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Stimulation of the natural fracture system by graded proppant injection
Journal of Petroleum Science and Engineering 111 (2013) 71–77
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Stimulation of the natural fracture system by graded proppant injection Aditya Khanna a, Alireza Keshavarz b, Kate Mobbs b, Michael Davis a, Pavel Bedrikovetsky b,n a b
School of Mechanical Engineering, The University of Adelaide, SA 5005, Australia Australian School of Petroleum, The University of Adelaide, SA 5005, Australia
art ic l e i nf o
a b s t r a c t
Article history: Received 22 May 2012 Accepted 15 July 2013 Available online 24 July 2013
Graded proppant injection i.e. the injection of particles of increasing size and decreasing concentration, is proposed as a method for stimulating natural fractures. Compared to the injection of mono-sized proppant, graded proppant injection would facilitate deeper percolation of proppant into the fracture system and enhanced well productivity. A mathematical model is developed which describes the injection stage and the capture kinetics of proppant particles in the natural fracture system. Based on the mathematical model, an injection schedule is developed which would result in the optimal placement of proppant in the fracture system. A case study is conducted to estimate the change in well productivity index due to the application of the graded particle injection method. & 2013 Elsevier B.V. All rights reserved.
Keywords: Coal bed methane Naturally fractured reservoirs Well stimulation Graded proppant injection Injection schedule Mathematical modelling
1. Introduction Naturally fractured reservoirs such as coal bed methane (CBM), tight and shale gas fields and geothermal reservoirs often require some form of stimulation in order to achieve economical production rates. Fluid injection at the wellbore prior to production is a widely adopted technology for stimulating natural fractures (Colmenares and Zoback, 2007). The injection of fluid without proppant may lead to an increase in permeability due to the shear induced opening of rough natural fractures (Hossain et al., 2002; Rahman et al., 2002). Injection of proppant on the other hand, leads to an increase in reservoir permeability by preventing the closure of the dilated natural fractures (Holditch et al., 1968; Warpinski et al., 2008). However, in the absence of a wide hydraulic fracture, the large proppant particles typically used in fracturing treatments can only stimulate a small region in the vicinity of the wellbore. This is due to the rapid decline in the aperture of stimulated natural fractures with increasing distance from the wellbore. In this paper, the method of graded proppant injection is proposed. Injecting small particles first followed by larger particles could increase the size of the stimulated zone, since the small particles are more likely to percolate deeper into the fracture system. The main parameters of interest in the problem of graded proppant injection are: the increase in well productivity and the
n
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proppant injection schedule i.e. the dependence of particle size and concentration on the injection time. Within this paper, a simplified engineering model is developed for calculating these parameters. The problem of proppant placement in a fracture system is formulated in Section 2. A mathematical model is then presented which describes: the opening of natural fractures due to fluid injection (Section 3.2), particle plugging in the opened fractures (Section 3.3), the optimal proppant placement in the fracture network (Section 3.4) and the injection schedule (Section 3.5). The change in well productivity is calculated in Section 3.6. A case study is conducted in Section 4 using typical reservoir data to determine the effect of injection rate, and size of stimulated zone on the change in well productivity index. 2. Problem formulation The proppant placement resulting from graded proppant injection is schematically presented in Fig. 1. It shows the straining of particles of different sizes in the fracture network. The basic features of the injection schedule which yields a dependence of the injected particle size and concentration upon the injection time, can be deduced from Fig. 1. During fluid injection at the wellbore, the fluid pressure and consequently, the opening of fractures diminishes with increasing distance from the well. This implies that smaller particles need to be injected before larger particles in order to achieve deeper percolation. In terms of the injection schedule, this implies that the injected particle radius
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combinations. For simplicity, the semi-analytical results obtained by Khanna et al. (2012) are utilized here. In the next section, the mathematical model for particle straining in fractured porous media is presented.
3. Mathematical model for flow in fractured porous media with strained particles 3.1. Modelling assumptions The following assumptions are utilized regarding graded proppant injection:
At the scale of the reservoir, the flow from the wellbore
Fig. 1. Plugging of the natural fracture system by graded proppant particles.
Normal stress
Fluid
Rigid proppant
Deformed flow channel
Normal stress Fig. 2. Two competitive factors affect the hydraulic resistance: (1) The deformation of the flow channel caused by rock stresses and (2) the additional tortuosity of the flow path due to the presence of the particles.
must increase with injection time. Also, a greater number of cleats are encountered by the flow with increasing distance from the wellbore. Thus, a greater concentration of small particles is needed to plug the filtration path compared to large particles. Consequently, the concentration of injected particles must decrease with injection time. There are two competitive mechanisms, shown in Fig. 2, which affect the permeability of the stimulated fracture system. These are: the reduction in the fracture aperture due to the confining stresses and the additional tortuosity of the flow path due to the presence of proppant particles. A dense packing of the proppant particles minimizes the deformation of the fracture and a sparse packing minimizes the additional tortuosity to the flow path. An intermediate proppant concentration exists, at which net effect of these two competitive mechanisms is minimum. At this optimum proppant concentration, the maximum conductivity of the fracture system is achieved. The optimal proppant concentration depends upon a number of parameter such as the reservoir stress, material properties of the rock and the strength of the proppant. In order to determine the value of the optimal proppant concentration and the fracture conductivity during production, experiments similar to Brannon et al. (2004) can be conducted for specific rock and proppant
towards the fracture system is axi-symmetric and can be described by Darcy's law for radial flow. The conductive fractures form a tree-like or dendritic structure with branches multiplying from the well towards the reservoir. During injection, the flow is steady state since the time of propagation of an elastic pressure wave in the stimulated zone is negligibly small compared to the total time of injection. The calculations are performed for constant injection rate at the wellbore. The naturally fractured reservoir is idealized by the matchstick model in which match-like blocks of rock matrix are separated by cleats or fractures. The height of the matrix blocks is equal to the height of the reservoir and the width of the blocks is equal to the average cleat spacing. The dependence of the permeability of the cleat system upon pressure is obtained from a widely used permeability model suggested by Palmer and Mansoori (1998). The terms within the Palmer–Mansoori model related to matrix swelling/shrinkage are neglected since the injection period is relatively short. In order to simplify the particle capture criterion, the presence of fracture wall asperities, in-fill material and tortuosity in the fractures are neglected. It is assumed that particle straining occurs when the particle diameter exceeds the pressuredependent opening of the fracture. The particles are assumed to travel at the same speed as the injection fluid, i.e. proppant settlement by gravity is neglected.
The mathematical model for water injection before particle plugging, based on the above assumptions, is presented in the next section. 3.2. Water injection stage In this section, the basic equations for water injection into naturally fractured system are presented. Darcy′s law for radial flow of an incompressible fluid through deformable rock is given by q kðpÞ dp ¼ 2πr μ dr
ð1Þ
Here q is the constant injection rate per unit thickness of the reservoir, pðrÞ is the pressure at a distance r from the wellbore, kðpÞ is the pressure dependent permeability of the fracture network and μ is the fluid viscosity. Assuming a given pressure dependence for permeability and separating variables in the ordinary differential equation (1) yields the implicit formula for pressure distribution around the well: Z p qμ r e ln ¼ kðpÞdp; ð2Þ 2π r pres
A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77
Here r e is the drainage radius at which the fluid pressure is equal to the initial reservoir pressure pres . The pressure–permeability relationship substituted into Eq. (2) is 3 C kðpÞ ¼ ko 1 þ ðppres Þ ð3Þ ϕo This expression comes from the model for permeability proposed by Palmer and Mansoori (1998). The constants ko and ϕo are the initial permeability and porosity of the reservoir, respectively and the constant C represents the reservoir compressibility under uniaxial strain. It can be expressed in terms of the Young′s modulus E and Poisson′s ratio ν of the reservoir as C¼
ð1 þ νÞð12νÞ 1 ð1νÞ E
ð4Þ
Substituting Eq. (3) into (2) and integrating in p yields the explicit formula for the radial pressure distribution during water injection: " 1=4 # ϕ 2qμC r e pðrÞ ¼ pres þ o ln þ 1 1 ; pðr w Þ ¼ pw ð5Þ πko ϕo C r The pressure at the wellbore radius r w is denoted by pw . The radial coordinate r can be normalized against the drainage radius of the reservoir to yield the dimensionless radial coordinate r D ¼ r=r e and the dimensionless wellbore radius r wD ¼ r w =r e . Further, a dimensionless injection rate εq can be defined as εq ¼
2qμC πko ϕo
ð6Þ
In terms of these dimensionless parameters, the expression for the radial pressure distribution can be re-written as " # ϕ 1 1=4 pðr D Þ ¼ pres þ o 1 þ εq ln 1 ; pðr wD Þ ¼ pw ð7Þ rD C By substituting Eq. (7) into (3), the permeability distribution around the well kðr D Þ can be obtained as 1 3=4 ð8Þ kðr D Þ ¼ ko 1 þ εq ln rD From the dimensionless analysis of the relationship between the rock permeability, mean aperture of fractures and matrix block size (Bear, 1972; Basniev et al., 1988) it follows that 3 k h ¼ ð9Þ ko ho where ko and ho are the initial permeability of the reservoir and the initial aperture of the fractures, respectively. Substituting Eq. (8) into relationship (9) results in an expression for the effective cleat opening given by hðr D Þ 1 1=4 hD ðr D Þ ¼ ¼ 1 þ εq ln ð10Þ ho rD Here the apertures h and ho are averaged openings incorporating the effect of surface asperities. From Eq. (10), it can be observed that a higher dimensionless rate εq results in a larger fracture opening at any given distance. Note that the initial fracture opening ho and the fractures spacing L can be related to the initial permeability and porosity of the fracture system by assuming a regular fracture arrangement in the reservoir. For a matchstick arrangement of matrix blocks, the formulae for permeability and porosity are given by (van GolfRacht, 1982) 3
ko ¼
ho ; 12L
ϕo ¼
2ho ; L
ð11Þ
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3.3. Size exclusion of particles during the injection In this section, a highly simplified model for proppant particle straining in the fractures is presented. It is assumed that particle straining occurs when the cleat aperture hðrÞ is twice the particle radius r s i.e. at the plugging moment: hðrÞ ¼ 2r s
ð12Þ
where the averaged cleat opening hðrÞ is given by Eq. (10). The size exclusion condition (12) neglects the asperities on the fracture walls as well as the tortuosity of the flow path in a network of fractures. An alternative size exclusion condition could be γhðrÞ ¼ 2r s , where γ o 1 is a shape factor. For hydraulic fractures, Valko and Economides (1995) suggest γ ¼ 1=3. For network of rough natural fractures, the shape factor can be determined either experimentally or by using computational fluid dynamics. However, determining such a shape factor is out of scope of the present work and instead, the simple criteria given by Eq. (12) is used. Once calculated, the shape factor can be readily incorporated in the present formulation. The size exclusion condition (12) can be used to calculate the plugging distance or the distance travelled by a particle of radius r s before being captured. The dimensionless plugging distance from the wellbore is given by ! 1ð2r Ds Þ4 r D ðr Ds Þ ¼ exp ; ð13Þ εq where r Ds ¼ r s =ho is the dimensionless particle size. The minimum proppant radius r o is simply obtained by substituting the radius of the stimulation zone r st into Eq. (13) and re-arranging to yield: 1 1 1=4 1 þ εq ln ð14Þ r o ¼ minðr Ds Þ ¼ 2 α where α ¼ r st =r e is the scaled radius of the stimulation zone. The size exclusion condition can also be used to calculate the travelling time of a particle before capture. The interstitial velocity U of the fluid in a cleat at a distance r from the well is simply given by U¼
q nðrÞhðrÞ
ð15Þ
where q is the injection rate per unit thickness of the reservoir, nðrÞ is the number of cleats through which the fluid flows and hðrÞ is the aperture of the cleats. The number of cleats nðrÞ encountered by the injected fluid increases with increasing distance from the wellbore. To obtain a relationship between the number of cleats and the distance from the wellbore, consider an idealized fracture system comprising of two perpendicular sets of evenly spaced fractures, also known as the matchstick model. A circle of diameter 2r >> L would be intersected by a set of evenly spaced fractures roughly 4r/L times. Hence, for two perpendicular sets of fractures, the number of cleats crossing a circle with radius r is approximated by nðrÞ ¼
8r L
ð16Þ
where L is the cleat spacing. Substituting (10) and (16) into (15) and separating the variables results in an expression for the particle trajectory, Z r qL r e 1=4 t¼ ρ 1 þ εq ln dρ ð17Þ 8ho ρ rw where t is the time taken by a particle to travel a distance r and ρ is the dummy integration variable. The above expression assumes that the particle velocity is the same as the velocity of the carrier fluid and no proppant settlement occurs. In terms of the
A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77
dimensionless radial coordinate r D ¼ r=r e , the travelling time is given by Z rD qL 1 1=4 t ¼ r 2e ℘ 1 þ εq ln d℘ ð18Þ 8ho ℘ rwD Substituting the expression for plugging distance given by (13) into Eq. (18) yields the settling time t s or the time it takes for the particle of size r Ds to strain in the thinning cleat, Z 8ho r 2e rD ðrDs Þ 1 1=4 ℘ 1 þ εq ln d℘ ð19Þ t s ðr Ds Þ ¼ ℘ qL rwD Eq. (19) can be used to calculate the total injection time i.e. the travelling time for the smallest particle t s ðr o Þ. It can be observed from Eq. (19) that a higher injection rate results in a shorter settling time i.e. faster arrival of a particle to its plugging site. The total injection time also decreases with increasing injection rate. 3.4. Optimal placement of proppant in the cleat system Let there be some averaged distance lðr s Þ between particles for which a parameter β can be defined as β¼
2r s lðr s Þ
ð20Þ
The parameter β is referred here as the packing ratio. A packing ratio equal to zero implies the absence of proppant particles in the cleat system and a packing ratio equal to unity implies that the proppant particles form a full monolayer in the cleat system. Thus, β represents the areal concentration of proppant in a partial monolayer. The separation of particles within the monolayer directly influences the permeability of the stimulated fracture. As discussed in Section 2 and shown in Fig. 2, there exists an optimal proppant concentration for which the fracture permeability is maximized. The optimal value of the parameter β is denoted by βn . For simplicity, we adopt the semi-analytical approach developed by Khanna et al. (2012) to calculate the optimal proppant concentration. However, it can also be determined by conducting experiments similar to Brannon et al. (2004) for specific rock and proppant combinations. The resultant permeability of the fractures within the stimulation zone is greater than the initial permeability ko , but less than the permeability kðr D Þ of the stimulated fractures before particle straining. The fracture system permeability inside the stimulation zone can be written as f kðr D Þ where the factor f represents a correction factor. In the semi-analytical model of Khanna et al. (2012) which is also used here, the factor f was determined as a function of the packing ratio β and the dimensionless stress in the reservoir εs ¼ sð1v2 Þ=E. The latter parameter incorporates
confining stress and the elastic properties of the rock. Classical Hertz contact theory and computational fluid dynamics were used to compute the function f ðβ; εs Þ. Fig. 3 shows the model setup and Fig. 4 shows the results. The optimal proppant concentration βn , for which the factor f is maximum, can be observed in Fig. 4. It can also be observed from Fig. 4 that the value of the optimal packing ratio βn increases with increasing value of dimensionless stress εs . This is because, a greater concentration of proppant particles is required to prevent the closure of the fracture at higher confining stress. In terms of the correction factor, the permeability of the fracture system during production is given by ( kpr ðr D Þ ¼
f ðβn ; εs Þkðr D Þ
r D ≤α
ko
rD 4 α
ð21Þ
where kðr D Þ is permeability of the fractures before plugging, given by (8) and α ¼ r st =r e is the scaled stimulation radius. For known reservoir stress and elastic properties, the optimal value of packing ratio βn and the corresponding value of the factor f ðβn ; εs Þ can be obtained from Fig. 4. During fluid injection, the hydraulic pressure keeps the fractures open i.e. f is a function of βn only. For the optimal value of packing ratio βn , the value of f ðβn ; 0Þ can be read from the εs ¼ 0 curve in Fig. 4. Thus, the permeability of the
0
1
CFD Results Best fit
10-4
0.8
Correction factor f(β,εσ)
74
0.6 Increasing dimensionless
-3
10
confining stress, εσ
0.4
10-2
0.2
0
0
0.2
0.4
0.6
0.8
1
Packing ratio ρ Fig. 4. Correction factor vs. packing ratio for various values of the dimensionless confining stress, values indicated above individual curves (Khanna et al., 2012).
l Flow streamlines l
Flow Inlet Outlet
Symmetry element
Measurement planes
Fig. 3. Semi-analytical approach to determine the optimal proppant concentration. (a) A regular arrangement of proppants between the fracture faces is considered. (b) The deformed fracture geometry is calculated using Hertz contact theory and the fracture conductivity is calculated by performing CFD analysis (Khanna et al., 2012).
A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77
fracture system during injection is given by ( f ðβn ; 0Þkðr D Þ r D ≤α kin ðr D Þ ¼ rD 4 α kðr D Þ
ð22Þ
75
The normalized productivity index is then given by J D ¼ J=J o . The normalized productivity index represents the ‘folds of increase’ in production due to graded proppant injection.
3.5. Calculation of the optimal injection schedule
4. Case study and numerical results
If the total injection time is taken as the settling time of the smallest particle, the injection time is given by
In order to quantify the requirements and benefits of the proposed stimulation method, a case study is conducted on a hypothetical coal seam gas reservoir using typical reservoir parameters provided by Moore (2012) and Laubach et al. (1998). At a dimensionless confining stress εs ¼ 103 , the value of the optimal packing ratio can be read from Fig. 4 as βn ¼ 0:24. The corresponding values of the correction factors are also read from Fig. 4 as f ðβn ; 0Þ ¼ 0:78 and f ðβn ; εq Þ ¼ 0:55. The calculations are performed for three values of the dimensionless injection rate. The chosen values of injection rates per unit thickness are 1, 5 and 10 bbl/min (where 1 bbl/min ¼2.65E 3 m3/s). The corresponding values of the dimensionless injection rate εq are approximately 2, 10 and 20. The dimensionless injection rate was calculated using Eq. (6). For the three different values of the dimensionless injection rate, the radial distribution of the cleat opening is calculated using Eq. (10). The results are shown in Fig. 5. The change in fracture
ð23Þ
Substituting (23) into (19), the time at which a particle of size r Ds should be injected can be obtained as Z 8ho r 2e α 1 1=4 ℘ 1 þ εq ln d℘ ð24Þ t in ðr Ds Þ ¼ ℘ qL rD ðrDs Þ The schedule for the injected particle radius vs. time r Ds ðt in Þ is obtained by inversion of the above function. The required number of particles of a given size, N p can be estimated as N p ðr Ds Þ ¼
H H 4r nðrÞ ¼ βn n L r Ds ho l ðr Ds Þ
ð25Þ
where βn is the optimal packing ratio, H is the height of the reservoir and L is the spacing between the cleats. Using particle radius r Ds as a parameter in Eqs. (25) and (24), the schedule for injected particle concentration vs. time, Np ðt in Þ can be obtained. The optimal injection schedule r Ds ðt in Þ and N p ðt in Þ depends on the dimensionless injection rate εq , the size of the stimulation zone r st =r e and the value of the optimal packing ratio βn . The value of optimal packing aspect ratio together with the effective permeability of the proppant monolayer allows determining well index during injection and production. 3.6. Calculation of injectivity and productivity index with plugged cleats
100
80
Cleat aperture h(r) (µm)
t in ðr Ds Þ ¼ t s ðr o Þt s ðr Ds Þ
The formula for the well injectivity index before particle straining is obtained by substituting Eq. (8) into (1) followed. Separating the variables and integrating both sides yields: "Z #1 1 εq q 2πko dr D πko Io ¼ ¼ ¼ 3=4 pw pres μ 2μ hD ðr wD Þ1 r wD r D ð1 þ εq lnð1=r D ÞÞ ð26Þ In the above expression, hD ðr wD Þ is the normalized opening at the wellbore. Similarly, the well injectivity after particles straining can be calculated by substituting (22) into (1). f ðβn ; 0Þεq πko 2μ ðhD ðr wD ÞhD ðαÞÞ þ f ðβn ; 0ÞðhD ðαÞ1Þ
2πko 1 μ lnð1=r wD Þ
0
0
100
200
300
400
500
Radial distance from the wellbore r (m) Fig. 5. The distribution of cleat opening as a function of the radial distance from the wellbore.
24
ð28Þ
When the entire flow region is stimulated, the productivity index after plugging is simply given by J ¼ f ðβn ; εq ÞI o . The productivity index can be normalized against the productivity index of an unstimulated reservoir: Jo ¼
ho
ð27Þ
When the entire flow region is stimulated, the fracture opening at the stimulation radius is equal to the initial fracture aperture, i.e. hD ðαÞ ¼ 1. In this case, the injectivity index after plugging is simply given by I ¼ f ðβn ; 0ÞI o . The productivity index during liquid (water) production can be calculated by substituting (21) into (1): f ðβn ; εq Þεq 2πko J¼ μ 4ðhD ðr wD ÞhD ðαÞÞ þ f ðβn ; εq Þεq lnð1=αÞ
40
20
ð29Þ
20
Settling time ts(r) (hours)
I¼
Increasing injection rate 60
16 12 Increasing injection rate
8 4 0
0
100
200
300
400
500
Radial distance from the wellbore r (m) Fig. 6. Particle settling time as a function of the radial distance from the wellbore.
76
A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77
14
No. of particles Np(rs) x107
Particle radius rs (µm)
50
40
30 Increasing injection rate
20
10
0
12 10 8 Increasing injection rate
6 4 2 0
0
5
10
15
20
25
0
5
Injection time tin (minutes)
10
15
20
25
Injection time tin (minutes)
Fig. 7. Proppant injection schedule for stimulating a 50 m radius around the wellbore: (a) particle radius vs. time and (b) particle concentration vs. time.
9
Normalized productivity index JD
aperture is very rapid near the wellbore at any given injection rate. The fracture opening increases with increasing injection rate. Next, the travelling time is calculated as a function of the radial distance using Eq. (18). The results are shown in Fig. 6. The slope of the curves increases with distance from the wellbore which implies that the interstitial velocity of the fluid in the cleats decreases with increasing distance from the wellbore. It is due to the branching structure of the fracture system. A higher injection rate results in the quicker arrival of particles at the plugging site. In the present example, a particle would take approximately 21 h to reach the drainage radius of 500 m at an injection rate of 1 bbl/ min, whereas it would take less than 4 h to reach the drainage radius at an injection rate of 10 bbl/min. The injection schedule is calculated based on a pre-determined size of the stimulation zone. In the present example, the radius of the stimulation zone is chosen to be 50 m. Hence, the value of the scaled stimulation radius is α ¼ r st =r e ¼ 50=500 ¼ 0:1. For α ¼ 0:1 and an optimal packing ratio βn ¼ 0:24, the injection schedule is calculated using Eqs. (24) and (25). The results are shown in Fig. 7. Fig. 7(a) shows the dependence of particle radius on injection time and Fig. 7(b) shows the dependence of particle concentration on the injection time. As discussed in Section 2, the particle radius increases with injection time. Conversely, the particle concentration decreases with injection time. The rapid increase in the particle size and the rapid decrease in the particle concentration near the end of the injection period are due to the change in fracture opening and fluid velocity near the wellbore. It can be observed that the injection rate significantly affects the injection schedule. With increasing rate, the capture time of the particles is reduced, hence the total time of injection decreases. In the present example, stimulating a 50 m radius around the wellbore requires approximately 22 min at an injection rate of 1 bbl/min but less than 4 min at an injection rate of 10 bbl/min. The required particle size increases with injection rate since larger fracture openings are produced. The injected particle concentration decreases with increase in injection rate since the inter-particle separation must increase with increasing particle radius to obtain the optimal value of packing ratio (see Eq. (20)). The productivity index is calculated using Eqs. (28) and (29). Fig. 8 shows the dependence of the normalized productivity index J D , which represents the ‘folds of increase’ in production, on injection rate and the radius of the stimulation zone r st . A higher injection rate results in a greater well productivity index for all values of the stimulation radius. The injection rate also controls the relative increment in productivity for a given increment in the
7
5 Increasing injection rate
3
1
0
100
200
300
400
500
Stimulation radius rst (m) Fig. 8. Normalized productivity index or folds of increase in production as a function of stimulation radius and injection rate.
radius of the stimulation zone. For example, at a low injection rate of 1 bbl/min, the productivity index does not increase significantly beyond a stimulation radius of 50 m. At a high injection rate of 10 bbl/min, the productivity index continues to increase with increasing size of the stimulation zone. It can be concluded from Fig. 8 that the graded proppant injection scheme is a promising technology for the stimulation of natural fractures. In the example considered here, a three-fold increase in production was obtained by stimulating a radius of 50 m around the wellbore at a rate of 5 bbl/min. The limitations and applicability of the current mathematical model are discussed in the next section.
5. Discussion Some radical simplifications were adopted to model the interaction of fluid with natural fractures as well as particle transport in the fracture. Some shortcomings of the present model and possible methods for improvement are:
Axi-symmetric flow geometry was assumed in the mathematical model (Eqs. (1) and (15)). The flow geometry simplification implies that the conductive fractures form a tree-like
A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77
(dendritic) structure with branches multiplying from the well towards the reservoir. This assumption is equivalent to parallel layers (tubes) for the plane-parallel flow, which is often used for estimating flow in porous media. A more precise description of flow in fractured media would require the application of percolation or effective media models. The detailed description of the proppant flow and capture in real fractured system certainly requires more elaborate modelling. Bedrikovetsky (2008) described the size exclusion of suspension flow in stochastic porous media. To the best of our knowledge, the model for suspension flow in fractured systems is not available in the contemporary literature. However, further development of the proposed technology must include modelling of the strainingdominated suspension transport in stochastic fractured systems (Wei et al., 2012). In order to apply the technique of graded particle injection in coal beds, the anisotropy of coal needs to be incorporated in the model to describe particle propagation in face and butt cleats. The elaborated theory must include the tensorial permeability of the cleat system and the capture criterion should also be direction-dependent. Non-elastic processes of the proppant crushing or embedment may result at high stresses if the rock is significantly harder or softer than the proppant. As a result, a partial proppant monolayer may provide inadequate fracture conductivity. To incorporate the effects of these phenomena on the optimal proppant concentration, experimental studies must be conducted on specific rock-proppant combinations under expected stress conditions. The injection of water changes the reservoir stress around the wellbore, often causing shear induced slip of the asperous fractures, which may lead to additional productivity enhancement. The effect of shear dilation must be included in the model for graded particle injection for stimulation of natural fractured systems.
6. Conclusion A simple mathematical model is proposed for studying the graded proppant injection in natural fracture systems, such as those in coal seam beds, shale gas reservoirs and geothermal fields. The case study conducted using typical reservoir data demonstrates that graded proppant injection may lead to the enhancement in well productivity. It was found that the most influential parameter affecting the proppant injection schedule
77
and the well productivity index is the fluid injection rate. A higher injection rate results in greater opening of the cleats, deeper percolation of the smallest particles, shorter injection time and a greater increase in well productivity. The applicability of the present model is limited due to a number of radical simplifications. Better estimates of increase in well productivity can be obtained by incorporating more advanced models for particle transport in fractured media.
Acknowledgements The authors wish to thank Prof. Sheik S. Rahman from The University of New South Wales and acknowledge the support of Santos Ltd. and ARC linkage grant LP110200799. References Basniev, K.S., Bedrikovetsky, P.G., Dedinez, E.N., 1988. Determination of the effective permeability for fractured porous medium. J. Eng. Phys. (Injenerno fizicheskii jurnal) 55 (6), 940–948. Bear, J., 1972. Dynamics of fluid in porous media. Elsevier. Bedrikovetsky, P., 2008. Upscaling of stochastic micro model for suspension transport in porous media. J. Transp. Porous Media 75 (3), 335–369. Brannon, H.D., Malone, M.R., Rickards, A.R., Wood, W.D., Edgeman, J.R., Bryant, J.L., 2004. Maximizing Fracture Conductivity with Proppant Partial Monolayers: Theoretical Curiosity or Highly Productive Reality? SPE90698. Colmenares, L.B., Zoback, M.D., 2007. Hydraulic fracturing and wellbore completion of coalbed methane wells in the Powder River Basin, Wyoming: implications for water and gas production. AAPG Bull. 91 (1), 51–67. Holditch, S.A., Ely J.W., Semmalbeck, M.E., Carter, R.H., Hinkel, J., Jeffrey, Jr. R.G., 1968. Enhanced Recovery of Coalbed Methane through Hydraulic Fracturing. SPE 18250. Hossain, M., Rahman, M., Rahman, S., 2002. Shear dilation stimulation model for production enhancement from naturally fractured reservoirs. Soc. Pet. Eng. 7 (2), 183–195. Khanna, A., Kotousov, A., Sobey, J., Weller, P., 2012. Optimal placement of proppant in narrow fracture channels. J. Pet. Sci. Eng. 100, 9–13. Laubach, S.E., Marrett, R.A., Olson, J.E., Scott, A.R., 1998. Characteristics and origins of coal cleat: a review. Int. J. Coal Geol. 35, 175–207. Moore, T.A., 2012. Coalbed methane: a review. Int. J. Coal Geol. 101, 26–81. Palmer, I., Mansoori, J., 1998. How Permeability Depends upon Stress and PorePressure in Coal Beds: A New Model. SPE 52607. Rahman, M.K., Hossain, M.M., Rahman, S.S., 2002. A shear-dilation-based model for evaluation of hydraulically stimulated naturally fractured reservoirs. Int. J. Numer. Anal. Methods Geomech. 26 (5), 469–497. Valko, P., Economides, M., 1995. Hydraulic Fracture Mechanics. Wiley, Chichester. van Golf-Racht, T.D., 1982. Fundamentals of Fractured Reservoir Engineering. Elsevier, The Netherlands. Warpinski, N.R., Mayerhofer, M.J., Vincent, M.C., Cipolla, C.L., Lolon, E.P., 2008. Stimulating Unconventional Reservoirs: Maximizing Network Growth while Optimizing Fracture Conductivity. SPE 114173. Wei, Z., Xiangyu, T., Weisbrod, N., Zhuo, G., 2012. A review of colloid transport in fractured rocks. J. Mount. Sci. 9, 770–787.
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Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Stimulation of the natural fracture system by graded proppant injection Aditya Khanna a, Alireza Keshavarz b, Kate Mobbs b, Michael Davis a, Pavel Bedrikovetsky b,n a b
School of Mechanical Engineering, The University of Adelaide, SA 5005, Australia Australian School of Petroleum, The University of Adelaide, SA 5005, Australia
art ic l e i nf o
a b s t r a c t
Article history: Received 22 May 2012 Accepted 15 July 2013 Available online 24 July 2013
Graded proppant injection i.e. the injection of particles of increasing size and decreasing concentration, is proposed as a method for stimulating natural fractures. Compared to the injection of mono-sized proppant, graded proppant injection would facilitate deeper percolation of proppant into the fracture system and enhanced well productivity. A mathematical model is developed which describes the injection stage and the capture kinetics of proppant particles in the natural fracture system. Based on the mathematical model, an injection schedule is developed which would result in the optimal placement of proppant in the fracture system. A case study is conducted to estimate the change in well productivity index due to the application of the graded particle injection method. & 2013 Elsevier B.V. All rights reserved.
Keywords: Coal bed methane Naturally fractured reservoirs Well stimulation Graded proppant injection Injection schedule Mathematical modelling
1. Introduction Naturally fractured reservoirs such as coal bed methane (CBM), tight and shale gas fields and geothermal reservoirs often require some form of stimulation in order to achieve economical production rates. Fluid injection at the wellbore prior to production is a widely adopted technology for stimulating natural fractures (Colmenares and Zoback, 2007). The injection of fluid without proppant may lead to an increase in permeability due to the shear induced opening of rough natural fractures (Hossain et al., 2002; Rahman et al., 2002). Injection of proppant on the other hand, leads to an increase in reservoir permeability by preventing the closure of the dilated natural fractures (Holditch et al., 1968; Warpinski et al., 2008). However, in the absence of a wide hydraulic fracture, the large proppant particles typically used in fracturing treatments can only stimulate a small region in the vicinity of the wellbore. This is due to the rapid decline in the aperture of stimulated natural fractures with increasing distance from the wellbore. In this paper, the method of graded proppant injection is proposed. Injecting small particles first followed by larger particles could increase the size of the stimulated zone, since the small particles are more likely to percolate deeper into the fracture system. The main parameters of interest in the problem of graded proppant injection are: the increase in well productivity and the
n
Corresponding author. Tel.: +61 8 8313 3082. E-mail address: [email protected] (P. Bedrikovetsky).
0920-4105/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.petrol.2013.07.004
proppant injection schedule i.e. the dependence of particle size and concentration on the injection time. Within this paper, a simplified engineering model is developed for calculating these parameters. The problem of proppant placement in a fracture system is formulated in Section 2. A mathematical model is then presented which describes: the opening of natural fractures due to fluid injection (Section 3.2), particle plugging in the opened fractures (Section 3.3), the optimal proppant placement in the fracture network (Section 3.4) and the injection schedule (Section 3.5). The change in well productivity is calculated in Section 3.6. A case study is conducted in Section 4 using typical reservoir data to determine the effect of injection rate, and size of stimulated zone on the change in well productivity index. 2. Problem formulation The proppant placement resulting from graded proppant injection is schematically presented in Fig. 1. It shows the straining of particles of different sizes in the fracture network. The basic features of the injection schedule which yields a dependence of the injected particle size and concentration upon the injection time, can be deduced from Fig. 1. During fluid injection at the wellbore, the fluid pressure and consequently, the opening of fractures diminishes with increasing distance from the well. This implies that smaller particles need to be injected before larger particles in order to achieve deeper percolation. In terms of the injection schedule, this implies that the injected particle radius
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combinations. For simplicity, the semi-analytical results obtained by Khanna et al. (2012) are utilized here. In the next section, the mathematical model for particle straining in fractured porous media is presented.
3. Mathematical model for flow in fractured porous media with strained particles 3.1. Modelling assumptions The following assumptions are utilized regarding graded proppant injection:
At the scale of the reservoir, the flow from the wellbore
Fig. 1. Plugging of the natural fracture system by graded proppant particles.
Normal stress
Fluid
Rigid proppant
Deformed flow channel
Normal stress Fig. 2. Two competitive factors affect the hydraulic resistance: (1) The deformation of the flow channel caused by rock stresses and (2) the additional tortuosity of the flow path due to the presence of the particles.
must increase with injection time. Also, a greater number of cleats are encountered by the flow with increasing distance from the wellbore. Thus, a greater concentration of small particles is needed to plug the filtration path compared to large particles. Consequently, the concentration of injected particles must decrease with injection time. There are two competitive mechanisms, shown in Fig. 2, which affect the permeability of the stimulated fracture system. These are: the reduction in the fracture aperture due to the confining stresses and the additional tortuosity of the flow path due to the presence of proppant particles. A dense packing of the proppant particles minimizes the deformation of the fracture and a sparse packing minimizes the additional tortuosity to the flow path. An intermediate proppant concentration exists, at which net effect of these two competitive mechanisms is minimum. At this optimum proppant concentration, the maximum conductivity of the fracture system is achieved. The optimal proppant concentration depends upon a number of parameter such as the reservoir stress, material properties of the rock and the strength of the proppant. In order to determine the value of the optimal proppant concentration and the fracture conductivity during production, experiments similar to Brannon et al. (2004) can be conducted for specific rock and proppant
towards the fracture system is axi-symmetric and can be described by Darcy's law for radial flow. The conductive fractures form a tree-like or dendritic structure with branches multiplying from the well towards the reservoir. During injection, the flow is steady state since the time of propagation of an elastic pressure wave in the stimulated zone is negligibly small compared to the total time of injection. The calculations are performed for constant injection rate at the wellbore. The naturally fractured reservoir is idealized by the matchstick model in which match-like blocks of rock matrix are separated by cleats or fractures. The height of the matrix blocks is equal to the height of the reservoir and the width of the blocks is equal to the average cleat spacing. The dependence of the permeability of the cleat system upon pressure is obtained from a widely used permeability model suggested by Palmer and Mansoori (1998). The terms within the Palmer–Mansoori model related to matrix swelling/shrinkage are neglected since the injection period is relatively short. In order to simplify the particle capture criterion, the presence of fracture wall asperities, in-fill material and tortuosity in the fractures are neglected. It is assumed that particle straining occurs when the particle diameter exceeds the pressuredependent opening of the fracture. The particles are assumed to travel at the same speed as the injection fluid, i.e. proppant settlement by gravity is neglected.
The mathematical model for water injection before particle plugging, based on the above assumptions, is presented in the next section. 3.2. Water injection stage In this section, the basic equations for water injection into naturally fractured system are presented. Darcy′s law for radial flow of an incompressible fluid through deformable rock is given by q kðpÞ dp ¼ 2πr μ dr
ð1Þ
Here q is the constant injection rate per unit thickness of the reservoir, pðrÞ is the pressure at a distance r from the wellbore, kðpÞ is the pressure dependent permeability of the fracture network and μ is the fluid viscosity. Assuming a given pressure dependence for permeability and separating variables in the ordinary differential equation (1) yields the implicit formula for pressure distribution around the well: Z p qμ r e ln ¼ kðpÞdp; ð2Þ 2π r pres
A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77
Here r e is the drainage radius at which the fluid pressure is equal to the initial reservoir pressure pres . The pressure–permeability relationship substituted into Eq. (2) is 3 C kðpÞ ¼ ko 1 þ ðppres Þ ð3Þ ϕo This expression comes from the model for permeability proposed by Palmer and Mansoori (1998). The constants ko and ϕo are the initial permeability and porosity of the reservoir, respectively and the constant C represents the reservoir compressibility under uniaxial strain. It can be expressed in terms of the Young′s modulus E and Poisson′s ratio ν of the reservoir as C¼
ð1 þ νÞð12νÞ 1 ð1νÞ E
ð4Þ
Substituting Eq. (3) into (2) and integrating in p yields the explicit formula for the radial pressure distribution during water injection: " 1=4 # ϕ 2qμC r e pðrÞ ¼ pres þ o ln þ 1 1 ; pðr w Þ ¼ pw ð5Þ πko ϕo C r The pressure at the wellbore radius r w is denoted by pw . The radial coordinate r can be normalized against the drainage radius of the reservoir to yield the dimensionless radial coordinate r D ¼ r=r e and the dimensionless wellbore radius r wD ¼ r w =r e . Further, a dimensionless injection rate εq can be defined as εq ¼
2qμC πko ϕo
ð6Þ
In terms of these dimensionless parameters, the expression for the radial pressure distribution can be re-written as " # ϕ 1 1=4 pðr D Þ ¼ pres þ o 1 þ εq ln 1 ; pðr wD Þ ¼ pw ð7Þ rD C By substituting Eq. (7) into (3), the permeability distribution around the well kðr D Þ can be obtained as 1 3=4 ð8Þ kðr D Þ ¼ ko 1 þ εq ln rD From the dimensionless analysis of the relationship between the rock permeability, mean aperture of fractures and matrix block size (Bear, 1972; Basniev et al., 1988) it follows that 3 k h ¼ ð9Þ ko ho where ko and ho are the initial permeability of the reservoir and the initial aperture of the fractures, respectively. Substituting Eq. (8) into relationship (9) results in an expression for the effective cleat opening given by hðr D Þ 1 1=4 hD ðr D Þ ¼ ¼ 1 þ εq ln ð10Þ ho rD Here the apertures h and ho are averaged openings incorporating the effect of surface asperities. From Eq. (10), it can be observed that a higher dimensionless rate εq results in a larger fracture opening at any given distance. Note that the initial fracture opening ho and the fractures spacing L can be related to the initial permeability and porosity of the fracture system by assuming a regular fracture arrangement in the reservoir. For a matchstick arrangement of matrix blocks, the formulae for permeability and porosity are given by (van GolfRacht, 1982) 3
ko ¼
ho ; 12L
ϕo ¼
2ho ; L
ð11Þ
73
3.3. Size exclusion of particles during the injection In this section, a highly simplified model for proppant particle straining in the fractures is presented. It is assumed that particle straining occurs when the cleat aperture hðrÞ is twice the particle radius r s i.e. at the plugging moment: hðrÞ ¼ 2r s
ð12Þ
where the averaged cleat opening hðrÞ is given by Eq. (10). The size exclusion condition (12) neglects the asperities on the fracture walls as well as the tortuosity of the flow path in a network of fractures. An alternative size exclusion condition could be γhðrÞ ¼ 2r s , where γ o 1 is a shape factor. For hydraulic fractures, Valko and Economides (1995) suggest γ ¼ 1=3. For network of rough natural fractures, the shape factor can be determined either experimentally or by using computational fluid dynamics. However, determining such a shape factor is out of scope of the present work and instead, the simple criteria given by Eq. (12) is used. Once calculated, the shape factor can be readily incorporated in the present formulation. The size exclusion condition (12) can be used to calculate the plugging distance or the distance travelled by a particle of radius r s before being captured. The dimensionless plugging distance from the wellbore is given by ! 1ð2r Ds Þ4 r D ðr Ds Þ ¼ exp ; ð13Þ εq where r Ds ¼ r s =ho is the dimensionless particle size. The minimum proppant radius r o is simply obtained by substituting the radius of the stimulation zone r st into Eq. (13) and re-arranging to yield: 1 1 1=4 1 þ εq ln ð14Þ r o ¼ minðr Ds Þ ¼ 2 α where α ¼ r st =r e is the scaled radius of the stimulation zone. The size exclusion condition can also be used to calculate the travelling time of a particle before capture. The interstitial velocity U of the fluid in a cleat at a distance r from the well is simply given by U¼
q nðrÞhðrÞ
ð15Þ
where q is the injection rate per unit thickness of the reservoir, nðrÞ is the number of cleats through which the fluid flows and hðrÞ is the aperture of the cleats. The number of cleats nðrÞ encountered by the injected fluid increases with increasing distance from the wellbore. To obtain a relationship between the number of cleats and the distance from the wellbore, consider an idealized fracture system comprising of two perpendicular sets of evenly spaced fractures, also known as the matchstick model. A circle of diameter 2r >> L would be intersected by a set of evenly spaced fractures roughly 4r/L times. Hence, for two perpendicular sets of fractures, the number of cleats crossing a circle with radius r is approximated by nðrÞ ¼
8r L
ð16Þ
where L is the cleat spacing. Substituting (10) and (16) into (15) and separating the variables results in an expression for the particle trajectory, Z r qL r e 1=4 t¼ ρ 1 þ εq ln dρ ð17Þ 8ho ρ rw where t is the time taken by a particle to travel a distance r and ρ is the dummy integration variable. The above expression assumes that the particle velocity is the same as the velocity of the carrier fluid and no proppant settlement occurs. In terms of the
A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77
dimensionless radial coordinate r D ¼ r=r e , the travelling time is given by Z rD qL 1 1=4 t ¼ r 2e ℘ 1 þ εq ln d℘ ð18Þ 8ho ℘ rwD Substituting the expression for plugging distance given by (13) into Eq. (18) yields the settling time t s or the time it takes for the particle of size r Ds to strain in the thinning cleat, Z 8ho r 2e rD ðrDs Þ 1 1=4 ℘ 1 þ εq ln d℘ ð19Þ t s ðr Ds Þ ¼ ℘ qL rwD Eq. (19) can be used to calculate the total injection time i.e. the travelling time for the smallest particle t s ðr o Þ. It can be observed from Eq. (19) that a higher injection rate results in a shorter settling time i.e. faster arrival of a particle to its plugging site. The total injection time also decreases with increasing injection rate. 3.4. Optimal placement of proppant in the cleat system Let there be some averaged distance lðr s Þ between particles for which a parameter β can be defined as β¼
2r s lðr s Þ
ð20Þ
The parameter β is referred here as the packing ratio. A packing ratio equal to zero implies the absence of proppant particles in the cleat system and a packing ratio equal to unity implies that the proppant particles form a full monolayer in the cleat system. Thus, β represents the areal concentration of proppant in a partial monolayer. The separation of particles within the monolayer directly influences the permeability of the stimulated fracture. As discussed in Section 2 and shown in Fig. 2, there exists an optimal proppant concentration for which the fracture permeability is maximized. The optimal value of the parameter β is denoted by βn . For simplicity, we adopt the semi-analytical approach developed by Khanna et al. (2012) to calculate the optimal proppant concentration. However, it can also be determined by conducting experiments similar to Brannon et al. (2004) for specific rock and proppant combinations. The resultant permeability of the fractures within the stimulation zone is greater than the initial permeability ko , but less than the permeability kðr D Þ of the stimulated fractures before particle straining. The fracture system permeability inside the stimulation zone can be written as f kðr D Þ where the factor f represents a correction factor. In the semi-analytical model of Khanna et al. (2012) which is also used here, the factor f was determined as a function of the packing ratio β and the dimensionless stress in the reservoir εs ¼ sð1v2 Þ=E. The latter parameter incorporates
confining stress and the elastic properties of the rock. Classical Hertz contact theory and computational fluid dynamics were used to compute the function f ðβ; εs Þ. Fig. 3 shows the model setup and Fig. 4 shows the results. The optimal proppant concentration βn , for which the factor f is maximum, can be observed in Fig. 4. It can also be observed from Fig. 4 that the value of the optimal packing ratio βn increases with increasing value of dimensionless stress εs . This is because, a greater concentration of proppant particles is required to prevent the closure of the fracture at higher confining stress. In terms of the correction factor, the permeability of the fracture system during production is given by ( kpr ðr D Þ ¼
f ðβn ; εs Þkðr D Þ
r D ≤α
ko
rD 4 α
ð21Þ
where kðr D Þ is permeability of the fractures before plugging, given by (8) and α ¼ r st =r e is the scaled stimulation radius. For known reservoir stress and elastic properties, the optimal value of packing ratio βn and the corresponding value of the factor f ðβn ; εs Þ can be obtained from Fig. 4. During fluid injection, the hydraulic pressure keeps the fractures open i.e. f is a function of βn only. For the optimal value of packing ratio βn , the value of f ðβn ; 0Þ can be read from the εs ¼ 0 curve in Fig. 4. Thus, the permeability of the
0
1
CFD Results Best fit
10-4
0.8
Correction factor f(β,εσ)
74
0.6 Increasing dimensionless
-3
10
confining stress, εσ
0.4
10-2
0.2
0
0
0.2
0.4
0.6
0.8
1
Packing ratio ρ Fig. 4. Correction factor vs. packing ratio for various values of the dimensionless confining stress, values indicated above individual curves (Khanna et al., 2012).
l Flow streamlines l
Flow Inlet Outlet
Symmetry element
Measurement planes
Fig. 3. Semi-analytical approach to determine the optimal proppant concentration. (a) A regular arrangement of proppants between the fracture faces is considered. (b) The deformed fracture geometry is calculated using Hertz contact theory and the fracture conductivity is calculated by performing CFD analysis (Khanna et al., 2012).
A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77
fracture system during injection is given by ( f ðβn ; 0Þkðr D Þ r D ≤α kin ðr D Þ ¼ rD 4 α kðr D Þ
ð22Þ
75
The normalized productivity index is then given by J D ¼ J=J o . The normalized productivity index represents the ‘folds of increase’ in production due to graded proppant injection.
3.5. Calculation of the optimal injection schedule
4. Case study and numerical results
If the total injection time is taken as the settling time of the smallest particle, the injection time is given by
In order to quantify the requirements and benefits of the proposed stimulation method, a case study is conducted on a hypothetical coal seam gas reservoir using typical reservoir parameters provided by Moore (2012) and Laubach et al. (1998). At a dimensionless confining stress εs ¼ 103 , the value of the optimal packing ratio can be read from Fig. 4 as βn ¼ 0:24. The corresponding values of the correction factors are also read from Fig. 4 as f ðβn ; 0Þ ¼ 0:78 and f ðβn ; εq Þ ¼ 0:55. The calculations are performed for three values of the dimensionless injection rate. The chosen values of injection rates per unit thickness are 1, 5 and 10 bbl/min (where 1 bbl/min ¼2.65E 3 m3/s). The corresponding values of the dimensionless injection rate εq are approximately 2, 10 and 20. The dimensionless injection rate was calculated using Eq. (6). For the three different values of the dimensionless injection rate, the radial distribution of the cleat opening is calculated using Eq. (10). The results are shown in Fig. 5. The change in fracture
ð23Þ
Substituting (23) into (19), the time at which a particle of size r Ds should be injected can be obtained as Z 8ho r 2e α 1 1=4 ℘ 1 þ εq ln d℘ ð24Þ t in ðr Ds Þ ¼ ℘ qL rD ðrDs Þ The schedule for the injected particle radius vs. time r Ds ðt in Þ is obtained by inversion of the above function. The required number of particles of a given size, N p can be estimated as N p ðr Ds Þ ¼
H H 4r nðrÞ ¼ βn n L r Ds ho l ðr Ds Þ
ð25Þ
where βn is the optimal packing ratio, H is the height of the reservoir and L is the spacing between the cleats. Using particle radius r Ds as a parameter in Eqs. (25) and (24), the schedule for injected particle concentration vs. time, Np ðt in Þ can be obtained. The optimal injection schedule r Ds ðt in Þ and N p ðt in Þ depends on the dimensionless injection rate εq , the size of the stimulation zone r st =r e and the value of the optimal packing ratio βn . The value of optimal packing aspect ratio together with the effective permeability of the proppant monolayer allows determining well index during injection and production. 3.6. Calculation of injectivity and productivity index with plugged cleats
100
80
Cleat aperture h(r) (µm)
t in ðr Ds Þ ¼ t s ðr o Þt s ðr Ds Þ
The formula for the well injectivity index before particle straining is obtained by substituting Eq. (8) into (1) followed. Separating the variables and integrating both sides yields: "Z #1 1 εq q 2πko dr D πko Io ¼ ¼ ¼ 3=4 pw pres μ 2μ hD ðr wD Þ1 r wD r D ð1 þ εq lnð1=r D ÞÞ ð26Þ In the above expression, hD ðr wD Þ is the normalized opening at the wellbore. Similarly, the well injectivity after particles straining can be calculated by substituting (22) into (1). f ðβn ; 0Þεq πko 2μ ðhD ðr wD ÞhD ðαÞÞ þ f ðβn ; 0ÞðhD ðαÞ1Þ
2πko 1 μ lnð1=r wD Þ
0
0
100
200
300
400
500
Radial distance from the wellbore r (m) Fig. 5. The distribution of cleat opening as a function of the radial distance from the wellbore.
24
ð28Þ
When the entire flow region is stimulated, the productivity index after plugging is simply given by J ¼ f ðβn ; εq ÞI o . The productivity index can be normalized against the productivity index of an unstimulated reservoir: Jo ¼
ho
ð27Þ
When the entire flow region is stimulated, the fracture opening at the stimulation radius is equal to the initial fracture aperture, i.e. hD ðαÞ ¼ 1. In this case, the injectivity index after plugging is simply given by I ¼ f ðβn ; 0ÞI o . The productivity index during liquid (water) production can be calculated by substituting (21) into (1): f ðβn ; εq Þεq 2πko J¼ μ 4ðhD ðr wD ÞhD ðαÞÞ þ f ðβn ; εq Þεq lnð1=αÞ
40
20
ð29Þ
20
Settling time ts(r) (hours)
I¼
Increasing injection rate 60
16 12 Increasing injection rate
8 4 0
0
100
200
300
400
500
Radial distance from the wellbore r (m) Fig. 6. Particle settling time as a function of the radial distance from the wellbore.
76
A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77
14
No. of particles Np(rs) x107
Particle radius rs (µm)
50
40
30 Increasing injection rate
20
10
0
12 10 8 Increasing injection rate
6 4 2 0
0
5
10
15
20
25
0
5
Injection time tin (minutes)
10
15
20
25
Injection time tin (minutes)
Fig. 7. Proppant injection schedule for stimulating a 50 m radius around the wellbore: (a) particle radius vs. time and (b) particle concentration vs. time.
9
Normalized productivity index JD
aperture is very rapid near the wellbore at any given injection rate. The fracture opening increases with increasing injection rate. Next, the travelling time is calculated as a function of the radial distance using Eq. (18). The results are shown in Fig. 6. The slope of the curves increases with distance from the wellbore which implies that the interstitial velocity of the fluid in the cleats decreases with increasing distance from the wellbore. It is due to the branching structure of the fracture system. A higher injection rate results in the quicker arrival of particles at the plugging site. In the present example, a particle would take approximately 21 h to reach the drainage radius of 500 m at an injection rate of 1 bbl/ min, whereas it would take less than 4 h to reach the drainage radius at an injection rate of 10 bbl/min. The injection schedule is calculated based on a pre-determined size of the stimulation zone. In the present example, the radius of the stimulation zone is chosen to be 50 m. Hence, the value of the scaled stimulation radius is α ¼ r st =r e ¼ 50=500 ¼ 0:1. For α ¼ 0:1 and an optimal packing ratio βn ¼ 0:24, the injection schedule is calculated using Eqs. (24) and (25). The results are shown in Fig. 7. Fig. 7(a) shows the dependence of particle radius on injection time and Fig. 7(b) shows the dependence of particle concentration on the injection time. As discussed in Section 2, the particle radius increases with injection time. Conversely, the particle concentration decreases with injection time. The rapid increase in the particle size and the rapid decrease in the particle concentration near the end of the injection period are due to the change in fracture opening and fluid velocity near the wellbore. It can be observed that the injection rate significantly affects the injection schedule. With increasing rate, the capture time of the particles is reduced, hence the total time of injection decreases. In the present example, stimulating a 50 m radius around the wellbore requires approximately 22 min at an injection rate of 1 bbl/min but less than 4 min at an injection rate of 10 bbl/min. The required particle size increases with injection rate since larger fracture openings are produced. The injected particle concentration decreases with increase in injection rate since the inter-particle separation must increase with increasing particle radius to obtain the optimal value of packing ratio (see Eq. (20)). The productivity index is calculated using Eqs. (28) and (29). Fig. 8 shows the dependence of the normalized productivity index J D , which represents the ‘folds of increase’ in production, on injection rate and the radius of the stimulation zone r st . A higher injection rate results in a greater well productivity index for all values of the stimulation radius. The injection rate also controls the relative increment in productivity for a given increment in the
7
5 Increasing injection rate
3
1
0
100
200
300
400
500
Stimulation radius rst (m) Fig. 8. Normalized productivity index or folds of increase in production as a function of stimulation radius and injection rate.
radius of the stimulation zone. For example, at a low injection rate of 1 bbl/min, the productivity index does not increase significantly beyond a stimulation radius of 50 m. At a high injection rate of 10 bbl/min, the productivity index continues to increase with increasing size of the stimulation zone. It can be concluded from Fig. 8 that the graded proppant injection scheme is a promising technology for the stimulation of natural fractures. In the example considered here, a three-fold increase in production was obtained by stimulating a radius of 50 m around the wellbore at a rate of 5 bbl/min. The limitations and applicability of the current mathematical model are discussed in the next section.
5. Discussion Some radical simplifications were adopted to model the interaction of fluid with natural fractures as well as particle transport in the fracture. Some shortcomings of the present model and possible methods for improvement are:
Axi-symmetric flow geometry was assumed in the mathematical model (Eqs. (1) and (15)). The flow geometry simplification implies that the conductive fractures form a tree-like
A. Khanna et al. / Journal of Petroleum Science and Engineering 111 (2013) 71–77
(dendritic) structure with branches multiplying from the well towards the reservoir. This assumption is equivalent to parallel layers (tubes) for the plane-parallel flow, which is often used for estimating flow in porous media. A more precise description of flow in fractured media would require the application of percolation or effective media models. The detailed description of the proppant flow and capture in real fractured system certainly requires more elaborate modelling. Bedrikovetsky (2008) described the size exclusion of suspension flow in stochastic porous media. To the best of our knowledge, the model for suspension flow in fractured systems is not available in the contemporary literature. However, further development of the proposed technology must include modelling of the strainingdominated suspension transport in stochastic fractured systems (Wei et al., 2012). In order to apply the technique of graded particle injection in coal beds, the anisotropy of coal needs to be incorporated in the model to describe particle propagation in face and butt cleats. The elaborated theory must include the tensorial permeability of the cleat system and the capture criterion should also be direction-dependent. Non-elastic processes of the proppant crushing or embedment may result at high stresses if the rock is significantly harder or softer than the proppant. As a result, a partial proppant monolayer may provide inadequate fracture conductivity. To incorporate the effects of these phenomena on the optimal proppant concentration, experimental studies must be conducted on specific rock-proppant combinations under expected stress conditions. The injection of water changes the reservoir stress around the wellbore, often causing shear induced slip of the asperous fractures, which may lead to additional productivity enhancement. The effect of shear dilation must be included in the model for graded particle injection for stimulation of natural fractured systems.
6. Conclusion A simple mathematical model is proposed for studying the graded proppant injection in natural fracture systems, such as those in coal seam beds, shale gas reservoirs and geothermal fields. The case study conducted using typical reservoir data demonstrates that graded proppant injection may lead to the enhancement in well productivity. It was found that the most influential parameter affecting the proppant injection schedule
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and the well productivity index is the fluid injection rate. A higher injection rate results in greater opening of the cleats, deeper percolation of the smallest particles, shorter injection time and a greater increase in well productivity. The applicability of the present model is limited due to a number of radical simplifications. Better estimates of increase in well productivity can be obtained by incorporating more advanced models for particle transport in fractured media.
Acknowledgements The authors wish to thank Prof. Sheik S. Rahman from The University of New South Wales and acknowledge the support of Santos Ltd. and ARC linkage grant LP110200799. References Basniev, K.S., Bedrikovetsky, P.G., Dedinez, E.N., 1988. Determination of the effective permeability for fractured porous medium. J. Eng. Phys. (Injenerno fizicheskii jurnal) 55 (6), 940–948. Bear, J., 1972. Dynamics of fluid in porous media. Elsevier. Bedrikovetsky, P., 2008. Upscaling of stochastic micro model for suspension transport in porous media. J. Transp. Porous Media 75 (3), 335–369. Brannon, H.D., Malone, M.R., Rickards, A.R., Wood, W.D., Edgeman, J.R., Bryant, J.L., 2004. Maximizing Fracture Conductivity with Proppant Partial Monolayers: Theoretical Curiosity or Highly Productive Reality? SPE90698. Colmenares, L.B., Zoback, M.D., 2007. Hydraulic fracturing and wellbore completion of coalbed methane wells in the Powder River Basin, Wyoming: implications for water and gas production. AAPG Bull. 91 (1), 51–67. Holditch, S.A., Ely J.W., Semmalbeck, M.E., Carter, R.H., Hinkel, J., Jeffrey, Jr. R.G., 1968. Enhanced Recovery of Coalbed Methane through Hydraulic Fracturing. SPE 18250. Hossain, M., Rahman, M., Rahman, S., 2002. Shear dilation stimulation model for production enhancement from naturally fractured reservoirs. Soc. Pet. Eng. 7 (2), 183–195. Khanna, A., Kotousov, A., Sobey, J., Weller, P., 2012. Optimal placement of proppant in narrow fracture channels. J. Pet. Sci. Eng. 100, 9–13. Laubach, S.E., Marrett, R.A., Olson, J.E., Scott, A.R., 1998. Characteristics and origins of coal cleat: a review. Int. J. Coal Geol. 35, 175–207. Moore, T.A., 2012. Coalbed methane: a review. Int. J. Coal Geol. 101, 26–81. Palmer, I., Mansoori, J., 1998. How Permeability Depends upon Stress and PorePressure in Coal Beds: A New Model. SPE 52607. Rahman, M.K., Hossain, M.M., Rahman, S.S., 2002. A shear-dilation-based model for evaluation of hydraulically stimulated naturally fractured reservoirs. Int. J. Numer. Anal. Methods Geomech. 26 (5), 469–497. Valko, P., Economides, M., 1995. Hydraulic Fracture Mechanics. Wiley, Chichester. van Golf-Racht, T.D., 1982. Fundamentals of Fractured Reservoir Engineering. Elsevier, The Netherlands. Warpinski, N.R., Mayerhofer, M.J., Vincent, M.C., Cipolla, C.L., Lolon, E.P., 2008. Stimulating Unconventional Reservoirs: Maximizing Network Growth while Optimizing Fracture Conductivity. SPE 114173. Wei, Z., Xiangyu, T., Weisbrod, N., Zhuo, G., 2012. A review of colloid transport in fractured rocks. J. Mount. Sci. 9, 770–787.