Ryan Sawyer Broussard Department of Petroleum Engineering Texas

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 Analytical Solutions for a Composite, Cylindrical Reservoir with a Power-Law Permeability Distribution in the Inner Cylinder

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 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 ●Summary and Final Conclusions

Problem Statement ■Multi-stage hydraulic fracturing along a horizontal well is the current 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 stimulation practice used in low permeability reservoirs

Problem Statement Cont. ■Hydraulic Fracturing Issues: Provided by: Microsoft 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 (US EIA 2012)

Problem Statement Cont. ■Proposed Stimulation Techniques: (Carter 2009) (Texas Tech University 2011) ■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 reservoirs? ■How does it perform compared to hydraulic fracturing? 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 ■How does this type of stimulation perform in low permeability

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- 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 fracture horizontal (MFH) to determine the PPR’s suitability to low permeability reservoirs

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) 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 (Freeman 2010)

Stimulation Concept: Power-Law Permeability ■ A hypothetical stimulation process creates an altered permeability zone surrounding the horizontal wellbore. 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 ■ The permeability within the altered zone follows a power-law function:

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 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 Space ■ Inverted numerically by Gaver. Wynn-Rho algorithm (Mathematica; Valko and Abate 2004)

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

Analytical Solution Derivation: Dimensionless Pressure ●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 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 ■ Region Interface: Continuous flux across the interface

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 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 and Hurst (1949))

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

Analytical Solution Derivation: Dimensionless Rate ■Dimensionless Variables: ■Inner Boundary: Constant pressure ■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 Slide — 14/40 presented a relationship between constant pressure and constant rate solutions

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 — 15/40 ● 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:

Solution Validation: Simulation Parameters and Gridding 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 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

PPR vs. MFH: MFH Gridding ■Take advantage of MFH symmetry ■Simulate stencil 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 ■ 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 Slide — 29/40 PPR vs. MFH: Comparisons ● xf = 75 ft. , wkf = 10 md-ft. , Fc. D = 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

PPR vs. MFH: Comparisons 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 ● xf = 75 ft. , wkf = 1 md-ft. , Fc. D = 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

PPR vs. MFH: Comparisons 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 ● xf = 75 ft. , wkf = 0. 1 md-ft. , Fc. D = 13. 33 ● PPR compares well with MFH, even slightly better

PPR vs. MFH: Comparisons ● xf = 50 ft. , wkf = 10 md-ft. , Fc. D = 2000 ● Reduction in stimulated volume has greatly affected 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 — 32/40 not so much the PPR ● Now 50 and 25 fracture case produce within the range of PPR

PPR vs. MFH: Comparisons 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 ● xf = 50 ft. , wkf = 1 md-ft. , Fc. D = 200 ● MFH performance from 10 to 1 md-ft. is small ● 50 and 25 fracture case produce within the range of PPR

PPR vs. MFH: Comparisons 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 ● xf = 50 ft. , wkf = 0. 1 md-ft. , Fc. D = 20 ● PPR performs better than the MFH

PPR vs. MFH: Comparisons ● xf = 25 ft. , wkf = 10 md-ft. , Fc. D = 4000 ● MFH rates dominated by low perm matrix at early times ● Rate decline follows closely to PPR ● PPR performs much better despite infinite conductivity 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 fractures

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 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 — 36/40 ●Conditions may exist where achieving high

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 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 — 37/40 ■Recommend the PPR when conditions exist that prevent optimal

Recommendations for Future Work ■Consider different permeability distributions: ■ Exponential permeability model (Wilson 2003) ■ Inverse-square permeability model (El-Khatib 2009) 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 ■ Linear permeability model

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