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P003 OPERATIVE ESTIMATION TECHNIQUE OF
HYDRAULIC FRACTURING EFFECTS
A.A. POZDNYAKOV & I.A. VINOGRADOVA
ООО «РНТЦ», Россия, 625000, г.Тюмень, ул.Республики 41
LS “Regional Scientific and Technological Centre” , Respubliki st. 41, 625000, Tyumen, Russia
Abstract
The calculation procedure for operative estimating of the fracture geometry and its influence
on well productivity is presented. It uses the data of the basic technological and geological
parameters for specific hydraulic fracturing treatment.
The mathematical model of hydraulic fracturing process is the base of the technique, which is
simple enough to provide convenience and calculation rapidity. The possibility of such model
creation is caused by quasistatic property of hydraulic fracturing process, which allows receiving
the finite (not differential) coupling between governing variables and parameters for each current
condition.
The mathematical model has three main components: balance ratio; physical relations for the
fracture characteristics resulted from the known elastic theory solutions; new representation of
the fracture net pressure as the function of the main governing parameters (taking into account
the similarity and dimensionality theory). The γ-coefficients are used in the model which
efficiency is proved by M.P.Cleary. The manner is offered to determine these coefficients.
The relationship between the productivity increasing after hydraulic fracturing and
geometrical and the conducting fracture characteristics is determined with using the model of
M.Prats.
The opportunities and examples of the concerned technique practical application are
presented.
1. The design model of hydraulic fracture and balance ratio
The vertical crack created at a
hydraulically fractured well is considered.
The crack is symmetric as regard to a well
and a median plane of a pay zone, which is
modelled as a horizontal layer of constant
thickness h0. The geometry of the created
crack is schematically pictured on fig.1,
where hm, Lm are the maximal height and
length of a crack wing, an axis z coincides
with a well axis. In the same figure the crack
area filling with proppant and adjoining to a
z
hm /2
h0 /2
0
L
Lm
x
-h0 /2
-hm /2
Fig. 1. The geometry of the one wing of the crack
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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well is shown. Its size L will termed the depth of proppant penetration in a crack.
The proppant filled volume of a crack at the moment of the injection shut-in can be calculated as
V = γV wm hm L
(1)
where wm is the maximal width of the created crack
wm << h0
(2)
γV - a numerical multiplier, taking into account the fracture geometry integrally (including
presence of two wings).
An efficiency of using of such generalized γ factors is proved in work [1].
It supposed, that after fixing a crack at proppant its length coincides with L, and the maximal
height and width accept values h and w. Thus the volume of the fixed crack becomes equal
Vf = γV w h L
(3)
The same as in (4) value of the factor γV is used here with the assumption of similarity of the
corresponding crack configurations (initial and fixed).
Volumes V and Vf are expressed through characteristics of the proppant contained in a crack
V = m / (c ρ)
(4)
Vf = m / (cf ρ)
(5)
where m is the weight, ρ is the mineral density, c, cf are the initial and final slurry average
volumetric concentrations of the proppant.
From (4), (5) follows
cf Vf = c V
(6)
Concentration c is expressed through used in practice parameter M - the average proppant
mass fraction in a fluid - by representation of volume V as the sum of proppant volume Vp and
carrying it fluid volume Vl
V = Vp + Vl = m / ρ + m / M = m (1 / ρ + 1 / M)
(7)
c = (M / ρ) / (1 + M / ρ)
(8)
Comparing with (4), we receive
Let's notice, that the leakoff volume doesn't appear in the balance ratio used above. In
particular, the value of M in (7) may be calculated directly as a time average proppant mass
fraction using the actual proppant transport diagram, since the leakoff intensity being
proportional t-1/2 is considerably reduced at last stage of process in a root zone of the crack,
where the subsequent proppant fixing occur.
2. Mechanical ratio
Maximal values of a crack width and height are defined by net pressure ∆p at the created
crack inlet.
According to classical solutions of elastic theory problems [2], [3] the following ratio are:
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h0
π ∆p
= cos(
)
2 σ
hm
wm = γ E
∆p
hm
E′
(10)
(11)
where σ is the difference in stress between the pay zone and the surrounding barrier layers,
E’ = E / (1 - ν2) is the plane strain modulus of the central layer, expressing through Young’s
modulus E and Poisson’s ratio ν, γE is the numerical factor describing the form of a crack crosssection.
With entering a designation for nondimencional pressure
Ψ = π ∆p / (2 σ)
(12)
hm = h0 / cos Ψ
(13)
wm = 2 γE (σ / E’) h0 Ψ / (π cos Ψ)
(14)
these formulas are presented as
The expression for the fixed crack length follows from (1), (4):
L=
1
m
cρ γ V wm hm
(15)
and after substitution (13), (14) becomes
L=
π
2γ V γ E
m E ′ cos 2 Ψ
Ψ
cρ h02 σ
(16)
Thus, the fracture geometrical characteristics are expressed through key parameters of
hydraulic fracturing technology and the environment elastic properties.
3. Net pressure in a crack
The development of net pressure ∆p is the most challenge problem in the modelling of
hydraulic fracturing process. For the model accepted in this work we use the approach similar to
one offered in [4].
As the hydraulic fracturing experience demonstrates, the value ∆p is approximately constant
at the basic stage of the fracture created quasistatic process, therefore it can be represented as
function of the basic governing parameters:
∆p = f (E’, σ, µQ, h0)
(17)
where Q is the fluid injection rate , µ is the fluid viscosity.
The type of function f can be established, using the methods of the similarity and
dimensionality theory [5]. In (17) there are 5 dimensional values and 2 independent dimensions
(lengths and pressure). Therefore for the dimensionless form of a ratio (17) it is possible to set
three independent dimensionless variables λ, ε and pressure, which kinds are chosen from
physical reasons:
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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λ=
E ′h03
,
µQ
ε=
σ
E′
∆p / σ = y (λ, ε)
(18)
(19)
Similary, according to physics of process, it is possible to set function y, for example, as:
y = γP / (1 + λα εβ)
(20)
where γP , α, β are some positive values.
The choice of such structure of the function (19) is caused by the following reasons. Net
pressure ∆p in a fracture shouldn't exceed the barrier stress: y < 1 according to (10). It should
grow with "injection power" µQ increasing (at the fixed values of other parameters) till approach
to asymptote: ∆p → σ. On the contrary, at µQ → 0 it is required that ∆p → 0. The rate of
respective ∆p alterations should be the lower, the more σ.
4. Determination of free model parameters
The mathematical model of hydraulic fracturing (12) - (14), (16), (18) - (20) contains 5 free
parameters (γV, γE, γP, α, β) and uses the minimal data set of geological and physical layer
properties (h0, E, ν, σ) and technological characteristics (m, ρ, M, µQ) of a fracturing process
(the parameter (12) is calculated from known function (19), which has, for example, a kind (20)).
The model makes possible to estimate the basic sizes of the fixed crack (the formula (13), (14),
(16)).
However, the free parameters of the model remain uncertain. Generally speaking, their values
for each specific well depend on the number of factors, which are not taken into account in the
model. Therefore the following procedure of determination of values γV, γE, γP, α, β is suggested
for the model becomes full closing.
Hydraulic fracturing treatments are now accomplish industrially, so there are enough data
about process including results of fracture design at the majority of operational objects
(reservoirs). Therefore for each object where representative sample of such data is realized it is
possible to carry out the proper statistical processing.
So, after building the diagram of a kind (1) for the group of wells, i.e. the dependence of
known for them values of complexes m (1 / ρ + 1/M) versus (wm hm L), it is possible to receive
with the aid of the least squares method an average slope of this dependence, which is interpreted
as value of parameter γV for conditions of the object exploited by this group.
Similarly, γE is determined on the diagram wm versus ∆phm/E’ (see (11)), and parameters γP,
α, β of basic function (20) are determined on the diagram ∆p / σ versus µQ / (E'h03) (see (18),
(19)).
After an ascertainment of free parameters values (an adjustment for object) the model is ready
completely for the calculations predicting the fracture sizes of the treatment projects.
5. An estimation of a well productivity after hydraulic fracturing
The conducting properties of a crack are characterized by dimensionless conductivity [6]:
FD = kf w h / (k L h0)
(21)
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where k is a layer permeability, kf is an average permeability of a crack cavity.
In (24) it is taken into account, that the fixed crack height exceeds the layer thickness (see
fig. 1). The permeability kf of fixing packing can be estimated through specific proppant
conductivity A and its mass m
kf = mA / Vf
(22)
FD = m A / (k h0) / (γV L2)
(23)
From (3), (21), (22) follows
i.e. dimensionless fracture conductivity is expressed through conducting characteristics of a layer
(kh0), the technological parameters (mA) and a crack geometry.
According to the model of M.Prats [7] the performance of a vertical crack of finite
conductivity, which drains the confined homogeneous layer of constant height, is characterized
by effective radius ref which is defined by universal function of parameter a :
a = π / (2 FD)
(24)
ref / L = α (a)
(25)
Further it is convenient to use the complex parameter la , entered in [8]:
⎛ 2 mA ⎞
⎟⎟
l a = ⎜⎜
⎝ πγ V kh0 ⎠
1/ 2
(26)
which assumes some characteristic length.
With the account (23), (26) the Prats’ parameter (24) and the dependence (25) become
a = (L / la)2
(27)
ref / la = Ra (a)
(28)
Ra (a) = a1/2 α (a)
(29)
where
is the universal function, which is shown on fig. 2.
It is approximated at the interval 10-2 ≤ a ≤ 102.5 by the dependence
Ra = 0.02398 + 0.1855 u – 0.2006 lg u,
u = a / 1.3839,
which has an inaccuracy less than 1% at 10-1 ≤ a ≤ 10 and about 7% at the interval borders.
Thus, with the known function (29), length L and parameter la of the fracture, it is possible to
determine its effective radius via (28), and then to calculate the fractured well productivity index
(PI):
J =
2πkh0
1
µ g B ln( R / ref )
(30)
where R is the drainage radius, µg is the reservoir fluids viscosity, B is the volumetric factor.
9th
European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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0.25
0.2
ref / l a
0.15
0.1
0.05
-2.50
-2.00
-1.50
-1.00
-0.50
0
0.00
0.50
1.00
1.50
2.00
2.50
lg(a)
Fig. 2. The schedule of function Ra (lg (a))
One of the basic parameters of technological efficiency is the multiplicity of a well PI
increasing after fracturing, which is designated further by symbol N. The most objective base
value for this characteristic calculation is the productivity index of an ideal unfractured well. It is
calculated with the well known Dupuit’s formula, similarly to (30) by substituting ref for a well
radius rW. Then
N=
ln( R / rW )
ln( R / ref )
(31)
Thus, determined the value of N assesses the "pure" fracturing efficiency received only due to
the main crack creation. An additional effect received due to the skin-factor removing, the well
performance optimization etc., is calculated via standard techniques.
6. An application of the operative estimation technique of hydraulic fracturing effects
Thus, the suggested technique for an individual well uses the
described model which is preliminary adjusted to reservoir
conditions on a representative sample of the actual hydraulic
fracturing treatments. If such sample is absent, the adjustments
received for the analogues objects are used.
With obtained values of the projected fracturing basic
technological parameters the crack sizes are predicted, the
fracture conducting characteristics and the multiplicity of a well
productivity index increasing after fracturing are determined
(see example in the table). Varying values of these parameters,
it is possible to estimate a degree of their influence on the
hydraulic fracturing result, to find their optimum set, which
providing achievement of the maximal value of N.
The wells most suitable for hydraulic fracturing treatment are
selected by testing the well fund with the aid of this procedure.
Example
Parameter
Input data
h0
k
E
ν
σ
2 rW
R
ρ
A
M
m
µ
Q
Output
Unit
Layer & Well
m
mD
MPa
MPa
mm
m
Proppant
kg/m3
mD/(kg/m3)
kg/m3
kg
Fracturing fluid
cp
m3/min
Value
10
15
1.8×104
0.25
4
88.9
250
2904.6
150
300
10000
250
3
Fracture
hm
w
L
ref
N
m
mm
m
m
31.4
4.6
82.1
8.76
2.58
The offered technique has been tested in practice. Some
results of its applying in the Western Siberia are described in
[9]. Also this technique is the base for calculation of hydraulic fracturing cost efficiency [10].
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References
1.
2.
3.
4.
5.
6.
7.
8.
9.
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European Conference on the Mathematics of Oil Recovery — Cannes, France, 30 August - 2 September 2004
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