CC BY
63UDC 519.6 10.23947/2587-8999-2018-2-2-133-143
Researching the mechanisms of fluid flow in the fracture-porous reservoir based on mathematical modelling *
Yu.O. Bobreneva **, A.A. Mazitov***, I. M. Gubaydullin
Institute of Petrochemistry and Catalysis of the Russian Academy of Sciences, Ufa, Russia Ufa State Petroleum Technological University, Ufa, Russia
Paper covers the researching the process of fluid flow in the reservoir in the fracture-porous reservoir based on mathematical modelling. The model of the dual porosity of Warren and Root was used. The model has two pore systems - a system of fractures and a system of matrices with different values of geometric dimensions and filtration-capacitive properties. The pressure distribution in the "system of fracture-matrix" system is described by the piezoconductivity equations. The paper presents a numerical solution of the problem under consideration. Рartial differential equations were approximated using an implicit difference scheme. The matrix sweep method was used for the calculation. Model pressure recovery curves were obtained. Analysis showed that the specific conductivity coefficient depends on the size of the matrix blocks and possible to evaluate the process of manifestation of the effect of dual porosity.
Keywords: fracture-porous reservoir, «system of fracture-matrix», pressure build-up curves, specific conductivity coefficient
Introduction. The role of carbonate reservoirs in the development of the oil industry in Russia is increased. Аs a rule, productive reservoir with dual porosity are not sufficiently studied in comparison with ordinary sandstone in terrigenous reservoirs [1,2,3,4]. Carbonate reservoirs is characterized by a number of specific features that are associated with the flow of fluid in environment with a dual porosity [5]. The development of methods for mathematical modeling of fluid flow in a given medium is an urgent problem. Due to their physicochemical properties, susceptibility to cracking, and recrystallization, carbonate reservoirs form a complex microstructure of the void space. The main characteristics of such rocks are fracture and cavernousness [6]. The main cause of the appearance of fractures in the rock is deformation phenomena when the stresses resulting from the action of mechanical loads of various nature, as well as tectonic movements and sedimentation processes, change. Fractures are violations of the continuity of the rock. Geometrically, they are characterized by a significant difference in dimensions in the fracture plane (width and length of fractures) and in the perpendicular direction (fracture opening or height). Fractures observed in carbonate rocks can be completely or partially filled with various mineral substances, for example carbonate or sulfates. Along with them, fractures that remain hollow or
* The reported study was funded by RFBR according to the research projects № 16-29-15116
** E-mail: yu.o.bobreneva@gmail.com.
*** E-mail: mazitov.ainur13@gmail.com.
open can be distinguished. Also, fractures can be filled with oil or bitumen. Disclosure of mineral fractures varies in very wide limits: from fractions of a millimeter to 1 cm or more. As a rule, openness of open fractures does not exceed 20-25 microns [1].
The appearance of a system of interconnected fractures in the rock can change the filtration properties of productive deposits [7, 8].
The technology for reservoir development with dual porosity can be implemented only on the basis of a comprehensive study of the mechanisms of filtration in heterogeneous fractured-porous reservoirs. Hydrodynamic methods for studying of fractured reservoirs due to strong heterogeneity differ significantly from traditional methods [9, 10]. Such reservoirs are characterized by an intensive exchange fluid flow between the fractures and porous blocks (matrix), which introduces certain corrections to known methods for determining reservoir parameters.
Fractured layer is characterized by the discreteness of the properties and parameters of the channels due to the presence of two types of voidness. The matrix has smaller pores and is more fine pores (or voids) and has a significant capacity, but low filtration properties. Fractured - low capacity and high filtration properties. Different authors studied the calculation of the flow characteristics under special conditions: Odeh, Kazemi, DeSwaan and Pollard, etc [13, 14, 15, 16]. The authors proposed various development methods based on simplified reservoir models. But, despite the great variety of approaches of different authors they all boil down to either particular cases or exceptions to the Warren-Ruth model. The Warren-Ruta model represents the general case and is the best method for describing the process of fluid filtration in a fractured formation under unsteady filtration conditions [17].
An analytical solution for the Warren and Root model can not be obtained in general form. There is an analytic expression that represents an approximate solution for some particular cases. In this paper, we consider the numerical solution of the filtration process in a fractured-pore-type collector based on the Warren and Root dual porosity model.
Problem statement.
The model considers a porous collector, schematized by the same rectangular parallelepipeds as shown in Figure 1. The collector or matrix has a high porosity and low permeability. The low-permeability matrix is divided by a system of natural fractures that have high permeability and low porosity. It is believed that the movement of fluid to the well is carried out through a system of fractures, and the matrix is a capacitance that continuously feeds the entire system of natural fractures. The redistribution of the fluid between the matrix and the fractures depends on the shape and size of the matrix blocks, the smaller the blocks, the easier the fluid flow between them [16, 17, 18]. The matrix and fracture have individual properties and are characterized by their own permeability, compressibility and porosity in the dual porosity model.
Fig. 1. The dual porosity model
To describe the filtering mechanism in the «system of fractures-matrix» system, the following equations of mathematical physics are used:
r)P k
m _ ç m i n p ^ Vmctm Qt S (Pf Pm)•
(c dPf = ld(kfrdPf)+Skm(p p)
(fCtf ~dr-Vd;(jr^) + ST(Pm-Pf)'
The following initial and boundary conditions are considered: Pfl =P0-AP, Pfl =P0,
J <r=0 0 J 'r=rp 0
(1)
PAt=0 = P0, Pmlt=0 = P0, (2)
^ \ or /r=rw
s = 4^n(n + 2)
L2
3• a^b • c
L = —t—T---, (4)
a^b + b^c + c^a
where, (fif - is the porosity of the natural fractures system, (m - is the porosity of the matrix, ctf -is the compressibility of the fractures system (1/Pa), ctm - is the compressibility of the matrix (1/Pa), kf - is the permeability of the fractures system (m2), km - is the permeability of the matrix (m2), ft — is the oil viscosity (Pa-s), Pf - is the formation pressure in the fractures system (MPa), Pm
- is the reservoir pressure in the matrix (MPa), h - is the effective thickness of the formation (m), q
- flow rate of liquid (m3/ day), n « 3.14159, S - is the coefficient of fractured rock (1/ m2), n - is
the number of mutually perpendicular fracture groups, L - is the block size (m), a - is the block side length (m), b - block side width m atrium (m), c - height of the matrix block side (m).
The dual porosity model is characterized by two additional parameters: storativity ratio (w) and transmissivity ratio (A). a> - is the fraction of fractures in the total formation system, the higher the coefficient, the greater the fracture-cavernous capacity in the reservoir.
VfCf
Ù)
(5)
<Pf cf + <P mc m
A - is the ability to filter from the matrix into fractures. This coefficient depends on the size and geometry of the matrix blocks. As the coefficient increases, the ability of the matrix to participate in filtering the system increases. Matrices of low permeability are characterized by lower values of the coefficient.
k
A = (6)
Kf
where, S - is the characteristic coefficient of the fractured rock (1/m2), rw - is the radius of the well (m).
Based on various calculations, the following order of magnitude of these parameters was established:
10-3 < A < 10-9 (7)
- corresponds to small values of S - blocks of large sizes, small values of km - impermeable matrix, and high values of kf - significant crack opening.
10-2 < A < 10-4 (8)
- corresponds to ^-cy >> (pmcm, and often (p f >> (pm
The limits of applicability of the parameters ro ^ 0, X ^ 0 and ro ^ 1, X ^ ro are due to the basic physical parameters, such as the voidness (porosity), permeability, crack density and block size. In some limiting cases, a system with a double porosity can be reduced to a system with one type of voidness.
Thus, the amount of calculations for the explicit scheme is significantly increased (approximately by 3 orders of magnitude) [19]. Using an implicit difference scheme [20] allows you to select an arbitrary grid, including an uneven grid.
pj + 1-pj b
= -/>{+') (I0)
= (r^l - (>•,+! - /r + + kT
(11)
To calculate the implicit scheme, the matrix sweep method was used [19, 20]. Matrix sweep refers to direct methods for solving difference equations. In comparison with other direct methods for solving difference problems, matrix sweeping is more universal, since it allows solving equations with variable coefficients and does not impose strong restrictions on the form of the boundary conditions.
The algorithm of matrix sweep. The system (10-11) can be reduced to a general form:
APi-i-CPi+BPi+i = -Fi
(12)
The system of linear algebraic equations with a block tridiagonal matrix needs to be solved. The solution of the system is found recursively by the formulas [21]:
ai = Co pi = C-1Fo;
ai+i = (Ci-Aiai)-1Bi, i = 1,2,...N-1 ßi+i = (Ci-Aiai)-1(Aißi + Fi), i = 1,2, ... N Pi = ai+iPi+i + ßi+i, i = N — 1,N — 2, ...1,0
PN = ßN + 1,
(13)
(14)
(15)
where a and fi are coefficients. The elements of the tridiagonal matrix are matrices (the dimension of the matrix in question is 2 * 2):
Ai =
0
0
0 -
1 kf
1
i
rft(pfctfh2 i-2
(16)
Ci =
1+
^^ T" ft(m Ctm
y ^^ T" WfCtf
Bi
y kk ^^ T"
ft(m ^tm
1+-
1 k
(r. i—r. l) +■
V ï J____ï J__
rft(fCtfh2 i+2 U2 WfCtf.
(17)
1
0 -
k
i
rft(fCtfh2
Fi =
P
1 77'
P
f
(18)
(19)
Results of numerical simulation. At the initial time, the production well was put into operation with a flow rate of 100 m3/day. After working out some period, at the time of 10 days the well is closed on the face. During stopping (q = 0 m3/day), the pressure in the formation begins to recover. The following initial and boundary parameters are set for the well and the region under consideration:
Table 1. The following initial and boundary parameters
1
0
0
1
Parameters Value Unit of measurement
oil viscosity, ft 2.2E-3 Pas
initial fracture pressure, Pf0 250.0E5 MPa
initial pressure in the matrix, Pm0 250.0E5 MPa
permeability of fractures, kf 1000.0E-15 m2
permeability of the matrix, km 1.0E-15 m2
compressibility of fractures, ctf 3.0E-9 1/Pa
compressibility of the matrix, ctm 3.0E-10 1/Pa
porosity of the natural fractures system, (pf 0.01
porosity of the matrix, (pm 0.10
number of mutually perpendicular fracture groups, n 3
block length, a 100 m
width of the block, b 100 m
block height, c 1 m
reservoir thickness, h 10 m
radius of the well, rw 0.10 m
well supply circuit, Re 100.0 m
Analysis of the modeling of hydrodynamic studies by the method of the pressure recovery curve in the production well was carried out [22]. The figures 2 below show the results of numerical simulation.
Fig. 2. Results of numerical simulation. Pressure dynamics
The figure 2 shows the calculations for two cases: for a block height of 1 m and for a block height of 40 m. Note that with a block height of 40 m, the pressure is restored more slowly. For the obtained pressure curves, derivatives were constructed.
The effect of dual porosity is manifested in the early times in a short period of time (a derivative jump downwards) due to the small size of the matrix blocks.
10
Q ^__i_i.......|_i_i.......|_i_i.......|_i_i.......|_i_i.......|_i_i.......|
0.001 0.01 0.1 1 10 100 1000
C 2 -C 1
Fig. 3. Log-log plot, block height 1 m and 40 m
For larger block sizes, the effect of dual porosity appears in later times.
Conclusion. The analysis showed that the transmissivity ratio (X) depends on the size of the matrix blocks, namely: as the size of the matrix block increases, the transmissivity ratio decreases, and the ability of the matrix to participate in filtering the system decreases accordingly.
It was also noted that of the three sides of the block, the height of the matrix block makes a greater contribution. The effect of dual porosity manifests itself in the earlier time region as it increases, and vice versa, the effect of dual porosity is manifested late when height decreases (redistribution of pressure between the crack and the matrix).
When the parameter ro is varied, i.e. change in porosity and compressibility of fractures and matrix, conclusions were drawn too. As this coefficient increases, the volume of fractured-cavernous capacity increases in the reservoir, respectively, the effect of dual porosity occurs later.
As a result of the conducted research on the basis of mathematical modeling, we obtained that the geometry of the conductive cracks distribution, the permeability must be take into account to predict the productivity of wells, the success of various geological and technological measures.
References
1. Denk S O 2004 Problemy treshchinovatyh produktivnyh ob"ektov (Perm': Ehlektronnye izdatel'skie sistemy)
2. Chernickij A V 2002 Geologicheskoe modelirovanie neftyanyh zalezhej massivnogo tipa v karbonatnyh treshchinovatyh kollektorah (Moscow: OAO «RMNTK Nefteotdacha»)
3. Charnyj I A 1963 Podzemnaya gidrogazodinamika (Moscow: Gostoptekhizdat)
4. Kotyahov F I 1977 Fizika neftyanyh i gazovyh kollektorov (Moscow: Nedra)
5. Nelson R A 2001 Geologic analysis of naturally fractured reservoirs 2nd ed (Woburn: Butterworth-Heinemann)
6. Narr W, Schechter D S and Thompson L B 2006 Naturally Fractured Reservoir Characterization (Society of Petroleum Engineers)
7. Barenblatt G I, Entov V M and Ryzhik V M 1984 Dvizhenie zhidkostej i gazov v prirodnyh plastah (Moscow: Nedra)
8. Smekhov E M 1987 Vtorichnaya poristost' gornyh porod-kollektorov nefti i gaza (Leningrad: Nedra)
9. Earlougher R C Jr 1977 Advances in Well Test Analysis 2nd printing (Dallas: Millet the Printer)
10. Rodriguez F, Arana-Ortiz V and Cinco-Ley H 2004 Well Test Characterization of Small- and Large- Scale Secondary Porosity in Naturally Fractured Reservoirs (Society of Petroleum Engineers)
11. Zheltov Yu P 1975 Mekhanika neftegazonosnogo plasta (Moscow: Nedra)
12. Kalitkin N N 2011 Chislennye metody (Saint Petersburg: BHV-Peterburg)
13. Odeh A S 1965 Unsteady-state behavior of naturally fractured reservoirs (Soc.Petrol.Eng.J.) pp 60-66
14. Kazemi H, Seth M S, Thomas G V 1969 The interpretation of interference tests in naturally fractured reservoirs with uniform fracture distribution (SPEJ) pp 463-472
15. DeSwaan A O 1976 Analytic solutions for determining naturally fractured reservoir properties by well testing ( SPEJ) pp 117-122
16. Van Golf-Racht T D 1982 Fundamentals of fractured reservoir engineering (Amsterdam: Elsevier Scientific Publishing Company)
17. Warren J E, Root P J 1963 The behaviour of naturally fractured reservoirs (Soc.Petrol.Eng.J.) pp 245-255
18. Bobrenyova Yu O, Davletbaev A Ya and Gubajdullin I M 2017 Proc. Int. Conf. «Differencial'nye uravneniya i ih prilozheniya v matematicheskom modelirovanii» (Saransk: Izd-vo Sredne-volzhskogo matematicheskogo obshchestva) pp 232-235
19. Samarskij A A and Nikolaev E S 1978 Metody resheniya setochnyh uravnenij (Moscow: Nauka)
20. Kuznecov G V and Sheremet M A 2007 Raznostnye metody resheniya zadach teploprovodnosti (Tomsk: TPU)
21. Samarskij A A and Gulin A V 1989 Chislennye metody (Moscow: Nauka)
22. Pollard P 1959 Evaluation of acid treatments from pressure build-up analysis (Trans.AIME.vol) pp 38-43.
Authors:
Bobreneva Yuliya Olegovna, Ufa State Petroleum Technological University (1 Kosmonavtov St., Ufa, Russian Federation)
Mazitov Aynur Asgatovich, Ufa State Petroleum Technological University (1 Kosmonavtov St., Ufa, Russian Federation)
Gubaydullin Irek Marsovich, Institute of Petrochemistry and Catalysis of the Russian Academy of Sciences (141 Oktyabrya avenue, Ufa, Russian Federation), Ufa State Petroleum Technological
University (1 Kosmonavtov St., Ufa, Russian Federation), Doctor of Science in Physics and Maths, Associate professor
УДК 519.6 10.23947/2587-8999-2018-2-2-133-143
Исследование механизмов фильтрации в коллекторе трещиновато-порового типа на основе математического моделирования*
Ю.О. Бобренёва**, А.А. Мазитов***, И.М. Губайдуллин
Институт нефтехимии и катализа РАН, Уфа, Россия
Уфимский государственный нефтяной технический университет, Уфа, Россия
Рассматривается процесс фильтрации жидкости в коллекторах трещиновато-порового типа. Фильтрация описывается с помощью модели двойной пористости, в которой присутствуют сеть естественных трещин и поровый пласт (матрица) с различными фильтрационно-емкостными свойствами. В результате решения задачи получены модельные кривые восстановления давления. Анализ результатов моделирования исследования методом кривой восстановления давления в добывающей скважине показал, что удельный коэффициент проводимости зависит от размеров матричных блоков. Проведен анализ для различных параметров системы, который позволил оценить процесс проявления эффекта двойной пористости.
Ключевые слова: коллекторы трещиновато-порового типа, модель двойной пористости, модельные кривые восстановления давления, удельный коэффициент проводимости
Авторы:
Бобренёва Юлия Олеговна, Уфимский государственный нефтяной технический университет (450062, Уфа, ул. Космонавтов, д. 1), аспирант
Мазитов Айнур Асгатович, Уфимский государственный нефтяной технический университет (450062, Уфа, ул. Космонавтов, д. 1), магистрант
Губайдуллин Ирек Марсович, Институт нефтехимии и катализа РАН (450075 Уфа, Проспект Октября, д. 141), Уфимский государственный нефтяной технический университет (450062, Уфа, ул. Космонавтов, д. 1), доктор физико-математических наук, доцент
* Работа выполнена при частичной поддержке гранта РФФИ № 16-29-15116.
** E-mail: vu.o.bobreneva@gmail.com.
*** E-mail: mazitov.ainur13@gmail.com.
