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Инвариантный интеграл физической мезомеханики как основа математической физики: некоторые приложения к проблемам космологии, электродинамики, механики и геофизики
Г.П. Черепанов
Нью-Йоркская академия наук, Нью-Йорк, 10007-2157, США
Инвариантный интеграл физической мезомеханики, основанный на законе сохранения энергии, вводится в настоящей работе с учетом гравитационной, космической, электромагнитной, упругой и термической энергии поля/материи, а также релятивистской энергии массы. Таким образом, физическая мезомеханика становится мегамеханикой, охватывающей почти все масштабы природы. Приведены несколько основных законов, вытекающих из данного инвариантного интеграла, в том числе: обобщенный закон Архимеда для тел, частично погруженных в жидкость, закон гравитации Ньютона, обобщенный на единое гравитационно-космическое поле, обобщенный закон Кулона для движущихся электрических зарядов и др. При помощи инвариантного интеграла изучен тепловой след движущихся трещин и дислокаций и найдено поле температуры в зависимости от скорости движения, а также от вязкости разрушения для трещин нормального разрыва или от трения скольжения для краевых дислокаций. Для пористых материалов, насыщенных жидкостью или газом, соответствующие инвариантные интегралы вводятся с использованием концепции двойного (бинарного) континуума. Применительно к горизонтальному бурению и гидроразрыву горных пород Земли определены поля давлений и скоростей ископаемого газа/нефти, а также дебит как для горизонтальных буровых скважин, так и для дискообразных трещин гидроразрыва, исходящих из скважин. Предложена теория фрекинга сланцевого газа/нефти для трех основных режимов проникания бурового раствора в разрушенную породу вблизи горизонтальной скважины. Определены форма и размеры разрушенной области вблизи скважины, а также дебит скважины в зависимости от горного давления, диаметра скважины, давления сланцевого газа в порах, а также давления и расхода бурового раствора в процессе фрекинга. Для решения указанных проблем, кроме инвариантного интеграла, использован метод пограничного слоя, а также метод функциональных уравнений в теории функций комплексного переменного.
Ключевые слова: инвариантный интеграл, закон Ньютона, закон Кулона, закон Архимеда, разрушение, фрекинг, горизонтальное бурение, добыча сланцевого газа/нефти, трещины и дислокации
The invariant integral of physical mesomechanics as the foundation of mathematical physics: some applications to cosmology, electrodynamics, mechanics and geophysics
G.P. Cherepanov
The New York Academy of Sciences, New York, 10007-2157, USA
The general invariant integral based on the energy conservation law is introduced into physical mesomechanics, with taking into account the cosmic, gravitational, mass, elastic, thermal and electromagnetic energy of matter. The physical mesomechanics thus becomes a mega-mechanics embracing most of the scales of nature. Some basic laws following from the general invariant integral are indicated, including Coulomb's law of electricity generalized for moving electric charges, Newton's law of gravitation generalized for coupled gravitational/cosmic field, the Archimedes' law of buoyancy generalized for bodies partially submerged in water, and others. Using the invariant integral the temperature track behind moving cracks and dislocations is found out, and the coupling of elastic and thermal energies is set up in fracturing and plastic flow, namely for opening mode cracks and edge dislocations. For porous materials saturated with a fluid or gas, the notion of binary continuum is used to introduce the corresponding invariant integrals. As applied to the horizontal drilling and hydrofracturing of boreholes in the Earth' crust, the field of pressure and flow rate as well as the fluid output from both a horizontal borehole and a disk-shape fracture issuing the borehole, are derived in the fluid extraction regime. A theory of fracking in shale gas/oil reservoirs is suggested for three basic regimes of the drill mud permeation into the multiply fractured rock region, with calculating the shape and volume of this region in terms of the geometry parameters and pressures of rock, drill mud and shale gas. The method of functional equations in the theory of a complex variable and the boundary layer method are also used to solve these problems.
Keywords: invariant integral, Coulomb's law, Newton's law, Archimedes' law, fracturing, horizontal drilling, shale gas/oil extraction, theory of fracking, cracks and dislocations
© Cherepanov G.P., 2015
1. Introduction
Let us consider physical fields that are stationary in the Cartesian frame of coordinates Ox1x2x3. It is assumed that the mathematical image of the matter under study is represented by the field and mass energy and by basic parameters of state. We confine ourselves by the combined field of cosmic, gravitational, thermal, elastic, and electromagnetic fields in dielectrics.
The field parameters under consideration are as follows: x1, x2, x3) is the potential of coupled gravitational/cosmic field (per unit mass), U(e j, Di, Bi, T) is the field energy per unit volume, T(x1, x2, x3) is the temperature, vi (xj, x2, x3) are the velocity components (i, j = 1, 2, 3), wi; j (xj, x2, x3) are the distortions, etj (xj, x2, x3) are the strains, Oy (xj, x2, x3) are the stresses, Ei, Dt, Ht, and Bt are the components of the vectors of electromagnetic field (some functions of xj, x2 and x3, and K(v) is the energy density of a moving mass, which is equal to the kinetic energy plus the Einstein constant when v << c, as follows 2
K (v) =
Pm c
(1)
K = pmc2 + j pmv2, when v << c.
Here pm is the mass density at rest, c is the speed of light in vacuum, and v is the value of the local velocity of matter in the Oxjx2x3 frame of coordinates.
The state equations of elastic dielectrics are
Z7 _ dU „ _dU =du
Ei =âD ' Hi=m ' ■
(2)
In linear approximation, function U(eij, Di, Bi, T) is a sum of some quadratic functions of eij, Dt, and Bi plus term Kb at eii (T - T0), where Kb is the bulk modulus, at is the coefficient of thermal expansion, and T0 is the reference temperature.
physical terms, quantity r(rj, r2, r3) called the driving force can be a real equivalent force upon the matter inside Q, or a configuration force, or an energy loss vector [1]. It measures the exchange of energy between the field at hand and other coupled fields, while the exchange taking place at field singularities.
If there are no field singularities inside Q, then rk = 0. In this case, equations (2) as well as Maxwell's equations and thermoelasticity equations, all of which can in this case be derived from Eq. (3) by using the divergence theorem, are valid at every point inside Q [1]. If there is a field singularity inside Q, then rk ^ 0 and r is called the singularity-driving force. By a field singularity, the coupling binds together the explicit energy expressed by a field equation and a latent energy hidden at the field singularity.
The common field singularities are point charges, linear currents, point holes, the front of cracks and dislocations, point inclusions, vacancies, concentrated forces, moments, and torques, etc. Making use of the procedure of the r-integration for divergent invariant integrals in Eq. (3) provides a very effective method to calculate the force upon a field singularity, and so, to get any physical law of the interaction forces [1]. In a sense, based on the mass-energy E* = M*c2 and space-time L* = cT* dualisms, mass and momentum are some forms of energy which, under common conditions, are independent of other forms of energy [1]. Here M*, E*, L*, and T* are the characteristic mass, energy, length, and time, respectively.
Let us provide some examples of the application of the general invariant integral in Eq. (3).
2.1. Newton s law of inertia
Suppose a rigid body moves along axis xj at some variable velocity v = v (xj). In this case, from Eqs. (3) and (1) we can find
Tj = | KnTdQ = j K ;IdV =
2. General invariant integral
The law of energy conservation for the combined field under consideration can be written in the shape of the general invariant integral as follows [1, 2]
I* = f[(8nG)-j(9,in -2^,-n-) + Aynk +
kt
+ Knk +PmChTnk + Unk + — Tinivk -
V
- °ijnjuik + EiDink + HiBink -- DtntEk - DnEk ]d&, i, j, k = 1, 2, 3.
(3)
Here rk is the external energy spent to move the matter inside surface Q on unit length along axis xk, Q is an arbitrary closed surface of integration, G is the gravitational constant, A is the cosmological constant, ch and kt are the specific heat and thermal conductivity of the matter. In
r 1 dK dv ^ d
= j---dV = —
^ v dv di di
f
m0 v
f-vv^c
(4)
where t is the time, V and Q are the volume and surface of the body, m0 is its mass at rest, and rj is the force upon the body. In Eq. (4), we used the following equality 1 dK dp = (vK) v dv dt c2 dt
The original Newton's law follows from Eq. (4) when v << c. Some basic invariant integrals of relativistic physics were presented in [3].
2.2. Generalized Coulomb s law for moving electric charges
Suppose two point electric charges q1 and q2 move along axis x1 at constant speed v in their own electromag-
netic field. In this case, from Eq. (3), it follows that the driving force upon the back charge is equal to [1, 4]
r _ v7a2 -1 1 1 , „,2 / 2 •
R 2 1 - v2/c 2
(5)
Here R is the distance between the charges in the proper coordinate frame, e0 is the dielectric constant, and a is the speed of light in the matter (a < c).
At v < a the opposite force of same value acts upon the front charge. At a < v < c the force upon the front charge equals zero, and the force upon the back charge changes its sign so that the driving force attracts the back charge to the front charge of same sign. These laws allowed us to explain some unusual features of the fracturing of various materials subject to the radiation of powerful relativistic electron beams [1, 4]. When v =0, equation (5) is Coulomb's law.
2.3. Generalized Newton s law of gravitation
Suppose two point masses m1 and m2 are on axis x1 at some distance R one from the other in their own coupled gravitational/cosmic field described by potential 9. In this case, from Eq. (3), it follows that the force upon mass m1 is equal to [1, 5]
^ Gmm 4n ^,
r, _-^ - — GAm,R.
1 R2 3 1
(6)
Here the first term describes the mass gravitation, and the second term the cosmic repulsion as a natural property of space. When A = 0, we arrive at Newton's law of gravitation.
In the scale of solar system, the second term 1023 times less than the first term; in the galactic scale, the second term 105 times less than the first term; and in the scale of supercluster of galaxies, the repulsion becomes essential. The cosmic component of the field is a carrier of negative mass-energy uniformly distributed in space (there are about 0.01 g of this substance in the Earth). The law in Eq. (6) allowed us to explain the accelerated expansion of our Universe and the singular density of matter at the center of galaxies [1, 5]. From Eq. (6), it follows that the orbital speed of stars in spiral galaxies is constant and equal to (Gkg )^2 =
= 250 km/s, where kg is the galactic constant that equals
21 g
10 kg/m [1, 5]. The extrapolation of Eq. (6) leads to the
conclusion that our Universe has a finite size and volume
about 1078 m3, and its total mass-energy equals zero so that
it represents a gigantic fluctuation that came from nothing
[1, 5].
2.4. Generalized Archimedes s law
Using the general invariant integral of hydrostatics the vertical force upon a body lying on the horizontal surface of a heavy fluid was found to be equal to [2]
r_rA ±YI «n^d*
J sin 9
(7)
Here rA is the Archimedes' force, y is the free surface energy of fluid per unit surface, ds is the element of the wetting contour of integration, a is the edge angle of non-wet-
ting, and 6 is the angle between the body surface and horizontal plane in the normal cross-section of the wetting contour. Sign plus is valid for hydrophobic fluids and sign minus for hydrophilic ones.
2.5. Dynamic cracks
Suppose an open mode crack front xl = x2 = 0 moves along axis x1 at a constant speed v < cT in an elastic isotropic homogenous material. In this case, from Eq. (3), it follows that the crack-driving force is equal to [1, 6, 7]
r _<f[K(v) + U-OijnjU^dQ :
_ (1+s)vf k - vL
8ER
(8)
Ra _V (1 - vT)(1 - vL) - (1 - - vT)2,
-2 v vT _
T _ — > vr _-c
1 - 25
2 - 28 •
CL cL
Here Kj is the stress intensity factor, E and 8 are the Young's modulus and Poisson's ratio, cL and cT are the longitudinal and transverse velocities of elastic waves.
From Eq. (8), it follows that r = K:2 E _1(1 -82) when v = 0, which is the Irwin's equation. The crack-driving force was found as well for inhomogeneous anisotropic elastic solids and for various structures of solids, shells, plates, and membranes [1, 2]. Many other singularities of elastic field were studied in [1, 2].
It is appropriate to mention here that the early forerunner of the general invariant integral in Eq. (3) is Eshelby's path-independent integral [8] used in [7].
3. Temperature track behind a moving crack or dislocation
Let us study the local stationary temperature field near the front of arbitrary cracks or dislocations. In this case, the field is assumed to be independent of x3 and a crack or dislocation in an elastic solid is at x2 = 0, x1 < 0 so that its front in the coordinate frame Ox1 x2 moves along x1 with respect to the solid. The value of force r1 driving a crack or dislocation turns into heat since the losses of energy for acoustic and electromagnetic radiation, for latent residual stresses, and for surface energy, are negligibly small. The work spent on local plastic deformations turns into heat. The moving front of a crack or dislocation is a heat source.
Driving force r1 in Eq. (8) can be calculated from the local elastic field of stresses ct - (x1, x2) and displacements Uj (x1, x2) near the crack front using the following formula [1]
n,.
r _ -lim[a2j (+e, 0)Uj (-e, 0)],
(9)
e ^ 0, j = 1, 2, 3. This equation is valid for dynamic and static cracks in arbitrary anisotropic linearly elastic materials and for interface
cracks. Besides, it is valid for nonlinear power-law hardening incompressible materials, with replacing coefficient ul2 by another coefficient dependent of power [1]. For a dislocation, equation Tj = a0jBj is valid, where Bj is the displacement discontinuity, and a0, are the stresses at the front location without the dislocation. The value of a21 for moving edge dislocations is known as Schmid's friction, while, to initiate the motion, this value in some metals can be considerably greater.
We come to the problem of a linear source of heat rj moving at constant speed vj = v along axis xj. From Eq. (3), it follows that the corresponding general invariant integral has the following form in this case k t
rj = f (p m chT«j +—Tn
(10)
Here T(Xj, x2) is the temperature increase due to the heat source at point O, and rj is the value of the force driving a crack or dislocation.
If there are no heat sources inside the integration contour, then so that applying the divergence theorem provides the following equation which is valid at all regular points of the stationary temperature field
kt = (Tjj + Tv) = -p mCh vTj. (U)
A similar equation emerged in a more complicated problem of heat/mass transfer in a fluid flow past a cylinder of an arbitrary cross-section [9]. The solution of Eq. (11), which is singular at point O, vanishes at infinity, and is an even function of x2, has the following form [9]
T = Ak-2eXx' K0(kr ), (12)
V 1 2 2kt Here A is a constant to be found, and K0 (kr) is the modified Bessel function which has the following asymptotes
K0(kr) ^ - ln| k^ |, when kr ^ 0,
(13)
K0 (kr) ^ J——e Xr, when kr ^ <» 2kr
Let us substitute function T(Xj, x2) in Eq. (10) by its asymptotic value for small Xr, see Eqs. (12) and (13). By taking a circle of infinitely small radius as the integration contour in Eq. (10) and by calculating the integral, we get
A=
vk2 2nk,
■rj.
(14)
The local temperature field produced by a moving crack or dislocation is
T =
Vrj -e"XlKo(kr),
2nkt
(15)
r=VX27XF, . V 1 2 2kt
The temperature has a logarithmic singularity at the front
of a moving crack or dislocation, with intensity, in the case
of cracks, being directly proportional to the square of fracture toughness.
This implies that the crack can grow due to the fluidiza-tion or vaporization of an infinitely small amount of the material at the crack tip. It provides an alternative to Griffith's view on the fracturing as a reversible exchange of elastic and surface energies. The irreversible exchange of elastic and thermal energies is a better choice. The growth of a through crack in a thin shell or membrane at low tensile loads due to a heat source at the crack tip produced by a laser beam is an example of cooperating effects of thermal and elastic energy in fracturing. Another example of similar phenomenon is the development of chains of islands in the Pacific Ocean as a result of rupture of the thin crust of the Earth by heat and pressure of magma.
4. Fluid/gas flow in porous materials: binary continuum
Let us model a stationary process of the flow of viscous fluid or gas in a porous material by a binary continuum so that two different continua are assumed to be at each point of space. One of them is an elastic solid characterized by the following general invariant integrals
(16)
rk = f (Unk -^ijnjui,k f (Pijnj +e p pn )dQ = and the other continuum is a fluid or gas flow characterized by the following general invariant integrals
rk = f (fktni -Pfvinivk)d^ fpf vini d^ = ^ f (f'kini -epPnk )d^ = 0
(17)
and by the following state equations for fluid and gas
fij =-pS j + n,(vt> j + v j i X pp^ = Cp. (18)
Here ep is the effective porosity, nf is the dynamic fluid/ gas viscosity, vi is the fluid/gas flow rate (real velocity equals vtjep), pf, p, and f are the density, pressure, and stresses in fluid/gas continuum, x and cp are constants characterizing the polytropic process for gas.
In Eqs. (16) and (17), the value of rk is the same due to the interconnection of both continua. The term ep pni of volume force follows from the tubular model of porous materials. The effective porosity accounts only for the volume of interconnected pores where the fluid flows. At regular points, the equations of the theory of elasticity and fluid or gas dynamics can be obtained from Eqs. (16)—( 18) by means of the divergence theorem [1]. Below we study some problems arising from the horizontal drilling of porous rocks.
4.1. Horizontal borehole in oil deposit: stationary extraction regime
Let axis x3 coincide with the axis of a vertical borehole so that plane xj x2 is parallel to the day surface. Suppose
there is also a horizontal cylindrical borehole along axis
x1 = x issuing from the vertical borehole. Let us use the
2 2 2
cylindrical coordinate frame Oxr where r = x2 + x3 and point O is the issue of the horizontal borehole. We assume that a horizontal borehole of radius r0 is embedded inside a fluid deposit which size is much greater than length lH of the horizontal borehole.
The porous rock is subject to stress a33 = -wr, which is equal to the weight of higher rocks per unit square and to stresses an = ct22 =-8Twr, where lateral thrust coefficient ST equals 8/ (1 -8) in terms of Poisson's ratio, in the planestrain model of rock structure. Since 1 > 8T > 0, fracking wins the best advantage from the horizontal drilling because fractures in rocks tend to grow along planes which are perpendicular to the day surface.
Let us find the fluid flow field ignoring elasticity of the porous medium. In this 3D problem there are two small dimensionless parameters and A2:
À! =
X2 =-p X1 << 1, X2 << 1.
(19)
Here kp is the permeability of the porous medium. Parameter X 2 is small, and the fluid transport through a horizontal borehole is much faster than through the porous rock.
These small parameters signal that there is a boundary layer in the domain 0 < x < lH, r0 < r < r*, where r0 is the thickness of the boundary layer [1, 10]. Calculating the general invariant integral in Eq. (17) over the surface of this boundary layer, the fluid pressure can be expressed as follows [1, 10]
P = Pb( x) +
P^- Pe(x) ln r ln(r»/ r0)
ln-,
kp dp kp dp
(20)
r|f dr r|f dx
Here vr and vx are the fluid flow rates, p^ is the initial pressure in the deposit, and PB (x) is the pressure in the horizontal borehole.
From here, we arrive at the following ordinary differential equations
dVe dx
= 2nr0 qe,
(21)
Vb(x) =nr4dPB, qe(x) = kP P~-Pe(x)
8^f dx ' v ' nf roln(r*/ro) Here VB (x) is the fluid flow rate through the horizontal borehole cross-section, and qB(x) is the inflow rate of fluid into the horizontal borehole.
The solution of Eqs. (21) can be written in the following form
P~
pb = Pe -
sh( lu/1 )
Pe sh -lH - x
l
kp P^- Pb
m ln(r»/r>) sh (lH/l)
-sh
Ih - x
(22)
Ve =
81% sh(lu/l)
Pb ch lH-x,
Here pb is the pressure at the issue of the horizontal borehole where x = 0.
The fluid output of the horizontal borehole without fractures per unit time is:
Q
2 %
( Y1/2 ( ln * ^
r0
(P^- Pb)cth -H-
(23)
To find r*/r0, it is convenient to use equality r*/r0 = = (lHlr0)a, where a is a fitting constant to be found from one numerical solution of the problem [1]. It is usually equal to about 0.7. This approach allowed us to get some high accuracy analytical solutions, see, e.g. [1, 10, 11].
4.2. Penny-shaped fracture in oil deposit: stationary extraction regime
Let a penny-shaped fracture of radius R0 be issuing from the horizontal borehole at x = x0 so that its center is on the x-axis and its plane is perpendicular to this axis. We move the frame Oxr along the x-axis to the center of the fracture, and designate it as O£r, where £ = x - x0.
The distance between opposite banks of the fracture at r0 < r < R < R0 can be taken constant equal to dp, where dp is the diameter of solid particles (proppants) in the drill mud used to make the fracture by fracking. The particles remain inside the open fracture after the mud is removed and the rock pressure is closing the opening. These particles keep the fracture open like a wedge does. The value of R is determined by the frackng process while the difference R0 - R can be found from the corresponding plane strain problem of fracture mechanics when R0 - R << R. We provide the result of its solution [1] Edp
2(1 -82^2n(R0 -R)
-8tn(R0 -R) = KIC
(24)
Here KIC is the rock fracture toughness. From Eq. (24) it follows that R0 - R is less than Edp[2n8Twr (1-82)] 1, i.e. less than about 0.1 m for sandstones at the depth 1 km and dp ~ 0.5 cm. Thus, R0 - R << R indeed. In what follows we assume that R >> r0; otherwise, the fracking has no advantage.
The fluid flow near the fracture has the structure of a boundary layer | £ |< x*, r0 < r < R, where A3 = x*f R << 1 and d2 >> kp so that Vf >> dpvr (Vf is the flow rate through the fracture cross-section per unit length). In the boundary layer, the previous approach provides the following basic equations
dr
(rVF) = rqF,
(25)
Vf = -
dp3 dPF 12nf dr
qF = 2
kp p„-p (r)
Here PF (r) and qF (r) are the pressure in and inflow rate into the fracture.
From Eqs. (25), it follows that
Ï = p.-PF(r)],
(26)
Pf =
ÍP., when r = R,
[A, when r = r0. Here, pb is the pressure at the issue of the fracture on the horizontal borehole.
The solution of the boundary problem in Eq. (26) can be written as follows
PF = p„- D( p„- Pb)x
R
where D =
K
R
- K
R
(27)
K
- K
R
-1
Here I0(r/b) is the modified Bessel function so that
"1/2 „
I0I — I ^ exp—I 2n—
when--
b
I01 1, when — ^ 0.
(28)
KbJ b
According to Eqs. (25) and (27) the fluid output from the fracture into the horizontal borehole per unit time is equal to
Qf =
nr0 dp 6bnf
R
D(p.- Pb) x
K,
+ Ko
R
(29)
Here Kj(r0/b) and I1(r0/b) are the corresponding modified Bessel functions.
For very large fractures, when R >> b >> r0, it is reduced to the simple equation
QF =
ndp( P.- Pb)
(30)
"F HHVro)
The oil output of a very large fracture significantly depends on dp, p„-pb, and nf, and much less on ft/r0 . The value of the latter fitting parameter is to be found from one numerical solution to the problem under some typical conditions.
Evidently, this extraction process can be productive only for large fractures in rocks of high permeability. In the case
of several fractures, when the distance between any two neighboring fractures is greater than x*, the total output is given by summation of Eqs. (23) and (29). The other case, when this distance is comparable with x*, requires more study.
5. The theory of fracking
Let us study the hydrofracturing in shale gas reservoirs [12]. Shales are characterized by high porosity, low permeability, and low fracture toughness. They are fractured by minor tensile stresses so that the shale destruction opens the way to extract gas stored in closed pores. Because of low permeability the fluid flow in rock beyond the fractured zone can be ignored. Horizontal boreholes in shales can be as long as 2 km. High pressure of the drill mud upon the horizontal borehole surface and chemicals dissolving links between the rock fragments at the front of fractures produce a well-fractured volume in the local vicinity of the horizontal borehole. Practically, all gas can be extracted from this volume. The capacity of the horizontal borehole depends on the volume of fractured rock. The fractures keep open using proppants embedded by the drill mud inside fractures during fracking.
When the drill mud pressure is low, the horizontal cylindrical channel in an elastic rock is subject to the following stresses in the surface layer of the channel
= - Pm,
ae = pm - wr[1 + ST + 2(1 -8T)cos(20)]. (31)
Here pm is the drill mud pressure, and 0 is the angle between the horizontal plane and radius in the polar system of coordinates Or0 in the cylinder cross-section with the center at the axis. The stresses far from this channel are
a11 =a22 =-8'
T wr,
CT33 = - wr.
(32)
The fracturing starts at the top point 0 = n/ 2, when
Pm > (38T - 1)wr-
Below we study three basic regimes of fracking. In the permeation regime, the drill mud penetrates everywhere inside fractures while in the nonpermeation regime it penetrates nowhere in the rock. The most practical regime is that of partial permeation. In all cases, the zone of fractured rock is assumed to enclose the horizontal borehole. We assume that many fractures issue from the horizontal borehole, all being radial, i.e. propagating along planes 0 = const. Evidently, in the cylinder cross-section the contour of the zone of fractured rock always represents an oval extended in the vertical direction, which supported as well by the solutions below at hand.
5.1. The permeation regime of fracking
The friable shale is fractured by minor tensile stresses caused by the pressure of the drill mud that permeates multiple fractures. As a result the hydrostatic pressure pm is setting in everywhere in the well-fractured rock so that
CT11 = CT22 = CT33 = - Pm inside ZF
(33)
(34)
Here ZF is the closed contour of fractured rock in the normal cross-section Ox2 x3 of the horizontal borehole. This stress state is similar to a specific fluidized state [13]. The rock outside ZF is intact and elastic, and is in a prefractured state on ZF so that
CTn = -Pm' CTt =-aS>
if CTt < 0, CTnt = 0 (on Zf). Here ctn, ctnt, and CTt are the normal, shear, and tangential stresses on ZF satisfying a failure criterion, e.g. von Mises criterion (ks is a constant)
(CTn -CTt)2 + (CTn -8twr)2 +
+ (CTt +8t Wr)2 + 6CTnt = 3k2. (35)
In the extreme case, ct t = 0 on ZF due to the effect of chemicals so that we can neglect the tensile strength of the rock.
It is required to find contour ZF and stresses outside ZF meeting these boundary conditions. This is an inverse problem of the theory of elasticity. Let us solve it. We apply the general invariant integrals in Eqs. (16) to an arbitrary elastic domain outside ZF and use the divergence theorem. As a result we have
=CTj =IF' ej = hUi'j+Uj'i). (36)
j j
This is the equation system of the theory of elasticity. In our plane strain case of the linearly elastic homogenous isotropic rock, the following representation is valid for stresses CT22, CT33, and CT23 outside ZF [1]
(37)
CT22 + CT33 = 4Re0 (z), z = x2 + ix-3, ct33 -ct22 + 2iCT23 = 2[z0'(z) + Y(z)]. Here the Kolosov-Muskhelishvili potentials O(z) and Y(z) are analytic functions outside contour ZF that is unknown beforehand and has to be found.
From Eqs. (32), (34) and (37), it follows that
40 (z) = -(1 + 8tK =- Pm-cts. (38)
Hence, the pressure of the drill mud necessary for this regime of fracking is
Pm = (1 + 8t)Wr -CTS. (39)
Using Eqs. (34), (37), (38), and the equation
CTt -CTn + 2iCTtn = e2'a (CT33 - CT22 + ^'^sX
we have the following boundary value problem
2ia
z) = Pm -CTS, z 6 Zf
(40)
Here a is the angle between the external normal to ZF and axis x2 being counted from the axis to the normal.
Let the conformal mapping of domain |ZI > 1 onto the domain outside ZF be provided by function z = ro(Q where Z is a new parametric complex variable. Since ro'(Z) e2ia = = -Z2ro'(Z) on |Z| = 1, the boundary condition (40) can be written as
(Pm -ct>'(z)+2Z V(zm®(0)=0, (41)
IZI = 1.
Using the method of functional equations introduced and developed in [14, 15], we get the solution to this boundary value problem in the following form
ffl(Z) = A z-z
^ 2
(Z)) = - ^ Pm -CTj^+^T 2 k+z2
(42)
Here A is an arbitrary constant of the dimension of length, and k is equal to 1-8
T - - - ~ - (43)
k = ■
1 + 8T
0 < k < 1
0 <8T <1.
The boundary of the fractured rock has a shape of ellipse which diameters in the vertical and horizontal directions are
DV = 2 A(1 + k), Dh = 2 A(1 - k). (44)
The output of shale gas in this regime is directly proportional to n(1 -k2) A2 Lh.
For A >> r0, the value of A2 is directly proportional to the volume of the drill mud pumped into the horizontal borehole divided by the length of the horizontal borehole. In this regime of fracking, the shale gas output is directly proportional to the drill mud volume pumped into the horizontal borehole. This regime is characteristic for the stage of well-developed fracking, while the pressure of gas liberated out of destructed pores of rock during the fracking process being ignored.
5.2. The nonpermeation regime of fracking
Let us also study an extreme case when the permeation of the drill mud into the fractured rock can be ignored. In the continuum approximation, for many radial fractures inside contour ZF, we get
CTr = - Pm
'00_ r
CTe = 0 CTre = 0 r >r0
(45)
Similarly to the previous problem, we use the conformal mapping of domain |ZI > 1 onto the domain outside ZF by function z = ro(Z) and arrive at the following boundary value problem when |ZI = 1
r0 Pm
4Re0(ro(Z)) = -
l®(Z)l
m(Z) d ffl'(Z) dZ
0(ro(Z)) + Y(ro(Z)) = -
r0Pm®(Z) (46)
2®(Z) l®(Z)l
The method of functional equations provides the following solution to this boundary value problem [14, 15]
®(Z) = B(2Z2-X)2 z-3,
2 (47)
1 z2
4 2Z2-X
0(ro(Z)) = Wr(1 + 8t)
^(®(Z)) = - - wr(1 + 8t) x
. Z4 (1 + Z 2 )[2(4 + X 2 ) Z 2 + X (4 - 3X 2 )] (2Z2 -X)3(2Z2 + 3X)
(48)
B =
r0 Pm
-, X3 + 4X = 8k.
(1 + 8 T)(4-X 2) wr Notice of erratum: the denominator of the second equation (5.2.23) in [15] should be equal to + - 2cts instead
of ^s - P.
Based on Eq. (47) contour ZF is presented by the following equations
x2 = B[4(1 - X) cos P + X2 cos(3P)], 2n > P > 0,
2 (49)
x3 = B[4(1 + X) sin P-X2 sin(3P)].
Here the vertical and horizontal diameters of the fractured zone are as follows
DV = 2B(2 + X)2, DH = 2B(2 - X)2. (50)
Contour ZF encloses the horizontal borehole when
Pm > (1 + 8T)^+X- (51)
wr 2 + X
It can be shown that the solution (47)-(50) is valid when
(52)
2 17
— > X > 0, i.e. when — < 8T < 1.
3 37 T
At X = 2/3, cusps appear at points x3 =± 1/2 DV of contour ZF (at this state DV = 4DH). This feature signals that for X>2/3 fractures grow in the intact rock from the cusps along the vertical plane x1 x2 because of the square-root singularity of tensile elastic stresses at the cusps.
According to Eqs. (49) the area of the cross-section of fractured zone is equal to
nB2(16-16X2 -3X4)-nr02. (53)
In the nonpermeation regime, parameters 8T and Pm/wr control the fracking process. When Pm >> wr, the volume of fractured rock in this regime is equal approximately to lHr02(Pm/wr )2. This regime is characteristic for the beginning stage of the process of fracking, with the pressure of gas liberated out of the fractured pores being ignored.
5.3. The general regime of fracking
The general regime of partial permeation occurs when the drill mud penetrates into some part of fractures at r0 < < r < r*, while the gas liberated out of fractured pores permeates all the remaining part of fractures. This regime is of most practical importance. The stresses in the rock between any two neighboring radial fractures meet the following equation
dCTr _„ _„
" (54)
, ar -ae n „ n
a - +-= 0, °re = 0,
dr r
e0 < e < e0 + Ae.
Since Ae << 1, we can put CTe = -Pm in the area where the drill mud wets the fracture surface, i.e. when r0 < r < r*. This is an axisymmetric area because it is determined by the axisymmetric conditions in the vicinity of the horizontal borehole. In the remaining part inside ZF when r > r*, we can put CTe = - pg , where PG is the pressure of shale gas liberated out of fractured pores.
From here and Eq. (54), in the continuum approximation for all fractured area inside ZF, we get
CTr = CTe = -Pm' CTre = 0 (55)
r0 < r < r* in ZF,
ar = -Pg + (Pg -Pm)—, cte = -pg> r
are = 0, r > r* in ZF.
(56)
Evidently, r* increases if Pm > pg , and r* decreases if pm < pG, with the boundary velocity dr*/di being much less than cT.
Let us assume that contour ZF embraces circle r = r*. In this case, the method of functional equations provides the following solution
z = ro(Z) = D(2Z -,) Z , IZI > 1 0(ro(Z)) = -Pg + ^(1 + 8T)Wr -
(wr +8Twr -2Pg)Z
D =-
2Z2 -,
( Pm - PG) r*
(57)
(58)
(4 -,2)(1 + 8t - 2pG )Wr
, + 4, = J-^, PG = ^
. _ (59)
1 + 8t -2Pg
Function Y(ro(Z)) coincides with that in Eq. (47) where factor wr(1 + 8T) is replaced by (1+ 8T)wr -2pG and X is replaced by
Contour Zf of the fractured zone and its vertical and horizontal diameters are provided by Eqs. (49) and (50) where X has to be replaced by | and B by D. It can be shown that the solution (55)-(59) exists for 2/3 >|> 0. At | = 2/3 cusps appear at points x3 =±D(2 + |)2 so that for |> 2/3 two fractures grow in the intact rock along plane x1 x2 issuing from those points.
In this general case, the cross-section area of fractured volume of rock is equal to
nD2(16 -16|2 - 3|4) - nr02. (60)
This regime of fracking is determined by four dimen-sionless parameters 8T, pm/ wr, pG/wr, and r*/r0 , which have to meet the following conditions
378t > 17 + 20pG, r*/r0 > 1, D(2-|)2 > r,. (61)
The fractured volume grows when pm > pG. When PG > Pm the gas pushes the drill mud out, and fractures close up to the level supported by proppants.
6. Conclusion
The basic invariant integral first introduced into fracture mechanics in [16], see also [7, 8, 17-23], is generalized by way of taking into account the mass-energy, cosmic/gravitational, elastic, temperature and electromagnetic fields of physical mesomechanics, thus supporting it as a mega-mechanics of nature. The invariant integrals are also offered for binary continua that model porous rocks saturated with gas or fluid. Beginning with a generalization of the fundamental laws of Newton, Coulomb and Archimedes, the invariant integral based method is used to find temperature induced by a moving crack or dislocation and then to derive the flow field around, and oil output from, horizontal boreholes with and without fractures produced by previous hydrofracturing of oil rock beds. As to shale gas reservoirs where the gas permeation through the intact rock can be neglected, the study is concentrated on the calculation of the shape and size of the local multiply-fractured region near a horizontal borehole for three basic regimes of the drill mud penetration into this region in the fracking process. The accurate analytical solution of the corresponding inverse problems of the theory of elasticity is achieved by using complex variables and the author's method of functional equations.
The author thanks Professor Ron Armstrong for the encouragement to write the present work.
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Поступила в редакцию 15.12.2014 г.
Сведения об авторе
Cherepanov Genady P., Prof., Hon. Life Member, The New York Academy of Sciences, USA, genacherepanov@hotmail.com
