Application of Digital Technology for Oil and Gas Fields
SALTANBEK MUKHAMBETZHANOV
Department of Computational Science and Statistics
al – Farabi Kazakh National University, Almaty, Al – Farabi Avenue, 71, KAZAKHSTAN
Abstract: The paper investigates the problem of isothermal filtration using an approximate method for solving the theory
of partial differential equations. The mathematical model under consideration is nonlinear and does not lend itself to
analytical methods of solution. The results obtained indicate the need for wide application in the development of oil and gas
fields in the Republic of Kazakhstan. In particular, the results of the study make it possible to solve the problems of adapting
mathematical models and evaluating changes in technological indicators, which are necessary attributes in the digital
technology "Information System for the Analysis of oil and Gas Field Development" (ISAR). Many problems and
mathematical problems of filtration theory arose while working on specific oil and gas fields in the western region of the
Republic of Kazakhstan. The above approximate solution methods have found applications not only in filtration theory, but
also in other problems (geophysics, ecology, etc.)
Key-Words: absolute permeability tensor, relative phase permeability, capillary pressure, viscosity, ISAR, Darcy's
laws, porous medium, technological data, saturation.
1 Introduction
Currently, the use of digital technology for oil and
gas fields is developing very intensively (“Eclipse”
and “Black Oil by Schlumberger”, “Tempest by
Roxar VIP by Landmark” and “TimeZY by Standard
Oil” and “Trust”). All the listed systems consist of
basic blocks: block - technological data, block -
engineering models and block - mathematical
models. For predictive calculations, there are mainly
block mathematical models. Approximate methods
for solving nonlinear partial differential equations are
mainly used to solve problems of filtration theory and
numerical implementation. The structure of the study
consists of the derivation of the equations of the
filtration theory, the application of the variational
method and the obtained scientific result formulated
in the form of theorems. Similar mathematical results
can be found in [1-5].
2 Problem Formulation
The theory of filtration of two immiscible liquids in
a porous medium is based on the following analogues
of Darcy's laws and continuity equations for each of
the phases:
i1,2, (1)
where and are, respectively, the phase
volume flow rates (filtration rates), pressure, density
and saturation, and - acceleration
vector of gravity; – porosity of the medium;
- phase permeability
tensors; – absolute permeability tensor;
– relative phase permeability; – phase
viscosity coefficients. In the case of non-
compressibility of the liquid, it is assumed
that the pressures differ by the amount of
capillary pressure:
(2)
Equations (1), (2) are a closed system of relatively
unknown functions. From equations (1), (2) by
introducing the following notation (see, for example,
[1]):
and as a new desired function, the reduced pressure:
hgg ;
f
G ;
)1(
2112
12
1
1
211
0
1
1
2
ff
hgdGP
GPGPhgd
k
k
s
P
PP
s
k
s
k
we obtain a system of equations with respect to
(3)
(4)
Received: April 19, 2024. Revised: September 11, 2024. Accepted: October 15, 2024. Published: November 6, 2024.
EARTH SCIENCES AND HUMAN CONSTRUCTIONS
DOI: 10.37394/232024.2024.4.17
Saltanbek Mukhambetzhanov
E-ISSN: 2944-9006
142
Volume 4, 2024
or an equivalent system for
, (5)
(6)
The filtration flow in a given bounded region with a
piecewise smooth boundary is considered
Let
- external normal to Г. It is
assumed that the boundary of the
filtration area may consist of three, and
corresponds to surfaces impervious to both liquids,
– production well, – injection well. Then the
conditions for both liquids on are equivalent:
ГТ
ГТ
ГТ
ГТ
ГТ
On you can specify the absence of a displaced
phase flow:
ГТ
ГТ (7)
and the total consumption:
ГТ
ГТ (8)
or the pressure of the displaced phase
ГТ (9)
At the same time, the most difficult is the formulation
of the boundary condition on
Based on the results of work [1] on until the
moment of breakthrough, it is possible to similarly
(7), (8) set the absence of a displacement fluid flow
and the total flow rate of the mixture:
ГТ
ГТ
ГТ (10)
moreover, the relation arising from (6) must be
fulfilled:
Г
Г
The resulting model is closed by setting the initial
condition with respect to saturation:
(11)
Algorithms for finding approximate solutions to the
problem are based on the variational principle (5) –
(8), (11). The application of duality principles makes
it possible to obtain dual functionals, which make it
possible to estimate the minimum values of the initial
functionals from below.
It is assumed that the movement of an incompressible
liquid in a porous medium occurs in a region of
arbitrary cross-section under the action of a pressure
gradient (according to Darcy's law) in the time
interval [0, T]. The equivalence of the formulation of
such a problem in terms of differential equations and
the variational principle is shown in [2, 3]. For slow
unsteady motion of the medium, the variation scheme
is as follows.
Let {}– splitting the segment [0, T]:
Δ
(12)
Let's introduce a sequence with respect to
saturation functions:
(12’)
such that minimizes the functionality:
Ф
(13)
where . Denote by
(14)
a piecewise linear t function. If the sequence of
functions (14) converges to some function at
any partition (12), then is called the solution
of a non-stationary problem.
For an approximate solution of the variational
problem (13), we construct a system of functions:
(15)
complete in the space
, on which it is natural to
consider the functionals (13), where the function ω(x)
is piecewise continuously differentiable and satisfies
the conditions:
EARTH SCIENCES AND HUMAN CONSTRUCTIONS
DOI: 10.37394/232024.2024.4.17
Saltanbek Mukhambetzhanov
E-ISSN: 2944-9006
143
Volume 4, 2024
It should be noted that in (13) the functionals are
undifferentiable. Therefore, based on the
regularization method and from (13), we obtain
finite-dimensional optimization problems. The
application of the latter approach is due to the fact
that the regularization method provides positive
certainty of the Hessian , where
Next, the sequence is considered
ℎ
(16)
where defines the direction; is the length of the
step in the direction . If the sequence (16)
minimizes the functionals (13), then it is called
relaxation, and the condition for stopping the
construction of the relaxation sequence is usually
considered as
Ф
Ф
(17)
which defines the function
. (18)
Minimizing sequences of the form are constructed:
ℎ
ℎ
ℎ
ℎ
ℎ
ℎ
ℎ
ℎ
Ф
ℎ
Ф
ℎ
(19)
The functions
are obtained from the
functions
by cutting off local maxima with
horizontal planes along h.
3 Main results
By virtue of (15) – (19), as well as the assumed
smoothness for the given problem (5) – (8), (11) fair
Theorem 1. If there is a limit to the sequence
for p→∞, independent of the method
of dividing the segment [0, T], then this limit is a
solution to a non-stationary problem and the estimate
is valid:
Finally, according to the scheme for obtaining dual
functionals for stationary problems and by
introducing the condition for defrosting differential
connections, the functionals (13) can be represented
as an upper bound for arbitrary smooth functions
and θ:
Ф
Ф
Ф
Ф
(20)
The given relations in (20) make it possible to
investigate the dual problem and obtain a final
statement regarding the original problem (5) – (8),
(11).
Theorem 2. If the statement of Theorem 1 is true and
the gradients of the functional
with
respect to μ and θ are nonzero, then the following
equality holds:
Ф
(21)
The proof is based on the construction of
approximate solutions in finite-dimensional spaces of
functions generated by systems of linear independent
elements, then minimizing sequences are constructed
using the Newton process.
Equality (21) allows us to obtain lower estimates of
the minimum values of the functionals
.
In addition, the statement of theorem 2 is a
convenient tool in the study of mathematical models
for anomalous liquids.
4 Conclusion
Further application of the minimum of functionals
(13) is carried out as follows. Until the moment of the
breakthrough, pressure and velocity are determined
according to Darcy's law, then saturation is specified
using the above method. This is how the task data is
adjusted. This method is also convenient to use in the
cyclic processing of wells. Numerical experiments
with specific technological indicators and forecast
calculations for real oil and gas fields in the western
region of the Republic of Kazakhstan have been
carried out. Due to limitations, the full results are not
given, but they are presented in the scientific project.
EARTH SCIENCES AND HUMAN CONSTRUCTIONS
DOI: 10.37394/232024.2024.4.17
Saltanbek Mukhambetzhanov
E-ISSN: 2944-9006
144
Volume 4, 2024
Acknowledgement:
The results obtained were carried out within the
framework of a scientific project (BR18574136)
“Development of deep learning methods and
intellectual analysis for solving complex problems of
mechanics and robotics” of the Ministry of Higher
Education of the Republic of Kazakhstan. The
specified scientific project is a source of funding.
References:
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[2] R. Burridge, J. B. Keller, Poroelasticity equations
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70, issue 4, 1981, pp. 1140 – 1146.
[3] E.Sanchez-Palencia, Non-homogeneous media
and vibration theory, Lecture Notes in Phys., 127,
Springer-Verlag, Berlin-New York, 1980.
[4] Mukhametzhanov S.T. Bektemesov M.A.,
Identifiability in the whole of the two – dimensinal
nonlinear equation of heat conductivity
//ABSTRACTS of the International Conference
“Inverse Problems: Modeling and Simulation” held
on June 07-12, 2004 at Fethiye, TURKEY. - Fethiye,
2004. - Р. 24 – 25.
[5] Orunkhanov M., Kornilov V., Mukhametzhanov
S.T. On the Inverse Problems of the Geoelectrics //
ABSTRACTS of the International Symposium on
Inverse Problems in Engineering Mechanics 2003:
18-21 February 2003, Nagano City, JAPAN. -
Nagano City, 2003. - Р. 148.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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EARTH SCIENCES AND HUMAN CONSTRUCTIONS
DOI: 10.37394/232024.2024.4.17
Saltanbek Mukhambetzhanov
E-ISSN: 2944-9006
145
Volume 4, 2024
