Study Of Viscoelastic Behaviors Of Rock And Creep Crack Propagation In Unconventional Reservoirs

With the prosperous development of unconventional oil and gas resources,the subject of labour is transferred from the classical sand and carbonate rock to shale,mud,coal,clay-rich sand,and clay-rich carbonate rock,which are unconsolidated,cracky,and soft hereditary materials.A lot of researches reveal that both unconsolidated sand and shale will show strong viscosity and time-dependent mechanical behaviors.Wellbore shrinkage caused by the creep of formation rock,will lead to borehole wall sloughing,casing collapse and sticking of the drill.During the fracturing operation,creep of the fracture will dissipate the elastic energy and prevent the further propagation of the fracture length.In the well production process,creep of the formation rock will lead to dynamic effects,such as,the fracture width decreasing of the non-propped fracture,which may decrease the conductivity of the fracture,and the relaxation of the closure stress of the rigid propped fracture,which may lead to delayed proppant flow back.There is a big difference between the viscoelasticity of gel,such as fracturing fluid and water shut off agent,and geomaterials.Creep and relaxation of geomaterials are much smaller than that of gel.Furthermore,the decay rates of the creep and relaxation of geomaterials are extremely slow and will show long-tail effect,when they step into the steady state.Hence,there is an advantage to fit the creep experimental data with power law model.However,the classical constitutive equation with integral order subject to exponential distribution,which cannot fit the creep experimental data accurately.Finding a new mathematical method to describe the viscoelastic behaviors of geomaterials with high fitting efficiency and clear physical meaning will benefit a lot.Fractional calculus is a powerful tool for dealing with the time-dependent,path-dependent and non-local problems.How to define the fractionally initial condition was avoided,when the Caputo fractional differential was put forward.Recently,fractional calculus was widely used in the description of the viscoelastic mechanical behaviors of rock and the simulations of rock engineering,due its good properties,such as clearer physical meaning,lower decay rate and better fitting efficiency.In this paper,the theory of fractional calculus and the principle of viscoelasticity were reviewed.Then,the creep experimental data of rock were studied fitted by fractional constitutive equation.A novel procedure for dealing with viscoelastic engineering problems analytically or semi-analytically based on fractional calculus was put forward.The phenomena of wellbore shrinkage,fracture width decreasing of non-propped fracture,and closure stress relaxation of rigid propped fracture were simulated.Some conclusions are as follow:(1)The creep experimental data of salt rock,shale and mud stone were fitted by classical and fractional constitutive equations.It is found that the fitting efficiency of fractional model is much higher than that of the classical model and the physical meaning of fractional model is also clearer.And the fractional order of fractional model is in direct proportion to the mass content of "soft" minerals,such as clay and kerogen.Fractional order is good metric for evaluating the viscoelasticity of geomaterials,since it increases with the increasing of stress level,temperature,and water content.(2)Two-parameter correspondence principle of viscoelasticity based on the Young's modulus and Poisson's ration,which represent by the axial and lateral creep compliances,was developed.The models of wellbore shrinkage,fracture width decreasing of non-propped fracture,and closure stress relaxation of rigid propped fracture were built and the simulated results were studied.If the fractional order is adjusted,the creep curve will change asymmetrically,which can be regulated by the speeding up of transient creep and lowering the rate of steady creep,which cannot be accomplished by adjusting one parameter in the classical models.(3)The stress and displacement distributions around the unit fracture model under local uniform inner pressure was derived.In conjunction with inner pressure discrete and superposition principle,a convenient and stable superposition model is developed to simulate the induced stress and displacement fields around fractures under arbitrarily distributed inner pressure.On the basis of the past research,a classical stress and displacement solution around a flat elliptical fracture was deduced,fulfilled and corrected.And the error caused by the plane strain assumption was corrected.(4)Constant external stress assumption was taken into consideration,and the corresponding J integral and C*integral in Laplace space were derived.Assuming that the delayed initiation of the creep fracture is controlled by the J integral criterion and the creep fracture propagation is controlled by the C*integral criterion,the model of delayed initiation and propagation of creep hydraulic fracture was developed.And the isochronally semi-analytical inverse Laplace transform method was put forward to solve the problem that Heaviside function cannot be inversed directly by the Stehfest numerical inverse Laplace transform method.(5)Based on the model of delayed initiation and propagation of creep hydraulic fracture,the influences of net pressure,operation time,viscosity of the fracturing fluid and formation rock on the creep fracture propagation were studied.It is found that rise the net pressure during the operation will increase the total fracture length most significantly.At last,a method was suggested to evaluate the upper and lower bonds of the shut in time.The lower bond of the shut in time should be when the value of J integral step into the steady state and the upper bond of the shut in time should be when the 95%of the potential creep width were closed of stationary fracture.