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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.

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

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  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

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