— In the practice of hydrodynamic calculations the linear flow law, commonly called Darcy's law, is now widely used. It is well known that it is violated at large pressure gradients. This means that there is a certain limit value of the pressure gradient ∆p* above which a deviation from the linear character of the flow law begins. This value of the pressure gradient is the upper limit of applicability. A method is p resented for the direct determination of the upper limit of the validity of the linear flow law (Darcy's law) for any porous media. The method is based on the principles of percolation modelling of fluid flows in porous media. The influence of the structur e of the pore space on the value of the boundary gradient is analysed. A qualitative comparison with the experimental data is performed. Keywords — Darcy's law, Upper limit, Linear flow, Pressure gradient, Fluid flow. I. INTRODUCTION The upper limit of applica bility of Darcy's law arises from the manifestation of inertial forces resulting from continuous changes in the direction and cross -section of pore channels [1 ]-[5]. Traditionally, to determine the upper limit of applicability of Darcy's law, a comparison of the flow Reynolds number (Re) with its critical value (Rec) has been proposed. This issue has been addressed in many literatures, beginning in the first half of the last century [6 ]-[10]. However, due to the diversity of the proposed expressions for cal culating the flow numbers Re and the large dispersion of their critical values, the application of this method in practice proves to be impossible. In this work, a different approach to solve this problem is proposed, based on the analysis of the fluid flo w in a porous medium at the micro level. In this case, the calculation of the flow in the model of a porous medium - the capillary grid - allows to take into account both linear hydraulic pressure losses due to the forces of viscous friction and pressure l osses at local resistances associated with the sudden narrowing and expansion of capillaries. II. METHODOLOGY Fluids move in productive formations along conductive channels formed by a system of interconnected pore channels (capillaries) between rock grains or a system of interconnected fractures, or a combination of such systems [11 ]-[14]. The nature of the formation of capillary channels, which are the narrowest elements in the structure of the pore space, varies in reservoirs of different types [15 ]-[18]. Fo r the sake of certainty, we will make further considerations for the case of a granular medium (Fig. 1), when the pore sizes are significantly larger than the transverse dimensions of the capillaries. ISSN No:1001-2400 http://xadzkjdx.cn/ Journal of Xidian University VOLUME 15, ISSUE 6, 2021 https://doi.org/10.37896/jxu15.6/029 277 Fig. 1 Schematic represe ntation of a granular porous medium: 1- capillary; 2 — pore The problem of the flow of a viscous Newtonian fluid of density ρ and viscosity µ in a model of a porous medium - a spatial lattice of capillaries - is considered. Such a model porous structure is characterized by the mean capillary length l, the function of the distribution density of capillaries along the radii f (r) (porosimetric curve) and the coordina tion number of the lattice z [19 ],[20]. Since the cross -sectional radii of capillaries in granular porous media are small compared to the characteristic p ore size, the hydraulic resistance in capillaries is much larger than in pores, and therefore the hydraulic losses in pores can be neglected. Based on the concept of the structure of an infi nite conducting cluster (IC) [19 ], we come to the problem of dete rmining the conductivity of its skeleton, which is responsible for the transport properties of both the infinite cluster itself and, accordingly, the porous medium as a whole. The IC skeleton consists of chains of hydraulically interconnected pore channels (capillaries) with different radii (Fig. 2). Fig. 2 Schematic representation of the formation of conducting r -chains The conductivity of each such chain is determined by the thinnest capillary contained in its structure. It is therefore natural to con sider its radius as the main characteristic of chains and to call chains in which the minimum radius of the capillaries forming them lays in the range r...r + dr "r -chains". 1 2 2 ISSN No:1001-2400 http://xadzkjdx.cn/ Journal of Xidian University VOLUME 15, ISSUE 6, 2021 https://doi.org/10.37896/jxu15.6/029 278 The pressure loss along the length, due to the forces of viscous friction, in eac h individual capillary of radius r is determined by Po iseuille's formula [21 ]-[24], q is the fluid flow in the capillary Pressure losses due to local resistances are associated with sudden contractions and expansions at the junction of cap illaries and pores and depend on their radii r 1 and r 2, the fluid density ρ, their viscosity μ, the flow rate q, and the characteristic value of shear stresses on the capillary walls τ 0: (1) This does not take into account that the area where the capillaries meet is not an ideal transition of the type of sudden expansion or contraction (Fig. 3). Fig. 3 Schematization of the connecting element of capillaries of diffe rent radii in the chain. To deter mine the value of Δp s, we use dimensional analysis. The functional dependence (1) contains six defining parameters, three of which (r 1, p, q) have independent dimensions. By the π -theorem, (1) can be represented in a dimensionless form Where Since the radius of the capillary r 1 is small relative to the radius of the pore r 2, the Reynolds numbers are quite large (Re > 500), while the Hedstrom numbers (He) are small (shear stresses τ 0 are usually neglected, i.e. He << 1). Assuming self -similarity in the parameters Re and He, we obtain ISSN No:1001-2400 http://xadzkjdx.cn/ Journal of Xidian University VOLUME 15, ISSUE 6, 2021 https://doi.org/10.37896/jxu15.6/029 279 Here ζ is the coefficient of local resistance expressed by the formulas Borda -Carnot and Zhukovsky for the expansion and contraction of the channel, respective ly: The total pressure losses are composed of the hydraulic losses and the local resistance losses: (2) Where i is the number of the capillary and j is the number of the next node after it. Next, consider a chain consisting of N capillaries of a given lattice with radii r greater than a given value r 1. After dividing (2) by the length of the chain Nl, we obtain It is assumed that t he length of the conduction chain Nl = L is large enough so that a transition from summation to integration is possible. The pore radii are independently distributed with distribution density f (r), and then the pressure gradient acting in the chain can be written as [20 ]. , Where r is the minimum radius of the capillaries contained in this chain. The values I 1 and I 2 are analogues of the hydraulic and local resistance coefficients for the porous structure. If we express q i n terms of ΔР / L, we obtain the relationship between the fluid flow in the r chain and the external pressure gradient acting on the macroscopic region under consideration: Using the Shklovsky -de Gen et model of the IC structure [19 ] and th e hierarchical model for the summation of th e conductivities of r -chains [19 ],[25], we obtain Newton's fluid flow law in a granular porous medium (3) ISSN No:1001-2400 http://xadzkjdx.cn/ Journal of Xidian University VOLUME 15, ISSUE 6, 2021 https://doi.org/10.37896/jxu15.6/029 280 Here, w is the modulus of the fluid flow rate, defined as the flow rate through a unit area of the flow cross section; ⎪ ∆P ⎪ is the modulus of the pressure gradient, and G (r) is the distribution density of the r - chains. Obviously, the behaviour of the function w (| ∆P |) depends essentially on the ratio of the parameters contained in (3). For the case where (r) >> 4 I 2(r) ⎪ ∆P ⎪, the second term in parentheses in the denominator of the integrand of equation 3 can be neglected. As a result, expression (3) takes the form (4) Consequently, the flow law (3) is a linear dependence (Darcy's law) at low pressure gradients. If << 4 I 2(r) ⎪ P ⎪, we can neglect the first term in the indicated parenthesis and write (3) in the form (5) The relation (5) is a quadratic flow law valid for large gradients. The linear law (4) starts to be violated (Fig. 4) when the inertial losses increase to the level of hydraulic losses. In th is case, the terms in brackets become quantities of the same order: Fig. 4 Dependence of the flow rate on the pressure gradient The limiting pressure gradient, above which the fluid flow is no longer describe d by the linear law, is thus (6) w ∆p* ∆p ISSN No:1001-2400 http://xadzkjdx.cn/ Journal of Xidian University VOLUME 15, ISSUE 6, 2021 https://doi.org/10.37896/jxu15.6/029 281 III. NUMERICAL SOLUTION The solution to the problem of determining the upper bound on the applicability of Darcy's linear law is reduced to the calculation of the coefficients I 1 and I 2 in expression (6). Having established the parameters of the fluid (density and viscosity), let us analyse how the structure of the pore space affects the location of this limit. The most importan t properties of the pore space are the density function of the distribution of the pore channels along the radii (porosimetric curve), as well as the lattice coordination number a nd the mean capillary length [26 ]-[30]. The porosimetric curve, and to some e xtent the lattice coordination number, characterize the degree of homogeneity (or heterogeneity) of the reservoir pore space. Let us first consider the variants of a homogeneous porous rock with capillary radii of the order of 10-6, 10-5, 10-4 m. The distr ibution functions considering the normalization condition In this case they have the form shown in Figure 5. Fig. 5 Model distribution functions for homogeneous porous media: r = 1.25•10-6 ± 0.25•10-6 m; r = 1.25•10-5 ± 0.25•10-5 m; r = 1.25•10-4 ± 0.25•10-4 m. The calculations were made for a fluid with a coefficient of viscosity µ = 5.10-3 Pa.s and a density p = 850 kg / m3. These calculations give the following values of the limiting pressure gradients for the corresponding distrib ution functions: 1. 2. 3. A qualitative picture of the influence of the characteristic radius of pore channels in homogeneous porous media on the boundary pressure gradient is shown in Figure 6. ISSN No:1001-2400 http://xadzkjdx.cn/ Journal of Xidian University VOLUME 15, ISSUE 6, 2021 https://doi.org/10.37896/jxu15.6/029 282 Fig. 6 Qualita tive curve of the dependence of the limiting pressure gradient on the radius of the capillaries for homogeneous porous media Consider next a real pore reservoir whose porosimetric curve is determined by the standard met hod on a core sample (Fig. 7) [31 ]-[34]. Fig. 7 Porosimetric curve of a real rock sample To make this distribution function more convenient for the calculations, we take its representation in the form of a linear spline (Fig. 8). The results of the calculations are shown graphi cally in Fig.9. Fig. 8 Approximation of a porosimetric curve by a linear spline ISSN No:1001-2400 http://xadzkjdx.cn/ Journal of Xidian University VOLUME 15, ISSUE 6, 2021 https://doi.org/10.37896/jxu15.6/029 283 Fig. 9 Curve of the dependence of the limiting pressure gradient on the radius of the capillaries for the porosimetric curve of the core sample Since the limiting pres sure gradient is a function of the capillary radius, each r chain will have its own local limiting gradient. A deviation from the linearity of the macroscopic flow law occurs when the applied external pressure gradient exceeds the smallest of the local boundary gradients. It is this smallest local boundary gradient that is naturally defined as the global value of the boundary gradient for a given porous medium. It is obvious that the smallest local boundary gradient occurs for the thickest r -chains of IC: t he so -called r c-chains. In [19], [20 ] it is shown that, with an accuracy of up to a few percent, the radius r c is given by the ratio The right -hand side of equation (7) is the limit of flow in a porous medium (D is the dimension of the pro blem), and the lower limit of integration is the critical radius r c. Accordingly, the global boundary gradient is determined by the r c value (Fig. 10). Fig. 10 Determination of the global limiting gradient along the critical radius r с ISSN No:1001-2400 http://xadzkjdx.cn/ Journal of Xidian University VOLUME 15, ISSUE 6, 2021 https://doi.org/10.37896/jxu15.6/029 284 IV. CONCLUSIONS An investigation has been made to find a method for the direct determination of the upper limit of the validity of the linear flow law (Darcy's law) for any porous media.  The proposed approach provides a sound methodology for directly dete rmining the upper bound of the validity of the linear flow law (Darcy's law) for any porous media. To apply the technique, it is necessary to know the porosimetric curve of the rock sample, the average capillary length and the coordination number of the gr id simulating the pore space.  Calculations have shown that Darcy's law works best in shallow - and medium -porosity reservoirs when the pore -channel radii do not exceed (2 ... 3) × 10 -5 m. In such media, the limiting gradient is about 108 Pa / m, which is we ll above the typical values of pressure gradients in the field.  It was found that for large -pore rocks (r ≥ 10 –4 m), Darcy's law is violated in almost the entire range of real gradients, since in this case the limiting gradient does not exceed 104 Pa / m.  Paragraphs 2 and 3 of these conclusions are in agreement with the experimental data.