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 Slide — 1/38

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 Slide — 2/40 ● 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 Outline

3. 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 Slide — 3/40 Problem Statement ■ Multi-stage hydraulic fracturing along a horizontal well is the current stimulation practice used in low permeability reservoirs

4. 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 Slide — 4/40 Problem Statement Cont. ■ Hydraulic Fracturing Issues: Provided by: Microsoft (US EIA 2012)

5. 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 Slide — 5/40 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)

6. 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 Slide — 6/40 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

7. 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 Slide — 7/40 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 (Valko: PETE 629 Lectures) (Freeman 2010)

8. 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 Slide — 8/40 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:

9. 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 Slide — 9/40 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)

10. 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 Slide — 10/40 ● Assumptions : ■ Slightly compressible liquid ■ Single-phase Darcy flow ■ Constant formation porosity and liquid viscosity ■ Negligible gravity effects ● Governing Equations: ■ Stimulated Zone: ■ Unstimulated Zone: Analytical Solution Derivation: Dimensionless Pressure

11. 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 Slide — 11/40 ● 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 Analytical Solution Derivation: Dimensionless Pressure

12. 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 Slide — 12/40 ● 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)) Analytical Solution Derivation: Dimensionless Pressure

13. 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 Slide — 13/40 ● Particular Solution ■ Stimulated Zone: ■ Unstimulated Zone: ■ Simplifying Notation: Analytical Solution Derivation: Dimensionless Pressure

14. 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 Slide — 14/40 ■ Dimensionless Variables: ■ Inner Boundary: Constant pressure ■ Van Everdingen and Hurst (1949) presented a relationship between constant pressure and constant rate solutions Analytical Solution Derivation: Dimensionless Rate

15. 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 Slide — 15/40 Solution Presentation ● Analytical Model Parameters

16. 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 Slide — 16/40 Solution Presentation:

17. 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 Slide — 17/40 Solution Presentation:

18. 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 Slide — 18/40 Solution Presentation:

19. 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 Slide — 19/40 Solution Presentation:

20. 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 Slide — 20/40 Solution Validation: Simulation Parameters and Gridding Radial grid increments = 2 cm.

21. 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 Slide — 21/40 Solution Validation:

22. 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 Slide — 22/40 Solution Validation:

23. 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 Slide — 23/40 Solution Validation:

24. 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 Slide — 24/40 Solution Validation:

25. 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 Slide — 25/40 Solution Validation:

26. 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 Slide — 26/40 Solution Validation:

27. 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 Slide — 27/40 PPR vs. MFH: Simulation Parameters

28. 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 Slide — 28/40 PPR vs. MFH: MFH Gridding ■ Take advantage of MFH symmetry ■ Simulate stencil ■ Quarter of the reservoir ■ Half of a fracture ■ x f = h f /2

29. 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 Slide — 29/40 PPR vs. MFH: Comparisons ● x f = 75 ft., wk f = 10 md-ft., F cD = 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

30. 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 Slide — 30/40 PPR vs. MFH: Comparisons ● x f = 75 ft., wk f = 1 md-ft., F cD = 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

31. 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 Slide — 31/40 PPR vs. MFH: Comparisons ● x f = 75 ft., wk f = 0.1 md-ft., F cD = 13.33 ● PPR compares well with MFH, even slightly better

32. 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 Slide — 32/40 PPR vs. MFH: Comparisons ● x f = 50 ft., wk f = 10 md-ft., F cD = 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

33. 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 Slide — 33/40 PPR vs. MFH: Comparisons ● x f = 50 ft., wk f = 1 md-ft., F cD = 200 ● MFH performance from 10 to 1 md-ft. is small ● 50 and 25 fracture case produce within the range of PPR

34. 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 Slide — 34/40 PPR vs. MFH: Comparisons ● x f = 50 ft., wk f = 0.1 md-ft., F cD = 20 ● PPR performs better than the MFH

35. 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 Slide — 35/40 PPR vs. MFH: Comparisons ● x f = 25 ft., wk f = 10 md-ft., F cD = 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

36. 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 Slide — 36/40 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.

37. 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 Slide — 37/40 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

38. 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 Slide — 38/40 Recommendations for Future Work ■ Consider different permeability distributions: ■ Exponential permeability model (Wilson 2003) ■ Inverse-square permeability model (El-Khatib 2009) ■ Linear permeability model

39. 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 Slide — 39/40 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. Rafiqul Awal, 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).