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 (USA) 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

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

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

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)

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)

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

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)

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:

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)

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

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

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

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

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

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

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:

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:

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:

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:

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.

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:

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:

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:

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:

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:

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:

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

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

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 = ● 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 Slide — 30/40 PPR vs. MFH: Comparisons ● x f = 75 ft., wk f = 1 md-ft., F cD = ● 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 Slide — 31/40 PPR vs. MFH: Comparisons ● x f = 75 ft., wk f = 0.1 md-ft., F cD = ● 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 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

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

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

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

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.

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

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

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: Bowman, F Introduction to Bessel Functions, first edition. New York, New York: Dover Publications Inc. Carter, E.E Novel Concepts for Unconventional Gas Development of Gas Resources in Gas Shales, Tight Sands and Coalbeds. RPSEA , Carter Technologies Co., Sugar Land, Texas (19 February 2009). El-Khatib, N.A.F Transient Pressure Behavior for a Reservoir With Continuous Permeability Distribution in the Invaded Zone, Paper SPE presented at the SPE Middle East Oil and Gas Show and Conference, Bahrain, Bahrain, March. SPE MS. Freeman, C.M 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 Wolfram Research, Champaign-Urbana, Illinois. Mursal 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 Dr. M. Rafiqul Awal, (accessed 31 October) van Everdingen, A.F. and Hurst, W The Application of the Laplace Transformation to Flow Problems in Reservoirs. J. Pet. Tech. 1 (12): SPE G. Wilson, B Modeling of Performance Behavior in Gas Condensate Reservoirs Using a Variable Mobility Concept. MS thesis, Texas A&M University, College Station, Texas (December 2003).