Multilateral wells and, more generally, multibranch wells have two or more well paths branching from a common main trunk as in Fig. 2-16. Dual completion strings may segregate production from the well paths, but this limits the number of branches. Otherwise, the flow from the branches is commingled in the main trunk. If the branch departures from the main trunk are separated, then the flow rate can be measured as
Figure 2-15. Selective inflow performance analysis.
p1
p2
p3
1/J1
1/J2
1/J3
Flow rate (B/D)
Datum pressure (psi)
Figure 2-16. Multibranch well with stacked horizontal branches commingled in an inclined trunk section.
Stacked lateral wells
previously discussed, by the difference above and below the branch or inside a sliding sleeve in the main trunk, where applicable. Alternatively, a test sequence analogous to the multilayer transient test with data acquired in the main trunk above each branch (Fig.
2-17) enables the analysis of each branch, with SIP analysis providing the branch productivities and tran-sient analysis providing a set of model parameters for each branch. If the branch departures are not sepa-rated, the flow rate measurement must be acquired in the branch. Permanent pressure and flow rate sensors installed in the branch could also provide a means to test the branch.
Model selection for a branch depends on the trajec-tory geometry of the branch, which can be vertical, slanted or horizontal. Karakas et al. (1991) published an interpretation for a series of tests in a bilateral well.
2-8. Permeability determination from a fracture injection test
†Fracture injection tests, called also calibration treat-ments, consist of injecting a known amount of the fracturing fluid into the formation, shutting down the pumps and observing the decline of the pressure in the wellbore. It is assumed that up to the end of injection time te, the injection rate i into one wing is constant. After injection, the pressure in the wellbore declines because the fluid is leaking off from the cre-ated fracture and the fracture faces are approaching
† This section by Professor Peter Valkó, Texas A&M University.
each other, relaxing the elastic force exerted on the formation. The decrease of the induced stress results in decline of the wellbore pressure. Because the whole process is controlled by fluid leakoff, pressure decline analysis has been a primary source of obtain-ing the parameters of the assumed leakoff model.
The polymer content of the fracturing fluid is partly intended to impede fluid loss into the reservoir. The phenomenon is envisioned as a continuous buildup of a thin layer (the filter cake) that manifests the ever increasing resistance to flow through the fracture face.
In reality, leakoff is determined by a coupled system, of which the filter cake is only one element. In the fol-lowing, pressure decline analysis is introduced as it pertains to the modeling of fluid loss (see Chapters 6 and 9) along with another method coupling filter-cake resistance and transient reservoir flow.
2-8.1. Pressure decline analysis with the Carter leakoff model
A fruitful formalism dating back to Howard and Fast (1957) is to consider the combined effect of the dif-ferent fluid-loss mechanisms as a material property.
According to this concept, the leakoff velocity uLis given by the Carter equation:
(2-30)
where CLis the leakoff coefficient in ft/min1/2and t is the time elapsed since the start of the leakoff process. The integrated form of Eq. 2-30 is
(2-31) where VLis the fluid volume that passes through the surface ALduring the time period from time zero to time t. The integration constant Spis called the spurt-loss coefficient. It can be considered the width (extent) of the fluid flowing through the surface instantaneously at the beginning of the leakoff process. The two coefficients, CLand Sp, can be determined from laboratory tests.
Application of Eqs. 2-30 and 2-31 during fracturing can be envisioned assuming that the given surface ele-ment “remembers” when it has been opened to fluid loss and has its own zero time, which may be different from location to location on a fracture surface.
Figure 2-17. Transient data acquisition in a multibranch well.
Paired flowmeters above and below branch
Pressure gauge
u C
L t
= L,
V
AL C t S
L
L p
=2 + ,
A hydraulic fracture injection may last from tens of minutes up to several hours. Points at the fracture face near the well are opened at the beginning of pumping, whereas points at the fracture tip are younger. Application of Eq. 2-31 necessitates track-ing the opentrack-ing time of the different fracture face elements. If only the overall material balance is con-sidered, then Carter’s concept is used:
(2-32) where Vi= it is the total injected volume for one wing with a fracture volume V, A is the surface area of one face of one wing, and rpis the ratio of permeable area to total fracture area (see Figs. 2-18 and 2-19 for detail).
The variable κis the opening-time distribution factor.
Clearly, the maximum possible value of κis 2. The maximum is reached if all the surface opens at the first moment of pumping.
Nolte (1979, 1986a) postulated a basic assumption leading to a remarkably simple form of material balance. Assuming that the fracture surface evolves according to a power law, then
(2-33) with the exponent αconstant during the injection period.
Considering the opening-time distribution factor, Nolte realized that it is a function of the exponent α, only
(2-34) The function g0(α) can be determined by an exact mathematical method and is given by Meyer and Hagel (1989):
(2-35)
where Γ(α) is the Euler gamma function. A remark-able fact concerning the g0(α) function is that its val-ues for two extremely departing valval-ues of the expo-nent α, namely at one-half and unity, differ only slightly: g0(1⁄2) = π/2 ≅1.57 and g0(1) = 4⁄3≅1.33.
Vi= +V 2Arp
(
κCL t+Sp)
,A A
t
e te
=
α
,
κ=g0
( )
α .g0
3 2
α α α π
( )
=( )
α +
Γ
Γ ,
Figure 2-18. Basic notation for PKN and KGD geometries. hp= permeable height, hf= fracture height and qL= rate of fluid loss.
2i
qL/2
qL/2 i
hp
hf
xf A = hfxf
A = hfxf rp = hp
hf
If the fracture area is assumed to remain constant after the pumps are stopped, at the time te+ ∆t the volume of the fracture is
(2-36) where dimensionless time is defined as
(2-37) and the two-variable function g0(α, ∆tD) is the fol-lowing mathematical expression (Valkó and Economides, 1995):
(2-38) The function F[a, b; c; z] is the hypergeometric function, available in tabular form (Abramowitz and Stegun, 1989) or computing algorithms.
The average fracture width at time ∆t after the end of pumping is
(2-39) Hence, time variation of the width is determined by the g(α, ∆tD) function, length of the injection period and leakoff coefficient but is not affected by the fracture area.
The fracture closure process (i.e., decrease of aver-age width) cannot be observed directly. However, from linear elastic theory the net pressure is known to be directly proportional to the average width as
(2-40) where pnet= p – pcand pcis the closure pressure. The significance of the closure pressure is described in Chapters 5, 6 and 9. The fracture stiffness sfis a pro-portionality constant for the fracture geometry mea-sured in psi/ft, and it plays a similar role as the con-stant in Hooke’s law. Its form depends on the frac-ture geometry, which may be PKN, KGD or radial (fracture geometries are described in Chapter 6). In petroleum engineering literature, its inverse 1⁄sfis
Figure 2-19. Basic notation for radial geometry.
2i
qL/2
qL/2 i
hp
Rf 1 – hp
2Rf hp
2Rf hp 2Rf 2 π
hp 2Rf
A=Rf2π 2 + arcsin rp =
1/2
Vte+∆t = −Vi 2A r Se p p−2A r ge p
(
α,∆t CD)
L te,.
e
w V
A r S r C t g t
t t i
p p p L e D
e+∆ = −2 −2
(
α,∆)
pnet =s wf ,
∆ ∆
t t
D t
e
=
g t
t t F t
D
D D D
α
α α α
, α
, ; ;
∆ .
∆ ∆ ∆
( )
= + + × +(
+)
+
−
4 2 1 1
2 1 1
1 2
1
also called the fracture compliance. Expressions of sffor common fracture geometries are in Table 2-2.
The combination of Eqs. 2-39 and 2-40 yields
(2-41) so that a plot of p versus g(α, ∆tD) has a slope mN
and intercept bNat g = 0.
Equation 2-41 suggests that the pressure variation in the shut-in period is governed mainly by the leak-off coefficient, and a plot of wellbore pressure versus g(α, ∆tD) values results in a straight line provided that the fracture area Ae, proportionality constant sf
and leakoff coefficient CLdo not vary with time. Under these assumptions the pressure behavior will depart from the linear trend only when the fracture finally closes.
The expression in Eq. 2-41 is the basis of Nolte’s pressure decline analysis. The technique requires plotting the wellbore pressure versus the values of the g-function, as first suggested by Castillo (1987).
The g-function values should be generated with the exponent αconsidered valid for the given model and rheology. Other choices for α(e.g., involving the estimated efficiency of the fracture) are discussed in Chapter 9. A straight line is fitted to the observed points.
For a plot of pressure falloff data from a fracture injection test versus the g-function, Eq. 2-41 implies that the closure pressure pcmust lie on the line fitted through the data. Hence, independent knowledge of the closure pressure, which can be determined from the step rate test described in Chapter 9, helps to identify which part of the falloff data to use for the straight-line fit.
The slope of the straight line is denoted by mNand the intercept by bN. From Eq. 2-41, the slope is re-lated to the unknown leakoff coefficient by
(2-42)
The intercept bNof the straight line at zero shut-in time provides an expression for the spurt-loss coeffi-cient:
(2-43) The first term in Eq. 2-43 can be interpreted as the gross width that would have been created without any fluid loss minus the apparent leakoff width wL. Depending on the fracture geometry, expressions for sfcan be substituted into Eqs. 2-42 and 2-43, result-ing in the expressions for CLin Table 2-3 and for sf
and wLin Table 2-4. Tables 2-3 and 2-4 show that calculation of the leakoff coefficient depends on the fracture geometry.
• PKN fracture geometry
For PKN geometry, the leakoff coefficient can be determined from Eq. 2-42 because sfis dependent on the fracture height, which is a known quantity.
Similarly, the spurt-loss coefficient and the appar-ent leakoff width can be computed from Eq. 2-43 using the expressions in Table 2-4.
• KGD and radial fracture geometries
For KGD geometry, sfis dependent on the fracture half-length xf; for radial geometry, it is dependent on the fracture radius Rf. If the spurt loss is negli-gible, then xfor Rfcan be determined from the expressions in Table 2-5, and, in turn, the leakoff coefficient can be computed from the appropriate expression in Table 2-3. This analysis procedure was introduced by Shlyapobersky et al. (1988a).
If the spurt loss cannot be neglected, the more detailed analysis procedures in Chapter 9 must be used.
If the closure pressure is not determined inde-pendently, straightforward analysis with the
Table 2-2. Proportionality constant sffor different fracture geometries.
Table 2-3. Leakoff coefficient CLfor different fracture geometries.
g-function plot relies on correct identification of the portion of the data through which the line should be fitted. As with Horner analysis of pres-sure buildup data, the success of this plot is under-mined if—for whatever reason—identification is not straightforward. More details on this analysis are provided in Chapter 9.
2-8.2. Filter-cake plus reservoir pressure drop leakoff model (according to Mayerhofer et al., 1993)
Carter’s bulk leakoff model is not the only possible interpretation of the leakoff process. Other models have been suggested, but one reason why such models have not been used widely is that it is difficult to design a calibration test interpretation procedure that is standardized (i.e., the results of which do not depend too much on subjective factors of the inter-preter). The Mayerhofer et al. (1993) method over-comes this difficulty. It describes the leakoff rate using two parameters that are physically more discernible than the leakoff coefficient to the petroleum engineer:
the reference resistance R0of the filter cake at a refer-ence time t0and the reservoir permeability kr. To obtain these parameters from an injection test, the reservoir pressure, reservoir fluid viscosity, formation porosity and total compressibility must be known.
Figure 2-20 is a schematic of the Mayerhofer et al.
model in which the total pressure difference between the inside of a created fracture and a far point in the
reservoir is shown with its components. Thus, the total pressure drop is
(2-44) where ∆pfaceis the pressure drop across the fracture face dominated by the filter cake, ∆ppizis the pres-sure drop across a polymer-invaded zone, and ∆pris the pressure drop in the reservoir.
In a series of experimental works using typical hydraulic fracturing fluids (e.g., borate- and zirconate-crosslinked fluids) and cores with permeability less than 5 md, no appreciable polymer-invaded zone was detected. At least for crosslinked fluids, the sec-ond term on the right side of Eq. 2-44 can be ignored:
(2-45) Using the Kelvin-Voigt viscoelastic model for description of the flow through a continuously depositing fracture filter cake, Mayerhofer et al.
(1993) gave the filter-cake pressure term as
Table 2-4. Spurt-loss coefficient and apparent leakoff width for different fracture geometries.
PKN KGD Radial
Table 2-5. Fracture exent from the no-spurt-loss assumption.
Figure 2-20. Filter-cake plus reservoir pressure drop in the Mayerhofer et al. (1993) model.
Pressure
(2-46) where R0is the characteristic resistance of the filter cake, which is reached during reference time t0. The flow rate qLis the leakoff rate from one wing of the fracture. In Eq. 2-46, it is divided by 2, because only one-half of it flows through area A.
The pressure drop in the reservoir can be tracked readily by employing a pressure transient model for injection into a porous medium from an infinite-con-ductivity fracture. For this purpose, known solutions such as the one by Gringarten et al. (1974) can be used. The only additional problem is that the surface area increases during fracture propagation.
Therefore, every time instant has a different fracture length, which, in turn, affects the computation of dimensionless time.
The transient pressure drop in the reservoir is
(2-47) where hfis the ratio of the leakoff area to the charac-teristic length (given as rphffor PKN and KGD geometries and as rpπRf/2 for radial geometry), µris the reservoir fluid viscosity, and pDis the dimension-less pressure function describing the behavior of the reservoir (unit response).
The factor 2 must be used in Eq. 2-47 in front of qjbecause q is defined as the leakoff rate from one wing. In petroleum engineering literature, how-ever, dimensionless pressure is defined using the total flow into (from) the formation.
Substituting Eqs. 2-46 and 2-47 into Eq. 2-45 obtains
(2-48) where the end of pumping is selected as the charac-teristic time for the filter-cake resistance. A simple rearrangement yields
(2-49)
Equation 2-49 can be used both in hydraulic frac-ture propagation and during fracfrac-ture closure. It allows determination of the leakoff rate at the time instant tn
if the total pressure difference between the fracture and the reservoir is known, as well as the history of the leakoff process. The dimensionless pressure solu-tion pD[(tj– tj – 1)D] has to be determined with respect to a dimensionless time that takes into account the actual fracture length at tn(not at tj).
The injection test interpretation processes data given as (tn, pn) pairs with n > ne, where neis the index of the first time point after shut-in. For the consideration of dimensionless pressure, the early-time approximation can be used for an infinite-conductivity fracture:
(2-50) The leakoff rates are strongly connected to the observed pressure changes according to
(2-51) for j≥ne+ 2.
Combining Eqs. 2-50 and 2-51 with Eq. 2-48 obtains for n > ne+ 2:
(2-52) During fracture propagation, the leakoff rates qj
for j = 1, . . ., ne+ 1 are not known exactly (nor are any values for tjin this period). Therefore, some kind of assumption is required to proceed.
The key assumption is that for these purposes the first ne+ 1 leakoff rates can be considered equal:
(2-53) for j = 1, . . ., ne+ 1.
In fact, it is more convenient to work not with the average leakoff rate but with the apparent leakoff width defined by
(2-54) where ηis the fluid efficiency.
The apparent leakoff width can be estimated from the Nolte-Shlyapobersky method as shown in Table 2-4. Then Eq. 2-52 leads to
(2-55) In terms of the apparent leakoff width and after rearrangement, Eq. 2-55 becomes
(2-56) Introducing the notation
Eq. 2-56 takes the form
(2-57)
The Mayerhofer et al. method is based on the fact that Eq. 2-57 can be written in straight-line form as
(2-58) for n > ne+ 2, where
(2-59)
(2-60) The coefficients c1and c2are geometry dependent and discussed later. Once the x and y coordinates are known, the (x,y) pairs can be plotted. The corre-sponding plot is referred to as the Mayerhofer plot.
A straight line determined from the Mayerhofer plot results in the estimate of the two parameters bMand mM. Those parameters are then interpreted in terms of the reservoir permeability and the reference filter-cake resistance. For the specific geometries, the coef-ficients c1and c2as well as the interpretation of the straight-line parameters are as follows.
• PKN geometry
• Radial geometry
• Example interpretation of fracture injection test Table 2-6 presents pressure decline data, and Table 2-7 presents reservoir and well information for this example. The closure pressure pcdetermined inde-pendently is 5850 psi.
Figure 2-21 is a plot of the data in Cartesian coordinates and also shows the closure pressure.
The g-function plot in Fig. 2-22 is created using α= 8⁄9, which is considered characteristic for the radial model.
From the intercept of the straight line is obtained the radial fracture radius Rf= 27.5 ft.
(The straight-line fit also provides the bulk leakoff coefficient CL= 0.033 ft/min1/2and fluid efficiency η= 17.9%.) The ratio of permeable to total area is rp= 0.76.
Figure 2-23 is the Mayerhofer plot. From the slope of the straight line (mM= 9.30 ×107) is obtained the apparent reservoir permeability kr,app= 8.2 md and the true reservoir permeability kr= 14.2 md. The resistance of the filter cake at the end of pumping (te= 21.2 min) is calculated from the intercept (bM= 2.5 ×10–2) as the apparent resistance R0,app= 1.8 ×104psi/(ft/min) and the true resistance R0= 1.4 ×104psi/(ft/min).
Table 2-6. Example pressure decline data.
∆∆t (min) pws(psi)
Table 2-7. Example reservoir and well information.
Permeable height, hp 42 ft
Reservoir fluid viscosity, µr 1 cp
Porosity, φ 0.23
Total compressibility, ct 2 ×10–5psi–1 Reservoir pressure, pr 1790 psi Plane strain modulus, E´ 8 ×105psi
Pumping time, te 21.75 min
Injected volume (two wing), 2Vi 9009 gal
Closure pressure, pc 5850 psi
Geometry Radial
Figure 2-21. Example of bottomhole pressure versus shut-in time.
Figure 2-22. Example g-function plot.
p (psi) 6500
8000 7500 7000
6000 5500
1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 g
Figure 2-23. Example Mayerhofer plot with radial geometry.
y
1.2 1.0 0.8 0.6 0.4 0.2 0
0 2 × 10–9 6 × 10–9 1 × 10–8 x
3-1. Introduction
The National Academy of Sciences defines rock mechanics as “the theoretical and applied science of the mechanical behavior of rock; it is that branch of mechanics concerned with the response of the rock to the force fields of its physical environment.” From this definition, the importance of rock mechanics in several aspects of the oil and gas industry can easily be understood. The fragmentation of rock governs its drillability, whereas its mechanical behavior influences all aspects of completion, stimulation and production.
However, not until recently has this particular aspect of earth sciences started to play a predominant role in energy extraction. The impetus was to explain, quali-tatively and quantiquali-tatively, the orientation of fractures (Hubbert and Willis, 1957), some unexpected reser-voir responses or catastrophic failures (e.g., less pro-duction after stimulation and pressure decline in wells surrounding an injection well; Murphy, 1982), casing shear failure (Nester et al., 1956; Cheatham and McEver, 1964), sand production (Bratli and Risnes, 1981; Perkins and Weingarten, 1988; Morita et al., 1987; Veeken et al., 1991; Kooijman et al., 1992;
Cook et al., 1994; Moricca et al., 1994; Geilikman et al., 1994; Ramos et al., 1994), rock matrix collapse during production (Risnes et al., 1982; Pattillo and Smith, 1985; Smits et al., 1988; Abdulraheem et al., 1992) and borehole stability problems (Gnirk, 1972;
Bradley, 1979; Guenot, 1989; Santarelli et al., 1992;
Ong and Roegiers, 1993; Maury, 1994; Last et al., 1995).
The significant contribution as far as the orientation of fractures is concerned was provided by the work of Hubbert and Willis (1957; see Sidebar 3A), which indicates ever-increasing differences between vertical and horizontal stresses within the earth’s crust. Until
The significant contribution as far as the orientation of fractures is concerned was provided by the work of Hubbert and Willis (1957; see Sidebar 3A), which indicates ever-increasing differences between vertical and horizontal stresses within the earth’s crust. Until