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Copyright 2007, Society of Petroleum Engineers This paper was prepared for presentation at the European Formation Damage Conference held in Scheveningen, The Netherlands, 30 May–1 June 2007. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, Texas 75083-3836 U.S.A., fax 01-972-952-9435.
Abstract
Hydraulic fracturing is one of the most common well stimulation techniques. Hence, considerable amount of efforts has been devoted to study their performance under different prevailing conditions. Description of hydraulically fractured wells using the existing commercial reservoir simulators requires the use of very fine grids, which is very cumbersome, costly and impractical. In this work a two dimensional single-well mathematical simulator has been developed, which is based on finite-difference methods, and simulates linear Darcy flow in a hydraulic fractured well. Based on the results of our in-house simulator different correlations have been developed for calculation of fracture (S
f
), face damage (S
fD
), and chocked damage (S
ck
) skin factors. These formulations benefit from relevant dimensionless numbers expressing the effect of geometry and those used to quantify the extent of damages occurred during fracturing operation. The results indicate that in-house S
f
values are more accurate than those available in the literature. S
fD
, reflecting the reduction of the permeability of the rock close to the fracture, is more realistically captured here when damaged thickness is assumed to decrease linearly towards the tip of the fracture. This is more superior to the uniform thickness assumption made in the literature, which overestimates S
fD
. It is also shown the available analytical expression for calculation of S
ck
, which is a reflection of reduction of fracture permeability, overestimate S
ck
hence, a more realistic correlation has been developed for this purpose. The skin correlations developed here enables the engineers to study the productivity of a fractured well using simple open-hole system. They can also be used for the optimum design of a hydraulic fracture system by minimizing flow resistance for a given fracture volume and fracture permeability. This can be achieved readily by conducting simple derivation of the mathematical expressions of skin formulations presented here.
Introduction:
Hydraulic fracturing is one of the well established well stimulation techniques especially for tight reservoirs. Because of the importance and wide applications of hydraulic fracturing, the study of pressure transient analysis and also productivity calculations in such systems has been the subject of interest for many investigators and lots of efforts have been directed to them
1-13
. All these studies can be classified into two categories. The first type of work focuses on the flow behavior and pressure changes at unsteady state conditions. The main objective of this type of work is presentation of well test analysis method for estimation of characteristics of an existing fracture and its associated conductivity. The second type of work, which is in line with this paper, investigates the flow behavior and pressure distribution around fracture at pseudosteady-state conditions. This is aimed at determination of improvement in well productivity or of optimized fracture design. Results of theses works are in the form of charts or correlations for calculating the well productivity, skin factor or effective well-bore radius. Skin factor is a useful tool for comparing the performance of a real system, which is often complex and difficult to replicate, with that of an open-hole system. In a Hydraulically Fractured System (HFS), the thickness of the fracture is usually less than 1 cm requiring fine grids for the fracture cells to accurately simulate fluid flow in these systems. In addition to the existence of this significant scale difference between fracture and porous media, fine grids are required around the fracture to capture the abrupt changes in flow parameters in this region. Accurate skin factors, on the other hand, can help a reservoir engineer to forecast the well productivity without using fine grid, which is costly and cumbersome. McGuire and Sikora
1
(1960) used an electric analogue computer to study the effect of finite-conductivity vertical fractures on the productivity of wells in expanding fluid-drive reservoirs. The result of their study was some curves demonstrating the productivity increase benefited from hydraulic fracturing. Prats
2
(1961) presented an analytical model for pseudosteady-state behavior of finite-conductivity vertical fractures. In this work Prats introduced the concept of effective well bore radius. He also showed that there is an
SPE 107634 New Mechanical and Damage Skin Factor Correlations for Hydraulically Fractured Wells
H. Mahdiyar, M. Jamiolahmady, and A. Danesh, Heriot-Watt U.
2 SPE 107634
optimum fracture design, length-width ratio, for a given fracture volume that maximizes productivity. Cinco-ley et al
3-8
(1977, 1978, 1981, 1982, 1987, 1989) studied HFS for both transient and pseudosteady-state. Some of their major contributions (regarding pseudosteady-state) are the introduction of fracture pseudo skin, an equation for estimating face damaged skin and also presenting a curve for estimating effective well bore radius. Valko et al
9-10
calculated pseudosteady-state productivity index of a fractured well and then presented some charts for optimum design of a fracture. Meyer and Jacot
11
(2005) presented a new model and an analytical solution for the calculation of dimensionless productivity index of a finite conductivity vertical fracture. In their work the well is located at the centre of a closed rectangular reservoir flowing under pseudosteady-state conditions. They also introduced a simple equation for estimating pseudo fracture skin when fracture length is negligible compared to reservoir length. The main objective of this paper is the investigation of single phase Darcy flow in HFSs. The result of this study is presented in the form of a number of skin correlations. These formulations express the effect of fractures on flow performance in a HFS compared to that of an equivalent open-hole system. There are also separate correlations to estimate the impact of fracture face and chocked damages. The introduced correlations are not only useful tools for simulating an open hole system instead of hydraulically fractured system but also can be used to optimize fracture design as will be discussed in the last part of this paper.
Simulation of a Hydraulically Fractured System
In commercial simulators the effects of stimulated or damaged zones on the well-bore flow rate are generally presented by a skin factor in the well block grid calculation. This enables a reservoir engineer to simulate a simple open-hole system instead of the complex real system. Flow around the well bore is usually stabilized very quickly hence, in commercial simulators flow from the well grid block into the well bore is calculated at steady state conditions. Because the main objective of this study is the development of skin factors correlations we simulated the flow around a HFS at steady state conditions. Fig. 1 schematically shows a hydraulically fractured system (HFS). In our model it is assumed that: a)
Width of the fracture is constant. b)
Fracture has penetrated vertically through the whole height of the reservoir. c)
Fracture has penetrated symmetrically in both directions. d)
Well-bore flow from the matrix is negligible, compared to the flow from the fracture to the well. e)
HFS has a square shape with constant pressure at the boundaries. f)
Flow is linear (Darcy flow). g)
Single-phase fluid under study is incompressible. Continuity equation and Darcy law can be combined to give:
0
=⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ∂∂∂∂+⎟ ⎠ ⎞⎜⎝ ⎛ ∂∂∂∂
y P k y x P k x
. (1) Boundary conditions required for solving this differential equation are: At x = 0
0
=∂∂
x P
, (1a) At y = 0
0
=∂∂
y P
, (1b) At x = X
e
P=P
e
, (1c) At y = X
e
P=P
e
, (1d) where, X
e
is the half length of the drainage area (the side length of a quarter of that).
Numerical Solution Technique
The Finite difference numerical method has been used for solving this partial differential equation. Because of the symmetry in the HFS, a quarter of this system, shown in Fig. 2, is simulated to save computational time. The model is divided into 1000 grid blocks, 40 blocks in x direction and 25 blocks in y direction. So there are 40 rows and 25 columns, as shown in Fig. 3. The grid blocks near the well and those near the tip of the fracture are finer than the remaining blocks to capture more accurately the abrupt changes in flow parameters in these areas especially near the well bore where flow velocity is at its maximum. This girding is the result of the previous studies in this department
12
as well as some recommendations available in the literature, Guppy et al.
13
(1982). Each block could have 2, 3 or 4 neighboring blocks, which have mass transfer to or from it. Fig. 4 shows schematically an inner block with its 4 neighboring blocks. Mass flow rate between two neighboring blocks is equal to driving force,
∆
P, divided by flow resistances between the centers of the two blocks. According to the mass conservation law, at steady state conditions, the algebraic summation of mass flows entering each block must be zero. That is,
0
),()1,(
),()1,(
),()1,(
),()1,(
),(),1(
),(),1(
),(),1(
),(),1(
=+−++−++−++−
−−++−−++
ji ji
ji ji
ji ji
ji ji
ji ji
ji ji
ji ji
ji ji
RY RY P P RY RY P P RX RX P P RX RX P P
. (2)
SPE 107634 3
In this equation RX
(i,j)
and RY
(i,j)
are flow resistances in the x- and y-directions, respectively. These values can be calculated as follows:
),(),(
),(),(),(),(
),(),(
22
ji ji
ji ji ji ji
ji ji
xhk y RY yhk x RX
∆∆=∆∆=
. (3) Eq. 2 is the finite difference form of Eq. 1 for inner blocks. For blocks neighboring the boundaries the shape of the equation is slightly different according to the boundary conditions expressed by Equations 1a-d. The solution of Eq. 2 together with the boundary condition expressed by Eq. 1 is the pressure distribution, which is used to calculate the well flow rste by the following equation:
)1,1(
)1,1(
4
RX P P q
W
−=
µ
. (4)
Comparison of In-House Simulator with ECLIPSE
Flow in a HFS with the same size and grid pattern as those in our in-house simulator was simulated by ECLIPSE commercial simulator. Here 64 injection wells with the bottom-hole pressure equals to P
e
are placed in all the boundary blocks and the bottom-hole pressure of all wells are set to constant value of P
e
. This makes the simulated model reach the steady state conditions rapidly with the boundary conditions similar to the boundary conditions imposed in the in-house simulator. The results of two simulators were compared for many prevailing conditions. The close agreement between the results confirmed the integrity of the structure of the in-house simulator. However, it is noted that the in-house simulator automatically generates mesh for the HFS with any geometry and fracture sizes. This allows us to study the HFS for many prevailing conditions and produce large data banks for developing the skin correlations. Doing the same job with commercial simulators is time consuming and cumbersome.
Skin Factors In Hydraulically Fractured Wells
Skin factor is generally defined by Eq. 5,
S r r Ln P khq
we
+⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ∆=
µ π
2
. (5) where, r
e
is the equivalent radius, equal to the radius of a cylindrical medium volumetrically equal to the square drainage area.
ee
X r
π
2
=
. (6) For hydraulically fractured systems Eq. 7 is used to define pseudo skin, denoted by S
’
,
'
ln2
S X r P khq
f e
+⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ∆=
µ π
. (7) Rearranging Eq. 7 in the form of Eq. 8 illustrates the physical meaning of pseudo skin:
fz mz f e
R R P khS X r kh P q
+∆=+⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ∆=
π π
2ln21
'
, (8) where,
R
mz
(ln(r
e
/X
f
)/2
π
kh) is the radial flow resistance in the matrix zone. Therefore
R
fz
(S
’
/2
π
kh)
mainly
represents the flow resistance in the fractured zone, although it also contains non-radial effect in the matrix zone. These regions are schematically shown in Fig. 5. It is noted that this definition of skin has a more realistic physical meaning compared to that expressed by Eq. 5 and also proved to give more efficient correlations. For Darcy flow systems with zero flow skin, the following simple relationship between skin and pseudo skin factors can be obtained by comparing Eq. 5 with Eq. 7.
w f
r X S S
ln
'
−=
. (9) In this paper pseudo mechanical skin is defined as the pseudo skin factor of a HFS containing single phase Darcy flow. This factor is the summation of pseudo fracture skin and damages skin factors, which will be discussed in the following sections.
Pseudo Fracture Skin Correlation
Pseudo fracture skin is defined as the skin factor of a no damaged HFS. As mentioned earlier this factor represents flow resistance in the fractured zone as well as the non-radial flow effect in the matrix zone around the fracture. Previous studies
4,11
have confirmed that flow resistance in the fractured zone is a function of dimensionless fracture conductivity,
4 SPE 107634
which is the ratio of conductivity in the fracture to that in the matrix inside the fractured zone.
f f f fD
kX wk C
=
. (10) On the other hand, considering the geometry of a fractured system it can be concluded that the non-redial flow effect in the matrix zone is affected by the relative size of the fracture compared to that of drainage area, expressed by the following dimensionless number:
f e X
X r I
=
. (11) Fig. 6 shows the variation of pseudo fracture skin with I
X
at three different dimensionless fracture conductivities (
C
fD
). Cinco-ley
4
has reported that the effect of I
X
on effective well bore radius, or pseudo fracture skin, is negligible when I
X
is bigger than 4.0. The results of Fig. 7 confirm his observation. This suggests that flow is almost radial for the region beyond I
X
>4. Meyer and Jacot
11
(2005) introduced the following equation for calculation of pseudo fracture skin for HFSs with I
X
>>1.0, where pseudo skin is only a function of C
fD
.
⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ +=
fD f
C S
π
2ln
'
. (12) Fig. 7 compares the results of our in-house simulator with the results of this equation with I
X
bigger than 4.0. A good agreement is noted between the results of this equation and those of the In-house simulator. It should be noted that Meyer has used semi-analytical solution technique to obtain Eq. 12 while our results have been obtained from our numerical simulator. For the general case including the cases that the effect of variation of non-radial flow in the matrix zone is important (I
X
is comparable with 1.0) we have developed the following correlation:
⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ +×=
fD f
C AS
π
2ln
'
, (13) where
⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ −−=
87.017.0ln
X
I e A
. (14) In this equation with increasing Ix, A approaches one and Eq. 13 converts to Eq. 12. For instance, when I
X
is equal to 4, A is 0.98 and the results of two Equations (12 and 13) are almost similar. The coefficients of Eq. 14 have been obtained, conducting a multi-regression exercise, for a wide range of variation of the pertinent parameters as follows: Fracture width (W
f
): 2, 6, 10, 14 mm Fracture permeability (k
f
): 10, 50, 90, 130, 170, 210 D Reservoir permeability (k): 0.1, 1, 9, 25, 48, 81 mD I
X
range: 1.2< I
X
< 10 C
fD
range: 0.2< C
fD
< 888 Number of data points: 15900 Fig. 8 verifies the accuracy of Eq. 13 by comparing its results with the outcome of the in-house simulator. Here the standard error of estimate is less than 0.016.
Skin Factors for Damages
Damage skin represents the difference between the resistance of the damaged layer and its resistance without any damage. Two most common damages in hydraulic fractured systems are fracture face and choked damages.
Fracture Face Damage
Fracture face damage expresses the permeability reduction normal to the fracture. Increasing the matrix resistance due to this damage can be represented with the fracture face damage skin factor, shown by S
fD
. Total flow resistance of a HFS (R
T
) can be calculated by the following equation:
( )
undamaged damaged f f e fD f f eT
R RS kh X r khS S X r kh R
−++=⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ++=
''
2ln2ln2
π µ π µ π µ
, (15) where R
damaged
is the damaged layer resistance and R
undamaged
is its resistance without damage. If it is assumed that the damaged layer has a constant thickness and permeability and flow from the matrix to the fracture is uniform and normal to the fracture face, we can write:
hkX w Rand h X k w R
f d undamaged f d d damaged
44
==
. (16) Combining Equations 15 and 16 will result in:

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