METHOD FOR QUANTITATIVE ANALYSIS OF DISTRIBUTION OF SUSPENSION SOID PARTICLES INVADED IN A POROUS MEDIUM DURING FILTRATION

Abstract

Solid particles are colored with a cationic dye and at least three calibration standards having different known mass concentrations of the colored particles are prepared. A digital analysis of the images of the calibration standards is carried out based on an additive RGB color model and intensity distribution profiles of red, green and blue colors are obtained for each calibration standard. A single norm function is selected to characterize quantitatively changes in the red, green and blue colors in all calibration standards. A single calibration curve is obtained by comparing values of the selected norm function for each calibration standard with a known mass concentration of the colored particles in this calibration standard. A suspension of the colored solid particles is injected through a porous medium sample and the sample is split into two parts along a suspension flow direction. A digital analysis of an image of the split of the porous medium sample is carried out based on the additive RGB color model and a two-dimensional intensity distribution of red, green and blue colors in the sample split is obtained. Reference intensities of red, green and blue colors are determined and based on the obtained two-dimensional intensity distribution of red, green and blue colors in the obtained image of the sample split, using the reference intensities of red, green and blue colors, the selected norm function and the obtained calibration curve a two-dimensional distribution of the mass concentration of the colored suspension solid particles is determined.

Claims

1. A method for quantitative analysis of a distribution of suspension solid particles invaded in a porous medium during filtration, the method comprising: coloring solid particles with a cationic dye and preparing at least three calibration standards having different known mass concentrations of the colored solid particles by mixing the colored solid particles with grains of a granular medium, said grains having a color similar to a color of the studied porous medium; photographing the prepared calibration standards and obtaining color images of all calibration standards; carrying out a digital analysis of the obtained images of the calibration standards based on an additive RGB color model and obtaining intensity distribution profiles of red, green and blue colors in each calibration standard; selecting a single norm function for quantitative characterization of variations of red, green and blue colors in all calibration standards based on an analysis of the obtained intensity distribution profiles of red, green and blue colors in the calibration standards; obtaining a single calibration curve by comparing the selected norm function with a known mass concentration of the colored particles in each calibration standard; preparing a suspension with the colored solid particles and injecting the suspension through a studied porous medium sample; after the injection of the prepared suspension of the colored solid particles dividing the porous medium sample into two parts along a suspension flow direction, photographing a sample split under the same conditions under which the images of the calibration standards were obtained, and obtaining an image of the split of the porous medium sample; carrying out a digital analysis of the split of the porous medium sample based on the additive RGB color model and obtaining a two-dimensional intensity distribution of red, green and blue colors in the sample split; determining reference intensities of red, green and blue colors, determining a two-dimensional distribution of mass concentration of the colored solid particles based on the obtained two-dimensional intensity distribution of red, green and blue colors in the obtained image of the sample split and using the determined reference intensities of red, green and blue colors, the selected norm function and the calibration curve.

2. The method of claim 1, wherein all images of all the standards are located on the same photograph.

3. The method of claim 1, wherein the images of the standards are located in different photographs obtained under the same conditions.

4. The method of claim 1, wherein the porous medium sample is a rock core, and the suspension is a drilling fluid.

5. The method of claim 1, wherein the reference intensities of red, green and blue colors are determined by a sample split image area where the penetrated suspension solid particles are absent.

6. The method of claim 1, wherein for determining the reference intensities of red, green and blue colors the split of the porous medium sample is combined with at least one split of another similar sample that does not contain penetrated suspension solid particles and is additionally photographed, and the reference intensities of red, green and blue colors are determined by the obtained image area that does not contain penetrated suspension particles.

7. The method of claim 1, wherein a two-dimensional distribution of a porous medium volume portion occupied by the suspension solid particles is calculated based on the obtained two-dimensional distribution of the mass concentration of the suspension solid particles in the sample split, according to the formula: $\sigma =\left(1-\varphi \right).Math.\frac{{\rho }_{\mathrm{skel}}}{{\rho }_s}.Math.C_s$ where σ is a porous medium volume portion occupied by the suspension solid particles, C.sub.s is the mass concentration of the suspension solid particles in the porous medium, ρ.sub.s is a density of a material of the suspension solid particles, ρ.sub.skel is a density of a material matrix of the porous medium, φ is a porosity.

8. The method to claim 7, wherein based on two-dimensional distribution of the porous medium volume portion occupied by the solid particles, a profile of the porous medium volume portion occupied by the suspension solid particles is determined by spatial averaging in a direction perpendicular to the flow direction of the suspension during injection.

9. The method of claim 7, wherein a two-dimensional distribution of the porous medium volume portion occupied by a package of the suspension solid particles is further calculated based on the obtained two-dimensional distribution of the porous medium volume portion occupied by the suspension solid particles, according to the formula: ${\sigma }_{\mathrm{fc}}=\frac{\sigma }{1-{\varphi }_{\mathrm{fc}}}$ where σ.sub.fc is the porous medium volume portion occupied by the package of the suspension particles, φ.sub.fc is an inherent porosity of the package of the suspension solid particles.

10. The method of claim 9, wherein based on the obtained two-dimensional distribution of the porous medium volume portion occupied by the package of the suspension solid particles, a profile of the porous medium volume portion occupied by the package of the suspension solids is further calculated by spatial averaging in the direction perpendicular to the flow direction of the suspension during injection.

11. The method of claim 1, wherein based on the obtained two-dimensional intensity distribution of red, green and blue colors in the sample split, intensity profiles of red, green and blue colors are determined along the sample split by spatial averaging in the direction perpendicular to the flow direction of the suspension during injection.

12. The method of claim 1, wherein based on the obtained two-dimensional distribution of the mass concentration of the colored suspension solid particles, profiles of the mass concentration of the suspension solid particles are determined along the sample split by spatial averaging in the direction perpendicular to the flow direction of the suspension during injection.

13. The method of claim 11, wherein empirical parameters describing a capture of particles in the porous medium are determined by achieving the best correspondence between the profile of the porous medium volume portion occupied by the solid particles, the profile being obtained by solving (analytically or numerically) an appropriate mathematical model of a filtration process of the suspension, and the profile determined based on the digital analysis of the image of the sample split.

14. The method of claim 13, wherein during injection of the suspension of the colored particles through the sample, a pressure drop along the sample and a flow rate of the injected suspension are measured, and a type of dependence between the porous medium permeability and the porous medium volume portion occupied by the suspension solid particles is determined based on the obtained empirical parameters describing the capture of particles in the porous medium by achieving the best correspondence between the pressure drop and the flow rate of the injected suspension, which were obtained by solving (analytically or numerically) an appropriate mathematical model of the filtration process of the suspension, and experimentally measured values.

15. The method of claim 14, wherein the obtained empirical parameters describing the capture of particles in the porous medium, and the type of dependency between the porous medium permeability and the porous medium volume portion occupied by the suspension solid particles are used to determine parameters of an internal filter cake and to predict a change in the properties of a near-wellbore area, the change being caused by penetrated components of the drilling fluid.

16. The method of claim 11, wherein during injection of the suspension of the colored particles through the sample, a pressure drop along the sample and a flow rate of the injected suspension are measured, and empirical parameters describing a capture of the particles in the porous medium and a type of dependence between porous medium permeability and the porous medium portion volume occupied by the suspension particles are determined by achieving the best correspondence simultaneously between the pressure drop and the flow rate of the injected suspension, which are obtained by solving (analytically or numerically) an appropriate mathematical model of a filtration process of the suspension, and experimentally measured values, and between the profile of the porous medium volume portion occupied by the solid particles, the profile is being obtained by solving (analytically or numerically) an appropriate mathematical model of the filtration process of the suspension particles, and the profile determined based on the digital analysis of the image of the sample split.

17. The method of claim 16, wherein the obtained parameters describing the capture of the particles in the porous medium, and the type of dependence between the porous medium permeability and the porous medium volume portion occupied by the suspension solid particles are used to determine parameters of an internal filter cake and to predict a change in the properties of a near-wellbore area, the change being caused by penetrated components of the drilling fluid.

18. The method of claim 1, wherein after injection of the suspension of the colored particles into the porous medium sample, the sample is dried to a complete removal of pore water, and a two-dimensional distribution of the porous medium volume portion occupied by the suspension particles is calculated based on the obtained two-dimensional distribution of the mass concentration of the suspension particle in the sample split, according to the formula: $\sigma =\gamma \left(1-\varphi \right).Math.\frac{{\rho }_{\mathrm{skel}}}{{\rho }_s}.Math.C_s$ where σ is the porous medium volume portion occupied by the suspension solid particles, C.sub.s is the mass concentration of the suspension particles in the porous medium, ρ.sub.s is a density of a material of the suspension particles, ρ.sub.skel is a density of a material matrix of the porous medium and, φ is a porosity, γ is a swelling factor.

19. The method of claim 18, wherein a two-dimensional distribution of the porous medium volume portion occupied by a package of the suspension solid particles is further calculated based on the obtained two-dimensional distribution of the porous medium volume portion occupied by the suspension solid particles, according to the formula: ${\sigma }_{\mathrm{fc}}=\frac{\sigma }{1-{\varphi }_{\mathrm{fc}}}$ where σ.sub.fc is the porous medium volume portion occupied by the package of the suspension particles, φ.sub.fc is an inherent porosity of the package of the suspension solid particles.

20. The method of claim 18, wherein based on the obtained two-dimensional distribution of the porous medium volume portion occupied by the solid particles, a profile of the porous medium volume portion occupied by the suspension solids is further calculated by spatial averaging in the direction perpendicular to the flow direction of the suspension during injection.

21. The method of claim 19, wherein based on the obtained two-dimensional distribution of the porous medium volume portion occupied by the package of the suspension solid particles, a profile of the porous medium volume portion occupied by the package of suspension solids is further calculated by spatial averaging in the direction perpendicular to the flow direction of the suspension during injection.

22. The method of claim 20, wherein empirical parameters describing a capture of the particles in the porous medium are further determined by achieving the best correspondence between the profile of the porous medium volume portion occupied by the solid particles, the profile being obtained by solving (analytically or numerically) an appropriate mathematical model of a filtration process of the suspension, and the profile determined based on the digital analysis of the image of the sample split.

23. The method of claim 22, wherein during injection of the suspension of the colored particles through the sample, a pressure drop along the sample and a flow rate of the injected suspension are measured, and a type of dependence between the porous medium permeability and the porous medium portion volume occupied by the suspension particle are determined based on the obtained empirical parameters describing the capture of the particles in the porous medium by achieving the best correspondence between the pressure drop and the flow rate of the injected suspension, which are obtained by solving (analytically or numerically) an appropriate mathematical model of the filtration process of the suspension, and experimentally measured values.

24. The method of claim 23, wherein the obtained parameters describing the capture of the particles in the porous medium, and the type of dependence between the porous medium permeability and the porous medium volume portion occupied by the suspension particles are used to determine parameters of an internal filter cake and to predict a change in the properties of a near-wellbore area, the change being caused by penetrated components of the drilling fluid.

25. The method of claim 20, wherein during injection of the suspension of the colored particles through the sample, a pressure drop along the sample and a flow rate of the injected suspension are measured, and empirical parameters describing a capture of the particles in the porous medium and a type of dependence between the porous medium permeability and the porous medium portion volume occupied by the suspension particles are determined by achieving the best correspondence simultaneously between the pressure drop and the flow rate of the injected suspension, which are obtained by solving (analytically or numerically) an appropriate mathematical model of a filtration process of the suspension, and experimentally measured values, and between the profile of the porous medium volume portion occupied by the solid particles, the profile is being obtained by solving (analytically or numerically) an appropriate mathematical model of the filtration process of the suspension particles, and the profile determined based on the digital analysis of the image of the sample split.

26. The method of claim 25, wherein the obtained parameters describing the capture of the particles in the porous medium, and the type of dependence between the porous medium permeability and the porous medium volume portion occupied by the suspension particles are used to determine parameters of an internal filter cake and to predict a change in the properties of a near-wellbore area, the change being caused by penetrated components of the drilling fluid.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0046] The disclosure is illustrated by drawings, where:

[0047] FIG. 1a shows intensity profiles for red (number 1), green (number 2), and blue (number 3) colors in nine color calibration standards (the first embodiment of the invention),

[0048] FIG. 1b shows the distribution of norm function S that quantitatively characterizes changes in red, green and blue colors in nine calibration standards,

[0049] FIG. 2 shows a calibration curve: a ratio between the value of the selected norm function S and the mass concentration C.sub.s of colored bentonite particles in nine calibration standards,

[0050] FIG. 3a shows the intensity profiles for red (number 1), green (number 2), and blue (number 3) colors for a sandstone sample Buff Berea after injection of a 1% solution of colored bentonite,

[0051] FIG. 3b shows the distribution profile of norm S for the sandstone sample Buff Berea after injection of the 1% bentonite solution,

[0052] FIG. 4 shows the profile of a volume portion of captured bentonite particles (dotted line) and a simulation result (solid line) for the sandstone sample Buff Berea after injection of the 1% colored bentonite,

[0053] FIG. 5 shows the profile of flow resistance (dotted line) and a simulation result (solid line) for the sandstone sample Buff Berea after injection of the 1% colored bentonite,

[0054] FIG. 6 shows the profile of a volume portion of captured bentonite particles in a sample of limestone Indiana Limestone after injection of the 1% colored bentonite (dotted line) and a simulation results obtained based on the model of two continua: the total profile (solid line), a separate profile for each continuum (dotted and dashed lines),

[0055] FIG. 7 shows the profile of flow resistance for the sample of limestone Indiana Limestone after injection of the 1% colored bentonite (dotted line) and simulation results obtained based on the model of two continua: the total profile (solid line), a separate profile for each continuum (dotted and dashed line),

[0056] FIG. 8a shows the intensity profiles for red (number 1), green (number 2), and blue (number 3) colors in ten calibration standards (the third embodiment of the invention),

[0057] FIG. 8b shows the distribution of norm S that quantitatively characterizes a change in red, green and blue colors, and

[0058] FIG. 9 shows a calibration curve which is a ratio between the value of the selected norm S and the mass concentration C.sub.s of colored bentonite particles in ten calibration standards (third embodiment of the invention).

DETAILED DESCRIPTION

[0059] The mathematical description of a non-stationary filtration process of suspension (i.e. liquid containing suspended solid particles, which called below “contaminant”) through a porous medium requires in addition to the equations of a mass balance of filtered fluids, the equation of transport and capture of the suspension particles in the porous medium (see. For example Theuveny, B., Mikhailov, D., Spesivtsev, P., Starostin, A., Osiptsov, A., Sidorova, M. and Shako, V. 2013. Integrated approach to simulation of near-wellbore and wellbore cleanup. SPE 166509).

[0060] The dynamics of capturing suspension particles in a porous medium is usually mathematically described (see, for example, Herzig J. P., Leclerc D. M. and Le Goff P., 1970, Flow of suspensions through porous media—Application to deep filtration Industrial and Engineering Chemistry Vol. 62 . . . No. 5. pp. 8-35) by the equation:

[00004]$\begin{array}{cc}\frac{\partial \sigma }{\partial t}={\lambda }_0.Math.F\left(\sigma \right).Math.\mathrm{wC}& \left(1\right)\end{array}$

where t is time; C is a volume concentration of mobile suspension particles; σ is a porous medium volume portion occupied by captured suspension particles (“volume portion”); w is the filtration rate of a carrier phase; λ.sub.0 is a capture factor; F(σ) is a correction function of dependence between the rate of capturing particles and the porous medium volume portion occupied by the captured suspension particles.

[0061] The filtration rate of carrier phase w is determined in accordance with Darcy's law:

[00005]$\begin{array}{cc}w\left(t\right)=\frac{k\left(\sigma \right)}{\mu }.Math.\frac{\partial }{\partial x}.Math.p& \left(2\right)\end{array}$

where μ is viscosity; p is pressure; k(σ) is the porous medium permeability that depends on the porous medium volume portion occupied by the captured components.

[0062] If only one saturated fluid is present, the equation describing the process of transporting particles in the porous medium is as follows:

[00006]$\begin{array}{cc}\frac{\partial }{\partial t}.Math.\varphi \left(\sigma \right).Math.C+w.Math.\frac{\partial }{\partial x}.Math.C=-\frac{\partial }{\partial t}.Math.\sigma & \left(3\right)\end{array}$

[0063] The capture of particles in the pore spaces reduces the porosity of the porous medium according to the relation

φ(σ)=φ.sub.0−σ  (4)

where φ is the current porosity, φ.sub.0 is the initial porosity of the porous medium.

[0064] If only one saturated fluid is present, at a small change in porosity φ ≈ φ.sub.0 and correction function F(σ)≈1, an analytical solution for the volume concentration of mobile contaminant particles (5) and the porous medium volume portion occupied by captured suspension particles (6) is as follows:

[00007]$\begin{array}{cc}C=\{\begin{array}{cc}{C}_0.Math.\exp \left(-{\lambda }_0.Math.x\right),& W_{\mathrm{inj}}\left(t\right)>x.Math..Math.{\varphi }_0\\ 0,& W_{\mathrm{inj}}\left(t\right)\le x.Math..Math.{\varphi }_0\end{array}& \left(5\right)\\ \sigma =\{\begin{array}{cc}{\lambda }_0.Math..Math.W_{\mathrm{inj}}\left(t\right)-x.Math..Math.{\varphi }_0.Math..Math.C_0.Math.\exp \left(-{\lambda }_0.Math.x\right),& W_{\mathrm{inj}}\left(t\right)>x.Math..Math.{\varphi }_0\\ 0,& W_{\mathrm{inj}}\left(t\right)\le x.Math..Math.{\varphi }_0\end{array}& \left(6\right)\end{array}$

where x is a spatial coordinate (along the axis of the porous medium sample), W.sub.inj=Q (t)/A is injected volume Q(t) referred to cross-sectional area A.

[0065] An analytical solution is also known for more complex dependence of the capture factor of particles on the portion of captured particles, such as F(σ)=1+bσ, at a low change in porosity (see, for example Shechtman Yu.M. Filtratsiya malokontsentrirovannyh suspenzii [Filtering of low-concentration suspensions], Moscow, Nedra, 1961, pp. 70-72; Civan F., “Reservoir formation damage: fundamentals, modeling, assessment and mitigation” Second Edition Gulf Publishing Company 2007. 1089 p, p. 796; Herzig J. P., Leclerc D. M., Le Goff P. 1970. Flow of suspensions through porous media—application to deep filtration Industrial and Engineering Chemistry Vol. 62. No. 5. pp. 8-35):

[00008]$\begin{array}{cc}C=\{\begin{array}{cc}{C}_0.Math.\frac{\exp \left({\lambda }_0.Math.C_0.Math.b.Math..Math.W_{\mathrm{inj}}\left(t\right)-x.Math..Math.{\varphi }_0.Math.\right)}{-1+\exp \left({\lambda }_0.Math.x\right)+\exp \left(-{\lambda }_0.Math.C_0.Math.b\left[W_{\mathrm{inj}}\left(t\right)-x.Math..Math.{\varphi }_0\right]\right)},& W_{\mathrm{inj}}\left(t\right)>x.Math..Math.{\varphi }_0\\ 0,& W_{\mathrm{inj}}\left(t\right)\le x.Math..Math.{\varphi }_0\end{array}& \left(7\right)\\ \sigma =\{\begin{array}{cc}-\frac{1}{b}.Math.\frac{1-\exp \left(-{\lambda }_0.Math.C_0.Math.b.Math..Math.W_{\mathrm{inj}}\left(t\right)-x.Math..Math.{\varphi }_0.Math.\right)}{1-\exp \left({\lambda }_0.Math.x\right)-\exp \left(-{\lambda }_0.Math.C_0.Math.b\left[W_{\mathrm{inj}}\left(t\right)-x.Math..Math.{\varphi }_0\right]\right)}& W_{\mathrm{inj}}\left(t\right)>x.Math..Math.{\varphi }_0\\ 0,& W_{\mathrm{inj}}\left(t\right)\le x.Math..Math.{\varphi }_0\end{array}& \left(8\right)\end{array}$

[0066] Conventional methods for determining a capture factor λ0 of particles in the pore space, and function F(σ) are based on the measurements of the dynamics of the dispersed phase concentration at the sample outlet (see, for example, Bai R., Tien C. Effect of deposition in deep-bed filtration: Determination and search of rate parameters// Journal of Colloid and Interface Science 2000. V. 231. pp. 299-311; Wennberg K. E., Sharma M. M. Determination of the filtration coefficient and the transition time for water injection wells// SPE 38181. 1987. pp. 353-364). In this case, the unknown parameters are determined either by using analytical solutions (5) or (7) for the volume concentration of mobile suspension particles or, if it is impossible to reproduce the measured dynamics of the dispersed phase concentration at the sample outlet by solutions (5) or (7), by using numerical solution of an equation system comprising the equation of capture of the suspension particle in the pore space (1) and the equation of transport of the particles in the porous medium (3).

[0067] A method of determining the capture factor by measuring the dynamics of a pressure drop in two core segments is known (Bedrikovetsky P., Marchesin D., Shecaira F., Souza A L, Milanez P V, Rezende E. Characterisation of deep bed filtration system from laboratory pressure drop measurements// Journal of Petroleum Science and Engineering. 2001. V. 32. No. 2-4. pp. 167-177).

[0068] However, if the rate of capturing particles is high, which is typical for the filtration of drilling fluids through a core sample, the concentration of particles at the outlet is small and cannot be detected with a good accuracy. In such a case, the method based on measuring a pressure drop in two different segments of the sample will be also complicated.

[0069] In this case, the empirical parameters describing the process of capturing particles in the porous medium (capture factor λ.sub.0 and correction function F(σ)) can be determined by reproducing the profile of invaded particles at a known volume of the injected suspension, using either analytical solutions of (6) or (8) for the porous medium volume portion occupied by captured suspension particles or using a numerical solution of an equation system including of the equation of capture of the suspension particles in the pore space (1) and the equation for transport of the particles in porous media (3).

[0070] At the next step, when capture factor λ.sub.0 and correction function F(σ) are determined, a type of dependence k(σ) between the porous medium permeability and the porous medium volume portion occupied by captured suspension particles are determined by reproduction of the dynamics of a pressure drop and the flow rate of the injected fluid, which were recorded in experiment.

[0071] The following types of dependency between the porous medium permeability and the porous medium volume portion occupied by captured particles are most frequently used (see, for example, Herzig J. P., Leclerc D. M., Le Goff P. Flow of suspensions through porous media—application to deep filtration// Industrial and Engineering Chemistry. 1970. V. 62. No. 5. pp. 8-35):

[00009]$\begin{array}{cc}k\left(\sigma \right)=\frac{k}{1+\mathrm{\beta \sigma }}.Math..Math.\mathrm{or}& \left(9\right)\\ k\left(\sigma \right)={\left(1-\frac{\sigma }{{\varphi }_0}\right)}^M& \left(10\right)\end{array}$

where β, M are empirical factors.

[0072] There is known a more complex relation (Sharma M. M., Pang S. Injectivity decline in water injection wells: an offshore Gulf of Mexico Case Study SPE 38180)

[00010]$\begin{array}{cc}k\left(\sigma \right)=\frac{k}{1+\mathrm{\beta \sigma }}.Math.{\left(1-\frac{\sigma }{{\varphi }_0}\right)}^M& \left(11\right)\end{array}$

[0073] If captured suspension particles form inside the porous space a package having its own considerable porosity, then the porous medium permeability depends on its volume portion occupied by the package of captured contaminant particles σ.sub.fc, but not by the volume portion occupied by captured particles σ, i.e. k=k(σ.sub.fc).

[0074] In this case, the porous medium volume portion σ.sub.fc occupied by the package of suspension particles is calculated according to the equation:

[00011]$\begin{array}{cc}{\sigma }_{\mathrm{fc}}=\frac{\sigma }{1-{\varphi }_{\mathrm{fc}}}& \left(12\right)\end{array}$

where σ.sub.fc is an inherent porosity of the package of the suspension particles (inherent porosity of an internal filter cake).

[0075] The above mathematical model describing the transport and capture of the suspension particles in the porous medium can be generalized.

[0076] For example, if porous medium is characterized by a complex structure of the pore space, the model may further comprise at least two pore continua having different properties (see, for example, Gruesbeck C., Collins R. E. Entrainment and deposition of fine particles in porous media// SPE 8430. 1982. pp. 847-856):

[00012]$\begin{array}{cc}\frac{\partial }{\partial t}\left[{\varphi }_i\left({\sigma }_i\right).Math.C_i\right]+w_i.Math.\frac{\partial }{\partial x}.Math.C_i=-\frac{\partial }{\partial t}.Math.{\sigma }_i& \left(13\right)\\ \frac{\partial }{\partial t}.Math.{\sigma }_i={\lambda }_{i.Math..Math.0}.Math.F_i\left({\sigma }_i\right).Math.C_i.Math.w_i& \left(14\right)\end{array}$

where subscript i numbers a group of pore, and φi=φi.sub.0−σi.

[0077] In this case, the rate of filtration is determined for each block according to Darcy's law:

[00013]$\begin{array}{cc}{w}_i=-\frac{k_i\left({\sigma }_i\right)}{{\mu }_0}.Math.\frac{\partial }{\partial x}.Math.p_i& \left(15\right)\end{array}$

[0078] As additional empirical parameters, there are added ratio αi of the initial porosity of each continuum to the total porosity:

[00014]$\begin{array}{cc}{\alpha }_i=\frac{{\varphi }_{i.Math..Math.0}}{{\varphi }_0}& \left(16\right)\end{array}$

where the total porosity is

[00015]${\varphi }_0=\underset{i}{.Math.}.Math..Math.{\varphi }_{i.Math..Math.0},$

and ratio {hacek over (ζ)}i of the initial permeability of each continuum to the total permeability:

[00016]$\begin{array}{cc}{\xi }_i=\frac{k_{i.Math..Math.0}}{k_0}& \left(17\right)\end{array}$

[0079] Empirical parameters (parameters of capture of the particles in each continuum, ratio αi of the porosity of each continuum to the total porosity, ratio {hacek over (ζ)}i of the permeability of each continuum to the total permeability, and type of dependence between the permeability of each continuum and the volume portion of captured particles in this continuum) are determined by reproducing the registered dynamics of a pressure drop and flow rate of the injected fluid, as well as the profile of the volume portion of invaded particles by using a numerical solution of the equation system.

[0080] If the suspension particles have substantially different properties (for example, there are several groups of particles with different geometric sizes), at least two types of particles can be introduced in the model (see, for example, Theuveny B., Mikhailov D., Spesivtsev P., Starostin A., Osiptsov A., Sidorova M., and Shako V. 2013. Integrated approach to simulation of near-wellbore and wellbore cleanup SPE 166509):

[00017]$\begin{array}{cc}\frac{\partial }{\partial t}\left[\varphi \left({\sigma }_{\Sigma }\right).Math.C_j\right]+w.Math.\frac{\partial }{\partial x}.Math.C_j=-\frac{\partial }{\partial t}.Math.{\sigma }_j& \left(18\right)\\ \frac{\partial }{\partial t}.Math.{\sigma }_j={\lambda }_{j.Math..Math.0}.Math.F_j\left({\sigma }_j\right).Math.C_j.Math.w& \left(19\right)\end{array}$

where subscript j numbers the type (group) of particles.

[0081] In this case, the porous medium volume portion occupied by captured particles of all types is defined as:

[00018]$\begin{array}{cc}{\sigma }_{\Sigma }=\underset{j}{.Math.}.Math..Math.{\sigma }_j& \left(20\right)\end{array}$

and a change in porosity is equal to:

φ(σ.sub.Σ)=φ.sub.0−σ.sub.Σ  (21)

[0082] The rate of filtration is determined according to Darcy's law

[00019]$\begin{array}{cc}w=-\frac{k\left({\sigma }_{\Sigma }\right)}{{\mu }_0}.Math.\frac{\partial }{\partial x}.Math.p& \left(22\right)\end{array}$

[0083] Empirical parameters (parameters of capture of each type particles and type of dependence between the porous medium permeability and the porous medium volume portion occupied by captured particles of all types) are determined by reproducing the recorded dynamics of a pressure drop and the flow rate of the injected fluid, and the profile of the volume portion of invaded particles by using numerical solutions of the equation system.

[0084] In general, when the structure of the pore space is complex, and the suspension particles have substantially different properties, the mathematical model of transporting and capturing particles can be generalized by introducing at least two continua of pores having different properties and at least two types of particles having different properties.

[0085] Based on the known porous medium volume portion occupied by captured particles, it is possible to calculate the mass concentration of captured particles.

[0086] By definition, the porous medium volume portion occupied by captured particles is equal to

[00020]$\begin{array}{cc}\sigma =\frac{V_s}{V_{\mathrm{bulk}}}& \left(23\right)\end{array}$

where V.sub.s is the volume of captured particles and V.sub.bulk is the elementary volume of porous medium.

[0087] The weight of captured particles is

M.sub.s=ρ.sub.sV.sub.s   (24)

where ρ.sub.s is a density of a material of which captured particles are composed.

[0088] The weight of the material composing a matrix is

M.sub.skel=ρ.sub.skelV.sub.skel=ρ.sub.skel(1−φ)V.sub.bulk   (25)

where ρ.sub.skel is a density of a material matrix of the porous medium, φ is a porosity, V.sub.skel is the volume occupied by the matrix of the porous medium (V.sub.skel=(1−φ) V.sub.bulk).

[0089] Taking into account (23), (24) and (25), it is possible to obtain a relation between the mass concentration C.sub.s of particles and the porous medium volume portion σ occupied by captured particles:

[00021]$\begin{array}{cc}{C}_s=\frac{M_s}{M_{\mathrm{skel}}}=\frac{1}{\left(1-\varphi \right)}.Math.\frac{{\rho }_s}{{\rho }_{\mathrm{skel}}}.Math.\sigma & \left(26\right)\end{array}$

or, conversely, a relation between the porous medium volume portion σ occupied by captured particles and the mass concentration C.sub.s of the particles:

[00022]$\begin{array}{cc}\sigma =\left(1-\varphi \right).Math.\frac{{\rho }_{\mathrm{skel}}}{{\rho }_s}.Math.C_s& \left(27\right)\end{array}$

[0090] If the suspension particles are swollen (e.g., clay), then the volume of the captured particles in the dried condition V.sub.s.sup.dry and the volume of the captured particles in the wet condition V.sub.s.sup.wet are linked by swelling factor γ:

V.sub.s.sup.wet=γV.sub.s.sup.dry   (28)

[0091] The ratio between the porous medium volume portion σ occupied by captured particles and the mass concentration C.sub.s of particles will be as follows:

[00023]$\begin{array}{cc}\sigma =\gamma \left(1-\varphi \right).Math.\frac{{\rho }_{\mathrm{skel}}}{{\rho }_s}.Math.C_s& \left(29\right)\end{array}$

[0092] Some examples of the implementation of the method are illustrated below.

[0093] The first example is filtration of bentonite particles through a sample of sandstone Buff Berea (initial porosity φ.sub.0=0.22, the initial water permeability k.sub.0=220 mD). Bentonite is one of the most commonly used components of drilling fluids, and its invasion in a reservoir results in a significant reduction in permeability. A direct detection of invasion bentonite particles in a sandstone sample is difficult for the following reasons: firstly, the color of bentonite is similar to the color of sandstone, and secondly, most of the natural sandstones comprise their own clays. The use of high-contrast cationic dye makes allows reliable detection of invaded bentonite particles and the determination of their location. Brilliant Green (triarylmethane dye, C29H37N2O4) was selected as a cationic dye.

[0094] Bentonite clay particles were colored as follows. The particles of bentonite clay were dispersed in an aqueous sodium chloride (18 g of NaCl and 10 g of bentonite were added to 1 liter of water, and the mixture was stirred at 24,000 rpm for 20 minutes), a dye in a concentration of 0.2 mg-eq./100 g (alcohol Brilliant Green) was added, and the clay mixture was stirred for 5 minutes at low speed of a mixer. The resulting solution was dried to obtain dry colored clay particles.

[0095] 9 “calibration standards” were prepared at different mass concentrations of colored particles (0%, 0.01%, 0.02%, 0.05%, 0.1%, 0.25%, 0.5%, 1%, 2%) by stirring bentonite particles with river sand.

[0096] The prepared “calibration standards” were photographed.

[0097] The obtained images of the calibration standards were subjected to digital analysis, using the software package MathLab®, imread function, based on the additive RGB (red, green, blue) color model, the intensity distribution of red, green and blue colors in the images of the calibration standards was determined and intensity profiles for red, green and blue colors were obtained for each calibration standard (FIG. 1a, where the intensity profile for red color is indicated as 1, the intensity profile of green color is indicated as 2, and the intensity profile of blue color is indicated as 3).

[0098] Based on the analysis of the distribution of red, green and blue colors in the images of the calibration standards, a norm function (FIG. 1b) was obtained to quantitatively characterize changes in red, green and blue colors:

S=|R R.sub.0|+|G G.sub.0|+|B B.sub.0|,

R+G+B≡1,

where R.sub.0, G.sub.0, and B.sub.0 are “reference” values of red, green and blue colors, which correspond to the calibration standards which are free of bentonite particles (mass concentration of colored bentonite particles is 0%).

[0099] The calibration curve (FIG. 2) was also obtained, i.e. the ratio between the value of the selected norm function S and mass concentration C.sub.s of bentonite colored particles in the calibration standards is:

C.sub.s=AS.sup.B,

where A≈30.81, B≈1.58.

[0100] Suspension of particles of the colored bentonite clay was prepared at a required concentration.

[0101] A filtration experiment for injection of the prepared colored bentonite suspension through a sample of sandstone Buff Berea ((φ.sub.0=0.22, k.sub.0=220 mD) was performed.

[0102] The sandstone sample was split (after injection of suspension) along the direction of the filtration.

[0103] The sandstone sample split was photographed under the same conditions (lighting, camera, shutter speed, aperture, etc.), under which the photographs of calibration standards were made.

[0104] Assuming that the parameters of the porous medium are the same in any plane perpendicular to the flow direction of the suspension during injection, i.e. any split of the sample is representative, the distribution of the mass concentration of colored suspension particles (or any other parameter) in the sample split reproduces the distribution of the mass concentration of the colored suspension particles (or any other parameter) in the porous medium.

[0105] The image of sandstone sample split was subjected to digital image analysis, using standards and the application software MathLab®, imread function, and the intensity distribution of red, green and blue colors in the image was obtained, and the intensity profiles of the red, green and blue colors along the photograph was determined by spatial averaging the obtained intensity distribution of the corresponding colors (FIG. 3a, where the intensity profile of red color is indicated as 1, the intensity profile of green color is indicated as 2, and intensity profile of blue color is indicated as 3).

[0106] An area was selected in the photograph of the sandstone sample split, which did not contain invaded suspension particles, and reference intensities of red, green and blue colors (i.e., values corresponding to the average intensity of red, green and blue colors in the sample without the invaded colored particles) were obtained.

[0107] A profile of the mass concentration of the penetrated particles was determined in the flow direction of the suspension during injection, based on the intensity profiles for red, green and blue colors along the photograph (FIG. 3a) by using the selected norm function (the distribution profile of the norm function along the photograph of the split is shown in FIG. 3b), and a calibration curve is plotted.

[0108] The profile of the volume concentration σ of the invaded particles was calculated based on the obtained profile of the mass concentration according to equation (24) comprising known density of the particle material p.sub.s, density of the matrix material of the porous medium ρ.sub.skel, porosity φ, and swelling factor γ (FIG. 4, dotted line):

[00024]$\sigma =\gamma \left(1-\varphi \right).Math.\frac{{\rho }_{\mathrm{skel}}}{{\rho }_s}.Math.C_s$

[0109] The obtained profile of the volume concentration of the invaded particles along the flow direction was used to determine parameters of capture of the particles (FIG. 4, solid line.): λ.sub.0=186 m.sup.−1, swelling factor: γ=5 for the sample of sandstone Buff Berea, after injection of the 1% colored bentonite solution.

[0110] Parameters of a reduction in permeability (11) were determined by reproducing the experimental profile of the fluid resistance (FIG. 5) at β=3574, M=2.6, for the sample of sandstone Buff Berea sample after injection of the 1% colored bentonite solution.

[0111] The second example is filtration of bentonite particles through a sample of limestone Indiana Limestone (φ.sub.0=0.16, k.sub.0=53 mD) that is characterized by a more complex structure of the pore space.

[0112] The sequence of operations, starting with the preparation of calibration standards and aqueous bentonite solution to obtaining the profile of the volume concentration of invaded particles along the flow direction is the same as in the first example.

[0113] A difference is in the procedure of determining the parameters of capture of the particles, wherein the determination is performed based on the obtained profile of the volume concentration of the penetrated particles (FIG. 6 dotted line). These parameters were determined by using a two-continuum model (13), (14), where i=1, 2, since the profile could not be reproduced in a model with a single continuum. The model parameters were as follows: φ.sub.1/φ.sub.0=0.55; λ.sub.1=79 min.sup.−1, λ.sub.2=907 m.sup.−1, swelling factor γ=2.

[0114] The simulation results are shown in FIG. 6, solid line. The simulation results of the parameters of a reduction in permeability are shown in FIG. 7, solid line (where the flow resistance was determined as the ratio of a pressure drop at a given time Δp(t) to the volume flow rate q(t) of the fluid at a given time). Parameters of the model were as follows: k.sub.01/k.sub.0=0.84; β.sub.1=2687, β.sub.2=1073; M.sub.1=3.4, M.sub.2=1.3.

[0115] The following type of dependence between the permeability of each continuum and the volume concentration of the penetrated particles in this continuum was obtained:

[00025]$k_i\left({\sigma }_i\right)=\frac{k_{i.Math..Math.0}}{1+{\beta }_i.Math.{\sigma }_i}.Math.{\left(1-\frac{{\sigma }_i}{{\varphi }_{i.Math..Math.0}}\right)}^{M_i},$

where β.sub.i, M.sub.i are empirical coefficients.

[0116] The third example of the invention is similar to the first one, but instead of nine calibration standards with the mass concentration of colored particles of 0%, 0.01%, 0.02%, 0.05%, 0.1%, 0.25%, 0.5%, 1%, 2%, ten calibration standards at mass concentrations of colored particles of 0%, 0.2%, 0.5%, 1%, 2%, 5%, 7.5%, 10%, 12.5%, 15% were prepared by mixing bentonite colored particles with river sand.

[0117] The prepared “calibration standards” were photographed.

[0118] The resulting image of calibration standards were subjected to digital analysis, using the application software MathLab®, imread function, based on the additive RGB (red, green, blue) color model, the intensity distribution of red, green and blue colors in the photography of the calibration standards was determined to obtain the intensity profiles of red, green and blue colors for each calibration standard (FIG. 8a, where the intensity profile of red color is indicated as 1, the intensity profile of green color is indicated as 2, and the intensity profile of blue color is indicated as 3).

[0119] A norm was obtained based on the analysis of the distribution of red, green and blue colors in the image of the calibration standards (FIG. 8b.) to quantitatively characterize changes in red, green and blue colors:

S=|R R.sub.0|+|G G.sub.0|+|B B.sub.0|,

R+G+B≡1,

[0120] where R.sub.0, G.sub.0, and B.sub.0 are “background” values of red, green and blue colors, which correspond to the “clean” site of the core (i.e., without penetrated bentonite).

[0121] The calibration curve (FIG. 9) was plotted, i.e. the ratio between the value of the selected norm function S and the mass concentration C.sub.s of the colored bentonite particles in the calibration standards

C.sub.s=AS.sup.B,

where A≈30.81, B≈1.58.

[0122] The remaining steps are similar to those in the first example.

[0123] Two examples of the procedure of coloring bentonite with different cationic dyes are given below. In both examples, the bentonite clay first was dispersed in 2% solution of sodium chloride in a blender, and then a required amount of dye in the form of an alcohol solution was added thereto (an aqueous solution, a solution in another solvent, or a dry dye may also be used) under vigorous stirring to obtain uniform coloration of bentonite particles.

[0124] In both examples, 5% bentonite suspension was prepared by using 50 g of bentonite and 950 g of the sodium chloride solution.

[0125] In the first example, the used dye was Brilliant green dye (C27H34N2O4S) in an amount of 0.4 mg-eq./100 g of bentonite (10 g of a 1% dye solution in alcohol per 1 kg of 5% bentonite suspension). When this dye was used, the bentonite clay was colored in green.

[0126] In the second example, the used due was Rhodamine 6G (C28H31N2O3C1), also in an amount of 0.4 mg-eq./100 g of bentonite (10 g of a 1% dye solution in alcohol per 1 kg of 5% bentonite suspension). When this dye was used, the bentonite clay was colored in pink.