# Similar constructing method for solving nonlinear spherical seepage model with quadratic pressure gradient of three-region composite fractal reservoir

- 26 Downloads

## Abstract

For the three-region composite reservoir, this paper establishes the nonlinear spherical seepage model of three-region composite fractal reservoir under three kinds of outer boundary conditions (infinite boundary, constant pressure boundary and closed boundary). The seepage model considers wellbore storage, effective radius and quadratic pressure gradient. First, the established seepage model is turned into boundary value problem of composite-modified Bessel equation in Laplace space by canceling out its dimensions and dealing it with Laplace transform. Second, using the similar constructing method to solve the nonlinear spherical seepage model, its analytic solution is obtained. Third, the expression of dimensionless bottom-hole pressure of the model in real space is obtained using the Stehfest numerical inversion equation to the solution in Laplace space. Finally, the corresponding-type curves of three-region composite reservoir with quadratics pressure gradient are mapped by programming. After that, sensitivity analysis of deferent parameters is carried out. Error analysis shows that the effects of quadratics pressure gradient should not be ignored.

## Keywords

Three-region composite fractal reservoir Nonlinear spherical seepage Quadratics pressure gradient Similar constructing method Similar structure of solution## List of symbols

*B*Formation volume factor (dimensionless)

*C*Well-bore storage coefficient (m

^{3}/MPa)- \( p_{\omega } \)
Bottom-hole pressure (MPa)

*q*Production rate (m

^{3}/d)- \( p_{0} \)
Initial reservoir pressure (MPa)

*R*The outer boundary radius (m)

*h*Reservoir thickness (m)

- \( \eta_{i} \)
Pressure transmitting coefficient (µm

^{3}MPa)*z*Laplace space variable (dimensionless)

- \( C_{{L_{i} }} \)
Fluid compressibility (MPa

^{−1})- \( r_{\text{we}} \)
Effective wellbore radius (m)

- \( C_{t} \)
Compressibility (MPa

^{−1})- \( k_{i} \)
Permeability (µm

^{2})- \( p_{i} \)
Reservoir pressure (MPa)

*r*Well-bore radius (m)

*S*Skin factor (dimensionless)

- \( \mu_{i} \)
Viscosity (MPa s)

- \( \phi_{i} \)
Porosity (dimensionless)

*t*Time (h)

- \( d_{{f_{i} }} \)
Fractal dimension (dimensionless)

- \( \theta_{i} \)
Fractal exponent (dimensionless)

## Superscript

- –
Laplace domain

## Subscripts

*D*Dimensionless

*w*Well-bore parameter

## Mathematics Subject Classification

35A25 35G30## Notes

### Funding

Project supported by Dynamic simulation of the mechanism of shale gas development and the optimal control of the district pollution, Applied fundamental research (Major frontier projects) of Sichuan Province (16JC0314).

## References

- Acuna JA, Yortsos YC (1995) Application of Fractal Geometry to the Study of Networks of Fractures and Their Pressure Transient. Water Resour Res 31(3):527–540CrossRefGoogle Scholar
- Acuna JA, Ershaghi I, Yortsos YC (1995) Practical application of fractal pressure transient analysis of naturally fractured reservoirs. SPE Form Eval 10(3):173–179CrossRefGoogle Scholar
- Al-Zainaldin S, Glover PWJ, Lorinczi P (2016) Synthetic Fractal Modelling of Heterogeneous and Anisotropic Reservoirs for Use in Simulation Studies: implications on Their Hydrocarbon Recovery Prediction. Transp Porous Media 116(1):1–32Google Scholar
- Beier RA (1994) Pressure-transient model for a vertically fractured well in a fractal reservoir. SPE Form Eval 9(2):122–128CrossRefGoogle Scholar
- Braeuning S, Jelmert TA, Sven AV (1998) The effect of the quadratic gradient term on variable-rate well tests. J Petrol Sci Eng 21(2):203–222CrossRefGoogle Scholar
- Camacho-Velázquez R, Fuentes-Cruz G, Vásquez-Cruz MA (2008) Decline-curve analysis of fractured reservoirs with fractal geometry. SPE Reserv Eval Eng 11(3):606–619CrossRefGoogle Scholar
- Chaidez-Félix JM, Velasco-Hernández JX (2018) An exploration of pressure dynamics using differential equations defined on a fractal geometry. Comput Appl Math 37(2):1279–1293MathSciNetCrossRefGoogle Scholar
- Chakrabarty C, Farouq Ali SM, Tortike WS (1993) Analytical solutions for radial pressure distribution including the effects of the quadratic gradient term. Water Resour Res 29(4):1171–1177CrossRefGoogle Scholar
- Chakrabarty C, Ali SMF, Tortike WS (2010) Analytical solutions for radial pressure distribution including the effects of the quadratic-gradient term. Water Resour Res 29(4):1171–1177CrossRefGoogle Scholar
- Chang J, Yotsors YC (1990) Pressure transient analysis of fractal reservoir. SPE Form Eval 5(1):31–38CrossRefGoogle Scholar
- Cossio M (2012) A semi-analytic solution for flow in finite-conductivity vertical fractures using fractal theory. In: Proceeding of the SPE annual technical conference and exhibition, San Antonio, Tex, USA, October 2012Google Scholar
- Debnath L (2003) Recent applications of fractional calculus to science and engineering. Int J Math Math Sci 2003(54):3413–3442MathSciNetCrossRefGoogle Scholar
- Dong XX, Li SC, Gui DD et al (2014) A study on solving the boundary value problem of three-region composite second-order linear homogeneous ODE. Energy Educ Sci Technol Part A Energy Sci Res 32(6):6035–6048Google Scholar
- Finjord J (1986) Effects of the quadratic gradient term in the infinite-acting period for two-dimensional reservoir flow. In: SPE 16451Google Scholar
- Finjord J, Aadnoy BS (1986) Effects of the nonlinear gradient term in exact analytical solutions of the radial flow equation for oil in a reservoir. In: SPE 15969Google Scholar
- Finjord J, Aadony BS (1989) Effects of quadratic gradient term in steady-state and quasi-steady-state solutions for reservoir pressure. SPE Form Eval 4(3):413–417CrossRefGoogle Scholar
- Flamenco-Lopez F, Camacho-Velazquez R (2003) Determination of fractal parameters of fracture networks using pressure-transient data. SPE Reservoir Eval Eng 6(1):39–47CrossRefGoogle Scholar
- Gaynor GC, Chang EY, Painter S et al (2000) Application of Lévy random fractal simulation techniques in modelling reservoir mechanisms in the Kuparuk River Field, North Slope, Alaska. SPE Paper 39739. SPE Reserv Eval Eng 3(3):263–271CrossRefGoogle Scholar
- Hu Y, Min C (2016) Identification and modelling of geochemical reactions occurring within the sandstone reservoir flooded by seawater. Pet Sci Technol 34(17–18):1595–1601CrossRefGoogle Scholar
- Hu Y, Mackay E (2017) Modelling of geochemical reactions occurring in the Gyda Field under cold-seawater injection on the basis of produced–water-chemistry data and implications for scale management. SPE Prod Oper 32(4):449–468Google Scholar
- Hu Y, Mackay E, Vazquez O, Ishkov O (2018) Streamline simulation of barium sulfate precipitation occurring within the reservoir coupled with analyses of observed produced water chemistry data to aid scale management. SPE Prod Oper 33(1):85–101Google Scholar
- Leont’ev NE (2013) Description of weakly compressible fluid flows in porous media for a nonlinear seepage law. Fluid Dyn 48(3):402–406MathSciNetCrossRefGoogle Scholar
- Nie RS, Ding Y (2010) Research on the nonlinear spherical seepage model with quadratic pressure gradient and its seepage characteristics. Nat Sci 2(2):98–105Google Scholar
- Odeh AS, Babu DK (1998) Comprising of solutions for the nonlinear and linearized diffusion equations. SPE Reserv Eng 3(4):1202–1206CrossRefGoogle Scholar
- Park HW, Choe J, Kang JM (2000) Pressure Behavior of Transport in Fractal Porous Media Using a Fractional Calculus Approach. Energy Sources 22(10):881–890CrossRefGoogle Scholar
- Park HW, Choe J, Kang JM (2001) Generalized Bottom-Hole Pressure with Fractality and Analyses of Three-Dimensional Anisotropic Fractal Reservoirs. Energy Sources 23(7):619–630CrossRefGoogle Scholar
- Poon D (1995) International meeting of the PetSoc. of. CIMSPE9-534Google Scholar
- Raghavan R (2011) Fractional derivatives: application to transient flow. J Petrol Sci Eng 80(1):7–13CrossRefGoogle Scholar
- Razminia K, Razminia A, Trujilo JJ (2015a) Analysis of radial composite systems based on fractal theory and fractional calculus. Signal Process 107:378–388CrossRefGoogle Scholar
- Razminia K, Razminia A, Torres DFM (2015b) Pressure responses of a vertically hydraulic fractured well in a reservoir with fractal structure. Appl Math Comput 257:374–380MathSciNetzbMATHGoogle Scholar
- Razminia K, Razminia A, Baleanu D (2015c) Investigation of the fractional diffusion equation based on generalized integral quadrature technique. Appl Math Model 39(1):86–98MathSciNetCrossRefGoogle Scholar
- Razminia K, Razminia A, Hashemi A (2016) Fractional-calculus-based formulation of the fractured wells in fractal radial composite reservoirs. Environ Earth Sci 75(22):1436CrossRefGoogle Scholar
- Stehfest H (1970) Algorithm 368: numerical inversion of Laplace transforms. Commun ACM 13(1):47–49CrossRefGoogle Scholar
- Wang W, Yuan B, Su Y et al (2017) A composite dual-porosity fractal model for channel-fractured horizontal wells. Eng Appl Comput Fluid Mech 12:1–13Google Scholar
- Xu P, Qiu SX, Yu BM, Jiang ZT (2013) Prediction of relative permeability in unsaturated porous media with a fractal approach. Int J Heat Mass Transf 64:829–837CrossRefGoogle Scholar