The Seepage Law Of Porous Media With High Permeability Based On Volume Averaging Method

In the reservoir production well or natural gas fracture,the fluid seepage velocity is very large,and the permeability in the seepage equation is relativehigh.In addition,in the gas reservoir with low permeability,as a result of the stress sensitivity of the formation permeability and the existence of start-up pressure gradient,the natural productivitiyof well is usually very low,and the hydraulic fracturing renovation measures must be taken to improve the production capacity.In the research of the fractured well production capacity,the nonlinear of the gas seepage must be considered.At this point,the inertia effect of flow cannot be ignored,and the linear Darcy's law can't be applied.In order to describe the nonlinear phenomenon of the single-phase seepage equation in the porous medium with high permeability,Forchheimer in 1901 proposed a one-dimensional classic high-speed flow seepage equationthroughexperiments:Forchheimer equation,which is added a quadratic correction with the velocity in the Darcy's law.Since the Forchheimer equation was proposed,the Forchheimer equation has been widely used in petroleum,chemical,environmental and other fields.However,the theoretical explanation for the Forchheimer equation has always been controversial.The main controversies are:1.The quadratic coefficient of Forchheimer equation cannot be derived from the porous medium structure.2.The perturbation expansion of theNavier-Stokes equations by homogenization shows that the first order nonlinear correction of the seepage equation is the third power of Reynolds number for isotropic porous media structure.3.The numerical results of the array of rules are shown that the Forchheimer equation does not apply to all regions of Re>0;In the contrary,for some special cases(the seepage direction is along the diagonal of the REV),the Forchheimer equation does not apply to any arbitrary Reynolds number interval which has been calculated.Therefore,it is of great significance to study the theory and numerical simulation of single-phase flow in high-permeability porous media.To solve these problems,this paper is written as the following five aspects:1.The momentum equation of macroscopic scale with inertial effect is derived by the volume averaging method from the micro-pore scale navier-stokes equation.In this paper,we take the series expansion form of the disturbance quantity and average velocity,and obtain the macroscopic seepage equation of the series form.The order coefficients in the series form can be calculated by the closed equations.And it can be proved that for the porous media structure of isotropic,the coefficient of the doublet in the seepage equation of the series form is 0.2.In order to solve the problem that the seepage equation of the series is only applicable to the small convergence radius of Reynolds number,the local expansion method of Reynolds number is proposed in this paper.By the theoretical derivation at arbitrary Reynolds number,the seepage law in a large Reynolds number range can be studied by the local expansion method through aiterative recursive way.Compared with the method of solving the Navier-stokes equations directly in nonlinear seepage flow problem,the calculation quantity is greatly reduced to some extent.3.In order to verify the validity of the seriesformseepage equation and the correctness of the local expansion method,the REVs of the ordered square and circular array are numerical simulated,and the results are compared with related references.The third power tensor obtained is compared with the results by solving Navier-Stokes equation directly at extremely low Re,and the validity of series form is verified.At the same time,the accuracy of the local expansion method is also verified by the calculation result of the variation of Reynolds number under the same condition.4.To study the physical mechanism of nonlinear intensity in the seepage equation with inertial effect,in this paper,the influence of different Reynolds number and seepage direction on the non-linearity of the seepage equation is compared by the local expansion method.At the same time,the results show the nonliearity of the filtration equation is quite related to the tortuosity of the transport region in the flow field.The variation between the nonlinearity of the seepage equation and Re shows that the Forchheimer equation is more suitable for the description of disordered structure arrays.5.The numerical simulation of ordered arrays shows that there is a angle difference between the pressure gradient and average velocity orientation.In general way,the pressure gradient direction is fixed.However,the region by the scaling rate between the nonlinearity of the seepage equation with Re cannot cover the case when the seepage direction along the diagonal of REV.While the filtration velocity orientation is fixed,the scaling rate of the pressure difference resistance in a large Reynolds number range is about 3.