A Novel Analytical Solution of Radial Diffusion Equation with Constant
Terminal Rate of Slightly Compressible Fluid
LUSJEN ISMAILI
Ministry of Infrastructure and Energy,
Tirana,
ALBANIA
Abstract: - In reservoir engineering, the flow and type of fluids crossing porous media, are associated with pressure
drops leading to variations on hydrodynamic parameters and the filtration stages which are of interest to be
assessed as they impact the production efficiency of oil and gas industry. These parameters are the key to the
practical solutions of numerous subjects faced during oil field exploitation. Dealing with the problem of sound field
development initiatives requires a rigorous examination of unsteady filtration of slightly compressible fluid in the
reserves layer. The state of the art of the proposed scientific work is to present an innovative mathematical
approach that gives unique results, in the mathematical connection and combination of the equations used versus
existing diffusion equation in the case of the constant terminal rate solution. This model helps the designers in the
field of oil and gas to better and faster evaluate the diffusivity of the pay zones, the different hydrodynamic
parameters, and the different variables that take part in the development of fluid filtration processes in the porous
medium expressed as in the dependence of time, distance, and other variables, all of which together impact the well
testing and long-term projections.
Key-Words: - compressibility, unsteady-state flow, reservoir, pay zone, hydrodynamic, parameter,
filtration, flow rate, porous media, equation.
Received: March 9, 2024. Revised: October 5, 2024. Accepted: November 9, 2024. Published: December 3, 2024.
1 Introduction
In the oil and gas industry, considering the cases
when the pay zone is still unopened, the pressure at
any point within the bed can be assumed as constant.
Analogously can be accepted for density as a state
parameter. With the opening of the pay zone from a
well, from the moment of well completion, it will
begin to drain and, due to the release of some fluid
from the producing formation, the pressure will begin
to drop. With the cleavage of the pay zone from a
well, starting from the well completion, draining
issues, and, due to fluid discharge from the producing
formation, the pressure will decrease in time. As
fluid withdrawal continues, the decrease in pressure
will propagate further from the well in the direction
of the reservoir boundary as stated by fluids law.
These fluid properties will have a direct impact,
especially in the field of oil and gas, and may
prohibit the exploitation of wells with high
efficiency. Practically, the filtration will be unstable
and the radius of influence of the well will constantly
increase as stated in the studies, [1], [2]. During this
time when the pressure changes at a certain rate, i.e.
it does not remain constant but is constantly
changing, the flow state is known to be unsteady-
state flow. In conditions where the flow is in an
unstable state, the flow rate into a representative
volume of a porous medium is not equal to the flow
rate leaving from this element of volume of porous
media. Based on these pressure changes and if the
probe radius of investigation has not reached the
boundary of the reservoir, i.e., the reservoir will act
as if it were of infinite size, it can be said that the
flow in the unsteady state is defined as the time
during which the boundary does not affect the
pressure behavior in the reservoir. The period, during
which the process of increasing the radius of
influence, approaching the radius of the contour (or
of the drainage area), is scientifically known as the
first stage of filtering for an unsteady-state flow
layer, [3]. Also, in cases when fluid quantity entering
from the feeding area to the zone of production is
less than what is leaving, the pressure of the layer (in
the contour) will begin to decrease, [4].
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Consequently, the filtering process is becoming
unsteady as theoretically expected. This fraction of
time is related to the second stage of filtration for an
unsteady-state flow, [1], [2]. Considering the first
case, the unsteady filtering stage can be considered as
a series of settled states and then move on to solving
the problem as mentioned in the respective studies,
[3], [4], [5]. On the other hand, when the
hydrodynamic study of the well refers to the
unsteady regime for a relatively short time of process
development; then the implementation of the
unsteady filtering replacement method with a series
of steady stages, leads to hefty errors, [3], [4]. In
such conditions, it is obligatory and indispensable to
delve into this concern in more detail by employing
new mathematical approaches with more parameters
of influence.
Strongly affected by the above-mentioned issues
and uncertainties, then the exploitation of a layer
from a central well with constant flow in the filtration
conditions for unsteady flow regimes is suggested.
Afterwards using a new mathematical technique, the
terminal constant rate solution of the radial diffusion
equation is established. The solution of the diffusion
equation with constant terminal rate taking into
account the entire flowing time was first presented in
1949 [6] using Laplace transforms for both the
constant terminal [7] rate and constant terminal
pressure cases, as well as by [8] for a well situated
within a no-flow boundary for each flow time value,
as well as for all the geometrical configurations. In
the solution presented by them, three conditions are
considered; the initial state in which the pressure
anywhere within the drainage volume is equal to the
initial equilibrium pressure p, as well as two
boundary conditions which are:
The first is the pressure at the outer, infinite
boundary that is not affected by the pressure
disturbance at the wellbore and vice versa, and the
second is the line source inner boundary condition.
They also use the Boltzmann transformation, the
diffusivity constant and the substitution of the
parameters taken into consideration by them.
The approach presented and the conclusions
obtained from our analysis are based on the initial
condition given in expression (i), the boundary
condition given in expression (ii), the piezometric
conductivity, the parameter x which is expressed as a
ratio that relates the two variables r and i which is
given in Eq 5, the parameter y which expresses the
change in pressure depending on the parameter x, as
well as the three variables
and
All these parameters taken into consideration and
their mathematical relationship expressed based on
the physical concept of fluid mobility in the porous
medium make it possible not only to solve the
diffusion equation in a different and simple
mathematical method, but also the variables that take
parts in this equation, which are expressed as a
function of different variables, help to solve many
problems encountered in the testing and
hydrodynamic analysis of wells as given in the
studies, [9], [10], [11], [12], [13].
From the literature we know that the diffusivity
equation is a combination of three physical
principles; the continuity equation, Darcy’s law and
the equation of state regarding a slightly
compressible liquid, [14], [15], [16]. Employing the
continuity equation, we can express velocities of the
flowing fluid for the case of three-direction system
(Eq.2):
(1)
The differential forms of the equation of motion
for the case of three dimensional can be given from
mathematical expression in Eq.2:
;
;
;
(2)
Likewise, the equation of state for the case of a
fluid is given and represented by the mathematical
expression in Eq.3:
(3)
On the other hand, the formulation of the
equation for the filtration of slightly compressible
fluids in isotropic porous media is reached and can be
represented by the mathematical expression in Eq.4
merging Eq.1, Eq.2 and Eq.3. [2], [4], [17].
(4)
where:
represents piezometric
conductivity.
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Based on Eq. 4, performing the transformation of
coordinates from polar to Cartesian (Laplace and
Fourier transforms) [18], [19] combining and
replacing different equations, as well as considering
two variables given in the following mathematical
expression below in Eq.5 and Eq.6 (represent my
assumptions) we can evaluate them for first and
second derivative as a function of :
;
(5)
and
(6)
Using the above assumptions we have succeeded
in solving the diffusion equation for the case of
constant terminal rate solution, applying a new,
simple, flexible, and mathematical technique never
applied in other oil and gas studies.
2 Methodology
Initially, we assume an oil-bearing bed with the same
thickness , having an infinite extent and initial
reservoir pressure . This layer is exploited by a well
with a constant flow rate “” with the focus of
examining the pressure distribution in space and
time.
In our approach as an initial condition governed
by the expression given in (i) is employed:
(i)
Afterwards the expression given in (ii) serves as a
boundary condition:
(ii)
Practically the radial flow conditions should be
expressed with cylindrical coordinates correlated to
cartesian coordinates as given in Figure 1, [2], [3].
Following the transformation from cylindrical
coordinates to cartesian the mathematical expressions
given in (iii) are carried out:
(iii)
Fig. 1: Cylindrical Coordinate to Cartesian
Coordinate. Adapted after [20]
From Eq. 4 and the transformation given in (iii),
the solution of the problem can be as given in Eq.7:
(7)
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Applying more transformation than the following
mathematical expression in Eq. 8 can be obtained:
(8)
Applying and merging the above expression with
their influencing parameters from (Eq.8) then the
expression Eq.9 is obtained.
(9)
Based on Eq.9 we simply write as following Eq.
10:
(10)
The equation of state for the case of a
compressible liquid as represented in Eq. 11 is used:
in respect to initial density values.
or
(11)
Applying the first derivative of the density
function with respect to r, we get the mathematical
expression as given in Eq.12:
(12)
Applying the second derivative of the above
function (Eq.12) to the mathematical expression we
get Eq.13.
(13)
By substituting the expression in Eq.13 in Eq. 10,
then the relationship as given in Eq.14 can be
achieved.
(14)
3 Results and Discussion
To simplify the analytical solution of the problem
raised, we have considered, without spoiling the
solution, that the radius of the well is inefficiently
small, thus having a suction point. Then we have
marked with x the ratio that connects the two
variables s and t and continuing with its substitution
in equation 14, the result was obtained as following
for three variables
and
(15)
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For
the mathematical relationship as
following in Eq.16 can be found.
(16)
For
the mathematical relationship as
following in Eq.17 can be carried out:
(17)
By substituting equations 15, 16 and 17 into equation
14, we get the result as following in Eq.18:
(18)
Then both sides are multiplied of the Eq.18 by
and considering that
than we can simply get
the result as given in Eq. 19:
(19)
Supposing the assumptions that
and
extending our calculation it is possible to get the
mathematical expression as given in Eq. 20:
(20)
Based on Eq. 20, performing the variable
separation and integrating, then the following
expression represented by Eq. 21 is carried out:
By integrating both sides of the pressure
equation, we simply get:
(21)
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For the term within the integral goes to
zero, so we will have the following
expression in
Eq.22:
(22)
Starting from continuity Eq. 23 and substitute it
further we get expression in Eq. 24:
(23)
Hence,
(24)
For
than we can get the flow rate
as given in expression in Eq.25.
(25)
As a conclusion the mathematical expression of
the pressure as a function of two variables as chosen
in the study, lead to the following expression in
Eq.26:
(26)
Integrating we can get expressions as following
in Eq.27
(27)
Substituting (Euler constant)
(
, as well as .
Assuming that our analysis and pressure values are
directly measured in the wellbore, and further
substituting the above values, then the mathematical
expression given in Eq. 28 regarding diffusivity
equation is carried out.
(28)
On the other hand, substituting
in Eq.
28 then the proposed mathematical model can be
carried out and represented by the following Eq.29:
(29)
If more transformations are performed, then
Eq.29 can be easily represented by the mathematical
expression in Eq.30.
(30)
The three forms of equations 28, 29, 30 are
called the basis of the diffusion equation, since using
dimensionless variables such as dimensionless
radius, dimensionless time and dimensionless
pressure, for a many reasons, as well as using
conversion to field unit and mechanical skin factor
[6], the equation's form will be change, [1], [4]. As
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can be seen, the transient solution is not true for the
entire drainage surface, as the reservoir appears to be
infinite in extent, at the moment we refer the
position of the well in relation to the contour, for a
short time when the equation is applicable, [1], [4].
The author in the study [8] suggested a method for
determination of average pressure in a bounded
reservoir. resolves the question on issues related to
drainage volumes with no pressure data. In that case
a plot of average pressure versus relative drainage
volume may allow the missing pressures to be
assessed, [8]. In the research work of [21] the flow of
a fluid with pressure-dependent viscosity through
variable permeability porous layer is performed. The
results showed that values of the permeability
proportionality constant have negligible or no effects
on flow characteristics.
4 Conclusion
As conclusion, the diffusivity equation presented
above, expresses a connection between the principles
and laws of physics and further structured to
mathematical analysis employing differential
equations, coordinate transformations, derivation,
and integration rules. Diffusion itself represents a
physical phenomenon of molecular movements,
usually manifested by the movement of liquids and
gases depending on the conditions and parameters
impacting it. For the case study, applicable to oil and
gas-bearing rocks, the pressure diffusion in the
reservoir, is mainly affected by the Darcy law
(filtration velocity of fluids in porous media), the law
of mass conservation, the equation of state for fluid
and rock structure as well. The study of the filtration
process of slightly compressible liquids in porous
media as well as the determination of its dynamics
pressure drop, and rates of exploitation time for a
given oil field, interconnected, and combined in the
diffusion equation, have a great importance in
practical applications during well testing. In this
research paper, the radial flow is treated in a layer to
an infinite extent exploited with a constant flow rate.
Further on it is mathematically represented by
solving the differential equation gathering different
variables we suggest the new mathematical technique
useful in both the theoretical and practical aspects
during well testing. The proposed method is simple
and applicable in the real conditions of fluid flow in
porous media and regardless of certain limitations
that exist during the solution of the diffusion
equation, all the mathematical relationships of the
different variables expressed above, which
are programmed in the corresponding software, not
only provide a quick and concise solution but help in
many situations in the calculation of various
parameters during the hydrodynamic study of wells.
5 Future Work
In the future, all these parameters taken into
consideration and their mathematical relationship
expressed based on the physical concept of fluid
mobility in the porous medium make it possible not
only to solve the diffusion equation in a different and
simple mathematical method but also the variables
that take parts in this equation, which are expressed
as a function of different variables, help to solve
many problems encountered in the testing and
hydrodynamic analysis of wells, leading to improve
the fuel extraction economy and fuel quality for a
better and safer environmental especially from
transport sector, [22], [23].
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