## 1. Introduction

The solution of global energy problems of mankind, first and foremost associated with the development of nuclear energy. Already by 2030 the share of nuclear power generation in total electricity production should be about 25-30% (today - 16%). Currently, the total amount of radioactive waste in Russia is estimated at 5 10^{8} m^{3}, the total β-activity of which is estimated at 7.3 10^{19} Bq. At the same time on the liquid radioactive waste (LRW) accounts for about 85% of total activity, and their treatment and disposal become the most important task of nuclear energy.

One of the safest ways of disposal of waste of nuclear and chemical production is injection of them into deep-seated subterranean formations. Therefore, an important issue is to study the processes of the joint heat and mass transfer during the injection of waste into a porous collector layer to predict and control the state of the areas covered by the influence of radioactive impurities. The above forecast is carried out mainly by calculations, since the possibility of experimentally sizing of deep zones of contamination is very limited.

The processes of mass transfer in porous media have long been the object of study for many researchers. Have become classics of the G.I. Barenblatt [1], Bear J. [2, 3, 4], Bachmat Y. [5], A. A. Ilyushin [6], V. M. Keyes [7], L. D. Landau [8], R. I. Nigmatullin [9], V. N. Nikolayevsky [10, 11], L. I. Sedov [12]. In the works of Prakash A. [13], A. A. Barmin [14], E. A. Bondarev [15], M. L. Zhemzhurov [16, 17], E. V. Venetsianov, R. N. Rubinstein [18] the problems of the filtration of solutions, taking into account the phenomenon of adsorption, are regarded. Fluid flow through porous materials [19-24] are coincided to be well studied. Study of models of multi-component flows is devoted to the work of R. E. Swing [25].

Problems of disposal of radioactive waste in geological formations and the resulting ecological problems discussed in works of A. S. Belitsky, E. Orlova [26], A. Rybalchenko, M. K. Pimenov [27]. Modeling of temperature and radiation fields examined in works of D. M. Noskov, A. D. Istomin, A. G. Kessler, A. Zhiganov [28-29] (Seversk Technological Institute), I. Kosareva, and E. V. Zakharova [30-31] (Institute of Physical Chemistry RAS), and other researchers. In the works of A. Lehova, Y. Shvarova studied the rate of radionuclides in groundwater, the behavior of radioactive waste in the earth's crust after the injection. At the same time remain relevant problem of determining the concentration dependence of the fields on the parameters of injection of radioactive impurities, injection technology on the parameters of layers, etc.

The study of filtration processes in multilayer formations, as well as any thermodynamic problems of contacts of the bodies and environments, leads to the necessity of solving the problems of conjugation. To solve these problems are widely used numerical methods. The analytical solutions are constructed only for simple cases, such as linear flow in mass-isolated formation [32, 33]. And as the disposal at the request of the IAEA carried out on the timing of the order of tens thousand years, in these circumstances, the porous layer, can hardly be considered mass-isolated.

In this paper, in example of study of the filtration process of radioactive solutions, represented a modification of the asymptotic method, allowing successfully construct approximated solutions to conjugacy problems.

## 2. The mathematical formulation of the problem of heat and mass transfer in fluid flow with radioactive contaminant in the deep layers

Let us consider problem of heat and mass transfer, which describe the interrelated fields of concentration and temperature of the radioactive contaminant in the porous layer, through which flows a liquid with impurities, and the covering and the underlying layers are waterproof.

Typically, in deep horizons an aqueous solution is injected. This solution consist of a different soluble chemical compounds formed during the acid treatment process of structural elements of reactors and other parts of the design (process waste), or in the decontamination of buildings, cars, clothing and so on (non-technological waste) and includes a mixture of various radioactive nuclides [34]. Quite naturally the initial density of the solution divided into two factions

where *k = 0*, we obtain the density of the solvent) *i*-th nuclide, *N*_{ch}, *N*_{rc} - the number of different non-radioactive and radioactive components in the solution, respectively.

Consider an arbitrary reaction volume *dS* reaction volume

where the first term takes into account the mass exchange with the environment through the diffusion and convection currents, the second term describes the rate of change of mass in chemical reactions, the third term takes into account the change in mass due to radioactive decay of radionuclides and the fourth term describes the mass transfer processes between the components of the solution and formation.

Denote by *p* the number of chemical reactions involving a *j*-component, and

where *j*-component.

The third term can be represented as

where *j* -th radionuclide, *j*-component of the radionuclide, or zero if otherwise.

We assume that the transition of the impurity molecules of the liquid in the skeleton and its transition from a skeleton into a liquid are determined by the chemical potentials

where *j*-component of the solution and the skeleton of rock,

Substituting (3) - (5) into (2) and transforming the surface integral into a volume integral, we obtain

By the arbitrariness of the reaction volume

The resulting equation is nonlinear, even in simple cases the values

Let us estimate in (7) the contribution of the second term. Obviously, the maximum change of mass in chemical reactions, while other things being equal, will be observed in the following two cases:

In both reactions the dissolved substances are excluded from consideration, which entails a decrease in the concentrations of the components of the solution. But this type of unpredictable chemical reactions creates the conditions for dangerous situations and in the deep burial of radioactive waste should be excluded. The chemical reaction scheme (acid-base and redox)

are valid and give a slight variation in the concentration of the solution, because typical of enthalpy

As shown in [35], the time of mass transfer between the fluid and the skeleton of the order of 0.1 s. Thus, the mass transfer, which is characterized by a concentration gradient, is almost instantaneous compared to the time of injection of pollutant that may be from several months to several years. Let us also neglect the processes of chemical compounds leaching from the porous rock to the solution, i.e. assume the condition

Based on the above, equation (7) takes the form

Divide the resulting equation into two components: non-radioactive and radioactive fractions

where the index *k* takes values *i* takes value

Because of the neglected mass changes in the course of chemical reactions and mass transfer processes in the equilibrium case, it follows that the concentration of impurities non-radioactive fraction with high accuracy can be taken as constant, i. e.

where

Write out the flow

According to the Onsager linear theory, the flow for a multicomponent mixture can be written as follows

where *j*-th radionuclide.

The real radioactive solutions, arriving at the burial in a deep-seated formations, depending on the half-life have a total volumetric activity of about 10^{-6} ~10^{1} Ci/l. Let us estimate the mass of radionuclides in solution. Strontium-90 from the volumetric activity 1 Ci/l has a mass of about 7.57 10^{-6} kg, and Ruthenium-106 is the same volumetric activity of the mass of the order of 0.3 10^{-6} kg. These estimates of the mass of radionuclides provide a basis for considering solution under investigation to be a very dilute solution (with respect to radionuclide fractions). Therefore, the correlation between the diffusion fluxes of components j and k will be negligible.

Thus, the assumption of a very dilute solution leads to the following representation of (13)

Introducing the notation

Relation (15) is known as Fick's first law, where *i*-th - radionuclide.

In many cases, the diffusion coefficient

Equation (16) is written for the porous layer, but it does not take into account the presence of porosity and sorption of radionuclides in the skeleton of the formation. To account for these effects, we introduce an auxiliary space-time function

Under the integral expression

Using the definition of an auxiliary function

Substituting (19) in equation (18), we obtain

(20) |

Again, because of the arbitrary choice of the reaction volume

We assume that the dependence of the impurity concentration in the skeleton of its concentration in the fluid is linear (Henry's isotherm) and does not depend on the volume activity, that is a good approximation for relatively small concentrations of fraction of radionuclide

Then the mass transfer equations take the form:

where the function *i*-th - radionuclide.

The final form of the equations of evolution of radionuclides in solution (liquid phase) in a porous layer, taking into account the porosity and adsorption on the skeleton, one can divide both sides of equation (23) by a factor

here *i*-th - radionuclide in a porous layer, (the rate of convective transport of radioactive contaminants).

Note that equation (24) is derived for the case when a radionuclide decaying, forms a non-radioactive nuclide. Possible decay scheme

i. e. when the decay product

where

The rate of filtration of snap motion of the liquid phases is determined by Darcy's law

In most common filtration processes, the deformation of the porous skeleton, compressibility, and associated with this changes in the temperature of liquids are small. The main effects that determine the motion of the system are the non-equlibrium joint motion of several liquid phases, molecular and convective diffusion of solute in the phases of the components, the absorption of the solid phase or sorption of the components, mass transfer between phases.

Thus, the system of equations describing the mass transfer during injection of liquid radioactive wastes in deep porous horizon is as follows:

For a complete statement of the problem requires knowledge of the radiochemical composition of the solution, flow rate of the injection, diffusion parameters and the geometry of the simulated porous layer. Note that if the injection rate is known, it is easy to determine the rate of filtration. Then integrating Darcy's equation, we can describe the pressure field in the formation.

The problem under consideration has cylindrical symmetry about the axis of the well, through which the liquid wastes are ejected; it is convenient to represent the system of equations (26) in a cylindrical coordinate system.

Writing the first equation (26) in a cylindrical coordinate system and, given that the liquid is distributed in the porous layer only in the radial direction, we obtain the equation for the velocity field:

solving this equation and applying the obvious boundary condition

Then the remaining equations of (26) using (27), and the anisotropy of diffusion coefficients and thermal conductivity in the directions *r* and *z* in a cylindrical coordinate system can be written as

(28) |

where *r*_{d} and *z*_{d}, respectively, and the multiplier

It is assumed that the real porous layer is represented by a multiphase system, where each phase consists of a sufficiently large number of randomly distributed small particles. Particle size, small in comparison with the basic physical quantities are assumed to be so large that within each particle condition of "local equilibrium" [38] and all the conservation laws are satisfied [9]. All contact surfaces of particles of different nature are surfaces of discontinuity of some physical fields. However, the above assumptions allow us in physically small volumes to define the space of continuous functions, carrying out the description of the fields of each phase. This determination is carried out by a predetermined method of averaging, from which, in general, depend on the results obtained in [9]. As with most occurring filtration processes, the deformation of the porous skeleton, compressibility and associated changes in temperature fluids rely small.

Given that the determining factor in the process of mass transfer is the concentration of the parent nuclide, confine ourselves to the problem for a single pollutant, which is radioactive and chemically active. The first equation (28) is represented as

Here we have introduced the notation

*D* is the effective diffusion coefficient in the layer. From (29) that in the equation describing the migration of contaminants, it is necessary to take into account the convective transport of pollutants, "complicated by" the presence of porosity in the skeleton and mass transfer processes occurring between the pollutant and the skeleton. Equation (29) to determine the rate of convective transport of pollutants in porous layer

The rate of convective transport of the impurity *R*_{d}, just as the filtration rate *R*_{w}. The position of the injected fluid front is determined from the mass balance of the injected fluid and, for the case of injection at a constant speed v_{0} into the layer through a cased hole of radius *r*_{0}, the corresponding expression is given by

## 3. The mathematical formulation of the problem of mass transfer

Fig. 1 shows the geometry of the problem in a cylindrical coordinate system whose axis coincides with the axis of the borehole. The environment is presented by three areas with flat boundaries. Injection of impurities into the area is out of the hole radius, covering and underlying layers are impermeable, middle area is a the porous region, all layers are considered homogeneous and anisotropic on the diffusion properties. Observation is carried out at a distance from the axis of the borehole

Through a hole of small (compared to the distance to the observation point) radius

The mathematical formulation of the problem of mass transfer for all areas involves the diffusion equation with taking into account the radioactive decay in the covering

and the underlying

layers, as well as the equation of convective diffusion, taking into account the radioactive decay in the porous layer

The conditions of conjugation represent the equality of densities and fluxes of dissolved substances at the interface of the layers

The density of pollutant at the entrance of porous layer assumed to be constant

Assuming that at the initial time the density of the of pollutant is equal to zero

In addition, at infinity the conditions of regularity

Let us turn then to the dimensionless quantities

We also introduce the analogue of the Péclet number

where

Let us estimate the ratio of the third and fourth terms in equation (43)

Boundary, initial conditions and conjugation conditions are not changed

The system of equations (44) - (50) defines a mathematical formulation of the problem of mass transfer.

## 4. Expansion of the solution to the problem of mass transfer on the asymptotic parameter

Let us consider the more general problem, which is obtained by introducing into the equations and boundary conditions of arbitrary asymptotic parameter

with boundary conditions

To find the solution to (51) - (57), one can represent the density function

Substituting expression (58) in (51) - (57) and grouping terms in powers of the expansion parameter

(61) |

(62) |

Analysis of the formulation of the problem shows that the factors of powers of ε in (61) contain the neighboring coefficients of the expansion, and in this sense, are linked. To solve the corresponding equations implemented decoupling procedure.

### 4.1. The mathematical formulation of the problem of mass transfer in a zero approximation

If we formally consider ε in equation (61) to be infinitely small, we obtain *r* and *t*. Consequently, in the zero approximation the density of the pollutant in each cylindrical cross section with the axis

Since

is also independent of z. Then (1) can be written as

Integrating successively, one can find the expression for the first derivative of the first coefficient

and the first coefficient of expansion in the form of a quadratic trinomial

with the functional coefficients to be determined. From the boundary conditions (62) with the cofactor

Hence, one can obtain an expression for the functional coefficients and through the traces of derivatives of the outer regions

Substituting (73) in (74), one can obtain the desired equation for the zero approximation of the density of impurities in the layer

The final statement of the problem in the zero approximation also includes the equations in the covering and underlying formations

and the appropriate initial and boundary conditions

Expressions (76) - (82) represent the boundary value problem for zero expansion coefficient

Finding the zero approximation of the density of the radioactive contaminant is important because just that approach arises in the zero approximation of the temperature problem.

### 4.2. The zero approximation of the problem of mass transport as the solution of the averaged problem

Let us average a parameterized problem (51) - (57) over *z* within the carrier layer according to

Successively averaging each term of equation (34)

one can obtain the following formulation of the averaged parameterized problem:

The resulting problem coincides with problem (83) - (89) for the zero approximation of the density of pollutant. The uniqueness of solutions implies that

### 4.3. The mathematical formulation of the problem of mass transfer in the first approximation

Equations (41) - (43) for the coefficients for

appropriate boundary and initial conditions are represented as

The solution to the problem is sought in the form of quadratic polynomial for *z* according to (71), where the auxiliary functions

here

Using (71) and (98), and linearity of the operator

Integrating the last expression over *z*

From the expression (101) and the boundary conditions (62) we have

From (90) and (91) one can get the equation for the definition of *Q*

The equation for the determination of first coefficient of expansion is obtained by substituting (57), (58) and (55) in (98) with using (100)

(105) |

The problem to determine the first coefficient of expansion also includes equations (90), (91) for the density field in the covering and underlying layers, respectively.

It is easy to see that the problem formulated by (90), (91), (105), (93) - (96) with the condition (97) has a trivial solution, so the condition (97) is redundant and should be weakened.

### 4.4. The problem for the remainder — Additional boundary condition

Substituting the expansion (58) with *n* = 1 in the parameterized problem (51) - (57), we obtain a problem for the remainder term

with the boundary conditions and conjugation conditions

Restrict our investigation of the problem, averaged over the thickness of the layer. By averaging the second derivative over the vertical coordinate, use the conjugation conditions (109)

The final formulation of the averaged problem for the remainder term represented as

(115) |

It is easy to show that the averaged problem (110) - (116) for the remainder term has a trivial solution if and only if

that is, when in the averaged problem for remainder term there are no sources. Averaging (107) with regard to (109), one can show that (122) is satisfied identically. Thus, in order to the averaged problem for the remainder term had a trivial solution it is sufficient for the average condition (121) to be satisfied. Therefore, in order to get the exact on the average solution of (44) - (50) on the field of density in the layer, in the formulation of the problem for the first coefficients of the asymptotic expansion (90) (91), (105) (95) - (97) the boundary condition (97) must be replaced by non-local (121).

## 5. Solution to the problem of mass transfer in the zero approximation

### 5.1. Solution to the problem in the zero approximation

In the image space of Laplace-Carson the problem (76) - (82) in the zero approximation is represented as

Taking into account the boundary conditions (126), as well as the fact that in the zero approximation the density of the pollutant in the porous layer is independent of *z* and is a function only of *r* and *t*, the solution of equations (123), (124) can be rewritten as follows:

These expressions allow us to determine the values of the traces of derivatives from the outer regions included in the equation for the layer, through the density of impurities in it

Substituting (131) into equation (125), after simple transformations we obtain an ordinary differential equation for the determination of

from which we finally get

Translation in the original space is carried out by the reference [51]. The expression for the density of radioactive contaminants for the porous layer in the original space is represented as

(134) |

Also transition is feasible in the original space for the coating (129) and underlying (130) layers

(135) |

(136) |

The first factor in the solution (134) - (136) describes the decrease in the density of the pollutant as a result of radioactive decay, the second - the Heaviside function, determines the radius of the spread of contamination zone and the third (the expression in curly brackets) takes into account changes of the density due to the diffusion of pollutants, and radioactive decay of diffusing nuclide. Since the contribution of radioactive decay is described by the factor of *e* due to the decay at the distances defined by the simple relation *R*_{e}=*h*

The resulting solution (134) contains the Heaviside function, which vanishes for *r* ≥

The most important physical results are described by the zero approximation of the asymptotic expansion, the first and the following coefficients determine the "correction term". In addition, due to the smallness of the diffusion coefficient (^{-9}÷10^{-11}), the spread of a contaminant in water-resistant layers in the vertical direction is negligible compared to the convective transport in a porous layer and has little effect on the size of the zone of contamination.

### 5.2. The solution to the problem of mass transfer in the first approximation

In the space transformations of Laplace-Carson, the problem (90), (91), (105), (95) - (97), (121) for the first coefficient of expansion is represented as

(138) |

The initial conditions and conjugation conditions at the boundaries are represented as

The operator

The action of this operator on the zero expansion in the image space is determined by the formula

The solution in the first approximation, according to (71), is sought in the form of quadratic polynomial

in which

and the function

Solutions of equations (139), (140) shall be as follows:

Let us find the traces of the outer regions of the right side of equation (138)

Note also that the action of the operator

according to (147), this leads to the following:

The final equation for determination of

It’s general solution is represented as

(155) |

Constant *C* is determined from the averaging (144)

Hence, one can obtain

As a result, the solution for the first coefficient of expansion in the images is represented as

Determination of the originals is carried out by help of the following correspondence:

Finally, we obtain for the porous layer

(161) |

for the covering layer

(162) |

and for the underlying layer

(163) |

Note that when *r* = 0 the first coefficient of expansion (158)

depends on the variable *z* and does not vanish, that is constructed solution does not satisfy the boundary condition (48). To eliminate the marked lack, it is necessary to build a boundary-layer functions in a neighborhood of *r* = 0.

### 5.3. Construction of boundary-layer solutions

Let us represent the solution to (78) - (82) as

where

in the following way:

(170) |

The problem for the zero coefficients has only the trivial solution

The problem for the first coefficient is divided into three independent parts. The first one is the problem for the boundary layer functions in the layer of the form

The solution to this problem found by the method of separation of variables and is defined by the formula

(179) |

Returning to the original, we obtain an expression for the first coefficient of expansion of the boundary-layer functions in the layer

(180) |

Equation (180) allows determining the boundary values of the first boundary layer coefficients for the surrounding half-spaces, the problem to determine which are the following:

Solutions to (181), (182) is represented through

which in the originals are represented as

(185) |

(186) |

The solution to the nonstationary problem (44) - (50) in the asymptotic approximation, taking into account the boundary-layer functions, represented as the sum of (134), (161) and (180) in the layer and, relevantly, for the surrounding area

## 6. Conclusion

Figure 2 shows the dependence (134), the density of radioactive contaminants on the radial coordinate in the zero approximation for different times of injection with (curves 1, 2, 3) and without (curve 4), radioactive decay, as well as in diffusional approximation (curve 5). In many technological and non-technological liquid wastes of atomic industry, as one of the component contains a radioactive isotope of strontium *h* = 1m, the diffusion coefficients *D*_{1z}= *D*_{2z} =10^{-11}m^{2}/c, *D*_{z}=10^{-9} m^{2}/c; borehall radius of 0.1 m, the polluter - Strontium 90; activity of the solution - 1 Ci / l (high level waste ), the initial density of radioactive substances in the solution 7.34 g/m^{3}; injection volume 100 m^{3}/day (At = 75.8, Pe =1.84 10^{7})

Figure 2 allows carrying out estimate of the contribution of diffusion and radioactive decay. The figure shows that the account of the radioactive decay (see curves 3 and 4) is the priority in comparison with the account of the mass transfer of the layer (curves 4 and 5) with the surrounding formations by diffusion. The figure also implies that the contribution of diffusion is essential in the front zone of radioactive contamination, where it is comparable with the magnitude of the density of matter in solution. With the zero approximation the size of the zone of contamination are determined.

On the fig. 3 a, b it is shown that taking into account the boundary layer solution eliminates the disadvantage of the first approximation, which consists in the fact that the density of radioactive contaminants exceeds one (curve 1). In the calculations agreed:

As the figure 3 b shows, on the border of the front distribution of contaminants occurs a sharp jump of density, indicating the presence of internal moving boundary layer, the patterns of which have not been investigated, and mathematical methods of eliminating it - not developed.