1 / 39

Ryan Sawyer Broussard Department of Petroleum Engineering Texas A&M University

Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder. Ryan Sawyer Broussard Department of Petroleum Engineering Texas A&M University College Station, TX 77843-3116 (USA) ryan.broussard@pe.tamu.edu. Outline.

adem
Download Presentation

Ryan Sawyer Broussard Department of Petroleum Engineering Texas A&M University

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder Ryan Sawyer Broussard Department of Petroleum Engineering Texas A&M University College Station, TX 77843-3116 (USA) ryan.broussard@pe.tamu.edu MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  2. Outline Problem Statement Research Objectives Stimulation Concepts: Hydraulic Fracturing Power-law permeability Analytical Model and Solution Derivations: Dimensionless pressure solution with a constant rate I.B.C Dimensionless rate solution with a constant pressure I.B.C. Presentation and Validation of the Solutions Power-Law Permeability vs. Multi-Fractured Horizontal Simulation Parameters and Gridding Comparisons Conclusions Summary and Final Conclusions MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  3. Problem Statement Multi-stage hydraulic fracturing along a horizontal well is the current stimulation practice used in low permeability reservoirs MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  4. Problem Statement Cont. Hydraulic Fracturing Issues: Provided by: Microsoft Provided by: Microsoft (US EIA 2012) MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  5. Problem Statement Cont. Proposed Stimulation Techniques: We are not proposing a new technique We evaluate a stimulation concept: Creating an altered permeability zone Permeability decreases from the wellbore following a power-law function How does this type of stimulation perform in low permeability reservoirs? How does it perform compared to hydraulic fracturing? (Carter 2009) (Texas Tech University 2011) MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  6. Research Objectives: Develop an analytical representation of the rate and pressure behavior for a horizontal well producing in the center of a reservoir with an altered zone characterized by a power-law permeability distribution Validate the analytical solutions by comparison to numerical reservoir simulation Compare the power-law permeability reservoir (PPR) to a multi-fracture horizontal (MFH) to determine the PPR’s suitability to low permeability reservoirs MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  7. Stimulation Concept: Multi-fracture horizontal • Pump large volumes of fluid at high rates and pressure into the formation • The high pressure breaks down the formation, creating fractures that propagate out into the reservoir • Direction determined by maximum and minimum stresses created by the surrounding rock • Process repeated several times along the length of the horizontal wellbore (Freeman 2010) (Valko: PETE 629 Lectures) MS Thesis Defense— Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  8. Stimulation Concept: Power-Law Permeability • A hypothetical stimulation process creates an altered permeability zone surrounding the horizontal wellbore. • The permeability within the altered zone follows a power-law function: MS Thesis Defense— Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  9. Analytical Model • Geometry • Composite, cylinder consists of two regions: • Inner region is stimulated. Permeability follows a power-law function. • Outer region is unstimulated and homogeneous. • Horizontal well is in the center of the cylindrical volume • Wellbore spans the entire length of the reservoir (i.e. radial flow only) • Mathematics • Solution obtained in Laplace Space • Inverted numerically by Gaver-Wynn-Rho algorithm (Mathematica; Valko and Abate 2004) MS Thesis Defense— Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  10. Analytical Solution Derivation: DimensionlessPressure Assumptions: Slightly compressible liquid Single-phase Darcy flow Constant formation porosity and liquid viscosity Negligible gravity effects Governing Equations: Stimulated Zone: Unstimulated Zone: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  11. Analytical Solution Derivation: DimensionlessPressure Initial and Boundary Conditions Initial Condtions: Uniform pressure at t=0 Outer Boundary: No flow Inner Boundary: Constant rate Region Interface: Continuous pressure across the interface Region Interface: Continuous flux across the interface MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  12. Analytical Solution Derivation: Dimensionless Pressure General Solutions in the Laplace Domain: Stimulated Zone: Solution from Bowman (1958) and Mursal(2002) Unstimulated Zone: Well known solution (obtained from Van Everdingen and Hurst (1949)) MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  13. Analytical Solution Derivation: DimensionlessPressure Particular Solution Stimulated Zone: Unstimulated Zone: Simplifying Notation: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  14. Analytical Solution Derivation: DimensionlessRate Dimensionless Variables: Inner Boundary: Constant pressure Van Everdingen and Hurst (1949) presented a relationship between constant pressure and constant rate solutions MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  15. Solution Presentation • Analytical Model Parameters MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  16. Solution Presentation: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  17. Solution Presentation: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  18. Solution Presentation: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  19. Solution Presentation: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  20. Solution Validation: Simulation Parameters and Gridding • Radial gridincrements = 2 cm. MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  21. Solution Validation: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  22. Solution Validation: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  23. Solution Validation: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  24. Solution Validation: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  25. Solution Validation: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  26. Solution Validation: MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  27. PPR vs. MFH: Simulation Parameters MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  28. PPR vs. MFH: MFH Gridding • Take advantage of MFH symmetry • Simulate stencil • Quarter of the reservoir • Half of a fracture • xf= hf/2 MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  29. PPR vs. MFH: Comparisons • xf= 75 ft., wkf = 10 md-ft., FcD= 1333.33 • See evacuation of near fracture, then formation linear flow • PPR Perm declines quickly, small surface area with high perm • MFH more favorable in all cases except 25 fracture case MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  30. PPR vs. MFH: Comparisons • xf= 75 ft., wkf = 1 md-ft., FcD= 133.33 • MFH early time rates reduced by an order of magnitude • Extended time to evacuate fracture and near fracture region • MFH more favorable in all cases except 25 fracture case MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  31. PPR vs. MFH: Comparisons • xf= 75 ft., wkf = 0.1 md-ft., FcD= 13.33 • PPR compares well with MFH, even slightly better MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  32. PPR vs. MFH: Comparisons • xf= 50 ft., wkf = 10 md-ft., FcD= 2000 • Reduction in stimulated volume has greatly affected MFH, not so much the PPR • Now 50 and 25 fracture case produce within the range of PPR MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  33. PPR vs. MFH: Comparisons • xf= 50 ft., wkf = 1 md-ft., FcD= 200 • MFH performance from 10 to 1 md-ft. is small • 50 and 25 fracture case produce within the range of PPR MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  34. PPR vs. MFH: Comparisons • xf= 50 ft., wkf = 0.1 md-ft., FcD= 20 • PPR performs better than the MFH MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  35. PPR vs. MFH: Comparisons • xf= 25 ft., wkf = 10 md-ft., FcD= 4000 • MFH rates dominated by low perm matrix at early times • Rate decline follows closely to PPR • PPR performs much better despite infinite conductivity fractures MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  36. PPR vs. MFH: Conclusions • The reduction in stimulated volume adversely affects the MFH more than the PPR: • Loss of high conductivity surface area • The PPR lacks the high permeability surface area that the MFH creates • Unless the fracture half-length is small or the fracture conductivity low, the PPR will not perform as well as the MFH • Conditions may exist where achieving high conductivity fractures is difficult. In these situations, the PPR may provide a suitable alternative in ultra-low permeability reservoirs. MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  37. Summary and Conclusions Introduced a stimulation concept for low perm reservoirs: Altered zone with a power-law permeability distribution Power-law is a “conservative” permeability distribution Derived an analytical pressure and rate solutions in the Laplace domain using a radial composite model Validated the analytical solutions using numerical simulation Compared the PPR stimulation concept to MFH, concluding that: The PPR does not perform as well as the MFH unless the fracture surface area is small and/or the fracture conductivity low The PPR does not provide adequate high permeability rock surface area Recommend the PPR when conditions exist that prevent optimal fracture conductivities MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  38. Recommendations for Future Work Consider different permeability distributions: Exponential permeability model (Wilson 2003) Inverse-square permeability model (El-Khatib 2009) Linear permeability model MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

  39. References Abate, J. and Valkó, P.P. 2004b. Multi-precision Laplace Transform Inversion. International Journal for Numerical Methods in Engineering. 60: 979-993. Bowman, F. 1958. Introduction to Bessel Functions, first edition. New York, New York: Dover Publications Inc. Carter, E.E. 2009. Novel Concepts for Unconventional Gas Development of Gas Resources in Gas Shales, Tight Sands and Coalbeds. RPSEA 07122-7, Carter Technologies Co., Sugar Land, Texas (19 February 2009). El-Khatib, N.A.F. 2009. Transient Pressure Behavior for a Reservoir With Continuous Permeability Distribution in the Invaded Zone, Paper SPE 120111 presented at the SPE Middle East Oil and Gas Show and Conference, Bahrain, Bahrain, 15-18 March. SPE-120111-MS. http://dx.doi.org/10.2118/120111-MS. Freeman, C.M. 2010. Study of Flow Regimes in Multiply-Fractured Horizontal Wells in Tight Gas and Shale Gas Reservoir Systems. MS thesis, Texas A&M University, College Station, Texas (May 2010). Mathematica, version 8.0. 2010. Wolfram Research, Champaign-Urbana, Illinois. Mursal. 2002. A New Approach For Interpreting a Pressure Transient Test After a Massive Acidizing Treatment. MS thesis, Texas A&M University, College Station, Texas (December 2002). Texas Tech University. 2011. Dr. M. RafiqulAwal, http://www.depts.ttu.edu/pe/dept/facstaff/awal/ (accessed 31 October) van Everdingen, A.F. and Hurst, W. 1949. The Application of the Laplace Transformation to Flow Problems in Reservoirs. J. Pet. Tech. 1 (12): 305-324. SPE-949305-G. http://dx.doi.org/10.2118/949305-G. Wilson, B. 2003. Modeling of Performance Behavior in Gas Condensate Reservoirs Using a Variable Mobility Concept. MS thesis, Texas A&M University, College Station, Texas (December 2003). MS Thesis Defense — Ryan Sawyer Broussard— Texas A&M University College Station, TX (USA) — 2 October 2012 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

More Related