horizontal wells outperform vertical wells: kv L ----- » k h h kh kv h ---- » -- kh L or k kh v Physically this means thin reservoir sections with long drainholes with decent ----- (0.05-0.1) Horizontal well outperforms vertical well when: Vertical well outperforms horizontal well when: kv L ----- » k h h kh kv L ----- « k h h kh (Seen a number of times in Prudhoe Bay) L kv kh , Sm k h h, S g L kv kh , S m k h h, S g Note: If the deviation <65 degrees, then treat as a vertical well. Pressure level in surrounding reservoir 1. Infinite reservoir • extrapolation of MTR for Pi • semi-infinite LTR for Pi (faults, kh changes, etc.) P*=Pi MTR P*=Pi LTR PWS MTR PWS ETR 1 tP + ∆ t log ---------------∆t ETR 1000 1 Pressure level in surrounding reservoir © 2000-2001 M. Peter Ferrero, IX tP + ∆ t log ---------------∆t 1000 Page 61
  62. 62. Pressure Transient Analysis 2. Depleted reservoir - average static drainage area pressure ( P ≠ P∗ ) P = stabilized pressure if well was SI in depleted field. A B C P P P L or distance P* method: Mathews-Brohs-Hazeroch (MBH) (pp. 35-38 of Lee) t + ∆t ∆t p 1. From Horner plot, extrapolate the MTR to P*, ∆ t → ∞ , ---------------- = 1 P*=Pi LTR P MTR ETR P WS tP + ∆ t log ---------------∆t 1 1000 2. Estimate drainage area shape and size (A) in ft2 ηt A p 3. Calculate ------- - use same tp used to construct Horner plot 4. Choose appropriate curve from figure 2.17 A-G (Lee) ηt A p 5. Enter plot on abscissa at ------- , go up to appropriate curve, read value of Pressure level in surrounding reservoir © 2000-2001 M. Peter Ferrero, IX Page 62
  63. 63. Pressure Transient Analysis 2.302 ( P∗ – P ) ------------------------------------ = P D MBH m Note: , calculate P . P∗ – P 2.302 ( P∗ – P ) ---------------------- = -----------------------------------q µβ m 70.6 ---------kh -----------------where m = horner MTR slope ( 162.6 q µβ ). Also, 70.6 q µβ = kh kh the derivative m. Advantages • does not require data beyond MTR. However, MTR MUST be present • applicable to wide variety of drainage shapes (well need not be centered) Disadvantages • requires knowledge of drainage area size and shape • not good for layered reservoirs • requires knowledge of fluid properties and porosity and ct Example 2.6 P* method Use data from examples 2.2-2.4 Well centered in square drainage area –4 tp = 13630 hours 2 ft ( 2.637 ×10 ) ( 7.65 ) η = ------------------------------------------------------------ = 3800 ----–5 hr ( 0.039 ) ( 0.8 ) ( 1.7 ×10 ) P* = 4577 psia m = 70 k = 7.65 mD A = (2640)2 = 6.97x106 ft2 (160 acres) ηtP ( 3800 ) ( 13630 ) -------- = --------------------------------------- = 7.45 6 A 6.97 ×10 From figure 2.17A, p. 36 PD MBH 2.302 ( P∗ – P ) = 5.45 = -----------------------------------m ( 5.45 ) ( 70 ) ∴P = 4577 – --------------------------- = 4411 psia 2.302 Modified Muskat Method (pp. 40-41, Lee) Pressure level in surrounding reservoir © 2000-2001 M. Peter Ferrero, IX Page 63
  64. 64. Pressure Transient Analysis Using superposition and PSS solution, the late time PBU can be approximated by:  – 0.00388 k ∆ t q µβ P – P WS = 118.6 ---------- exp  ---------------------------------  2 kh  φµ c t r e   k∆t  q µβ log ( P – P WS ) = log  118.6 ---------  – 0.00168  ----------------  kh   φµ c t r 2 e log ( P – P WS ) = A + B ∆ t Therefore, plot log ( P – PWS ) vs.∆t. If correct, P will plot as a straight line. Data must be in following time range: 2 2 250φµ c t r e 750φµ c t re -------------------------- ≤ ∆ t ≤ -------------------------k k or 2 2 ( 0.51 re ) ( 0.89 r e ) ----------------------- ≤ ∆ t ≤ ----------------------4η 4η Modified Muskat Method Procedure 1. Assume a value of P 2. Plot log ( P – P WS ) vs.∆t 3. If line is straight - correct P If line curves upward - P too large If line curves downward - P too small Too large log ( P – PWS ) Too small ∆t 4. Try another P using above guidelines until line is straight Restrictions • method fails if well not centered in drainage area • requires long shut-in (needs to reach PSS) • difficult to pick correct straight line Pressure level in surrounding reservoir © 2000-2001 M. Peter Ferrero, IX Page 64
  65. 65. Pressure Transient Analysis Advantages • requires no knowledge of reservoir properties (A, φ, ct, etc.) • works for hydraulically fractured wells (assuming radial flow is established) Drill Stem Tests (DST) • • • • • • • • performed through dedicated test string (3.5-4.5) valves are annulus or tubing operated (perform poorly in mud due to solids) downhole shut-in a plus to minimize wellbore storage must kill well to recover string and gauges normally only done on exploration or appraisal wells requires rig to trip test string can perforate tubing conveyed perforators or wire line need cushion to bring well on (seawater, diesel, nitrogen) Flow/PBU periods • oil: 24 hour stable flow after cleanup (defined as basic sediment and water < 5%) 36 hour PBU • gas: 3-4 rate test after cleanup, 8 hours per test (single rate, same as oil test, if D not required) 36 hour PBU Pressure measurement • memory gauges • surface readout Conducting Well Tests Completed Wells (development scenario) • wells completed with final tubing string 1. Run memory gauges (2) on SL or use surface readout gauges (PLT) on EL • quartz gauges with lithium battery (temperatures to 350F) • Run gauges with well flowing or prior to opening up well. Must record flowing pressures prior to PBU for skin calculation • Obtain accurate rate measurement (history) • Do not move gauge or wireline, or perform any well operation during PBU! • Place gauge as close to perforations as possible to minimize phase segregation effects • Make static gradient mm while pulling out of hole at conclusion of PBU to verify wellbore fluid composition • Memory gauges must be programmed on surface prior to running in hole on SL Drill Stem Tests (DST) Page 65 © 2000-2001 M. Peter Ferrero, IX
  66. 66. Pressure Transient Analysis • Rate of pressure measurement can be modified with surface readout gauges (EL) 2. Softset gauges - not good with sand production! • Run tandem gauges on SL and set on bottom • Retrieve days, weeks, months later and download/analyze data • Need accurate rate measurement to take full advantage of pressure measurement • Many short PBU’s will occur over weeks, months • Use gap capacitance/quartz gauges 3. Permanent gauge installation • Install 2 gauges (quartz) in mandrels • Need electrical cable run to surface (similar to ESP), as well as, data transmission cable (pressure, time temperature) • Excellent for remote locations where wire line intervention is difficult. Negates the need to run wire line gauges • Payout over life of well (cost ≅ U$150,000) 4. Gauge/flowmeter installation • Exal/Expro permanent gauge/flowmeter • Quartz gauge • Flowmeter, venturi effect, estimate flow rate based on ∆P (Bernoulli’s principle) • Remote locations, subsea applications where a dedicated flowline per well is not feasible • Only good for single phase flow • An example of such an installation is the BP-Amoco/Shell/Marathon Troika project Wellbore Effects Phase segregation • Need two or more phases • If gradient changes between gauge and perforations during PBU due to phase segregation (water falling/oil rising, water falling/gas rising), the pressure data will be corrupted until phase segregation is complete • If gauge is above the interval and phase segregation occurs during PBU, the pressure is greater than pure reservoir response • Gas humping: Water falling back/imbibing into formation during PBU. Can be especially severe in low permeability gas reservoirs. Remedy: Place gauges as close as possible to top of perforations or within perforated interval or below interval (within 50 feet should be OK) Wellbore Effects Page 66 © 2000-2001 M. Peter Ferrero, IX
  67. 67. Pressure Transient Analysis Index ù ÷ õ ñ í ë ã ! # ' ) Buildup Test Solutions 28 Continuity equation (cylindrical coordinates) 7 Darcy’s Law 8 Derivative Analysis (Drawdown case) 30 Drawdown test 4 Drill Stem Tests 65 Drillstem test (DST) 5 Falloff test 5 Flow efficiency 13 Flow Regimes & Model Recognition 34 Horizontal wells 60 Horner’s Approximation 28 Injection test 4 Interference/pulse test 5 Isotropic 8 Mathews-Brohs-Hazeroch 62 Modified Muskat Method 63 Multiple Rate Testing 45 Multi-rate Drawdown Test Analysis 56 Odeh-Jones Analysis 56 Pressure buildup test 4 Pseudo (Y(P)) Equation Development 38 Radius of Investigation 21 Skin (drawdown) 31 Index Page 67 © 2000-2001 M. Peter Ferrero, IX
  68. 68. Pressure Transient Analysis 1 Skin and D for Gas Wells 45 Skin Development 12 Solutions to the Diffusivity Equation 11 Exact solution 11 Infinite reservoir, line source 11 Line source solution 11 Van Everdingen & Hurst terminal rate solution 11 Superposition 22 Wellbore Effects 66 Wellbore Solutions 16 Ideal reservoir (no skin) 16 Solution at sandface (including skin) 16 Wellbore Storage 16 Competely liquid filled wellbore 19 Determining the end of WBS 20 Gas-liquid interface 17 Index Page 68 © 2000-2001 M. Peter Ferrero, IX

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