Future Work - PDF Mathematical Modeling of Hydraulic Fracturing In Shale Gas Reservoirs

 Develop a laboratory experimental setup to investigate the effect of various parameters on the fracture geometry and compare with the results obtained from the PKN-C model.

REFERENCES

Gomaa, A. M. Qi Qu, Maharidge, R., Nelson, S., and Reed, T., 2014. New Insights into Hydraulic Fracturing of Shale Formations, Presented at the International Petroleum Technology Conference 2014, IPTC 2014, IPTC 17594, Doha, Qatar January 20-22, 2014.

Valko, P., and Economides, M. J. 1995. Hydraulic Fracture Mechanics. Chichester, England: John Wiley & Sons.

American Petroleum Institute. (2013, October). Shale Energy: 10 Points everyone should know. Retrieved February 5, 2014, from API.org:

http://www.api.org/~/media/Files/Policy/Hydraulic_Fracturing/Hydraulic- Fracturing-10-points.pdf

Economides, M. J., and Nolte, K. G. 1989. Reservoir Stimulation, Second edition.

Houston, TX: Schlumberger Educational Services.

Guo, B., Lyons, W., & Ghalambor, A. (2007). Petroleum production engineering (1st ed.). Burlington, MA: Gulf Professional Pub.

Nolen-Hoeksema, R. (2013). Elements of Hydraulic Fracturing. Oilfield Review, 25(2), 51.

Geertsma, J. 1989. Two-Dimensional Fracture-Propagation Models. In: Recent Advances in Hydraulic Fracturing, Monograph Vol. 12, Gidley, J. L. et al. (Eds.).

Richardson, TX: SPE.

Xiang, J. 2011. A PKN Hydraulic Fracture Model Study and Formation Permeability Determination, MS thesis, Texas A&M University, College Station, TX.

Rahman, M. M. and Rahman, K. K. 2010. A Review of Hydraulic Fracture Models and Development of an Improved Pseudo-3D Model for Stimulating Tight Oil/Gas Sand. Energy Sources, Part A 32:1416–1436.

Smrecak, T. A., & PRI Marcellus Shale Team, 2011. Jointing and Fracturing in the Marcellus Shale. The Marcellus Papers, August 2011 (5). Retrieved from:

http://cce.cornell.edu/EnergyClimateChange/NaturalGasDev/Documents/PRI%20Pa pers/Marcellus_issue5.pdf.

Syed, A. H. 2010. Fracture initiation and propagation in poroelastic medium and investigation of potential for introducing a secondary fracture treatment, MS thesis, University of New South Wales, Australia.

Allen, T. O., & Roberts, A. P. 1989. Production Operations, Well Completions, Workover, and Stimulation (3^rd ed.). Tulsa: Oil & Gas Consultants International.

Mohanty, K. K., 2009. Improvement of Fracturing in Gas Shales, Technology Status Assessment. Retrieved from:

http://www.rpsea.org/media/files/project/91bafd02/07122-38-TS- Improvement_Fracturing_Gas_Shales-05-26-09_P.pdf.

Fisher, M., & Warpinski, N. (2012). Hydraulic-Fracture-Height Growth: Real Data. SPE Production & Operations, 27(01), 8-19. doi:10.2118/145949-PA

Newman, M. S., Pavloudis, M., & Rahman, M. M. 2009. Importance of Fracture Geometry and Conductivity in Improving Efficiency of Acid Fracturing in

Carbonates. Presented at the Canadian International Petroleum Conference, Paper 2009-146. Calgary, Canada.

Cinco-Ley, H., & Samaniego-V., F. (1981). Transient Pressure Analysis for Fractured Wells. Journal of Petroleum Technology, 33(09), 1749–1766.

NSI Technologies. Inc. (2001). Optimum Fracture Design. NSI Technologies Frac Tips, 3 (1). Retrieved from http://www.nsitech.com/wp-

content/uploads/2014/06/nsi_fractip_optimumfracture.pdf

Clark, C., Burnham, A., Harto, C., and Horner, R., 2013, Hydraulic Fracturing and Shale Gas Production: Technology, Impacts, and Regulations, ANL/EVS/R-12/5, Argonne National Laboratory, Argonne, IL.

Amadei, B. (n.d). Lecter Note 5: Deformability Properties of Rocks and Masses, Retrieved from: http://ceae.colorado.edu/~amadei/CVEN5768/PDF/NOTES5.pdf Total S.A. (n.d). Three Main Sources Of Unconventional Gas. Retreived from:

http://www.total.com/en/energies-expertise/oil-gas/exploration-production/strategic- sectors/unconventional-gas/presentation/three-main-sources-unconventional-gas.

34 APPENDIX

APPENDIX 1: Gantt Chart

 The following chart illustrates the timeline and key milestones of the project during FYP I

Table 4 Gantt Chart FYP I

 The following chart illustrates the timeline and key milestones of the project during FYP II.

Table 5 Gantt Chart FYP II

APPENDIX II: Young’s Modulus for Various Rocks

Table 6. Range of Young's modulus for various rocks. Adapted from Deformability Properties of Rocks and Masses, by B. Amadei (n.d).

APPENDIX III: Poisson’s Ratio for Various Rocks

Table 7.Range of Poisson’s Ratio for various rocks. Adapted from Deformability Properties of Rocks and Masses, by B. Amadei (n.d)

APPENDIX IV: Permeability of Different types of Gas Reservoirs

Figure A1 Permeability for different types of gas reservoir. Adapted from Three Main Sources Of Unconventional Gas, by Total S.A. (n.d)

APPENDIX V: Mathematical Computer Code for PKN-C Design Mode

% read from excel file filename = 'myExample.xlsx';

% CL = first col

% Sp = second col

% hf = third col

% dE = forth col

% mu = fifth col

% i = sixth col

% xf = seventh col data = xlsread(filename);

[EW, tsolved , W_W0, P_nw] = myfun(data(:,1),data(:,2),data(:,3),data(:,4),data(:,5),data(:,6),data(:,7));

results = [EW,tsolved,W_W0,P_nw];

Names_var = {'EW'; 't'; 'W_W0';'P_nw'};

%write data in csv format

fid = fopen('results_April_15-2014.csv','w');

fprintf(fid, [Names_var{1} sprintf(',%s',Names_var{2:end}) '\n']);

fclose(fid)

dlmwrite('results_April_15-2014.csv', results, '-append', ...

'delimiter', ',','precision','%.10f');

function [EW, tsolved , W_W0, P_nw] = myfun(CL,Sp,hf,dE,mu,i,xf)

EW = 2.05*((mu.*i.*xf)./dE).^0.25;

n = size(CL,1);

tsolved = zeros(n,1);

for k = 1:n syms t

tsolved(k) = solve(xf(k) == ((EW(k)+2*Sp(k))*i(k))/(4*CL(k)^2*pi*hf(k))*...

(exp((2*CL(k)*sqrt(pi*t)/(EW(k)+2*Sp(k)))^2 )*erfc((2*CL(k)*sqrt(pi*t))/(EW(k)+2*Sp(k)))+...

(2*(2*CL(k)*sqrt(pi*t))/(EW(k)+2*Sp(k)))/(sqrt(pi))-1),t);

end

W_W0 = 3.27*((mu.*i.*xf)./dE).^0.25;

P_nw = (dE./(2.*hf)).*W_W0;

figure;

subplot(2,2,1) plot(mu,EW) title('mu vs EW') subplot(2,2,2) plot(mu,P_nw) title('mu vs pressure') subplot(2,2,3) plot(P_nw,EW) title('P vs width ') subplot(2,2,4) plot(P_nw,EW) title('P vs xf')

end

APPENDIX VI: Mathematical Computer Code for PKN-C Simulation Mode

% read from excel file filename = 'myExamplexf.xlsx';

data = xlsread(filename);

[EW, xfsolved , W_W0, P_nw] = myfun_xf(data(:,1),data(:,2),data(:,3),data(:,4),data(:,5),data(:,6),data(:,7));

results = [EW,xfsolved,W_W0,P_nw];

Names_var = {'EW'; 'xf'; 'W_W0';'P_nw'};

%write data in csv format fid = fopen('Results_xf.csv','w');

fprintf(fid, [Names_var{1} sprintf(',%s',Names_var{2:end}) '\n']);

fclose(fid)

dlmwrite('Results_xf.csv', results, '-append', ...

'delimiter', ',','precision','%.10f');

function [EW, xfsolved , W_W0, P_nw] = myfun_xf(CL,Sp,hf,dE,mu,i,t)

n = size(CL,1);

xfsolved = zeros(n,1);

EW = zeros(n,1);

for k = 1:n syms xf EW

[xfsolved(k), EW(k)] = solve(xf == ((EW+2*Sp(k))*i(k))/(4*CL(k)^2*pi*hf(k))*...

(exp((2*CL(k)*sqrt(pi*t(k))/(EW+2*Sp(k)))^2 )*erfc((2*CL(k)*sqrt(pi*t(k)))/(EW+2*Sp(k)))+...

(2*(2*CL(k)*sqrt(pi*t(k)))/(EW+2*Sp(k)))/(sqrt(pi))-1),EW == 2.05*((mu(k).*i(k).*xf)./dE(k)).^0.25, xf, EW);

EW = 2.05*((mu.*i.*xfsolved)./dE).^0.25;

W_W0 = 3.27*((mu.*i.*xfsolved)./dE).^0.25;

P_nw = (dE./(2.*hf)).*W_W0;

end figure;

subplot(2,2,1) plot(xfsolved,dE) title('dE vs length') subplot(2,2,2) plot(EW,dE) title('dE vs width') subplot(2,2,3) plot(xfsolved,t) title('xf vs t') subplot(2,2,4) plot(hf,t) title('hf vs t') end

APPENDIX VII: Mathematical Computer Code for KGD-C Design Mode

% read from excel file

filename = 'myExamplekgd.xlsx';

data = xlsread(filename);

[EW, tsolved , W_W0, P_nw] = myfunkgd(data(:,1),data(:,2),data(:,3),data(:,4),data(:,5),data(:,6),data(:,7));

results = [EW,tsolved,W_W0,P_nw];

Names_var = {'EW'; 't'; 'W_W0';'P_nw'};

%write data in csv format fid = fopen('KGD_results.csv','w');

fprintf(fid, [Names_var{1} sprintf(',%s',Names_var{2:end}) '\n']);

fclose(fid)

dlmwrite('KGD_results.csv', results, '-append', ...

'delimiter', ',','precision','%.10f');

function [EW, tsolved , W_W0, P_nw] = myfunkgd(CL,Sp,hf,dE,mu,i,xf)

EW = 2.53*((mu.*i.*xf.^2)./(hf.*dE)).^0.25;

n = size(CL,1);

tsolved = zeros(n,1);

for k = 1:n syms t

tsolved(k) = solve(xf(k) == ((EW(k)+2*Sp(k))*i(k))/(4*CL(k)^2*pi*hf(k))*...

(exp((2*CL(k)*sqrt(pi*t)/(EW(k)+2*Sp(k)))^2 )*erfc((2*CL(k)*sqrt(pi*t))/(EW(k)+2*Sp(k)))+...

(2*(2*CL(k)*sqrt(pi*t))/(EW(k)+2*Sp(k)))/(sqrt(pi))-1),t);

end

W_W0 = 3.22*((mu.*i.*xf.^2)./(hf.*dE)).^0.25;

P_nw = (dE./(4.*xf)).*W_W0;

figure;

subplot(2,2,1) plot(mu,EW) title('mu vs EW') subplot(2,2,2) plot(mu,P_nw) title('mu vs pressure') subplot(2,2,3) plot(P_nw,EW) title('P vs width ') subplot(2,2,4) plot(P_nw,EW) title('P vs xf')

end

APPENDIX VIII: Mathematical Computer Code for KGD-C Simulation Mode

% read from excel file

filename = 'myExamplexfkgd.xlsx';

data = xlsread(filename);

[EW, xfsolved , W_W0, P_nw] = myfun_xfkgd(data(:,1),data(:,2),data(:,3),data(:,4),data(:,5),data(:,6),data(:,7));

results = [EW,xfsolved,W_W0,P_nw];

Names_var = {'EW'; 'xf'; 'W_W0';'P_nw'};

%write data in csv format

fid = fopen('Results_xf_KGD.csv','w');

fprintf(fid, [Names_var{1} sprintf(',%s',Names_var{2:end}) '\n']);

fclose(fid)

dlmwrite('Results_xf_KGD.csv', results, '-append', ...

'delimiter', ',','precision','%.10f');

function [EW, xfsolved , W_W0, P_nw] = myfun_xfkgd(CL,Sp,hf,dE,mu,i,t)

n = size(CL,1);

xfsolved = zeros(n,1);

EW = zeros(n,1);

for k = 1:n syms xf EW

[xfsolved(k), EW(k)] = solve(xf == ((EW+2*Sp(k))*i(k))/(4*CL(k)^2*pi*hf(k))*...

(exp((2*CL(k)*sqrt(pi*t(k))/(EW+2*Sp(k)))^2 )*erfc((2*CL(k)*sqrt(pi*t(k)))/(EW+2*Sp(k)))+...

(2*(2*CL(k)*sqrt(pi*t(k)))/(EW+2*Sp(k)))/(sqrt(pi))-1),EW ==

2.53*((mu(k).*i(k).*xf.^2)./(hf(k).*dE(k))).^0.25, xf, EW);

EW = 2.53*((mu.*i.*xfsolved.^2)./(hf.*dE)).^0.25;

W_W0 = 3.22*((mu.*i.*xfsolved.^2)./(hf.*dE)).^0.25;

P_nw = (dE./(4.*xfsolved)).*W_W0;

end figure;

subplot(2,2,1) plot(xfsolved,dE) title('dE vs length') subplot(2,2,2) plot(EW,dE) title('dE vs width') subplot(2,2,3) plot(xfsolved,t) title('xf vs t') subplot(2,2,4) plot(hf,t) title('hf vs t') end

APPENDIX IX: Input and output parameters for examining effect of rockstiffness on length and width

Table 8. Input and out parameters for examining effect of rock stiffness

mmPaPa.s 3 s - PammmPa9.84E-06051.81.9E+09 0.20.0662120000.11.92E+09 0.01741754.13810.027772514471.79.84E-06051.83.1E+09 0.20.0662120000.13.13E+09 0.015741821.36620.025109758182.49.84E-060 51.85.05E+09 0.2 0.0662120000.1 5.1E+09 0.014228893.53990.02269511170149.84E-06051.88.23E+09 0.20.0662120000.18.31E+09 0.012856970.81990.02050616451509.84E-06051.81.34E+10 0.20.0662120000.11.35E+10 0.0116121053.3310.01852224221609.84E-06051.82.19E+10 0.20.0662120000.12.21E+10 0.0104841141.1520.0167243564825

APPENDIX X: Input and out parameters for KGD-C and PKN-C width andlength comparison

Table 9. Input and out parameters for KGD-C and PKN-C width and length comparison

mmPa.s 3 s Pamm

PKN-CKGD-CPKN-CKGD-C

9.84E-06051.80.20.066270001.38E+100.0104550.019115704.4876419.25159.84E-060 51.80.20.066280001.38E+100.0107210.019968779.1449457.5349

9.84E-06051.80.20.066290001.38E+100.0109620.020753851.377494.1799

9.84E-06051.80.20.0662100001.38E+100.0111810.02148921.5048529.42729.84E-060 51.80.20.0662110001.38E+100.0113820.02216989.779563.4618

9.84E-06051.80.20.0662120001.38E+100.0115690.0227991056.401596.4298

APPENDIX XI: Input and out parameters for investigating effect of height

Table 10. Input and out parameters for investigating effect of height on length

mmPa.s 3 s PammmPa

Fracture height = 40 m9.84E-060 400.20.066220002.02E+100.008029358.97520.01280732340159.84E-060 400.20.066240002.02E+100.009166609.73760.0146236919999.84E-060 400.20.066260002.02E+100.009898829.26680.01578939870299.84E-060 400.20.066280002.02E+100.010451030.2190.01666942092849.84E-060 400.20.0662100002.02E+100.0108971218.1570.01738243893659.84E-060 400.20.0662120002.02E+100.0112751396.1870.0179854541630

Fracture height = 80 m9.84E-060 800.20.066220002.02E+100.006962202.91380.01110414020839.84E-060 800.20.066240002.02E+100.007941343.48480.01266615992749.84E-060 800.20.066260002.02E+100.00857466.0830.01367117260839.84E-060 800.20.066280002.02E+100.009044578.00250.01442618214969.84E-060 800.20.0662100002.02E+100.009427682.44790.01503818987309.84E-060 800.20.0662120002.02E+100.009751781.20810.0155551963982

APPENDIX XII: Input and out parameters for investigating effect of Leak off

Table 11. Input and out parameters for investigating effect of Leak off on length

mmPa.s 3 s PammmPa

0.0009440450.20.066220002.02E+100.00397921.660340.00634714247530.0009440450.20.066240002.02E+100.00434530.796180.00693115557780.0009440 450.2 0.066260002.02E+100.00457437.813510.00729616377030.0009440450.20.066280002.02E+100.00474343.732590.00756616983400.0009440450.20.0662100002.02E+100.00487948.949270.00778217468670.0009440 450.2 0.0662120002.02E+100.00499253.666720.0079631787515

0.0018880450.20.066220002.02E+100.00335710.977930.00535612021360.0018880 450.2 0.066240002.02E+100.00366315.560340.00584413116810.0018880450.20.066260002.02E+100.00385519.077990.00614913802460.0018880450.20.066280002.02E+100.00399722.044160.00637514310230.0018880 450.2 0.0662100002.02E+100.00411 24.657790.00655614716750.0018880450.20.0662120002.02E+100.00420527.020940.0067081505735

APPENDIX XIII: Input and out parameters for calculating Dimensionlessfracture conductivity and Folds of increase

Table 12. Input and out parameters for calculating Dimensionless fracture conductivity and Folds of increase

mmPa.s 3 s Pamm- -

PKN-CKGD-CPKN-CKGD-CPKN-CKGD-CPKN-CKGD-C

9.84E-060450.2 0.066220002.02E+100.0078370.012764325.8576210.916119.2396148.414545.5881894.540717

9.84E-060450.2 0.066240002.02E+100.0089450.016024553.1803332.404512.9365138.565348.0947715.850029

9.84E-060450.2 0.066260002.02E+100.0096590.018301752.0675433.568910.2748333.7677110.662267.015166

9.84E-060450.2 0.066280002.02E+100.0101970.020108934.0443523.41548.73356530.7331813.887978.149347

9.84E-060450.2 0.0662100002.02E+100.0106330.02163 1104.176605.67077.70350728.5701417.948669.297453

9.84E-060450.2 0.0662120002.02E+100.0110010.0229581265.291682.33316.95543726.9173622.1325310.37877

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