Skip to main content
SpringerLink
Log in

A Thermal–Hydraulic–Mechanical–Chemical Coupling Model for Acid Fracture Propagation Based on a Phase-Field Method

  • Original Paper
  • Published:
Rock Mechanics and Rock Engineering Aims and scope Submit manuscript

Abstract

Acid fracturing is a technique to enhance productivity in carbonate formations. In this work, a thermal–hydraulic–mechanical–chemical (THMC) coupling model for acid fracture propagation is proposed based on a phase-field approach. The phase-field variable is utilized as an indicator function to distinguish the fracture and the reservoir, and to track the propagation of the fracture. The resulting system is a nonstationary, nonlinear, variational inequality system in which five different physical modules for the displacement, the phase-field, the pressure, the temperature, and the acid concentration are coupled. This multi-physical system includes numerical challenges in terms of nonlinearities, solution coupling algorithms, and computational cost. To this end, high fidelity physics-based discretizations, parallel solvers, and mesh adaptivity techniques are required. The model solves the phase-field and the displacement variables by a quasi-monolithic scheme and the other variables  by a partitioned schemes, where the resulting overall algorithm is of iterative coupling type. In order to maintain the computational cost low, the adaptive mesh refinement technique in terms of a predictor-corrector method is employed. The error indicators are obtained from both the phase-field and concentration approximations. The proposed model and the computational robustness were investigated by studying fourteen cases as well as some mesh refinement studies. It is observed that the acid and thermal effect increase the fracture volume and fracture width. Moreover, the natural fractures and holes affect the acid fracture propagation direction.

Highlights

  • thermal–hydraulic–mechanical–chemical coupling system was established for acid fracture propagation based on a phase-field method.

  • The acid fluid equations including diffusion, transport and reaction were derived. The penalization method was introduced based on physical reality.

  • Acid fracture problem is a kind of dynamic heterogeneous problem. The adaptive mesh refinement was extended to help researchers get smooth simulation results and save computation costs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Data availability

The authors confirm that the data supporting the findings of this study are available within the article and data will be made available on request.

Abbreviations

\(\Lambda\) :
\(\partial \Lambda\) :

Boundary of computational domain

t :

Computational time interval

T :

Final computational time

f :

Lower dimensional fracture

\(\Omega _F,\Omega _R\) :

Fracture and reservoir sub-domains in \(\Lambda\)

\(\varepsilon\) :

Half thickness of the fracture edge area

\(\partial F\) :

Fracture interface in \(\Lambda\)

\(\phi\) :

Phase-field variable

\(\textbf{u}\) :

Vector-valued displacement

p :

Scalar-valued pressure

\(\Theta\) :

Scalar-valued temperature

c :

Scalar-valued acid concentration

\(\rho _F,\rho _R\) :

Fracture and reservoir densities

\(\rho _{Ff}\) :

Fluid densities in fracture

\(\rho _{Rf},\rho _{Rr}\) :

The rock and fluid densities in reservoir

\(\varphi _F^*, \varphi _R^*\) :

Porosity of fracture and reservoir

\({\texttt {Cap}}_F, \texttt{Cap}_R\) :

Heat capacities of fracture and reservoir

\(K_{Fd}, K_{Rd}\) :

Thermal diffusion coefficient of reservoir and fracture

\(V_F,V_R\) :

Fluid velocities in the fracture and reservoir

\(q_{F\Theta }\), \(q_{R\Theta }\) :

Source terms of temperature equation

\(q_{L }\) :

Leak-off terms of pressure equation

M :

Biot’s modulus

\(\alpha\) :

Biot’s coefficient

\(K_{tc}\) :

Thermal expansion coefficient of pores

\({K_{etch}}\) :

Acid etching coefficient

\(R_c\) :

Acid–rock reaction term

\(t_{etch}\) :

Etching time at any fixed points

\(\eta _F,\eta _R\) :

Fluid viscosities in fracture and reservoir

\(K_F, K_R\) :

Permeabilities in fracture and reservoir

\(K_{Ffc},K_{Rfc}\) :

Fluid compressibilities in fracture and reservoir

\({q_{Ff}},{q_{Rf}}\) :

Fluid source terms

\({K_{Fc}},{K_{Rc}}\) :

Acid diffusion coefficients

\({q_{Fc}},{q_{Rc}}\) :

Acid source terms

\({K_{ar}}\) :

Acid–rock reaction rate speed

\({p_r}\) :

Reference pressure for acid reaction

Ea :

Activation energy of reaction equation

R :

Gas constant.

\({\Theta _T}\) :

Absolute temperature (unit in Kelvin)

\({G_F}\) :

Critical energy release rate

\(\sigma (\mathbf{{u}})\) :

Linear elastic stress tensor

\(\sigma _{eff}\) :

Poroelastic stress tensor

\(e(\mathbf{{u}})\) :

Strain tensor

\(\lambda , G\) :

Lame coefficients

I :

Second-order identity tensor

E :

Young’s modulus

\(\nu\) :

Poisson’s ratio

\(3{\alpha _\Theta }\) :

Thermal expansion coefficient.

k :

Regularization parameter

\({C_\Theta }\) :

Temperature relation constant near the fracture

\(\sigma ^ +\) :

Tensile stress

\(\sigma ^ -\) :

Compression stress

n :

The outward normal vector of the domain \(\Lambda\)

\({f_p}, f_\Theta , f_c\) :

Given boundary values

\(M_h\) :

The mesh family, \({h > 0}\) is the mash size

\(\chi _F, \chi _R\) :

The indicator functions for subdomain

\(D_f, D_r\) :

The distinguishing indexes for fracture and reservoir

\(c_s\) :

Saturation concentration of acid

\(\gamma _0, \gamma _{s}\) :

Penalization terms

\(T{H_\phi }, T{H_{c}}\) :

Thresholds for adaptive mesh refinement

\(Res( \cdot )\) :

Residual of each PDE system

\(\mathrm {TOL_{s}}, {TOL_{fs}}\) :

The solver tolerance

THMC :

Thermal–hydraulic–mechanical–chemical

QOI :

Quantities of interest

FV :

Fracture volume

COD :

Crack opening displacements

AC :

Acid consumption

References

  • Adachi J, Siebrits E, Peirce A et al (2007) Computer simulation of hydraulic fractures. Int J Rock Mech Min Sci 44:739–757

    Article  Google Scholar 

  • Aldakheel F, Noii N, Wick T et al (2021) A global-local approach for hydraulic phase-field fracture in poroelastic media. Comput Math Appl 91:99–121

    MathSciNet  Google Scholar 

  • Aljawad MS, Schwalbert MP, Zhu D et al (2020) Improving acid fracture design in dolomite formations utilizing a fully integrated acid fracture model. J Petrol Sci Eng 184:106481

    Article  CAS  Google Scholar 

  • Ambati M, Gerasimov T, De Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55(2):383–405

    Article  MathSciNet  Google Scholar 

  • Ambrosio L, Tortorelli V (1990) Approximation of functionals depending on jumps by elliptic functionals via \(\gamma\)-convergence. Comm Pure Appl Math 43:999–1036

    Article  MathSciNet  Google Scholar 

  • Ambrosio L, Tortorelli V (1992) On the approximation of free discontinuity problems. Boll Un Mat Ital B 6:105–123

    MathSciNet  Google Scholar 

  • Arndt D, Bangerth W, Davydov D, et al (2017) The deal. ii library, version 8.5. J Numer Math 25(3) :137–145

  • Arrhenius S (1889) Über die dissociationswärme und den einfluss der temperatur auf den dissociationsgrad der elektrolyte. Zeitschrift für physikalische Chemie 4(1):96–116

    Article  Google Scholar 

  • Bai B, He Y, Li X et al (2017) Experimental and analytical study of the overall heat transfer coefficient of water flowing through a single fracture in a granite core. Appl Thermal Eng 116:79–90

    Article  CAS  Google Scholar 

  • Bing H, Dai Yifan FM, Kunpeng Z et al (2022) Numerical simulation of pores connection by acid fracturing based on phase field method. Acta Petrolei Sinica 43(6):849

    Google Scholar 

  • Bourdin B, Francfort GA (2019) Past and present of variational fracture. SIAM News 52 (9)

  • Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826

    Article  ADS  MathSciNet  Google Scholar 

  • Bourdin B, Francfort G, Marigo JJ (2008) The variational approach to fracture. J Elasticity 91(1–3):1–148

    MathSciNet  Google Scholar 

  • Bourdin B, Chukwudozie C, Yoshioka K (2012) A variational approach to the numerical simulation of hydraulic fracturing. SPE Journal, Conference Paper 159154-MS

  • Bueckner HF (1971) Crack problems in the classical theory of elasticity (in sneddon and m. lowengrub)

  • Castonguay S, Mear M, Dean R, et al (2013) Predictions of the growth of multiple interacting hydraulic fractures in three dimensions. SPE-166259-MS pp 1–12

  • Chong ZR, Yang M, Khoo BC et al (2016) Size effect of porous media on methane hydrate formation and dissociation in an excess gas environment. Ind Eng Chem Res 55(29):7981–7991

    Article  CAS  Google Scholar 

  • Chukwudozie C, Bourdin B, Yoshioka K (2019) A variational phase-field model for hydraulic fracturing in porous media. Comput Methods Appl Mech Eng 347:957–982

    Article  ADS  MathSciNet  Google Scholar 

  • Coussy O (2004) Poromechanics. John Wiley & Sons

    Google Scholar 

  • Dai Y, Hou B, Zhou C et al (2021) Interaction law between natural fractures-vugs and acid-etched fracture during steering acid fracturing in carbonate reservoirs. Geofluids 2021:1–16

    Google Scholar 

  • Diehl P, Lipton R, Wick T et al (2022) A comparative review of peridynamics and phase-field models for engineering fracture mechanics. Comput Mech 69:1259–1293

    Article  MathSciNet  Google Scholar 

  • Du J, Guo G, Liu P et al (2022) Experimental study on the autogenic acid fluid system of a high-temperature carbonate reservoir by acid fracturing. ACS Omega 7(14):12066–12075

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Fan C, Luo M, Li S et al (2019) A thermo-hydro-mechanical-chemical coupling model and its application in acid fracturing enhanced coalbed methane recovery simulation. Energies 12(4):626

    Article  CAS  Google Scholar 

  • Fei F, Costa A, Dolbow JE, et al (2023) Phase-field simulation of near-wellbore nucleation and propagation of hydraulic fractures in enhanced geothermal systems (egs) . In: SPE Reservoir Simulation Conference, OnePetro

  • Francfort G, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342

    Article  ADS  MathSciNet  Google Scholar 

  • Ghidelli M, Sebastiani M, Johanns KE et al (2017) Effects of indenter angle on micro-scale fracture toughness measurement by pillar splitting. J Am Ceramic Soc 100(12):5731–5738

    Article  CAS  Google Scholar 

  • Gou B, Zhan L, Guo J et al (2021) Effect of different types of stimulation fluids on fracture propagation behavior in naturally fractured carbonate rock through ct scan. J Petrol Sci Eng 201:108529

    Article  CAS  Google Scholar 

  • Guo J, Liu H, Zhu Y et al (2014) Effects of acid-rock reaction heat on fluid temperature profile in fracture during acid fracturing in carbonate reservoirs. J Petrol Sci Eng 122:31–37

    Article  CAS  Google Scholar 

  • Gupta P, Duarte C (2014) Simulation of non-planar three-dimensional hydraulic fracture propagation. Int J Numer Anal Meth Geomech 38:1397–1430

    Article  CAS  Google Scholar 

  • Gupta P, Duarte CA (2016) Coupled formulation and algorithms for the simulation of non-planar three-dimensional hydraulic fractures using the generalized finite element method. Int J Numer Anal Methods Geomech 40(10):1402–1437

    Article  Google Scholar 

  • Gupta S, Deusner C, Haeckel M et al (2017) Testing a thermo-chemo-hydro-geomechanical model for gas hydrate-bearing sediments using triaxial compression laboratory experiments. Geochem Geophys Geosyst 18(9):3419–3437

    Article  ADS  CAS  Google Scholar 

  • Heider Y (2021) A review on phase-field modeling of hydraulic fracturing. Eng Fracture Mech 253:107881

    Article  Google Scholar 

  • Heider Y, Markert B (2017) A phase-field modeling approach of hydraulic fracture in saturated porous media. Mech Res Commun 80:38–46

    Article  Google Scholar 

  • Heider Y, Reiche S, Siebert P et al (2018) Modeling of hydraulic fracturing using a porous-media phase-field approach with reference to experimental data. Eng Fracture Mech 202:116–134

    Article  Google Scholar 

  • Heister T, Wick T (2020) pfm-cracks: A parallel-adaptive framework for phase-field fracture propagation. Softw Impacts 6:100045

    Article  Google Scholar 

  • Heister T, Wheeler MF, Wick T (2015) A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach. Comput Methods Appl Mech Eng 290:466–495

    Article  ADS  MathSciNet  Google Scholar 

  • Hou B, Zhang R, Chen M et al (2019) Investigation on acid fracturing treatment in limestone formation based on true tri-axial experiment. Fuel 235:473–484

    Article  CAS  Google Scholar 

  • Hou B, Dai Y, Zhou C et al (2021) Mechanism study on steering acid fracture initiation and propagation under different engineering geological conditions. Geomech Geophys Geo-Energy Geo-Resources 7(3):73

    Article  Google Scholar 

  • Jiang C, Hong C, Wang T et al (2023) N-side cell-based smoothed finite element method for incompressible flow with heat transfer problems. Eng Anal Boundary Elements 146:749–766

    Article  MathSciNet  Google Scholar 

  • Jiawei K, Yan J, Kunpeng Z et al (2020) Experimental investigation on the characteristics of acid-etched fractures in acid fracturing by an improved true tri-axial equipment. J Petrol Sci Eng 184:106471

    Article  CAS  Google Scholar 

  • Ju M, Li J, Yao Q et al (2019) Rate effect on crack propagation measurement results with crack propagation gauge, digital image correlation, and visual methods. Eng Fracture Mech 219:106537

    Article  Google Scholar 

  • Khoei A, Moallemi S, Haghighat E (2012) Thermo-hydro-mechanical modeling of impermeable discontinuity in saturated porous media with x-fem technique. Eng Fracture Mech 96:701–723

    Article  Google Scholar 

  • Kuhn C, Müller R (2010) A continuum phase field model for fracture. Eng Fracture Mech 77(18):3625–3634

    Article  Google Scholar 

  • Lee S, Wheeler MF (2021) Modeling interactions of natural and two-phase fluid-filled fracture propagation in porous media. Comput Geosci 25:731–755

    Article  MathSciNet  Google Scholar 

  • Lee S, Mikelić A, Wheeler MF et al (2016) Phase-field modeling of proppant-filled fractures in a poroelastic medium. Comput Methods Appl Mech Eng 312:509–541

    Article  ADS  MathSciNet  Google Scholar 

  • Lee S, Reber JE, Hayman NW et al (2016) Investigation of wing crack formation with a combined phase-field and experimental approach. Geophys Res Lett 43(15):7946–7952

    Article  ADS  Google Scholar 

  • Lee S, Wheeler MF, Wick T (2016) Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model. Comput Methods Appl Mech Eng 305:111–132

    Article  ADS  MathSciNet  Google Scholar 

  • Lee S, Wheeler MF, Wick T (2017) Iterative coupling of flow, geomechanics and adaptive phase-field fracture including level-set crack width approaches. J Comput Appl Math 314:40–60

    Article  MathSciNet  Google Scholar 

  • Lee S, Wheeler MF, Wick T et al (2017) Initialization of phase-field fracture propagation in porous media using probability maps of fracture networks. Mech Res Commun 80:16–23

    Article  Google Scholar 

  • Lee S, Mikelić A, Wheeler M et al (2018) Phase-field modeling of two phase fluid filled fractures in a poroelastic medium. Multiscale Modeling Simul 16(4):1542–1580

    Article  MathSciNet  Google Scholar 

  • Lee S, Min B, Wheeler MF (2018) Optimal design of hydraulic fracturing in porous media using the phase field fracture model coupled with genetic algorithm. Comput Geosci 22:833–849

    Article  MathSciNet  Google Scholar 

  • Lei J, Jia J, Liu B et al (2021) Research and performance evaluation of an autogenic acidic fracturing fluid system for high-temperature carbonate reservoirs. Chem Technol Fuels Oils 57:854–864

    Article  CAS  Google Scholar 

  • Li S, Ma H, Zhang H et al (2018) Study on the acid fracturing technology for high-inclination wells and horizontal wells of the sinian system gas reservoir in the sichuan basin. J Southwest Petroleum Univ (Sci Technol Ed) 40(3):146

    Google Scholar 

  • Liu Z, Chen M, Zhang G (2014) Analysis of the influence of a natural fracture network on hydraulic fracture propagation in carbonate formations. Rock Mech Rock Eng 47(2):575–587

    Article  ADS  Google Scholar 

  • Luo Z, Zhang N, Zhao L et al (2020) An extended finite element method for the prediction of acid-etched fracture propagation behavior in fractured-vuggy carbonate reservoirs. J Petrol Sci Eng 191:107170

    Article  CAS  Google Scholar 

  • Malagon C, Pournik M, Hill AD (2008) The texture of acidized fracture surfaces: implications for acid fracture conductivity. SPE Prod Oper 23(03):343–352

    CAS  Google Scholar 

  • Melendez MG, Pournik M, Zhu D, et al (2007) The effects of acid contact time and the resulting weakening of the rock surfaces on acid-fracture conductivity. In: European Formation Damage Conference, OnePetro

  • Miehe C, Mauthe S (2016) Phase field modeling of fracture in multi-physics problems. part iii. crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media. Comput Methods Appl Mech Eng 304:619–655

    Article  ADS  MathSciNet  Google Scholar 

  • Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field fe implementations. Int J Numer Methods Eng 83(10):1273–1311

    Article  MathSciNet  Google Scholar 

  • Miehe C, Hofacker M, Schaenzel LM et al (2015) Phase field modeling of fracture in multi-physics problems. part ii. coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic-plastic solids. Comput Methods Appl Mech Eng 294:486–522

    Article  ADS  MathSciNet  Google Scholar 

  • Miehe C, Mauthe S, Teichtmeister S (2015) Minimization principles for the coupled problem of darcy-biot-type fluid transport in porous media linked to phase field modeling of fracture. J Mech Phys Solids 82:186–217

    Article  ADS  MathSciNet  Google Scholar 

  • Miehe C, Schaenzel LM, Ulmer H (2015) Phase field modeling of fracture in multi-physics problems. part i. balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids. Comput Methods Appli Mech Eng 294:449–485

    Article  ADS  MathSciNet  Google Scholar 

  • Mikelić A, Wheeler MF (2012) Convergence of iterative coupling for coupled flow and geomechanics. Comput Geosci 17(3):455–462

    Article  MathSciNet  Google Scholar 

  • Mikelić A, Wheeler M, Wick T (2013) A phase-field approach to the fluid filled fracture surrounded by a poroelastic medium, iCES Report 13-15

  • Mikelić A, Wheeler MF, Wick T (2015) A phase-field method for propagating fluid-filled fractures coupled to a surrounding porous medium. SIAM Multiscale Model Simul 13(1):367–398

    Article  MathSciNet  Google Scholar 

  • Mikelić A, Wheeler MF, Wick T (2015) A quasi-static phase-field approach to pressurized fractures. Nonlinearity 28(5):1371

    Article  ADS  MathSciNet  Google Scholar 

  • Min L, Wang Z, Hu X et al (2023) A chemo-thermo-mechanical coupled phase field framework for failure in thermal barrier coatings. Comput Methods Appl Mech Eng 411:116044

    Article  ADS  MathSciNet  Google Scholar 

  • Nguyen CL, Heider Y, Markert B (2023) A non-isothermal phase-field hydraulic fracture modeling in saturated porous media with convection-dominated heat transport. Acta Geotechnica

  • Noii N, Wick T (2019) A phase-field description for pressurized and non-isothermal propagating fractures. Comput Methods Appl Mech Eng 351:860–890

    Article  ADS  MathSciNet  Google Scholar 

  • Ogata S, Yasuhara H, Kinoshita N et al (2018) Modeling of coupled thermal-hydraulic-mechanical-chemical processes for predicting the evolution in permeability and reactive transport behavior within single rock fractures. Int J Rock Mech Min Sci 107:271–281

    Article  Google Scholar 

  • Peaceman DW (1978) Interpretation of well-block pressures in numerical reservoir simulation (includes associated paper 6988). Soc Petrol Eng J 18(03):183–194

    Article  Google Scholar 

  • Peaceman DW (1983) Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability. Soc Petrol Eng J 23(03):531–543

    Article  Google Scholar 

  • Peirce A, Detournay E (2008) An implicit level set method for modeling hydraulically driven fractures. Comput Meth Appl Mech Eng 197:2858–2885

    Article  Google Scholar 

  • Pham TV, Trang HT, Nguyen HMT (2022) Temperature and pressure-dependent rate constants for the reaction of the propargyl radical with molecular oxygen. ACS Omega 7(37):33470–33481

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Poulet T, Karrech A, Regenauer-Lieb K et al (2012) Thermal-hydraulic-mechanical-chemical coupling with damage mechanics using escriptrt and abaqus. Tectonophysics 526:124–132

    Article  ADS  Google Scholar 

  • Pournik M (2008) Laboratory-scale fracture conductivity created by acid etching. Texas A &M University

  • Rabczuk T, Bordas S, Zi G (2010) On three-dimensional modelling of crack growth using partition of unity methods. Comput Struct 88(23–24):1391–1411

    Article  Google Scholar 

  • Santillan D, Juanes R, Cueto-Felgueroso L (2017) Phase field model of fluid-driven fracture in elastic media: Immersed-fracture formulation and validation with analytical solutions. J Geophys Res 122 (4) :2565–2589. 2016JB013572

  • Schrefler BA, Secchi S (2012) A method for 3-d hydraulic fracturing simulation. Int J Fract 178:245–258

    Article  Google Scholar 

  • Schröder J, Wick T, Reese S et al (2021) A selection of benchmark problems in solid mechanics and applied mathematics. Arch Computat Methods Eng 28:713–751

    Article  MathSciNet  Google Scholar 

  • Shiozawa S, Lee S, Wheeler MF (2019) The effect of stress boundary conditions on fluid-driven fracture propagation in porous media using a phase-field modeling approach. Int J Numer Anal Methods Geomech 43(6):1316–1340

    Article  Google Scholar 

  • Shu B, Zhu R, Elsworth D et al (2020) Effect of temperature and confining pressure on the evolution of hydraulic and heat transfer properties of geothermal fracture in granite. Appl Energy 272:115290

    Article  Google Scholar 

  • Sneddon IN, Lowengrub M (1969) Crack problems in the classical theory of elasticity. SIAM series in Applied Mathematics, John Wiley and Sons, Philadelphia

    Google Scholar 

  • Suh HS, Sun W (2021) Asynchronous phase field fracture model for porous media with thermally non-equilibrated constituents. Comput Methods Appl Mech Eng 387:114182

    Article  ADS  MathSciNet  Google Scholar 

  • Takabi B, Tai BL (2019) Finite element modeling of orthogonal machining of brittle materials using an embedded cohesive element mesh. J Manuf Mater Process 3(2):36

    CAS  Google Scholar 

  • Tran D, Settari AT, Nghiem L (2013) Predicting growth and decay of hydraulic-fracture witdh in porous media subjected to isothermal and nonisothermal flow. SPE-J pp 781–794

  • Wang X, Li P, Qi T et al (2023) A framework to model the hydraulic fracturing with thermo-hydro-mechanical coupling based on the variational phase-field approach. Comput Methods Appl Mech Eng 417:116406

    Article  ADS  MathSciNet  Google Scholar 

  • Wang Y, Zhou C, Yi X et al (2020) Research and evaluation of a new autogenic acid system suitable for acid fracturing of a high-temperature reservoir. ACS Omega 5(33):20734–20738

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Weng X (2015) Modeling of complex hydraulic fractures in naturally fractured formation. J Unconventional Oil Gas Resources 9:114–135

    Article  Google Scholar 

  • Wheeler MF, Wick T, Wollner W (2014) An augmented-Lagrangian method for the phase-field approach for pressurized fractures. Comput Methods Appl Mech Eng 271:69–85

    Article  ADS  MathSciNet  Google Scholar 

  • Wheeler MF, Wick T, Lee S (2020) IPACS: Integrated Phase-Field Advanced Crack Propagation Simulator. An adaptive, parallel, physics-based-discretization phase-field framework for fracture propagation in porous media. Comput Methods Appl Mech Eng 367:113124

  • Wick T (2020) Multiphysics Phase-Field Fracture: Modeling, Adaptive Discretizations, and Solvers. De Gruyter, Berlin, Boston

    Book  Google Scholar 

  • Wick T, Singh G, Wheeler M (2016) Fluid-filled fracture propagation using a phase-field approach and coupling to a reservoir simulator. SPE J 21(03):981–999

    Article  CAS  Google Scholar 

  • Wilson ZA, Landis CM (2016) Phase-field modeling of hydraulic fracture. J Mech Phys Solids 96:264–290

    Article  ADS  MathSciNet  Google Scholar 

  • Wu JY, Nguyen VP, Nguyen CT et al (2020) Phase-field modeling of fracture. Adv Appl Mech 53:1–183

    Article  Google Scholar 

  • Wu L, Hou Z, Xie Y et al (2023) Fracture initiation and propagation of supercritical carbon dioxide fracturing in calcite-rich shale: a coupled thermal-hydraulic-mechanical-chemical simulation. Int J Rock Mech Min Sci 167:105389

    Article  Google Scholar 

  • Xu H, Cheng J, Zhao Z et al (2021) Coupled thermo-hydro-mechanical-chemical modeling on acid fracturing in carbonatite geothermal reservoirs containing a heterogeneous fracture. Renew Energy 172:145–157

    Article  CAS  Google Scholar 

  • Xue Y, Liu S, Chai J et al (2023) Effect of water-cooling shock on fracture initiation and morphology of high-temperature granite: application of hydraulic fracturing to enhanced geothermal systems. Appl Energy 337:120858

    Article  Google Scholar 

  • Yoshioka K, Bourdin B (2016) A variational hydraulic fracturing model coupled to a reservoir simulator. Int J Rock Mech Min Sci 88:137–150

    Article  Google Scholar 

  • Yu H, Taleghani AD, Lian Z (2019) On how pumping hesitations may improve complexity of hydraulic fractures, a simulation study. Fuel 249:294–308

    Article  CAS  Google Scholar 

  • Zhou B, Jin Y, Xiong W et al (2021) Investigation on surface strength of acid fracture from scratch test. J Petrol Sci Eng 206:109017

    Article  CAS  Google Scholar 

  • Zhou S, Zhuang X, Rabczuk T (2018) A phase-field modeling approach of fracture propagation in poroelastic media. Eng Geol 240:189–203

    Article  Google Scholar 

  • Zhou S, Zhuang X, Rabczuk T (2019) Phase-field modeling of fluid-driven dynamic cracking in porous media. Comput Methods Appl Mech Eng 350:169–198

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to The National Key Research and Development Program of China under Grant (NO. 2022YFE0129800) and The National Natural Science Foundation of China (No. 52074311 and No.U19B6003-05). The first author is supported by a CSC scholarship. Moreover, the first author gratefully acknowledges the hospitality and computing resources at the Institute of Applied Mathematics during his CSC research stay.

Funding

No funding was received for conducting this study.

Author information

Authors and Affiliations

  1. National Key Laboratory of Petroleum Resources and Engineering, China University of Petroleum(Beijing), Fuxue road 18 Changping, Beijing, China

    Yifan Dai & Bing Hou

  2. Ministry of Education Key Laboratory of Petroleum Engineering, China University of Petroleum(Beijing), Fuxue road 18 Changping, Beijing, China

    Yifan Dai

  3. Department of Petroleum Engineering, China University of Petroleum(Beijing) at Karamay, Karamay, 834000, Xinjiang, China

    Bing Hou

  4. Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL, 32306-4510, USA

    Sanghyun Lee

  5. Institut für Angewandte, Mathematik, AG Wissenschaftliches Rechnen, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Yifan Dai & Thomas Wick

Authors
  1. Yifan Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bing Hou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sanghyun Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Thomas Wick
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bing Hou.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dai, Y., Hou, B., Lee, S. et al. A Thermal–Hydraulic–Mechanical–Chemical Coupling Model for Acid Fracture Propagation Based on a Phase-Field Method. Rock Mech Rock Eng (2024). https://doi.org/10.1007/s00603-024-03769-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00603-024-03769-x

Keywords

Search

Navigation