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УДК 550.8.013
Permeability Anisotropy of Fractured Reservoirs
Valery M.Kiselev* Anatoly V.Chashkov^
Siberian Federal University, Svobodny 79, Krasnoyarsk, 660041
Russia
Received 12.08.2009, received in revised form 30.09.2009, accepted 20.10.2009 A mathematical model of permeability in fractured reservoir beds which have block structure has been presented. The functional dependence of permeability index for arbitrary direction on geometrical size of the block has been obtained.
Keywords: permeability anisotropy, permeability index, reservoir beds, oil and gas deposits.
Introduction
The permeability index K is one of the most important features of a reservoir bed, which reflects the ability of a bed to filter the fluids under the pressure gradient. Keeping this feature in mind is an essential condition for the correct operation of the oil and gas deposits.
For the usual, the so called porous, isotropic reservoirs with intergrain permeability, the coefficient K is determined by the Darcy filtration law [1]:
u = Q = K (1)
S uL
in which u is filtration velocity, Q is fluid expenditure volume, S is the area of filtration, AP is the pressure differential over the distance L, u is the coefficient of dynamic viscosity of the fluid. The permeability measure unit is 1 um2. The permeable reservoirs are described as K > 10-2 Um2, the impermeable ones are described as K < 10 4 um2.
The discovery of oil deposits in West Texas, in the Middle East and in the southern regions of the USSR gave rise to a heightened interest to fractured reservoirs in the 30-50s of the last century. It turned out that the permeability anisotropy, conditioned by fractureability of reservoirs, significantly influences upon the character of hydrodynamic processes, taking place in the working seam. There are numerous examples of the distinct difference in work modes of producing wells when they are located at an equal distance from the pumping well.
The majority of the research of the fractured reservoirs has been carried out overseas, though there are also significant works by the scientists of our country [1]. A sufficiently good survey of the existing information on the fractured reservoirs is presented in the monograph by T.D.Golf-Raht
[2]. In the recent decades, when the main interest was focused on the deposits in Western Siberia, little attention was paid to the study of fractured reservoirs in Russian’s oil-field geophysics. Also it should be noted that it was possible to describe more or less successfully the discovered and
* e-mail: kvm@akadem.ru ^e-mail: chashkov@kgf.ru © Siberian Federal University. All rights reserved
developed reserves (both in carbonate and in terrigenous stratum) using the framework of the reservoir pseudoisotropical permeability subordinate to the Darcy Law [2]. But recent research of perspective deposits in Eastern Siberia shows that the reservoir beds of these deposits are typical fractured beds with regular fracture formation. This calls for an appropriate modeling of the hydrodynamic processes in such beds considering their permeability anisotropy. Anisotropy permeability data can help essentially in matching production history, while interpreting the results of interference test and hydrodynamic research.
1. Statement of Problem
We shall treat the settled laminar flows in a fractured bed in which there are permeable fractures with flatly parallel walls and impermeable blocks. The flow laminarity is conditioned by the smallness of the Reynolds number Re=p-u-2g/u in narrow fractures with the typical opening value g^100 um and the flow velocity u^ 1 m/s. The weight force compared to the pressure gradient VP is also neglected in equations of Navier-Stokes. Then the average velocity qk of Poiseuille’s plane-parallel flow in fracture k with opening gk can be determined as follows:
= - iu-Vp. (2)
The basic ideas of the fluid flow modeling through the fracture system in a bed belong to S.E.Romm [1]. According to [1] the fluid flow velocity in k direction is computed by means of the formula 3
Uk = 9k fkqk = - jtt- fk(VP -k) -k, (3)
where fk is the linear density of fractions (the number of fractions per linear unit in the direction perpendicular to the selected), k is the unit vector in the k direction. The overall flow velocity Uk is the sum of velocities in n fractures. Introducing the permeability tensor K of the bed, the equation (3) can be written in the following form [1]
P
u = -K------, (4)
u
where the tensor K has the components
1 n
Kij = 12 9kf k (Sij — aik ■ ajk ). (5)
k=1
In the equation (5) Sij is the Kronecker symbol, aik are the cosines of the angles between the coordinate axes and the directions of the unit vectors perpendicular to vector k. If the coordinate axes coincide with the main axes of the permeability tensor, then tensor K assumes a diagonal shape.
Equations (4) and (5) are, as a matter of fact, a formal solution of the problem of permeability anisotropy of fractured reservoirs with impermeable matrix. But it is hard to use it in practice as it requires exhaustive information on the fractured reservoir (dip angles and strike azimuth, their linear density and what is more important their opening in different directions).
This type of information can be obtained with the help of fullbore formation microimager FMI. However, in the overwhelming majority of East Siberian cased boreholes the FMI was not applied. That is why one has to use other data, such as, the results of core macro fissuring analysis,
the well logging interpretation, seismic survey data, as well as the well testing permeability Kweii test determined by the results of hydrodynamic well tests. In [2] it is stated that Kweii test can be assumed to be the average for fractured rock. Estimation of fracture opening 9k according to core seems incorrect, as the gk in the core raised to the day surface differs from their opening under the bed conditions [3].
The purpose of this work is to express the permeability anisotropy of fractured reservoirs by means of experimentally determined linear sizes of the bed by well testing results.
2. Permeability Coefficient Computing
Suppose that the reservoir is an aggregate of a large number of the same size and equally oriented in space rectangular blocks. We will orient the Cartesian system of coordinates along the sides of the blocks. Let xo, yo, zo be the block sizes along the corresponding axes, L be the typical linear size of the bed and L3^ xo-yo ■zo. Then the number of fractions parallel to OX axis will be
L L L2 N L
Nx = — ■ — = --------. Hence the linear fraction density along the OX axis is fx = = -----.
yo zo yozo L yozo
By analogy for the OY and OZ axes we obtain fy = —^ = ----------------, fz = = -. Let’s
L xozo L xoyo
assume that all the fracture openings are the same and equal to g, and the pressure gradient in
all directions is constant and equal to AP/L. Then according to (3)
93 L AP 93 L AP 93 L AP
ux = —^-----------Ti uy = —^--------------T'> uz = —^--------------------------T-
12 yozo uL 12 xozo uL 12 xoyo uL
Anisotropy coefficients along the selected axes are:
Kx = g3L , Ky = 9l , Kz = 9l .
12yozo 12xozo 12xoyo
Let’s introduce the bed anisotropy parameters:
ux Kx xo b ux Kx xo
uy Ky yo uz Kz zo
(6)
Let at a certain moment at the origin of the selected system of coordinates a fluid pressure gradient be created in the bed. If the medium is isotropic in permeability, then the flow front equation at the moment of time t may be written down as
\Jx? + y2 + z2
t = ---------------- = const,
u
where u is the flow velocity. The flow front has the shape of a sphere. If the fluid spreads only along the fractures, then the flow front equation from the spot source at the moment t will have the following form:
x y z
t =------1------1---= const,
ux uy uz
or taking into account (6)
x + a ■ y + b ■ z = const. (7)
If the distance to the front exceeds the linear sizes of impermeable blocks, then in (7) we can proceed to increments:
dx + a ■ dy + b ■ dz = 0. (8)
In the spherical coordinates (r, 9, p) the equation (8) can be written as follows
dr sin 9 • sin p — a • sin 9 • cos p ^ cos 9 • cos p + a • cos 9 • sin p — b • sin 9 ^
r sin 9 • cos p + a • sin 9 • sin p + b • cos 9 sin 9 • cos p + a • sin 9 • sin p + b • cos 9
Hence
1 + tg2 —
r(9’p> = CX(9,p) ■«»,*) = (9)
where C is the integration constant,
x(9, p) = sin 9 • cos p + a • sin 9 • sin p + b • cos 9, (10)
^(9,p) = sin 9 • (l + 2 • a • tg pp — tg2 pj + b • cos 9 • (l + tg2 pj , (11)
1 + tg2 —
w(9,p)= x(9, p) • ^(9,p)' (12)
According to (9) the flow front geometry is determined by functional form w(9, p), which, in its turn, depends on anisotropy parameters of the bed a and b. The obtained solution (9) makes physical sense only for p G [0,n/2], 9 G [0, n/2]. The solution for other directions can be easily
constructed proceeding from the flow symmetry relative to the XOZ, XOY and YOZ planes.
The considered solution peculiarities are typical for the spherical coordinates system, in which the cross points of Cartesian system of coordinates with a sphere are singular points.
i 1 + a?
The absolute minimum w(9, p) corresponds to the polar angle 9m '
1 + a2 + b2
and to the azimuth angle pmin = arcsin . . In the direction of OX, OY, OZ axes the
V 1 + a2
(n \ /nn\ 1 1
functions w(9, p) are: w —, 0 =1, w —, — =^r, w(0, p) = —.
2 2 2 a2 b2
We will find the integration constant C in equation (9) proceeding from the fitting condition with the average permeability data Kweii test according to well test results. The bed is supposed to be isotropic in these measurements, that is why the seeming position of the flow front rT at the time t in concordance with (1) is determined by the equation
AP
rT |u| • t Kwell test • T~ • T C1 • Kwell test• (13)
yU,L
On the other hand, the value rT computed by means of the equation (9) corresponds to the
average value of the radius vector, which determines the position of the real flow front, namely
n 2n
rT = —2 / / r(9, p)d9dp = C •w, (14)
where
2n2
0 0
n 2n
w = i// dedp (15>
00
For fractured ground at the same time t
AP
r(9, p) = K(9, p) —— • t = Ci • K(9, p) = C • w(9, p) . (16)
UL
Thus, from (13), (14) and (16) we obtain the system of equations
j Cl • Kweii test = C • w,
\ Cl • K(0,?) = C • w(0,?).
After eliminating the constants C and C1 we get
K(0,?) = Kwelltest • ^4^ . (17)
w
The equation (17) is the solution of the posed problem. The total permeability Kweii test is determined by the results of hydrodynamic survey of the wells, the w(0, ?) and w functions are computed by means of the equations (9) and (15) and depend only on the geometric sizes of the blocks. The fracture opening value is hard to determine experimentally, but we are not dealt with this value in the equation (17). It is also important to mention that the permeability in fractured beds is proportional to the cube of fracture opening.
3. Model Example of Permeability Anisotropy
To demonstrate how much Kweii test may differ from the real bed permeability in the given directions let’s consider the following example. Let the block geometry be given by a = xo/yo = 2,b = x0/z0 = 5. According to (15) w = 0,07. Impermeable blocks are stretched out along the OX axis, the maximum linear fracture density is perpendicular to it, that is why Max(K/Kweii test) = 14, 29 is achieved along the OX axis (0 = n/2,? = 0). Min(K/Kweii test) = 0, 47 is for the direction 9min = 24°. Therefore, Kmax/Kmin = 30, 4. From this example it is evident that in certain directions the bed permeability may exceed the one measured by well test approximately in single order or more.
16
0 30 60 90 120 150 180
Angle, degree
-------Kxy-'Kwell test - - - ■ &:z-Kv,rll test------------KyzKwen test
Fig. 1. Angle dependence of permeability in XOY, XOZ, YOZ planes (a=2, b=5)
The relative permeability changes in three main planes of the Cartesian system of coordinates for this case are presented in fig. 1.
The ratio Kmax/Kmin can be treated as the fractured bed’s permeability anisotropy feature. The changes of this value depending on the value of parameter b are presented in fig. 2. The value of parameter a was fixed and equal to 1. In this case
Kmin w(Omin Turnin')
2 + b2 ^min a/2
Fig. 2. Permeability anisotropy depending on b parameter (a= 1)
Kmax 1
where Omin
According to computations for a=1, b=1 the ratio K/Kweii test changes from 0,66 to 1,98, for a=1, b=10 the ratio K/Kwell test changes from 0,19 to 19,61. In other words, the permeability of fractured beds in certain directions may be several times less, as well as dozens of times more than the permeability measured by the well test results.
4. Transformation of Coordinates
The geophysical survey is carried out and interpreted in the system of coordinates OX’Y’Z’ connected with the day surface: OZ’ axis goes vertically down, OX’ axis runs to the north, OY’ axis to the east. To obtain type function w(O, p) in the system of coordinates OX’Y’Z’, it is necessary to turn the original system of coordinates OXYZ, connected with the system of orthogonal fractures around the general point of origin using the Euler angles ^o, Oo and po [4]. If the OX axis is oriented along the line of knots, then these three angles in our case can be defined as follows: the angle of precession ^0 is the azimuth of the dip angle of the fractures oriented along the OZ axis; the nutation angle Oo is the supplementary angle to the dip angle of these fractures; the pure rotation angle po is the dip angle of the fractures oriented along the OX axis (the strike azimuth angle of these fractures is ^o = 180°). The rotation matrix of the OXYZ system in the OX’Y’Z’ position around the general point of origin has the following components, expressed in terms of Euler angles [4]:
cii = cos ^o cos po — sin ^o sin po cos Oo; C21 = — cos ^o sin po — sin ^o cos po cos Oo;
C12 = cos ^o sin po + sin ^o cos po cos Oo; C22 = — sin ^o sin po + cos ^o cos po cos Oo; , ,
C13 = sin po sin Oo; C23 = cos po sin Oo;
C31 = sin ^o cos Oo; C32 = — cos ^o sin Oo; C33 = cos Oo.
Then the equation of transformation of coordinates (O, p) in the OXYZ system into the , p') system in the matrix type will look as follows:
sin O' cos p' \ I C11 C12 C13 \ I sin O cos p
sin O' sin p' = C21 C22 C23 • sin O sin p | . (19)
cos O' / \ C31 C32 C33 ) \ cos O
5. Conclusion
The main feature of the model of the fractured reservoir’s permeability anisotropy presented above is that the fracture opening value that is hard to determine in experiments has not been dealt with. The permeability of the reservoir can be computed within the frames of this model in any selected direction. It is not difficult to compute the diagonal components of permeability tensor, if this is required by the need to use the obtained results in the standard simulation program of the hydrodynamic processes in the reservoirs. It is worthwhile to mention once again those simplifications which have been made regarding a real fractured reservoir. Firstly, the spatial orientation and linear sizes of impermeable blocks were taken to be constants, and the blocks were presented as rectangular parallelepipeds. Secondly, the fracture permeability in three orthogonal directions was taken to be a constant. Thirdly, we neglected the cavernous and intergrain permeability.
The permeability model presented above was used in computing for one of the Yurubchenskoe oil field deposit sites. It is expected that the peculiarities of geological, geophysical and fluidodynamic characteristics of this deposit allow us to apply simplified mathematical models in estimation of the physical properties of reservoirs. It has been illustrated by means of this example that in certain directions permeability can exceed more than one order.
The permeability computed by the results of well-logging, and the permeability anisotropy (maximal permeability ratio to the minimal one) may be more than one hundred. All these circumstances have to be taken into consideration while preparing this deposit for development.
References
[1] T.D.van Golf-Racht, Fundamental of fractured reservoir engeneering, Amsterdam, Elsevier, 1982.
[2] E.S.Romm, Fluid flow in fractured rocks, Moscow, Nedra, 1966 (in Russian).
[3] V.M.Dobrynin, Deformations and physical changes of oil and gas reservoirs, Moscow, Nedra, 1970 (in Russian).
[4] A.I.Borisenko, I.E.Tarapov, Vector analysis and principles of tensor calculus, Kharkov, Vi-shcha shkola, 1986 (in Russian).
