The article discusses the problem of approximating solutions of differential equations describing the process of a two-dimensional fluid flow in porous media. The approximation is presented as a combination of radial basis functions on the basis of the Green’s function is used to solve the Poisson equation with variable coefficients in the case of steady state filtration and parabolic equations in the transient regime. To illustrate the effectiveness of the proposed approximation obtained by the field pressure distribution in the reservoir with a network of injection and production wells. Compare approximated pressure and design points to a satisfactory accuracy of the results

A. M. Gubanov

K. A. Sidelnikov

V. E. Lyalin

English

Vibroengineering PROCEDIA

 164 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
Radial basis approximation of single-phase flow in 
porous media based on the Green’s functions 
K. A. Sidelnikov1, A. M. Gubanov2, V. E. Lyalin3 
1JSC “Izhevsk Petroleum Research Center”, Izhevsk, Russia 
2Institute of Oil and Gas, Udmurt State University, Izhevsk, Russia 
3Kalashnikov Izhevsk State Technical University, Izhevsk, Russia 
3Corresponding author 
E-mail: 1kasidelnikov@udmurtneft.ru, 3velyalin@mail.ru 
(Received 5 August 2016; accepted 8 August 2016) 
Abstract. The article discusses the problem of approximating solutions of differential equations 
describing the process of a two-dimensional fluid flow in porous media. The approximation is 
presented as a combination of radial basis functions on the basis of the Green’s function is used 
to solve the Poisson equation with variable coefficients in the case of steady state filtration and 
parabolic equations in the transient regime. To illustrate the effectiveness of the proposed 
approximation obtained by the field pressure distribution in the reservoir with a network of 
injection and production wells. Compare approximated pressure and design points to a satisfactory 
accuracy of the results. 
Keywords: approximation, Green’s function, two-dimensional fluid flow, porous medium. 
1. Approximation of the steady-state fluid flow in porous media 
Let us consider the problem of approximating solutions of differential equations describing 
two-dimensional fluid flow through a porous medium. To describe steady-state two-dimensional 
fluid flow we study the following boundary value problem: 
∇ ቀℎ(ܚ)ߣ(ܚ)൫∇ܲ − ߛ∇ܼ(ܚ)൯ቁ = −ℎ(ܚ) ෍ ݍ௟(ݐ)ߜ(ܚ − ܚ௟)
௅
௟ୀଵ
, (1)
with Neumann boundary condition ∂ܲ(ܚ) ∂ܖ⁄ |୻ = 0,  where λ(ܚ) = ܓ(ܚ) ߤ⁄ ,  ߛ = ߩ݃,  
ܚ ∈ Ω ⊂ Ωഥଶ; ܲ(ܚ) – pressure at point ܚ = (ݔ, ݕ) ∈ Ω; Ωഥ – computational domain with boundary 
Γ; ܖ – outer-pointing normal; ܓ = diag(݇ଵ, ݇ଶ) – diagonal tensor of absolute permeability; ℎ – 
reservoir thickness; ܼ  – reservoir depth; ߤ  – fluid viscosity; ߩ  – fluid density; ݃  – gravity 
acceleration; ܠ௟ – coordinates of ݈th well locations with flow rate ݍ௟, ݈ = 1, ܮ; ܮ – number of wells; 
ߜ(⋅) – generalized two-dimensional Dirac delta function. 
Eq. (1) is Poisson’s equation with variable coefficients. For ordinary Poisson’s equation: 
∇ଶΦ = ∂
ଶΦ
∂ݔଶ +
∂ଶΦ
∂ݕଶ = −ܳ. (2)
ܚ = (ݔ, ݕ) ∈ ܦ, where ܦ denotes the entire plane, we can obtain the solution with Green’s 
function ܩ(ܚ, ߩ)[1]: 
Φ(ܚ) = න ܩ(ܚ, ߩ)ܳ(ߩ)݀ߪ(ߩ)
஽
, (3)
where: 
ܩ(ܚ, ߩ) = 12ߨ ln
1
|ܚ − ߩ|. (4)
RADIAL BASIS APPROXIMATION OF SINGLE-PHASE FLOW IN POROUS MEDIA BASED ON THE GREEN’S FUNCTIONS.  
K. A. SIDELNIKOV, A. M. GUBANOV, V. E. LYALIN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533 165 
Such representation of solution to Eq. (2) in the presence of point sources and sinks is 
examined in [2], and it is known as well interference problem. Under constant coefficients and 
isotropy in Eq. (1) the solution Eq. (3) is given by: 
Φ(ܚ) = ݇ܲߤ =
1
2ߨ ෍ ݍ௟ln|࢘ − ߩ௟|
௅
௟ୀଵ
, (5)
where ݍ௟ – well flow rate.  
In case of variable coefficients Eq. (5) leads to significant errors in solution. We therefore 
propose the use of Green’s function Eq. (4) as radial basis functions ܩ(|ܚ − ߩ|) to approximate 
Eq. (1) based on their linear combination. Coefficients of linear combination can be calculated 
through the numerical solution obtained by finite difference method. Let us consider rectangular 
domain ܦ = ሾܽ, ܾ; ܿ, ݀ሿ  which is divided into finite blocks Δݔ = ܾ − ܽ ݊௫⁄ , Δݕ = ݀ − ܿ ݊௬⁄ , 
where ݊௫ × ݊௬ – grid size. 
Let us denote the numerical solution to Eq. (1) obtained by finite difference method as ௜ܲ௝,  
݅ = 1, ݊௫ ; ݆ = 1, ݊௬ , and its approximation as ௜ܲ௝௔ , ݅ = 1, ݊௫ ; ݆ = 1, ݊௬ . The approximation is 
constructed using linear combination of radial functions ܩ(|ܚ − ߩ|) . The radial function is 
undefined at a block contained source when ܚ = ߩ. Therefore, its value at this point is set equal to 
a constant determined by the size of the block [3], e.g. ܩ௟௟ = −ln൫ඥΔݔΔݕ൯/2ߨ. 
Then the approximate expression can be written as: 
௜ܲ௝௔ = ௞ܲ௔ = ෍ ൤ߙ௟ߜ௟௟
ܩ௟௟
ߣ௟ + ߚ௟(1 − ߜ௟௞)
ܩ௞௟
ߣ௞ ൨
௅
௟ୀଵ
ݍ௟ΔݔΔݕ, (6)
where: 
ܩ௞௟ =
1
2ߨ ቐ
−ln(ඥΔݔΔݕ, ݈ = ݇,
ln 1|ܚ௞ − ρ௟| , ݈ ≠ ݇,
ߜ௟௞ = ൜1, ݈ = ݇,0, ݈ ≠ ݇. (7)
The sources can be with different signs. Here it is assumed that injection well is a source with 
positive sign and production well is a source with negative sign. We introduce also index ݇ which 
corresponds to the point ݎ௜௝, ݅ = 1, ݊௫; ݆ = 1, ݊௬; ߙ௟, ߚ௟ – unknown coefficients that is determined 
through the condition ∑ ( ௞ܲ − ௞ܲ௔)ଶே௞ୀଵ → minఈ,ఉ , where ܰ = ݊௫݊௬. 
To take into account Neumann boundary conditions Eq. (6) should be completed with 
additional image sources located outside of the computational domain. Each source ݍ௟  is 
associated with four image sources ݍ௟௠, ݈ = 1, ܮ; ݉ = 1,4. Their coordinates are given by: 
ߩ௟ଵ = (ܽ − ݔ௟, ݕ௟),    ߩ௟ଶ = (ݔ௟, 2݀ − ݕ௟), ߩ௟ଷ = (2ܾ − ݔ௟, ݕ௟), ߩ௟ସ = (ݔ௟, ܿ − ݕ௟), 
thus we can rewrite Eq. (6) as: 
௞ܲ௔ = ෍ ቈߙ௟ߜ௟௟
ܩ௟௟
ߣ௟ + ߚ௟(1 − ߜ௟௞)
ܩ௞௟(|ܚ௞ − ߩ௟|)
ߣ௞ ቉
௅
௟ୀଵ
ݍ௟ΔݔΔݕ
      + ෍ ෍ ߚ௟(1 − ߜ௟௞)
ܩ௞௟(|ܚ௞ − ߩ௟௠|)
ߣ௞
௅
௟ୀଵ
ݍ௟ΔݔΔݕ
ସ
௠ୀଵ
.
We use the following notation: 
RADIAL BASIS APPROXIMATION OF SINGLE-PHASE FLOW IN POROUS MEDIA BASED ON THE GREEN’S FUNCTIONS.  
K. A. SIDELNIKOV, A. M. GUBANOV, V. E. LYALIN 
166 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
ۯ = ൤ߜ௟௟ܩ௟௟ߣ௟௟ ൨ ,     ۰ = ቈ(1 − ߜ௟௞)
ܩ௞௟(|ܚ௞ − ߩ௟|)
ߣ௞ ቉ ,    ݈ = 1, ܮ;     ݇ = 1, ܮ,     ܌ = ൥
ଵܲ
⋮
ேܲ
൩ ,     ܆ =
ۏ
ێ
ێ
ێ
ێ
ۍߙଵ⋮
ߙ௅
ߚଵ
⋮
ߚ௅ ے
ۑ
ۑ
ۑ
ۑ
ې
, 
۱ = ቈ(1 − ߜ௟௞)
ܩ௞௟(|ܚ௞ − ߩ௟|)
ߣ௞ + ෍ (1 − ߜ௟௞)
ܩ௞௟(|ܚ௞ − ߩ௟௠|)
ߣ௞
ସ
௠ୀଵ
቉ , ݈ = ܮ, 2ܮ, ݇ = ܮ, ܰ, 
where ܌ – vector of known values; ܆ – vector of unknown coefficients; ۱ – calculated matrices. 
From ۯ, ۰, ۱ we construct block matrix: 
۵ = ቂۯ ۰0 ۱ቃ = ሾ݃௞௟ሿ,    ݇ = 1, ܰ, ݈ = 1,2ܮ.
Here ۯ, ۰ are square matrices, ۱, ۵ are rectangular matrices (ܰ > 2ܮ).  
According to the minimum condition of standard deviation we obtain matrix equation: 
۵܆ = ܌. (8)
We multiply Eq. (8) from left by the transposed matrix ۵்: ۵்۵܆ = ۵்܌. 
Then we calculate inverse of the matrix ۵்۵  and obtain expression to find unknown 
coefficients: 
܆ = (۵்۵)ିଵ۵்܌. (9)
The operation (۵்۵)ିଵ۵் = ۵ା is called pseudoinversion [4], and the approximation based 
on radial basis function is known as radial basis neural network. 
Then radial approximation appears as follows: 
۾௔ = ۵܆. (10)
Coefficients ܆ are calculated for some preselected source locations and their flow rates. Then 
the approximation Eq. (10) is used for arbitrary source locations and their flow rates. The number 
of sources should be less than or equal to ܮ. For a smaller number of sources, the flow rates of 
extra sources are set to zero. According to the Neumann boundary condition the material balance 
that accounts for source terms should be zero [5, 6].  
For illustration, let us consider the following case study. The domain ܦ = ሾܽ, ܾ; ܿ, ݀ሿ, ܽ = 0; 
ܾ = 1000 m; ܿ = 0; ݀ = 1000 m was divided into 256×256 blocks of finite difference grid. 
Reservoir thickness was changed according to ℎ(ܚ) = 10(1 + 0.1sin(4ߨ(ݔ + ݕ))),  fluid 
mobility was changed according to ߣ(ܚ) = 10ିଽ(1 + 0.2sin(2ߨݔ) + 0.15cos(2ߨݕ)). There were 
10 wells (5 production wells and 5 injection wells) in the domain. The absolute values of flow 
rates of all sources were the same and equal to ܳ = 50 m3/s. The sources were randomly located 
in the domain. To calculate pressure in blocks multigrid method with 7 nested grids was used.  
Fig. 1 demonstrates distribution of the normalized pressure ݌ = (ܲ − ܲ୫୧୬) (ܲ୫ୟ୶ − ܲ୫୧୬)⁄  
in the domain, where ܲ୫ୟ୶, ܲ୫୧୬ – maximum and minimum pressure values. The distribution of 
the approximate pressure is shown in Fig. 2. 
As shown in Fig. 3, comparison of the pressure values in midsection of the domain 
demonstrates relatively sufficient approximation. 
Then we changed locations of the sources and their flow rates in random manner. Comparison 
of the numerical pressure values with their approximate values in midsection of the domain for 
different source distribution is shown in Fig. 4. 
RADIAL BASIS APPROXIMATION OF SINGLE-PHASE FLOW IN POROUS MEDIA BASED ON THE GREEN’S FUNCTIONS.  
K. A. SIDELNIKOV, A. M. GUBANOV, V. E. LYALIN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533 167 
As can be seen, the radial network is well trained and provides sufficient approximation of the 
data that is independent of the training set. Fig. 5 demonstrates comparison of the numerical 
pressure values with their approximate values for all blocks of the domain. 
The standard deviation of the approximate pressure values from their numerical values is 0.95 %. 
The plus signs in Fig. 5 depict pressure values in blocks contained sources. The maximum deviations 
are observed in these particular blocks where pressure changes more rapidly. The results show that 
the Neumann boundary conditions are satisfied. It should be noted that Neumann boundary 
conditions are the most difficult case study especially when they are applied for the entire boundary. 
 
Fig. 1. Pressure distribution obtained  
by finite difference method 
 
Fig. 2. Approximate pressure distribution 
 
 
 
Fig. 3. Comparison of the numerical pressure  
values with their approximate values  
in midsection of the domain 
 
Fig. 4. Comparison of the numerical pressure values 
with their approximate values in midsection of the 
domain in case of different source locations 
 
Fig. 5. Comparison of the numerical pressure values  
with their approximate values for all blocks of the domain 
2. Approximation of the nonstationary fluid flow in porous media 
For the case of compressible fluid flow and rock porosity depending on the pressure Eq. (1) 
becomes parabolic differential equation: 
∇ ቀℎ(ܚ)ߣ(ܚ)൫∇ܲ − ߛ∇ܼ(ܚ)൯ቁ = ℎ(ܚ)ߚ(ܚ) ∂ܲ∂ݐ − ℎ(ܚ) ෍ ݍ௟(ݐ)ߜ(ܚ − ܚ௟)
௅
௟ୀଵ
, (11)
RADIAL BASIS APPROXIMATION OF SINGLE-PHASE FLOW IN POROUS MEDIA BASED ON THE GREEN’S FUNCTIONS.  
K. A. SIDELNIKOV, A. M. GUBANOV, V. E. LYALIN 
168 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
where ߣ(ܚ) = ܓ(ܚ) ߤܤ⁄ ;  ߚ(ܠ) = ߮(ܚ) ܿℓ ܤ°⁄ + ߮°(ܚ) ܿఝ ܤ⁄ ;  ߩ = ߩ°൫1 + ܿℓ(ܲ − ܲ°)൯,  
ܤ = ܤ° 1 + ܿℓ(ܲ − ܲ°)⁄ ,  ߤ = ߤ° 1 − ܿఓ(ܲ − ܲ°)⁄ ,  ߮(ܚ) = ߮°(ܚ) ቀ1 + ܿఝ(ܲ − ܲ°)ቁ , ߮(ܚ)  – 
rock porosity; ܤ  – fluid formation volume factor; ܲ°  – reference pressure. Eq. (11) is 
complemented by the initial condition: 
ܲ(0, ܚ) = ଴ܲ(ܚ). (12)
For parabolic model equation: 
∇ଶΦ = ߙଶ ∂ܲ∂ݐ + ݂(ݐ, ܚ), Φ(0, ܚ) = Φ଴(ܚ), (13)
there is Green’s function solution as well [1]: 
Φ(ݐ, ܚ) = න නܩ(ݐ, ܚ; ߬, ߦ)݂(߬, ߦ)ܸ݀߬݀(ߦ)
௏
௧
଴
− ߙଶ නܩ(ݐ, ܚ; 0, ߦ)Φ଴(ߦ)ܸ݀(ߦ)
௏
, (14)
where ܩ(ݐ, ܚ; ߬, ߦ) – Green’s function, which can be constructed from eigenfunctions ߰௞  and 
eigenvalues ߣ௞ of Helmholtz equation: 
∇ଶ߰(ܚ) + ߣ߰(ܚ) = 0,   ܚ ∈ ܸ, ܩ(ݐ, ܚ; ߬, ξ) = − 1ߙଶ ෍ ݁
ିఒೖఈమ(௧ିఛ) ത߰௞(ߦ)߰௞(ܚ)௞ . (15)
Eq. (14) and Eq. (15) can be used as a basis for approximate solution to Eq. (11). It is important 
to note that Green’s function (15) can be represented as product of two functions (separation of 
variables): the first one depends exponentially only on time variable, the second one depends only 
on spatial variables. 
Assume first that flow rate of sources is time-independent. Let us restrict ourselves to the one 
summand in Eq. (15). Then Eq. (15) can be rewritten as: 
ܩ(ݐ, ܚ; ߬, ߦ) = − 1ܽଶ ݁
ି ఒఈమ(௧ିఛ)Ψ(ߦ, ܚ),
where Ψ(ߦ, ܚ) – some function sought for approximation. 
Time integration of Eq. (14) gives: 
Φ(ݐ, ܚ) =
൬1 − ݁ି
ఒ
ఈమ௧൰
ߣ නΨ(ߦ, ܚ)݂(ߦ)ܸ݀(ߦ)௏ + ݁
ି ఒఈమ௧ නΨ(ߦ, ܚ)Φ଴(ߦ)ܸ݀(ߦ)
௏
. (16)
It follows from Eq. (16) that solution to nonstationary equation Eq. (13) contains steady-state 
component: 
Φௌ(ܚ) = නΨ(ߦ, ܚ)݂(ߦ)ܸ݀(ߦ)
௏
.
To which solution Eq. (16) converges asymptotically as ݐ → ∞ . The second term 
݁ିఒ௧ ఈమ⁄ ׬ Ψ(ߦ, ܚ)Φ଴(ߦ)ܸ݀(ߦ)௏  corresponds to the initial condition as ݐ → 0 . Therefore, 
approximation problem is solved in two steps. At first we seek an approximant to a steady-state 
solution as ݐ → ∞. For this purpose the radial function Eq. (7) denoted as ܩௌ(ߦ, ܚ) is used. For the 
second term in Eq. (16) we take unit integral transform to satisfy the given initial condition. Then 
RADIAL BASIS APPROXIMATION OF SINGLE-PHASE FLOW IN POROUS MEDIA BASED ON THE GREEN’S FUNCTIONS.  
K. A. SIDELNIKOV, A. M. GUBANOV, V. E. LYALIN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533 169 
the parameter nonstationarity ߣ is adjusted. For best approximation on time several terms of series 
can be used in Eq. (15), so Green’s function can be presented as: 
ܩ(ݐ, ܚ; ߬, ߦ) = − 1ܽଶ ෍ ݁
ିఒೖఈమ(௧ିఛ)
௞
Ψ(ߦ, ܚ),
and solution Eq. (16) can be written as: 
Φ(ݐ, ܚ) = ෍
ቆ1 − ݁ି
ఒೖ
ఈమ௧ቇ
ߣ௞௞ නܩ௦(ߦ, ܚ)݂(ߦ)ܸ݀(ߦ)௏ + ෍ ݁
ିఒೖఈమ௧
௞
Φ଴(ܚ).
(17)
It should be noted that the underlying considerations are not strictly mathematically sound and 
needed only to choose basis functions when constructing the approximant. 
Let us consider the case where ∑ ℎ(ܚ௟)ݍ௟௅௟ୀଵ = ߜݍ ≠ 0. The steady-state solution to Eq. (11) 
as ∂ܲ ∂ݐ⁄ → 0 doesn’t exist subject to the Neumann boundary condition. To construct function 
ܩௌ(ߦ, ܚ) we introduce background sources: 
Δݍ௜௝ = −
∑ ℎ(ܚ௟)ݍ௟௅௟ୀଵ
(ܾ − ܽ)(݀ − ܿ) Δݔ௜Δݕ௝, ݅ = 1, ݊௫, ݆ = 1, ݊௬,
for each grid blocks of the computational domain. Now we can define the corresponding steady 
state and fit the function ܩௌ(ߦ, ܚ) which describes this state. Eq. (17) is now as follows: 
Φ(ݐ, ܚ) = ෍
ቆ1 − ݁ି
ఒೖ
ఈమ௧ቇ
ߣ௞௞ නܩ௦(ߦ, ܚ)݂(ߦ)ܸ݀(ߦ)௏ + ෍ ݁
ିఒೖఈమ௧
௞
Φ଴(ܚ) +
ߜݍ
ߙଶ ݐ.
If the flow rate of a source is a function of time ݂(ݐ, ܚ) then the nonstationary process is divided 
into time steps of length Δݐ௡ by analogy with finite-difference representation, so the approximate 
expression for pressure is written as: 
ܲ௔(ݐ௡ାଵ, ܚ) 
      = ෍
൬1 − ݁ି
ఒೖ
ఈమ௧൰
௡ାଵ
ߣ௞௞ නܩ௦(ߦ, ܚ)݂(ݐ
௡ାଵߦ)ܸ݀(ߦ)
௏
+ ෍ ൬݁ି
ఒೖ
ఈమ௧൰
௡ାଵ
௞
ܲ௔(ݐ௡, ܚ) + ൬ߜݍߙଶ ݐ൰
௡ାଵ
, 
(18)
where ߙଶ = ℎ(ܚ)ߚ(ܚ); ݂(ݐ, ܚ௟) = ℎ(ܚ) ∑ ݍ௟(ݐ)ߜ(ܚ − ܚ௟)௅௟ୀଵ .  
For one summand in sum Eq. (18) can be written in the form: 
ܲ௔൫ݐ௡, ݎ௜௝൯ = ߠ଴൫ݐ௡, ݎ௜௝൯ + ߠଵ൫ݐ௡, ݎ௜௝൯݁
ି ఒఈ೔ೕమ
௧೙
.
According to the condition ∑ ൫ܲ௔(ݐ௡, ݎ௜௝) − ܲ(ݐ௡, ݎ௜௝)൯ଶ௜,௝,௡ → minఒ  we obtain the expression 
for coefficient ߣ based on the calculation of ௜ܲ௝௡ by finite-difference method: 
ߣ = − 1ܰ݊௫݊௬ ෍
ܽ௜௝ଶ
ݐ௡ ln
௜ܲ௝௡ − ߠ଴௜௝௡
ߠଵ௜௝௡௜௝௡
where ௜ܲ௝௡ – calculated pressure values; ܰ – number of time steps. The results showed that in most 
RADIAL BASIS APPROXIMATION OF SINGLE-PHASE FLOW IN POROUS MEDIA BASED ON THE GREEN’S FUNCTIONS.  
K. A. SIDELNIKOV, A. M. GUBANOV, V. E. LYALIN 
170 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
practical cases ߣ ≈ 0.25. 
Let us consider the case study under the same conditions as in the steady-state case. In addition, 
we assumed that porosity ߮° = 0.15 , and coefficients ܿఝ = 10-4; ܿ௟ = 2∙10-4; ܿఓ = 0. The 
stationary function ܩௌ(ρ, ܚ) was taken as a result of previous training. The material unbalance was 
ܳ = 39 m3/s. The calculated value of ߣ was ߣ = 0.27. Fig. 6 compares numerical and approximate 
pressures for different time steps in midsection of the domain. 
The solid lines correspond to pressure obtained numerically by finite-difference method, and 
the circles correspond to approximate pressure. The comparison of the normalized numerical and 
approximate pressure values for all the grid blocks is shown in Fig. 7. 
The standard deviation of the approximate pressure values from their numerical values is 
1.84 %, that is approximately twice as high as for steady-state flow. 
Fig. 6. The comparison of the numerical and 
approximate pressures in midsection of the domain 
in case of nonstationary flow 1 – ݐ̅ = 0.2;  
2 – ݐ̅ = 0.4; 3 – ݐ̅ = 0.6; 4 – ݐ̅ = 0.8 
 
Fig. 7. Comparison of the normalized  
numerical and approximate pressures 
3. Conclusions 
1) The developed model of radial approximation of the steady-state equation of the fluid flow 
in porous media based on basis Green’s functions provides that standard deviation of the 
approximate pressure from its numerical values will be within the range 0.95-1.05 %. Such basis 
functions are used to approximate solution of the nonstationary multiphase flow in porous media 
as well. 
2) To approximate solution of the nonstationary fluid flow in porous media Green’s function 
is used. It is represented as a product of two terms (separation of variables): the first one 
exponentially depends only on time; the second one depends only on spatial coordinates. The use 
of background sources allows the construction of approximants which are applicable for 
time-varying flow rates of the sources and unbalanced flow rates. 
References 
[1] Korn G. A., Korn T. M. Mathematical Handbook for Scientists and Engineers; Definitions, 
Theorems, and Formulas for Reference and Review. McGraw-Hill, New York, 1968, p. 1151.  
[2] Basniev K. S., Kochina I. N., Maksimov V. M. Underground Hydrodynamics. Nedra, Moscow, 1993, 
p. 416, (in Russian). 
[3] Kanevskaya R. D. Mathematical modeling of hydrodynamic processes of hydrocarbon deposit 
development. Izhevsk, 2003, p. 128, (in Russian). 
[4] Osowski S. Neural Networks for Information Processing. Finance and Statistics, Moscow, 2002, 
p. 344, (in Russian). 
[5] Chanson H. Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows. CRC Press, 
Taylor & Francis Group, Leiden, The Netherlands, 2009, p. 478. 
[6] Economides Michael J., Nolte Kenneth G. Reservoir Stimulation. Wiley, USA, 2000, p. 856. 


Radial basis approximation of single-phase flow in porous media based on the Green’s functions

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Radial basis approximation of single-phase flow in porous media based on the Green’s functions

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