Green functions of mass diffusion waves in porous media

A formulism of frequency-domain mass diffusion-waves in porous media is derived by means of Fourier transform. In analogy to conventional thermal-wave fields, internally consistent Green functions in Cartesian coordinates for linear mass diffusion-waves is also presented for infinite, semi-infinite and finite-size domains in three-dimensional spaces. The Green functions are utilized to analyze the response of a particular type of mass diffusion physical system to any arbitrary tracer source distributions. This method allows the introduction of frequency-dependent physically intrinsic properties of porous media. The Green functions presented in this letter may significantly advance understanding in linear mass diffusion-wave physics in porous media, and can be applied to retrieve spatial-temporal diffusive-wave fields from ambient mass fluctuations in geological reservoirs.


Introduction
Green functions are used to describe the response of a particular type of physical system to a point source in spatial-temporal domains [1]. Therefore, the physical field response to any arbitrary source distribution can be found by a convolution integral of the distribution with the Green function over the source volume [2]. Green functions in diffusive-wave physics have been proposed by Mandelis [3]. Green function techniques have been widely applied in thermal-wave physics [4][5][6][7]. Mandelis [8] has presented method and Green functions of diffusion-wave fields. Diffusion waves arise from the classical parabolic diffusion equation with an oscillatory force function in homogeneous media [9]. Diffusion-wave methods have been developed in the study of heat transfer [10], diffusive neutron waves [11], diffusive viscosity waves [12], pressure diffusion in porous media [13][14][15][16] and mass transport [17]. Diffusion waves are heavily damped with relatively slow velocity and short wavelength. The penetration depth and complex wavenumber describe diffusion-wave behavior. Diffusion waves in porous media obey an accumulation-depletion law [14].
Recently diffusion-waves of mass transport have been receiving much attention in geochemistry, and found applications in volcanic eruption linked by radioactive element transfer [18] and organic compound migration [19] in soil. Linear mass diffusion-wave mathematical formalisms are based on harmonic wave solutions, or Laplace transform methods [20]. These theories have introduced theoretical treatments of mass diffusion-wave fields. Green functions can be used to model the response of a system to a prescribed excitation with knowing the internal properties of the system. Temporal-spatial Green functions for mass diffusion-waves have been presented by Paterson [21]. Mass field response to any arbitrary mass source distribution can be represented as a convolution integral of the distribution with a Green function over source coordinates. An important question arises over the existence of the spectrum of chemical tracer concentrations observed in a geological reservoir, which encompasses the entire spectral bandwidth. To analyze the conventional mass diffusion-wave behavior, the wideband spectral concentration measurements must be reduced to a single spectral component. The Green functions of frequency domain must be used to calculate mass-diffusion wave fields.
In this letter, we present a formulism of frequency-spatial mass diffusion in porous media, and present Cartesian-coordinate mass diffusion-wave Green functions for infinite, semi-infinite, and finite-size domains in three dimensional spaces. The linear mass diffusion-wave fields are mathematically analyzed with Green functions, which can be used for precisely predicting the radioactive element transfer (dispersion-decay) in the large-scale nuclear geological reservoirs. This method allows the introduction of frequency-dependent material properties of geological systems. By virtue of the uniqueness, rapidity of convergence, and closed-form representations of Green functions, mass diffusion will be precisely predicted for all physically acceptable boundary conditions. It is hoped that methodologies of Green functions will form a mathematically rigorous and useful reference for enhancement of shale gas, oil recovery, and prediction of underground radioactive element leakage. Furthermore, Green functions can be used to inversely determine the mass diffusivity, mass source and decay constant of radioactive elements in the Earth.
2. Mathematical methods and green functions 2.1. Green function of mass diffusion wave Based on the well-known diffusion theorem [9], a mass diffusion (dispersion-decay) wave [22] generated by a source function r q t , ( ) in porous media is given by where D eff is the mass effective diffusivity in a porous medium and the function of porosity, and l presents a decay constant of a radioactive tracer. r t , J( )denotes mass concentration. The mass diffusion equation (1) is essentially a modified heat diffusion equation [9] with the lambda term, r t , , lJ( ) corresponding to the depletion. The underlining physical phenomenon of the depletion term includes the geological decay of radioactive tracers, and adsorption of the mobile tracer or solute to the surface of porous media. Using Fourier transform [2], the mass diffusion-wave obeys, in the frequency domain, the following equation It notes that the derivation of equation (2) from equation (1) requires that the mass concentration is bounded.
The solution to the homogeneous (dispersion-decay) diffusion wave equation (2) is expressed as ) is a spectrum of the source distribution r q t , . ( ) r, F w ( )presents the mass concentration at location r due to the random forcing r Q , . w ( ) The Green function (see appendix) for the linear mass (dispersion-decay) diffusion waves in three-dimensions is given by equation (4): The Green function (4) is the solution to the mass diffusion equation (1) when the source is a delta function. Where r r o ( )is the coordinate of the observation (source) point with respect to the origin, t t o ( )presents the observation (source appearance) time, and H denotes the Heaviside function H t t t t t t Green function of equation (4) satisfies , , here the Green function for parabolic equations in three variables r r , describes the response of the system at the point r to a point source located at r . o The point source is given by ) the product of Dirac delta functions. 2  is the Laplace operator. Taking the temporal and using method presented by Mandelis [8,26], we find r r r r r r Use has been made of the notation [8] r r r r G e G t , ; The Green function can be written by By interchanging the coordinates r r o « in equations (2) and (9), multiplying equation (2) Applying a reciprocity property [1] of Green functions r r )and the shifting property of Delta function [8], carrying out an integration by parts in the region of the source volume, and using Green's theorem [2,8], it gives r r r r r r r r r S The time modulation factor e j t 0 w is implicitly complied in equation (9). Here , .
o o ( | ) The equations of (2), (9) and (10) are the fundamental formula for the dispersion-decay diffusion-wave field in porous media. It points out that 8 the final expression of mass (dispersion-decay) diffusion fields must be multiplied by e . j t 0 w

Three dimensional infinite Green functions
Appendix describes the Green function of mass diffusion-waves (dispersion-decay) for infinite threedimensional spaces is the modified mass diffusion-wave field wavenumber.

Three dimensional semi-infinite green functions
If a linear dispersion-decay diffusion field is divided by a surface at the observation coordinate z 0, = a semiinfinite geometry is considered. The Green function can be obtained directly from equation (11) by the method of images presented by Mandelis [26]. In mass diffusion wave fields, the complex wavenumber is

Three dimensional green functions for finite geometry
Morse and Feshbach [1] has presented method of images, where image sources must be located at source coordinates as shown in figure 2 of reference [26]. Green functions satisfy either Dirichlet or Neumann boundary conditions at z L 0, . o = For this case, Green function is known for an infinite geometry and it is required for a geometry with finite boundaries. Using the method similar to derivation of equations (13) and (14), the Green function is given by Here±corresponds to Neumann or Dirichlet boundary conditions. With the array of image sources found in The exact behavior of mass diffusion waves depends on the boundary conditions. The influence of different boundaries is clearly shown from equation (15).

Application of green functions
We consider the semi-infinite region with mass concentration specified by r , t From equation (13), it follows that The mass diffusion-wave field represented by equation (22) is in a compact form, which can be separated out two components including the real and imaginary parts. The real k r and imaginary k im square root of The physical mass-wave field is the real part of equation (24). The dispersion-decay of radioactive elements (rubidium) is simulated in a nuclear geological reservoir [18]. Figure 1 depicts the depth-dependent amplitude of the mass-wave field (22) with time-modulation angular frequency as a parameter. It shows that the slope of the spatial decay remarkably increases with increasing .
w The mass diffusion length, , known as the penetration depth, describes the root mean square depth [24] of the diffusion-wave penetration into the domain z 0.
> As a frequency increases, a mass penetration depth decreases, and the diffusion-wave attenuation accelerates. The imaginary part of the wavenumber (see equation (23)) can attain enough energy to appear, as relatively high frequencies are reached [25]. It proves that the mass diffusion waves are dispersive, and internally damped. Figure 2 illustrates similar depth dependence at a constant modulation frequency of 10 −10 Hz, with domain mass diffusivity as a parameter. The effect of increasing mass diffusivity on the slope of the decay curves is seen to be similar to that of decreasing frequency, as expected from the structure of equation (22). Such phenomena can be explained on the physical grounds. As mass diffusivity increases, the penetration depth accordingly increases, a fact that corresponds to an increase in the degree of mass transfer. The diffusivity is a transport property of the porous media, which is linearly proportional to the porosity [22]. It indicates that effects of porosity on behaviors of mass diffusion-wave fields become similar to that of the corresponding diffusivity. Figure 3 shows the effects of various values of the parameter λ on the amplitude of the mass diffusion-wave depth profile. It is seen that with increasing λ the slope of the spatial delay increases. The differences in amplitude are sensitive to the value of λ. As λ decreases, the degree of mass decay decreases, and the mass diffusion-wave attenuates slowly, leading to mass accumulation within a porous medium. As shown in figures 1-3, amplitude profiles clearly exhibit the periodical behaviors. The wave-front is well behaved and captured. The periodically forced function with internal damping has remarkable influences on mass diffusion-wave motions. The mass diffusion-wave fields obey the field-gradient-driven accumulation-depletion rules [9]. The application example is directed to geophysical phenomena. However the chemical reactions in geological reservoirs are nonlinear. Future works could be needed to extend the application of Green functions to nonlinear problems [28].

Conclusions
We propose a formulism of frequency-domain mass diffusion-waves in porous media, and derive internally consistent Cartesian-coordinate mass-wave Green functions for infinite, semi-infinite, and finite-size domains in three-dimensional spaces. It notes that the three-dimensional semi-infinite Green function is of practical importance, because it describes the exact behavior of diffusive mass excited by an arbitrary source in geological reservoirs. The integral expressions for propagating mass diffusion-wave fields are presented in homogeneous systems and in practically geological geometries. A specific application is explicitly implemented in terms of Dirichlet boundary conditions in a case of semi-infinite region with mass impulse function prescribed over the interface plane. The periodically forced function with internal damping has remarkable influences on mass diffusion-wave motions. The mass diffusion-wave fields obey the field-gradient-driven accumulation-depletion rules. Green functions obtained for mass diffusion-waves can be used for all physically acceptable boundary l =´--conditions under fixed geometries. The exact behavior of mass diffusion waves depends on the boundary conditions. It is hoped that Green functions presented in this article will form a mathematically rigorous and useful reference for enhancement of shale gas or oil recovery, and migration of radioactivity in nuclear geology. Furthermore, Green functions can be used to determine the mass diffusivity and fluid properties of geological reservoirs.  With the similar methodologies, the one-dimensional Green function is given by The method of images [1] is used to derive the temporal-spatial Green function for semi-infinite domains