PETE 662 Production EngineeringSlide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Chapter 2Production from Undersaturated Oil ReservoirsPETE 662Production EngineeringUndersaturated Oil Reservoirs  pe,pwf> pbubble point single phase flow in reservoir rockUndersaturated Oil Reservoirs Darcy’s Law for Single Phase Radial Flow)12( −−−−−−−−−−=drdpkAqµ wellofcenter from distance radial :pressure : viscosityfluid :flow lar toperpendicuarea sectional-cross :typermeabili : rateflow :rpAkqµwhere:tpcrprkrr ∂∂=∂∂∂∂ρφµρ1 Diffusivity equation describes the pressure profile in the reservoir during production.Undersaturated Oil Reservoirs Diffusivity Equation:where: time: wellofcenter from distance radial :pressure : viscosityfluid :porosity :typermeabili : ilitycompressib system :density fluid :trpkcµφρSingle-phase Flow under Steady-State Condition• A well draining from a reservoir with an open boundary where the fluid withdraw from the well is balanced exactly by fluid entry across the open boundary.Undersaturated Oil Reservoirs )42(ln2−−−−=−wwfrrkhqppπµSkin Effects – The Near-Wellbore Damage)52(2−−−−−=∆ skhqpsπµThe additional pressure drop required to achieve a given flow rate caused by the skin effect can be expressed asIncluding skin effect with oil field units, Eq. 2-4 becomes)82(ln2.141−−−−+=− srrkhqBppwewfeµVariable Oilfield Units SI Conversion (Multiply Oilfield Unit)Area acre m24.04 × 103Compressibility psi-1Pa-11.45 × 10-4Length ft m 3.05 × 10-1Permeability md m29.9 × 10-16Pressure psi Pa 6.9 × 103Rate (oil) STB/d m3/s 1.84 × 10-6Rate (gas) MSCF/d m3/s 3.28 ×10-4Viscosity cp Pa-s 1 × 10-3Table 1-1 Typical Units for Reservoir and Production Engineering CalculationsSkin Effects – Effective Wellbore Radius)82(ln2.141−−−−+=− srrkhqBppwewfeµ)112(ln2.141−−−−=−⇒−swewfeerrkhqBppµ)122('−−−−−−−−−=−swwerrDefine effective wellbore radius, as ,'wrEq. 2-11 can be written as =−'ln2.141wewferrkhqBppµ.105.4,105'wwrrs−×=⇒=.403,6'wwrrs =⇒−=Example:In contrast,)122('−−−−−−−−−=−swwerrImpact of Skin Effects on Effective Wellbore Radius Positive skin factor decreases effective wellbore radius  Negative skin factor increases effective wellbore radiusImpact of Skin Effects on Well Productivity)61( −−−−−=−=DrwfJBkhppqJµα)142(ln1ln1'−−−−−−−=+=weweDrrsrrJFor steady-state flow to a vertical well,Oil well productivity index is defined as Positive skin factor decreases effective wellbore radius  Smaller effective wellbore radius results in lower well productivity Stimulation(e.g., acidizing, fracturing, …, etc.) reduces skin, thereby increasing effective wellbore radius and well productivityUndersaturated Oil Reservoirs Well Flowing Pressure Profile under Constant Rate ProductionSingle-phase Flow under Transient Condition• During the initial pressure decline from p = pi, the pressure response in the reservoir is not affected by the outer boundary• The system appears infinite in extentUndersaturated Oil Reservoirs For radial flow of a slightly compressible, constant viscosity fluid in an infinite-acting reservoir, the pressure profile can be described by)202(122−−−−−−−∂∂=∂∂+∂∂tpkcrprrptφµ.Single-phase Flow under Transient ConditionUndersaturated Oil Reservoirs )202(122−−−−−−−∂∂=∂∂+∂∂tpkcrprrptφµSolving Eq. 2-20 for constant flow rate (in field units) yields2162.6log log 3.23) (2 6)wf itwqB kpp tkh c rµφµ= − + − −−− −----------(2-24))252()23.3loglog6.162)(12−−−−−−+−=−wtwfircktBppkhqφµµFor constant bottom-hole-pressure (in field units):)262(87.023.3loglog6.162)(12−−−−−+−+−=−srcktBppkhqwtwfiφµµIncorporating skin factor yieldsUndersaturated Oil Reservoirs Single-phase Flow under Pseudosteady State Condition• The total effect of outer boundary has been felt• Usually applicable to reservoirs with no-flow boundaries which have been producing for some timeFrom the radial diffusivity equation, the pressure p at any point r in a reservoir of radius reunder pseudosteady state is given byUndersaturated Oil Reservoirs Single-phase Flow under Pseudosteady State Condition).282(2ln2.14122−−−−−+=ewwfrrrrkhqBppµ)292(21ln2.141−−−−−+=wewferrkhqBppµ)342(472.0ln2.141−−−−+=− srrkhqBppwewfµAt r = re,,pSince peis usually unknown and the average reservoir,can be obtained from pressure buildup tests, a more useful expression for pseudosteady state with skin effect, s, included would beWells Draining Irregular PatternsUndersaturated Oil Reservoirs )442(4ln212.1412−−−−+=− srCAkhqBppwAwf㵓Shape factors”, CA, accounts for irregular drainage shapes or asymmetrical positioning of wellsFor pseudosteady state flowUndersaturated Oil Reservoirs • IPR is the relationship between bottomhole pressure, pwf, and well flow rate, q• IPR combines what the reservoir can deliver (q → pwf= 0) and what the imposed wellbore hydraulic would allow (q→ pwf> 0)Inflow Performance Relationship (IPR))82(ln2.141−−−−+=− srrkhqBppwewfeµ)342(472.0ln2.141−−−−+=− srrkhqBppwewfµ)262(87.023.3loglog6.162)(12−−−−−+−+−=−srcktBppkhqwtwfiφµµTransient IPR (in field units):Pseudosteady State IPR (in field units):Steady-State IPR (in field units):,19.0=φ( ))502(25.4log565116.2−−−−+−=tpqwfExample 2-7 Transient IPRUsing the well and reservoir data in Appendix A, construct transient IPR curves for 1,6, and 24 months. Assume zero skin and transient flow.Solution:For transient flow,From Appendix A,k = 8.2 md, h = 53 ft, pi= 5651 psi, B = 1.1 rb/STB, μ = 1.7 cp, ct= 1.29 ×