Pressure Behavior of Partially Penetrating Multiple Hydraulic Fractures

This paper introduces a new technique for analyzing the pressure behavior of a horizontal well with multiple vertical and inclined partially penetrating hydraulic fractures. The hydraulic fractures in this model could be longitudinal or transverse, vertical or inclined, symmetrical or asymmetrical. The fractures are propagated in isotropic or anisotropic formations and considered having different dimensions and different spacing. This technique, based on pressure and pressure derivative concept, can be used to calculate various reservoir parameters, including directional permeability, fracture length and percentage of penetration. The study has shown that the pressure behavior of small penetration rate is similar to the horizontal wells without hydraulic fractures.


Introduction
Several factors control the final output of the hydraulic fracturing process.Fracture dimensions (half fracture length, fracture width, and fracture height) are of great importance in the performance as are the orientation of the fractures as well as the rock and fluid properties.Typically, it is preferred that the fracture height be equal to the formation height, where fully-penetrating fractures can be produced.Unfortunately, the fractures can not always penetrate totally the formation where partially penetrating fractures may be produced.Partially penetrating hydraulic fractures are undesirable stimulation process due to the possibility of reducing the expected production rate of the fractured formation.However, fully penetrating fractures in a reservoir with water and oil in contact may lead to an early or immediate water production.Therefore, partially penetrating fractures may be the only way to prevent the production of unwanted water.
Great attentions have been focused to model the pressure transient behavior for either horizontal or vertical wells, with or without hydraulic fractures.As a result several models were developed based on the using of the source solution and Green's function to solve unsteady-state flow problem in the reservoir.Gringarten and Ramey (1973) used source function and Newman product method for solving transient flow problem.Although this approach is extremely powerful in solving two and three dimensions problem, it has some limitations such as incorporating the influence of storage and skin effects.The transient flow solutions have been extended to predict the behavior of the infinite conductivity vertical fracture in homogenous formations or in dual-porosity media.Ozkan (1988) presented an extensive library of different solutions for diffusivity equation in terms of the Laplace transform variable to reduce the limitations in the source solution presented by Gringarten and Ramey.He considered a wide variety of wellbore configurations, different bounded systems, and homogeneous or double-porosity reservoirs.Benjamin (1978) used a finite element model to study pressure behavior of a well intersecting a vertical fracture at the center of closed square reservoir.Wong et al (1985) analyzed the data using type curve matching and pressure and pressure derivative for cases of vertical fractured wells with no skin and no wellbore storage and cases with both skin and wellbore storage during the bi-linear flow period.Cinco-Ley and Meng (1988) studied the results obtained from the transient behavior of a well intersected by a vertical fracture in a double porosity reservoir.They introduced two models; the first one was a general semi-analytical and the second one was a simplified fully analytical model Raghavan et al. (1997) developed a mathematical model to discern the characteristic response of multiply-fractured horizontal wells.Three significant flow periods have been observed based on their model; the early time period in which the system behaved like the one with n-layers, the intermediate time period in which the system reflected the interference between the fractures, and late time period in which the system behaved as a single fracture horizontal well with length equal the distance between the outermost fractures.Wan and Aziz (1999) developed general solution for horizontal wells with multiple fractures.They showed that four flow regimes can be observed; the early linear, transient, late linear, and late time radial flow.Zerzar et al. (2003) combined the boundary element method and Laplace transformation to present a comprehensive solution for multiple vertical fractures horizontal wells.Seven flow regimes have been noticed; bilinear, first linear, elliptical, radial, pseudo-radial, second linear, and pseudo-steady state.Al-Kobaisi and Ozkan (2004) presented a hybrid numerical-analytical model for the pressure transient response of horizontal wells intercepted by a vertical fracture.Anh and Tiab (2009) solved the analytical model presented by Cinco-Ley (1974) for the pressure transient behavior caused by an inclined fracture associated with vertical wellbore.The model used the uniform flux and infinite conductivity fracture solution for different inclination angles from the vertical direction.Both type curve and TDS technique have been used to estimate the formation parameters such as permeability, skin factor, and fracture length.
Eventhouth, great attentions were focused on the study of pressure transient analysis of hydraulically fractured wells; there are few studies about the effects of the partially penetrating fractures.Raghavan et al (1978) were the first presented an analytical model that examines the effect of the fracture height on the pressure behavior of single vertical fracture.Their model was derived based on the solution technique presented by Gringarten and Ramey (1973).Rodriguez and Cinco-Ley (1984) developed semi-analytical solution for the transient flow behavior of a reservoir with a well intersecting a partially-penetrating single vertical fracture of both finite and infinite conductivity cases.The results of this study explained that the flow behavior of partially penetrating fracture during the early time period is equivalent to that of totally penetrating fracture.Alpheous and Tiab (2008) studied the effect of the partial penetrating infinite conductivity hydraulic fractures on the pressure behavior of horizontal well extending in naturally fractured formation.They stated that the duration of early linear flow regime is a function of the hydraulic fractures height.

Mathematical Models
The analytical model for the pressure behavior of a horizontal well intersecting with partially penetrating multiple vertical and inclined hydraulic fractures can be derived based on the solution for the diffusivity equation in the porous media.The following facts would be important to be noticed:

•
The flow from the reservoir to the wellbore sections between fractures is negligible as compared with the flow from the reservoir to the fracture plane.

•
Fluid flows from the reservoir to the well through planar inclined and vertical fractures.
• A first approximation of the behavior of the system is the uniform flux fracture case.It is assumed that fluid enters the fractures at a uniform rate per unit area of the fracture face.
The following assumptions are important for the derivation of the model: 1-The reservoir is homogenous, having constant and uniform thickness with two impermeable layers at the top and bottom of the formation.
2-Constant porosity and permeability in each direction, but the formation is anisotropic.
3-Gravitational and frictional effects are negligible.

4-
The well is extending in the midpoint of the formation height (symmetrical).
5-Single phase fluid of small and constant compressibility, constant viscosity, and formation volume factor, flows from the reservoir to the wellbore.
6-Reservoir pressure is initially constant.
(1) 7-The pressure at the outer boundaries of the reservoir is assumed to be constant and equal to the initial reservoir pressure. (2) 8-The pressure at the upper and lower impermeable boundaries is assumed to be constant so that: (3) (4) Figure 1.Horizontal well intersected by partially penetrating multiple vertical hydraulic fractures Consider a horizontal well with partially penetrating vertical transverse hydraulic fractures in an infinite, homogenous, isotropic or anisotropic, horizontal slab reservoir as shown in Figure 1.Each fracture is considered as a single plane of length (2x f ), width (w), height (h f ).The spacing between fractures is (D).If we assume that all fluid withdrawal will be through the fractures, the fractures are partially penetrating the formation, the fractures can be simulated as inclined plane sources.The unsteady state pressure drop created by these planes at any point (x m , y m , z m is: (5) The model for pressure response of horizontal wells intersecting by multiple partially penetrating vertical hydraulic fractures in dimensionless form is: (6)

Horizontal well intersected by partially penetrating multiple inclined hydraulic fractures
For partially penetrating multiple inclined fractures as shown in Figure 2, the model for pressure behavior in dimensionless form is: (7)

Pressure Behavior
The penetration ratio (the ratio of the fracture's height to the formation's height) has significant influence on the pressure behavior at the wellbore and flow regimes in the vicinity of the wellbore.A set of type-curve matching plots will be presented in this paper to reflect the compound effect of the penetration ratio, the number of fractures, the spacing between fractures as well as fracture dimensions and inclination angle from the vertical axis.Seven flow regimes may develop for different cases: -First linear flow: At early time, reservoir fluid flows linearly and directly from the formation to the individual fractures in the XZ plane.
-Second Linear Flow: When the pressure pulse reaches the upper and lower boundary, reservoir fluid continues flowing linearly and directly from the formation to the fractures in the XZ plane.
-Third Linear Flow: This flow regime develops for short spacing, large number of hydraulic fractures and large half fracture length.In this case, pressure behavior can be considered similar to the pressure behavior of long horizontal wells.
-Early radial flow: Early radial flow regime represents the radial flow around each fracture in the YZ plane.Typically, this flow is observed when the penetration ratio is small (h hfD <0.5) and the spacing between fractures is long (D D >5).
-Intermediate radial flow: Intermediate radial flow regimes develop for long spacing between fractures when there is sufficient time for reservoir fluid to flow radially in the XY plan to each individual fracture.
-Pseudo-radial flow: Pseudo-radial flow regime is the dominant flow for all cases at late time when reservoir fluids flow in the XY plane radially toward the fractures.
-Elliptical flow: This flow regime indicates elliptical flow toward the fractures.
The following responses are easy to identify based on different penetration ratios and different half fracture lengths:

Large Penetration Ratio (h hfD >0.5)
Because of the penetration ratio, the pressure behavior in this case tends to be similar to the fully penetrating fractures where other factors such as the number of fractures, spacing between them, fracture dimensions, and inclination angle have the main influence.
1) For a small number of hydraulic fractures (less than five) and short spacing, first linear, transition, second linear, transition and pseudo-radial flow are observed as shown in Figure 3 and Figure 4.
2) For a small number of hydraulic fractures (less than five) and long spacing, first linear, transition, second linear, intermediate radial, transition and pseudo radial flow are observed as shown in Figure 5 and Figure 6.
3) For a large number of hydraulic fractures (more than five) and small spacing, first linear flow, transition flow, second linear flow, third linear flow, transition flow, and pseudo-radial flow regimes are observed such as in Figure 7 and Figure 8.
4) For a large number of hydraulic fractures (more than five) and long spacing, first linear, transition, second linear, intermediate radial, elliptical, transition and pseudo radial flow regimes are observed as shown in Figure 9 and Figure 10.

Intermediate radial flow
Pseudo-radial flow flow 1) For a small number of hydraulic fractures (less than five) and short spacing, first linear flow, transition flow, second linear flow, transition flow, and pseudo-radial flow regimes are observed as shown in Figure 11 and Figure 12.
2) For a small number of hydraulic fractures (less than five) and long spacing, first linear flow, transition flow, second linear flow, intermediate radial flow, transition flow, and pseudo radial flow regimes are observed as shown in Figure 13 and Figure 14.
3) For a large number of hydraulic fractures (more than five) and small spacing, first linear flow is not observed.Therefore, second linear flow, third linear flow, transition flow, and pseudo-radial flow regimes are the only flow regimes that are observed such as in Figure 15 and Figure 16.
4) For a large number of hydraulic fractures (more than five) and long spacing, first linear flow also is not observed.Second linear flow, intermediate radial flow, elliptical flow, transition flow, and pseudo radial flow regimes are observed as shown in Figure 17 and Figure 18.

Transition flow
Pseudo-radial flow Sixteen vertical transverse hydraulic fractures, hxfD=2, DD=8 Sixteen vertical transverse hydraulic fractures, hxfD=8, DD=8 1) For a small number of hydraulic fractures (less than five) and short spacing, first linear flow can not be observed.Early radial flow, second linear flow, transition flow, and pseudo-radial flow regimes are observed as shown in Figure 27 and Figure 28.The behavior in these two cases is similar to horizontal wells with short to moderate wellbore length.
2) For a small number of hydraulic fractures (less than five) and long spacing, first linear flow can not be observed.Early radial flow, second linear flow, transition flow, intermediate radial flow, transition flow, and pseudo radial flow regimes are observed as shown in Figure 29 and Figure 30.
3) For a large number of hydraulic fractures (more than five) and small spacing, neither first linear flow nor early radial flow can be observed.Second linear flow, third linear flow, transition flow, and pseudo-radial flow are the only flow regimes that are observed such as in Figure 31 and Figure 32.The behavior in these two cases is similar to a single vertical hydraulic fracture.
4) For a large number of hydraulic fractures (more than five) and long spacing, neither first linear flow nor early radial flow can be observed also.Second linear flow, intermediate radial flow, elliptical flow, transition flow, and pseudo radial flow regimes are observed as shown in Figure 33 and Figure 34.The behavior in these two cases is similar to multiple hydraulic fractures.

Effect of Inclination Angle
The inclination angle from the vertical axis has a similar effect on pressure behavior of partially penetrating hydraulic fractures as the penetration ratio.It can be explained by the reduction in the fracture height which produces a reduction in the penetration ratio, when the fractures are inclined from the vertical direction.As fractures propagate in inclined directions rather than the vertical one, the probability for partially penetrating fractures to occur is reasonable.Figure 35 and Figure 36 represent pressure behaviors for two partially penetrating inclined hydraulic fractures for different inclination angles.While Figure 37 and Figure 38 represent pressure behaviors of ten partially penetrating inclined hydraulic fractures for different inclination angles.For all cases, the early radial flow develops when the inclination angle from the vertical direction increases.

Application of Type Curve Matching
As shown on the plots in Appendix (B), the pressure and pressure derivative have different shapes for each combination of penetration rate, half fracture length, number of fractures, spacing between fractures, and inclination angle from the vertical axis.Type-curve matching can provide a quick estimation for reservoir and fractures parameters.
The following information is associated with each type curve: penetration rate (h hfD ), half fracture length to fracture height ratio (h xfD ), dimensionless spacing between fractures (D D ), number of fractures (n), and inclination angle ( ).Thus, the following information can be obtained from the type curve matching process: (P D ) M , (ΔP) M , (t D ) M , (Δt) M , ( ) M , (h xfD ) M , (D) M , (h hfD ) M , (n) M .The following steps illustrate how type curve matching is used to determine reservoir characteristics such as: permeability, inclination angle, spacing, pseudo-skin factor, fracture half length, and number of fractures.
Step-2 Obtain the best match of the data with one of the type curves.
Step-3 Read from any match point: .
4-Fracture height and penetration ratio.

Solution
Step-1 Plot ( vs. ) and ( vs. ) on log-log paper as shown in Figure 39.
Step-2 Obtain the best match of the data with one of the type curves as shown in Figure 40.Step-3 Read from any match point: Step-4 Calculate half fracture length from Equation 12.
Table 1 summarizes the input data and the resulted values of Example-1.

Example -2
Pressure drawdown test data of a hydraulically fractured horizontal well is given in Table (C-2) in Appendix (C).Sixteen vertical hydraulic fractures have been designed with a half fracture length (310 ft).Other known reservoir and well data are: q = 100 STB/D  = 0.04  = 0.8 cp c t = 1.0x10 -6 psi -1 h = 10 ft r w = 0.5 ft p i = 10000 psi B = 1.1 bbl/STB

Solution
Step-1 Plot ( vs. ) and ( vs. ) on log-log paper as shown in Figure 41.
Step-2 Obtain the best match of the data with one of the type curves as shown in Figure 42.Step-3 Read from any match point: Step-4 Calculate from Equation 8.

Conclusions
1) An early radial flow regime is expected to be noticed for the case of the partially penetrating hydraulic fractures where the fluid flows radially in the parallel plane to the wellbore toward each individual fracture.This type of flow regime can be used as an indication for the uncompleted penetration or the fracture height is less than the formation height.
2) Second linear flow represents fluid's linear flow from the formation toward each fracture in the normal plane to the wellbore.This flow regime develops shortly after the upper and lower boundaries have been reached.
3) Third linear flow regime develops for short spacing partially penetrating hydraulic fractures where the fluid flows linearly in a parallel plane to the wellbore.
4) Intermediate radial flow appears for the case of wide spacing between fractures where the radial flow in the horizontal plane toward each fracture is developed.Intermediate radial flow can be used as an indication for serious production problem when the fractures do not perform properly.
5) For small penetrating ratio and large number of hydraulic fractures, the pressure behavior is similar to the behavior of long horizontal wells.
6) The inclination angle from the vertical direction has the same impact of the partial penetrating on pressure behavior of hydraulically fractured horizontal wells.considered as a single plane of length (2x f ), width (w), height (h f ).The spacing between fractures is (D).If we assume that all fluid withdrawal will be through the fractures, the fractures are partially penetrating the formation, the fractures can be simulated as inclined plane sources.The unsteady state pressure drop created by these planes at any point (x m , y m , z m ) is:

Nomenclatures
where; S xyz is the instantaneous source function for an inclined plane source in an infinite slab reservoir and (q) is the fluid withdrawal per unit fracture surface area per unit time.

Figure
Figure 3. Pressure behavior of two partially penetrating vertical hydraulic fractures Figure 4. Pressure behavior of two partially penetrating vertical hydraulic fractures

Figure 7 .
Figure 7. Pressure behavior of sixteen partially penetrating vertical hydraulic fractures Figure 11.Pressure behavior of two partially penetrating vertical hydraulic fractures

Figure 17 .
Figure 17.Pressure behavior of sixteen partially penetrating vertical hydraulic fractures

Figure 19 .
Figure 19.Pressure behavior of two partially penetrating vertical hydraulic fractures

Figure 21 .
Figure 21.Pressure behavior of two partially penetrating vertical hydraulic fractures

Figure 27 .Figure 31 .
Figure 27.Pressure behavior of two partially penetrating vertical hydraulic fractures

Figure 35 .
Figure 35.Pressure behavior of two partially penetrating inclined hydraulic fractures

Figure 39 .
Figure 39.Pressure and pressure derivative plot Example -1 Figure 40.Type-curve matching plot for Example -1

Figure 41 .
Figure 41.Pressure and pressure derivative plot Example -2 Figure 42.Type-curve matching plot for Example -2 Consider a horizontal well with partially penetrating vertical transverse hydraulic fractures in an infinite, homogenous, isotropic or anisotropic, horizontal slab reservoir as shown in Figure A-1.Each fracture is Figure A-1.Horizontal well intersected by partially penetrating multiple hydraulic fractures Figure A-3.Schematic diagram of partial penetrating hydraulic fracture Figure A-4.Horizontal well intersected by partially penetrating multiple inclined hydraulic fractures

Figure B- 7 .
Figure B-7.Pressure and pressure derivative plot Figure B-8.Pressure and pressure derivative plot

Figure B- 13 .
Figure B-13.Pressure and pressure derivative plot Figure B-14.Pressure and pressure derivative plot

Figure B- 31 .
Figure B-31.Pressure and pressure derivative plot Figure B-32.Pressure and pressure derivative plot Figure B-37.Pressure and pressure derivative plot Figure B-38.Pressure and pressure derivative plot

Table 1 .
Summary of results of Example-1

Table - 2
summarizes the input data and the resulted value for Example-2.