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A well simulator for homogeneous reservoirs based on formulations of the isogeometric boundary element method

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Abstract

The development of a simulator for homogeneous reservoirs with application in producer wells (represented by a sink) and the aquifer analysis is obtained by combining the Boundary Element Method (BEM), the Isogeometric Formulation using NURBS (Non-Uniform Rational B-Spline) as shape functions, and also the Axisymmetric Formulation. The Isogeometric Formulation makes the discretization of geometric model (mesh generation), which is the step of numerical analysis that is more time-consuming for the engineer, be no longer necessary, since the same functions that describe the geometry also approximate the field variables in the BEM. In other words, the same discretization used in the geometric model, generated in CAD (Computer Aided Design) modeling programs, also is used by the BEM. The oil and water reservoirs, as simplified models for validation of the new mathematical methodology, can be fully represented by the analysis of a plane passing through the axis of rotational (axial) symmetry. The dimension of the problem is reduced from three to two dimensions: radial and axial directions only, and all variables in the circumferential direction are assumed to be constant. When the geometry and the problem variables are both axisymmetric, then the problem is considered fully axisymmetric. The isogeometric and axisymmetric formulations are coupled to obtain the well simulator for the single and double phase case, i.e., one or two incompressible fluids inside the reservoir. The determination of boundary conditions for the model, including the analysis of fluids interface movement, is also presented. The final code is a new tool for the analysis of gas/water coning phenomenon and quick drawdown problem in homogeneous reservoirs, as validation models. Validation of the results is carried out by comparing with others numerical methods and analytical results.

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Abbreviations

\(\mathbf{A }\) :

Unknown variables vector

A :

Cross-sectional area

\(\mathbf{b }\) :

Unknown variables vector

B :

Control points

b :

Thickness

c :

Multiplying constant

C :

Curve function

\(\mathbf{e }\) :

Unit normal vector

\(\mathbf{E }\) :

Transformation matrix

F :

Interface function

g :

Gravity acceleration

\(\mathbf{G }\) :

Matrix

\(\mathbf{H }\) :

Matrix

H :

Height

K :

Hydraulic conductivity

L :

Length

m :

Parameter

\(\mathbf{n }\) :

Normal vector

n :

Porosity

N :

Basis function

p :

Pressure; Load point

\(\mathbf{q }\) :

Apparent velocity vector

q :

Volumetric flow rate per unit cross-sectional area

Q :

Volumetric flow rate; Field point

r :

Radial coordinate; Radius

R :

Rational basis function

\(\mathbf{s }\) :

Sink vector

t :

Time; Curve parameter

\(\mathbf{u }\) :

Real average velocity vector;

u :

Knot vector values

\(\mathbf{U }\) :

Knots vector

\(\mathbf{x }\) :

Variable vector; Position vector

\(\mathbf{V }\) :

Translation vector

x :

Ordinate axis

y :

Abscissa axis

z :

Vertical and axial coordinate

\(\alpha \) :

Specific mass ratio

\(\beta \) :

Weighting factor

\(\Gamma \) :

Boundary

\(\delta \) :

Dirac delta

\(\Delta \) :

Finite difference

\(\nabla \) :

Mathematical operator

\(\eta \) :

Free surface slope angle

\(\theta \) :

Tangential direction

\(\kappa \) :

Absolute permeability

\(\lambda \) :

Interface height function

\(\mu \) :

Fluid dynamic viscosity

\(\rho \) :

Fluid specific mass

\(\Phi \) :

Velocity potential (piezometric head)

\(\omega \) :

Weight function

axi :

Relative to the axial

d :

Relative to the source point

e :

Relative to the elliptic; relative to the external

g :

Relative to the gas

i :

Relative to the specific control point

j :

Relative to the quantity of sink

k :

Relative to the curve order

m :

Relative to the time step

o :

Relative to the oil

p :

Relative to the pump

s :

Relative to the sink

t :

Relative to the top; relative to the total

w :

Relative to the well; relative to the water

x :

Relative to the arbitrary control point

z :

Relative to the height of interface

2D :

Relative to the two-dimensional

3D :

Relative to the three-dimensional

c :

Relative to the control point

References

  1. Dake LP (1983) Fundamentals of Reservoir Engineering, vol 8, 1st edn. Elsevier, Amsterdam

    Google Scholar 

  2. Fortaleza ELF, Filho JOAL, Gontijo GSV, Albuquerque EL, Solimões RDP, Soares MM, Miranda MER, Abade GC (2019) Analytical, numerical and experimental study of gas coning on horizontal wells. J Braz Soc Mech Sci Eng 12:1783–1810

    Google Scholar 

  3. Ramsak M, Skerget L (2014) A highly efficient multidomain BEM for multimillion subdomains. Eng Anal Bound Elem 43:76–85

    Article  MathSciNet  Google Scholar 

  4. Bear J (1972) Dynamics of fluids in porous media. Elsevier, Amsterdam

    MATH  Google Scholar 

  5. Ligget JA, Liu PLF (1983) The boundary integral equation method for porous media flow. George Allen and Unwin, Crows Nest

    Google Scholar 

  6. Lin F (1996) NURBS in CAD and Computer Graphics, 1st edn. Springer, New York

    Google Scholar 

  7. Piegl L, Tiller W (1996) The NURBS book. Springer, Berlin

    MATH  Google Scholar 

  8. Rogers DF (2000) An introduction to NURBS: with historical perspective. Elsevier, Amsterdam

    Google Scholar 

  9. Farin GE (1999) NURBS for curve & surface design: from projective geometry to practical use. A K Peters/CRC Press, Boca Raton

    Book  Google Scholar 

  10. Piegl L (1991) On nurbs: a survey. IEEE Comput Gr Appl 11(1):55–71

    Article  Google Scholar 

  11. Lamousin H, Waggenspack N (1994) Nurbs-based free-form deformations. IEEE Comput Gr Appl 14(6):59–65

    Article  Google Scholar 

  12. Hughes T, Reali A, Sangalli G (2010) Efficient quadrature for nurbs-based isogeometric analysis. Comput Methods Appl Mech Eng 199(5–8):301–313

    Article  MathSciNet  Google Scholar 

  13. Temizer I, Wriggers P, Hughes T (2011) Contact treatment in isogeometric analysis with nurbs. Comput Methods Appl Mech Eng 200(9–12):1100–1112

    Article  MathSciNet  Google Scholar 

  14. Nielsen PN, Gersborg AR, Gravesen J, Pedersen NL (2011) Discretizations in isogeometric analysis of Navier–Stokes flow. Comput Methods Appl Mech Eng 200(45):3242–3253

    Article  MathSciNet  Google Scholar 

  15. Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid-structure interaction: theory, algorithms, and computations. Comput Mech 43(1):3–37

    Article  MathSciNet  Google Scholar 

  16. Bazilevs Y, Akkerman I (2010) Large eddy simulation of turbulent Taylor–Couette flow using isogeometric analysis and the residual-based variational multiscale method. J Comput Phys 229(9):3402–3414

    Article  MathSciNet  Google Scholar 

  17. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: Cad, finite elements, nurbs, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195

    Article  MathSciNet  Google Scholar 

  18. Cabral JJSP, Wrobel LC, Brebbia CA (1990) A bem formulation using b-splines: I-uniform blending functions. Eng Anal Bound Elem 7(3):136–144

    Article  Google Scholar 

  19. Cabral JJSP, Wrobel LC, Brebbia CA (1991) A BEM formulation using b-splines: Ii-multiple knots and non-uniform blending functions. Eng Anal Bound Elem 8:51–55

    Article  Google Scholar 

  20. Simpson RN, Bordas SPA, Trevelyan J, Rabczuk T (2012) A two-dimensional isogeometric boundary element method for elastostatic analysis. Comput Methods Appl Mech Eng 209:87–100

    Article  MathSciNet  Google Scholar 

  21. Simpson RN, Bordas SPA, Lian H, Trevelyan J (2013) An isogeometric boundary element method for elastostatic analysis: 2d implementation aspects. Comput Struct 118:2–12

    Article  Google Scholar 

  22. Wang D, Zhang H (2014) A consistently coupled isogeometric-meshfree method. Comput Methods Appl Mech Eng 268:843–870

    Article  MathSciNet  Google Scholar 

  23. Shaw A, Roy D (2008) Nurbs-based parametric mesh-free methods. Comput Methods Appl Mech Eng 197(17):1541–1567

    Article  Google Scholar 

  24. Kim HJ, Youn SK (2012) Spline-based meshfree method. Int J Numer Methods Eng 92(9):802–834

    Article  MathSciNet  Google Scholar 

  25. Campos LS, Albuquerque EL, Wrobel LC (2017) An aca accelerated isogeometric boundary element analysis of potential problems with non-uniform boundary conditions. Eng Anal Bound Elem 80:108–115

    Article  MathSciNet  Google Scholar 

  26. Lucas SK, Blake JR, Kucera A (1991) A boundary-integral method applied to water coning in oil reservoirs. J Aust Math Soc 32:261–283

    Article  MathSciNet  Google Scholar 

  27. Bakr AA (1986) The boundary integral equation method in axisymmetric stress analysis problems. Springer, Heidelberg

    Book  Google Scholar 

  28. Becker AA (1992) The boundary element method in engineering: a complete course. McGraw-Hill, New York

    Google Scholar 

  29. Katsikadelis JT (1984) The boundary element method for engineers and scientists—theory and applications, 2nd edn. Springer, New York

    Google Scholar 

  30. Zhang H, Barray DA, Hocking GC (1999) Analysis of continuous and pulsed pumping of a phreatic aquifer. Adv Water Resour 22(6):623–632

    Article  Google Scholar 

  31. Rafiezadeh K, Ataie-Ashtiani B (2014) Transient free-surface seepage in three-dimensional general anisotropic media by BEM. Eng Anal Bound Elem 46:51–66

    Article  MathSciNet  Google Scholar 

  32. Hall H (1955) An investigation of steady flow toward a gravity well. La Houille Blanche 10:8–35

    Article  Google Scholar 

  33. Taylor G, Luthin J (1969) Computer methods for transient analysis of water-table aquifers. Water Resour Res 5:144–152

    Article  Google Scholar 

  34. Cooley R (1983) Some new procedures for numerical solution of variably saturated flow problems. Water Resour Res 19:1271–1285

    Article  Google Scholar 

  35. Ruiz JAM, Chen B, Trevelyan J, Gourgiotis PA, Albuquerque EL (2019) Enriched BEM for water coning. In: Proceedings of the UKBIM12 conference, Oxford, UK, pp 1–8

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Authors and Affiliations

  1. Department of Mechanical Engineering, Federal University of Espirito Santo, 29075-910, Vitoria, ES, Brazil

    Lucas S. Campos

  2. Durham University, DH1, Durham, UK

    Jon Trevelyan

  3. Department of Mechanical Engineering, University of Brasilia, 70910-900, Brasilia, DF, Brazil

    Lúcio G. Nascimento, Gustavo S. V. Gontijo, Éder L. Albuquerque, Lucas S. Campos, Jon Trevelyan & Eugênio L. F. Fortaleza

Authors
  1. Lúcio G. Nascimento
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  2. Gustavo S. V. Gontijo
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  3. Éder L. Albuquerque
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  4. Lucas S. Campos
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  5. Jon Trevelyan
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  6. Eugênio L. F. Fortaleza
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Correspondence to Lúcio G. Nascimento.

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Technical Editor: Celso Kazuyuki Morooka.

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The Authors would like to acknowledge the Brazilian institutions: CNPq, CAPES, FINEP and MCT for supporting the present study.

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Nascimento, L.G., Gontijo, G.S.V., Albuquerque, É.L. et al. A well simulator for homogeneous reservoirs based on formulations of the isogeometric boundary element method. J Braz. Soc. Mech. Sci. Eng. 43, 206 (2021). https://doi.org/10.1007/s40430-021-02924-7

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  • DOI: https://doi.org/10.1007/s40430-021-02924-7

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