Simulation and Mathematical Modeling of Stimulated Shale Gas

Simulation and Mathematical Modeling of Stimulated Shale Gas...

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Simulation and Mathematical Modeling of Stimulated Shale Gas Reservoirs Samarth D. Patwardhan,*,† Fatemeh Famoori,‡ Radhika G. Gunaji,‡ and Suresh Kumar Govindarajan† †

Department of Ocean Engineering, Petroleum Engineering Programme, Indian Institute of Technology Madras, Chennai, India Department of Petroleum Engineering, Maharashtra Institute of Technology, Pune, India

‡

ABSTRACT: Gas from shale reservoirs is diﬃcult to produce, unless they are eﬀectively stimulated. Production from wells completed in these quad-porosity reservoirs is dependent on the placement of hydraulic fractures and their degree of connectivity to the existing natural fractures. These propped fractures and their eﬀectiveness is a direct function of the in situ stress in the formation. Furthermore, geochemical diagenesis in the created fractures signiﬁcantly impacts fracture conductivity. This paper utilizes a fracture-completed horizontal well in diﬀerent conﬁgurations of quad-porosity shale gas reservoir models to assess the eﬀect of gas ﬂow and storage in these systems on production parameters. Furthermore, sensitivity analysis is carried out on critical parameters to observe its impact on well performance. This work will help to provide a better understanding of hydraulic fracturing treatments and its eﬀect on the forecast of a stimulated well with reasonable certainty.

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the reservoir that contains nanopores, the ﬂow that takes place is predominantly diﬀusion-based and non-Darcian in nature. However, it can be noted that Darcy’s law is valid between speciﬁed lower and upper limits of intrinsic permeability values for a given pressure gradient, and Darcy’s law is supposed to maintain a linear relationship between the ﬂuid ﬂow rate and the pressure gradient, which cannot be expected while describing ﬂuid ﬂow through nanopores associated with an insigniﬁcant intrinsic permeability. In addition, this is where the ﬂuid ﬂow through shale reservoirs becomes non-Darcian in the lower limit, while the same non-Darcian ﬂuid ﬂow in the upper limit is associated with inertial eﬀects, as well as turbulence. A conventional single-porosity model based on a single continuum generally is unable to capture this type of nonDarcian ﬂuid ﬂow. Along with the diﬀusion-based ﬂow, the conventional concept of applying no-slip boundary conditions along the solid boundary is highly remote, while slip ﬂow at the boundary of solid matrix is also observed,2 which is diﬀerent from conventional Darcy ﬂow of gas. Unlike the migration of hydrocarbons that take place from source rock to the reservoir rock, shale acts both as a source as well as reservoir rock. Storage of hydrocarbons in conventional reservoirs takes place in the pore space, whereas in shale reservoirs, the storage is much more complex and is characterized by the coupled eﬀect of diﬀusion and adsorption, because of the size and structure of the pore connectivity. The most accurate way to characterize a shale reservoir is to experimentally determine the pore size distributions and permeability values, followed by its validation using mathematical and simulation models.3 It is believed that the gas in a shale

INTRODUCTION The complexity of ﬂow and storage mechanisms in shale gas reservoirs make it diﬃcult for the reservoir to produce gas. These reservoirs consist of very low porosity and nano-Darcy permeability. Although natural fractures exist in shale gas reservoirs, they still exhibit very low permeability to gas ﬂow. The only way to commercially produce gas from these reservoirs is to hydraulically fracture them. Hydraulic fractures tend to connect a larger surface area of the reservoir to the wellbore and, at the same time, assist in connecting the natural fractures. Figure 1 shows a chart that lists the various scales of

Figure 1. Chart for estimating porosity and permeability. (Reproduced with permission from ref 1. Copyright 2010, Society for Petroleum Engineers, Richardson, TX.)

Special Issue: Energy System Modeling and Optimization Conference 2013

porosity and permeability for conventional, tight gas, and shale gas reservoirs.1 Simulation of shale gas reservoirs that consist of nanopores is performed using either a single-porosity model or a dual-porosity model. The physics of ﬂow through these nanopores is diﬀerent than that occurring in regular-, medium-, and high-permeability sandstone reservoirs. In the section of © 2014 American Chemical Society

Received: Revised: Accepted: Published: 19788

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reservoir is stored both as free and adsorbed gas.4 It is seen that shale reservoirs consist of a combination of organic matter, inorganic matter, natural fractures, and hydraulic fractures, if induced. Figure 2 shows that gas occupies space in the

Figure 3. Scanned electron microscopy (SEM) image of a shale reservoir. (Reproduced with permission from ref 8. Copyright 2010, Society for Petroleum Engineers, Richardson, TX.)8

Figure 2. Schematic of adsorbed and dissolved gas molecules in a grain pore system of shale reservoir. (Reproduced with permission from ref 2. Copyright 2009, SPE Canada, Calgary, Alberta, Canada.)

simulator, the secondary fractures are usually placed in a direction normal to the created primary/hydraulic fractures. In reality, the direction of these fractures in a shale reservoir varies, depending on the in situ stress as well as the overburden. Furthermore, a horizontal well is usually placed in a shale reservoir such that the created hydraulic fractures intersect with as many natural fractures as possible, thereby ensuring maximum connectivity for the gas to ﬂow through this complex network, into the wellbore. These natural fractures are also called as secondary fractures, and production performance from such reservoirs is highly dependent on the extent and density of the fracture network. When the wells completed in such reservoirs are fracture-treated, the primary fractures connect with the natural fractures and proppant placement takes place in the secondary fractures as well. Fracture length, width, spacing, and orientation are important parameters that characterize an induced fracture, while existing natural fractures in the reservoir are characterized by their density. The width and conductivity of both the primary and secondary fractures are a function of the closure/in situ stress in the formation. Proppants that are mixed with the fracturing ﬂuids are injected to resist this stress and hold the fractures open. Certain properties need to be considered while selecting a particular fracturing ﬂuid, which include (but are not limited to) low leakoﬀ rate, the ability to carry the propping agent, low pumping friction loss, compatibility with the natural formation ﬂuids, minimal damage to the formation permeability, etc. The selection of the proppant for hydraulic fracturing treatment depends on the proppant crush strength, permeability, availability, and cost.9

adsorbed form, as well as the dissolved state within the kerogen, along with being present within the natural fractures and nanopores. A sequence of how gas production takes place from these complex reservoirs is given in closely related work.2 Because of pressure drawdown, gas ﬂow is initiated, which is followed by gas desorption, and the last process to follow is gas diﬀusion. Work was performed5 that reported the presence of porous networks within the organic material of the shale reservoirs. As a part of the work, the authors discussed organic material, natural fractures, hydraulic fractures, and mineral matrix as the four types of porosity present. Inside the organic matter, kerogen is divided into two categories: porous space and kerogen bulk. Gas is known to be stored in the nanopores of this kerogen, as well as in a dissolved state within the kerogen bulk.6 It is important to consider the porosity in all of these individual subsystems, because it aﬀects the original gas in place (OGIP) calculations. Shale gas reservoir can be accurately represented if the porosity is represented by a quad-porosity model,7 as given in eq 1: ϕsh = ϕm + ϕorg + ϕ2 + ϕhf

(1)

Equation 1 deﬁnes the complete porosity of a shale gas reservoir as a combination of matrix porosity, organic matrix porosity, natural fracture porosity, and hydraulic fracture porosity. The organic matter within the shale reservoir contains nanopores, as evidenced by a Barnett shale sample, as shown in Figure 3. Gas in the shale reservoir desorbs from the organic matter surface and ﬂows through the nanopores and the naturally existing fractures through the phenomenon of diﬀusion and convection. This ﬂow through nanopores takes diﬀusive ﬂow into account, along with the viscous ﬂow, for which Darcy law is applicable. Characterization of such extremely low-permeability, organics-rich, fractured reservoirs using these models does not aptly capture the physics and complexities of gas storage and ﬂow through their nanopores. As mentioned above, for production to be economic from these unconventional shale gas reservoirs, a stimulation treatment is deemed necessary. When shale gas reservoirs are treated with a hydraulic fracturing treatment, proppants are usually placed in the primary fractures. These reservoirs have natural/existing fractures, which are also called as secondary fractures, and they act as a source of ﬂow for gas. In a numerical

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BACKGROUND As mentioned in the Introduction section, in situ stress in these unconventional reservoirs is known to aﬀect the performance of the fractures over time. The stress leads to crushing, embedment, and stress cycling of the proppant. The permeability of the proppant placed is an integral part of the fracture conductivity, and its decrease leads to a decrease in the fracture conductivity and, hence, the overall productivity of the well. Similar to the eﬀect of stress on reduction of natural fracture permeability, several authors10,11 have looked at the eﬀect of in situ stress on the permeability reduction in the created fractures. Most of the work is a result of extensive 19789

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in. for 4000 psi vs 0.0016 in. for 10 000 psi). This leads to a reduction in conductivity by ∼17%. Some studies15 have discussed the combined eﬀect temperature and closure stress has on the fracture conductivity of a propped fracture. It is noted that proppants with a slightly larger sieve distribution were slightly more conductive at stresses below 10 000 psi and the smaller sieve distribution had higher conductivity at stresses above 10 000 psi, as given in Figure 6. Experiments were conducted16 that involved ﬂowing

experimental studies and simulations which validate the experimental ﬁndings. As time progresses, because of production, the conductivity of the secondary fractures is bound to reduce.10 The eﬀect of stress on natural fracture porosity reduction and its eﬀect on permeability has been investigated.11−13 Long-term eﬀects of closure and temperatures on sands, resin-coated sands, and ceramic proppants between monel shims and medium-hard sandstones were studied, and it was concluded that fracture ﬂow capacities and productivity increases are a magnitude higher than the actual results, mainly because of ﬂuid rheology and leakoﬀ.14 Figure 4

Figure 6. Fracture conductivity versus stress and reservoir temperature for 20/30 Carbo-Prop HC. (Reproduced with permission from ref 15. Copyright 1987, Society for Petroleum Engineers, Richardson, TX.)

Figure 4. Conductivity and permeability of a 2 lb/ft2 20/40 Jordan sand. (Reproduced with permission from ref 14. Copyright 2001, Society for Petroleum Engineers, Richardson, TX.)

fracturing ﬂuids through a proppant pack, while allowing ﬂuid to leak oﬀ through a core. Results indicate that the fracturing ﬂuid ﬁltrate leakoﬀ in the formation leads to an increase in the water saturation up to gas saturation in the fracture wall, thereby inhibiting gas ﬂow. Post-hydraulic fracture treatment cleanup operations are performed, but a part of the gel (called gel residue) remains in the formation, which signiﬁcantly aﬀects the fracture conductivity.17 In the literature, this phenomenon is commonly referred to as fracture face skin, and there are other reasons for its occurrence. Similar experiments were conducted,9 and it was concluded that, in some cases, the permeability of the proppant pack was reduced by as much as 90%. Experiments on a setup called the Dynamic Compression Device (DCD) were conducted,18 to look at the eﬀect on proppant pack permeability. Studies conducted19 propose that the closure stress in a fracture on a proppant could be signiﬁcantly higher than common estimations, because of the inﬂuence of the bounding layers and the elastic response of the formation that acts on the proppant. Ottawa sand (20/40 mesh) at a temperature of 200 °F was used to investigate the eﬀect of this extra closure stress on fracture conductivity. Compared to the stress that the formation exhibits after closing in after a mini-fracture test has been determined using only the ﬂuid, the closure stress magnitude for a situation where proppant is used is observed to be higher. A method was presented20 to observe fracture width changes with the help of pressure transients in naturally fractured reservoirs, and it was concluded that the fracture permeability decreases as the pore pressure depleted under isotropic stress conditions. Correlations that evaluate productivity losses in reservoirs with stress-dependent natural fracture permeability have been presented.11 Furthermore, a numerical model is used that coupled geomechanical and single-phase ﬂuid ﬂow aspects

shows the permeability and conductivity of a 2 lb/ft2 20/40 Jordan Sand, when it was subjected to increasing closure stress and temperature conditions. It can be seen that the permeability of the sand falls below 100 Darcies at 4000 psi, and reduces to half its value with an increase in stress by 2000 psi. Drawdown over a long period of time is a critical factor responsible for the conductivity of the propped fractures to reduce. Figure 5 discusses the stress-induced embedment of proppant on the fracture walls. The ﬁgure shows an SEM micrograph of embedment observed in Ohio sandstones at various closure stresses. The depth of embedment of a 20/40 sand increases with increasing pressure (embedment of 0.001

Figure 5. SEM micrographs depicting proppant embedment. (Reproduced with permission from ref 14. Copyright 2001, Society for Petroleum Engineers, Richardson, TX.) 19790

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Table 1. Correlation for Evaluating Productivity Lossesa correlation

1

reference

⎡ ⎛ ⎞⎤ ϕ k f = k fi exp⎢c1⎜⎜ − 1⎟⎟⎥ ⎢⎣ ⎝ ϕi ⎠⎥⎦

Rutqvist et al. (2002)

with ϕ = ϕr1 + (ϕi − ϕr1) exp(a1ΔP )

2

k = kiFk 3

with Fk =

{bmax2⎡⎣exp(d2σy′) − exp(d2σyix)⎤⎦}

Rutqvist et al. (2002) + {bmax2⎡⎣exp(d2σz′) − exp(d2σzi′ )⎤⎦}3

bi 23 + bi 23

3

k f = k fi exp(− df 3ΔP )

Raghavan and Chin (2002) (Rock Type I)

4

k f = k fi(1 − mf 4 ΔP )

Raghavan and Chin (2002) (Rock Type II)

5

⎛ ϕ ⎞n f k f = k fi⎜⎜ ⎟⎟ ⎝ ϕfi ⎠

Raghavan and Chin (2002) (Rock Type III)

with ϕfn + 1 = 1 −

6

1 − ϕfn exp(Δεν)

and Δεν =

α5ΔP (1 + ν5)(2ν5 − 1) E5(1 − ν5)

⎛ ϕ ⎞n f k f = k fI ⎜⎜ ⎟⎟ ⎝ ϕfI ⎠

Raghavan and Chin (2002) and Cells et al. (1994)

with ϕf = ϕfI exp(df 8ΔP )

7

⎛ ϕ ⎞n f k f = k fi⎜⎜ ⎟⎟ ⎝ ϕfi ⎠

Raghavan and Chin (2002) and Rutqvist et al. (2002)

with ϕf = ϕfiFϕ , Fϕ =

8

b1 + b2 + b3 b1i + b2i + b3i

and b = bi7 + bmax7⎡⎣exp(df 7σn′) − exp df 7σni′ )⎤⎦

(

⎛ ϕ ⎞n f k f = k fi⎜⎜ ⎟⎟ ϕ ⎝ fi ⎠

Raghavan and Chin (2002) and Minkoﬀ et al. (2003)

with ϕf = ϕfi(1 − mf 8ΔP )

9

⎛ ϕ ⎞n f k f = k fi⎜⎜ ⎟⎟ ⎝ ϕfi ⎠ with ϕ = 1 −

Raghavan and Chin (2002) and Minkoﬀ et al. (2003) 1 − ϕi exp(εν)

and εν =

α9ΔP (1 + ν9)(2ν9 − 1) E9(1 − ν9)

a

Reproduced, with permission, from ref 21. Copyright 1997, Society of Petroleum Engineers, Richardson, TX. References cited here are given in the work from which this table was reproduced.

factors in unconventional gas reservoirs and how various hydraulic fracturing parameters and nonideal reservoir behaviors aﬀect the horizontal well completion design. Short-term productivity scenarios resulting from a relation between fracture parameters and treatment design were evaluated.29 Single porosity and dual-porosity models are compared30 for assessing scenarios of a horizontal well producing gas and water, and geomechanical aspects were evaluated31 for their impact on well performance. Although in situ stress aﬀects the matrix and natural fracture permeability, it is also known to aﬀect the created hydraulic fracture permeability.14 The decrease in permeability is due to the drawdown that the system experiences when the well is producing over a period of time. Transport of gas in shale reservoirs is characterized by diﬀusion in the nanopore spaces of the matrix and adsorption of the gas on the matrix surface. Diﬀusion and desorption,

to correlate productivity reduction with time, for a wide range of operating conditions and reservoir properties. These correlations were developed by taking into consideration three common types of stress-dependence correlations on permeability. Table 1 gives a summary of some of these correlations from the literature. Since most of these correlations are applicable to matrix permeability, the study21 validated the use of these correlations for shale natural fracture permeability. This was done by matching experimental fracture permeability data with speciﬁc correlation data. Their study involved examining the eﬀects of pressure-dependent natural fracture permeability on production from shale gas wells by selecting speciﬁc correlations from Table 1, and applying them to synthetic as well as speciﬁc ﬁeld data of wells completed in Barnett and Haynesville shale. Authors28 have used a numerical reservoir simulation study to develop simple correlations that quantify what fracture spacing is necessary to optimize recovery 19791

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along with viscous ﬂow in natural and hydraulic fractures, together comprise the combined transport mechanism in a shale reservoir. Furthermore, ﬂuid ﬂow mechanisms in the organic matter of the shale gas reservoir are delineated as free gas ﬂow, desorption, diﬀusion, and imbibition suction. Figure 7

important parameters that aﬀect the productivity of the well. For these simulations, the reservoir model is considered to be a dual-porosity model. Second, a new model is built that incorporates the quad-porosity system, in which additional ﬂow mechanisms are incorporated. Simulations are run to compare the results with those of the previous runs. The eﬀect on production, OGIP, and pressure is studied by considering the ﬂow mechanisms of adsorption, slippage, and diﬀusion. Simulation Study. First, similar to the earlier work that involved natural fractures,11,20,21 this work involves application of a constantly decreasing permeability to the created hydraulic fractures. The decrease of permeability in the placed fractures is simulated using an exponential approach, as well as a linear approach. Figure 8 shows how the permeability is varied inside

Figure 7. Dissociation of experimental data3 into four ﬂow regimes: fractures, matrix (nanopores), desorption, and diﬀusion from kerogen bulk. (Reproduced with permission from ref 6. Copyright 2013, Society for Petroleum Engineers, Richardson, TX.)

(reproduced from ref 6) separates the four ﬂow regimes and their contribution and further supports the theory of diﬀusion for gas ﬂow in shale gas reservoirs. Historically, the rate of diﬀusion from kerogen bulk to shale pores is considered negligible, whereas theoretically it is shown23 that the eﬀect of incorporation of diﬀusive ﬂow and other ﬂow regimes on the pressure transients and cumulative recovery behavior is signiﬁcant. In addition, the adsorbed gas on organic matter could have adverse eﬀects on the permeability, because of increased drag on gas molecules. In a nonporous organic medium, diﬀusion plays an important role, whereas, in a porous organic medium, the role of diﬀusion is markedly reduced.5 Experiments are carried out22 that quantify the contribution of self-diﬀusion on the basis of laboratory experiments and emphasize further consideration of the same for shale gas reservoirs. Gas diﬀusion from kerogen is an important source of gas to be added for the quantiﬁcation of OGIP in shales. Multiple studies23−27 suggest that the adsorption, as well as diﬀusion, of gas leads to signiﬁcant increases in the amount of gas stored in the organic matter, thereby leading to an increase in the OGIP value.

Figure 8. Decreasing permeability with distance from the wellbore.

the created fractures within the model. Simulations were run by varying multiple parameters, to observe the impact of this decrease on the overall productivity of the well. These declining trends were chosen because they are convenient for incorporating in a shale gas fractured horizontal well model, and because of the availability of data. Investigations were conducted to examine the combined eﬀect of reduction in secondary fracture permeability, as well as proppant pack permeability due to pressure, and its eﬀect on productivity of a well. For the assessment of this impact of fracture permeability reduction with drawdown, accurate reservoir simulation models that represent a realistic reservoir were built and run. Figure 9

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OBJECTIVES As evidenced by the work of multiple authors,11,12 the permeability of secondary fractures has a tendency to decrease as the stress increases. Similarly, stress also acts on the proppants that are pumped in the created hydraulic fractures, to keep them open and create a pathway for the hydrocarbons to ﬂow to the wellbore. The proppant pack permeability has a tendency to decrease as the stress increases, as a result of the drawdown being applied. To investigate the impact of this change in permeability within the fracture on well productivity, this paper takes a two-pronged approach. First, a simulation study is conducted where a linearly and exponentially decreasing permeability within the created hydraulic fracture is considered, and its eﬀect on the well performance is assessed. This is done by building and running accurate reservoir simulation models that represent a realistic reservoir. Furthermore, this paper also investigates the impact of other

Figure 9. Fracture completed horizontal well used in the model.

shows the model, with a single horizontal producer and four equally spaced hydraulic fractures in the reservoir. Hydraulic fractures are placed 700 ft apart and are generated in a transverse direction, perpendicular to the horizontal wellbore. Apart from the runs where the fracture spacing is the sensitivity parameter, the spacing has been kept constant. The model used here is based on standard assumptions of a common fractured well that produces gas from a shale gas reservoir. In this model, secondary fractures are considered to be orthogonal to the placed hydraulic fractures. Coarse grids are selected for the reservoir, whereas local grid reﬁnement (LGR) has been done 19792

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3500 ft × 2500 ft. The depth to which a horizontal well is placed is 8000 ft in the vertical direction, and then extended 3300 ft laterally. The reservoir is assumed to be a dry gas reservoir, without the production of any water, and consists of both primary as well as secondary fractures. The main (primary) fractures represent the created hydraulic fractures; these are placed in a direction perpendicular to the horizontal wellbore and are assumed to be propped with proppant. These fractures are connected to the wellbore, and the gas ﬂow takes place through them. Primary fracture spacing, secondary fracture spacing, fracture conductivity, reservoir permeability, and permeability anisotropy in the reservoir and half-length are the main parameters that aﬀect the productivity of a fracture completed well. A parametric investigation is done by building models that depict the above-mentioned factors. Furthermore, a sensitivity analysis is performed to observe the eﬀect of a change in the above parameters on the well performance. The secondary (natural) fractures in this model are considered to be unpropped. The productivity of a shale gas well is highly dependent on the complex interactions between the propped fractures with the natural fractures. The model considers uniform spacing between the created hydraulic fractures. Simulations with these models assume the permeability in the created hydraulic fractures to be decreasing linearly as well as exponentially from the wellbore toward the tip. In the following ﬁgures, we investigate the eﬀect that each parameter has on the productivity of a well. Reservoirs with three diﬀerent permeability values (0.0001, 0.001, and 0.01 mD) are considered here. Eﬀect of Half-Length. Half-length of the placed fracture plays an important role in the ﬁnal recovery of the hydrocarbons from the reservoir. Higher half-length connects with a larger reservoir area, and with more natural fractures, thereby impacting productivity. Figure 11 shows the diﬀerence in the 30-year cumulative gas production for a well completed in reservoirs with three diﬀerent permeability values, when the half-length is varied to 250, 500, and 750 ft, respectively. It can be seen from Figure 11 that the reservoir permeability plays a critical role, even if the half-lengths are increased for better reservoir contact. It can be seen that, because of the initially high production in the 0.01 mD case, a half-length of 750 ft achieves a higher cumulative production in 10 years than what the same half-length achieves in 30 years, for the 0.001 mD reservoir case. This denotes the fact that, although the permeability inside the fracture is exponentially decreasing, thereby aﬀecting the ﬂow of ﬂuid toward the wellbore, the ﬂow of ﬂuid from the reservoir to the fractures is a function of the reservoir permeability. Figure 12 shows the cumulative gas production curves for the linearly decreasing permeability case. Although we see a large diﬀerence in the cumulative gas production for the base models in Figure 10, the diﬀerence in the cumulative values here is slightly more than 1 bcf (e.g., a well with a half-length of 750 ft with a linearly decreasing permeability produces 13.3 bcf of gas, while the same well with an exponentially decreasing permeability produces 12.2 bcf of gas). Eﬀect of Primary Fracture Conductivity. Figure 13 shows the eﬀect of the primary fracture conductivity runs by changing the conductivity to 4, 40, 100, 200, 300, and 400 mD ft, respectively. The model consisted of fractures spaced 700 ft apart, with a total stimulated reservoir volume (SRV) of 180 × 106 ft3. It should be noted that the primary fractures extended to a distance of 750 ft from the wellbore, and the secondary

for the created hydraulic fractures. Higher spatial discretization of the space conﬁning the hydraulic fractures leads to moreaccurate computation of the pressure, rates, and saturations at these points. Although computationally intensive, capturing these pressure and rate transients help ascertain the deliverability of the well in a better manner. Fractures are not usually smooth and consist of minerals that lend a surface roughness to them, thereby aﬀecting the fracture aperture. Both the natural and placed fractures in this model do not include a detailed characterization of the eﬀect of mineralization, because they are primarily used to establish a functional relationship between pressure and fracture permeability. Furthermore, since the objective of the paper was not to come up with a new correlation for stress-dependent permeability change for a speciﬁc shale reservoir, linearly and exponentially varying correlations were used, which have been presented in the literature, as mentioned in the Background section of this paper. Figure 10 shows the eﬀect of a linearly and exponentially

Figure 10. Eﬀect of decrease in permeability in hydraulic fractures well used in the model.

varying permeability in the created hydraulic fractures on the cumulative production of the modeled well. This ﬁgure shows that the exponential decrease severely hampers the cumulative gas production. The cumulative gas produced from the linearly decreasing case is 8.27 bcf, while that from the exponentially decreasing case is 13.3 bcf. [Note that bcf stands for billions of cubic feet.] Also, it is observed that the exponential decrease aﬀects the initial productivity of the well. This work includes using these correlations for placed fractures and observing its eﬀect. Furthermore, a sensitivity study is performed on various critical parameters, to determine which parameter the productivity is most sensitive to. The main reservoir parameters that are used for this study are given in Table 2. The size of a grid block in all the three directions is assumed to be 100 ft. With only one vertical layer modeled, the size of the reservoir in this model is Table 2. Reservoir Properties property

value(s)

matrix permeability reservoir pressure net pay porosity water saturation gas gravity reservoir temperature

0.0001 mD, 0.001 mD, and 0.01 mD 3600 psi 300 ft 3.6% 30% 0.6 106 °F 19793

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Figure 11. Eﬀect of half-length (exponential).

Figure 12. Eﬀect of half-length (linear).

Figure 13. Eﬀect of primary fracture conductivity (exponential).

Figure 14. Eﬀect of primary fracture conductivity (linear).

However, note that, although we see the gap between the 4 mD ft curve and the 40 mD ft curve decreasing, the gap between the 40 mD ft curve and the 100 mD ft curve remains almost constant. The gap between the 200 mD ft curve and the 400 mD ft curve also remains constant. This denotes that the percentage/incremental increase in cumulative gas production remains constant, even at higher primary conductivity values. Although the reservoir permeability is responsible for a faster recovery from the reservoir, the primary fracture conductivity

fracture spacing was kept constant at 100 ft. It can be seen from this ﬁgure that, for the lowest reservoir permeability case (0.0001 mD), the cumulative gas production curve is almost linear in nature, and this linearity goes on changing as the reservoir permeability changes by an order of magnitude. This indicates that the initial gas production rate increases as the reservoir permeability increases. It can also be seen that the initial gas rates in the case of 0.01 mD reservoir case are signiﬁcantly higher than those in other reservoir cases. 19794

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Figure 15. Eﬀect of primary fracture spacing (exponential).

Figure 16. Eﬀect of primary fracture spacing (linear).

Figure 17. Eﬀect of secondary fracture spacing (exponential).

case of linearly decreasing permeability, where the cumulative values are higher than those in the exponential case. Eﬀect of Secondary Fracture Spacing. In the following runs, the primary fracture spacing is assumed to be 600 ft, and they are assumed to be 100% eﬀective. The secondary fracture spacing for all the previous runs was assumed to be 100 ft, which is now varied from 50 ft to 300 ft, to observe its impact on the productivity of the well. A smaller spacing means a tighter network of fractures, which are more closely interconnected. The more closely spaced the natural fractures are, the better the ﬂow capacity of the ﬂuids in the reservoir, as well as from the interconnected hydraulic fractures toward the wellbore. Figure 17 shows that the incremental diﬀerence in cumulative gas production that is observed by reducing the secondary fracture spacing continues to decrease with an increase in reservoir permeability. In the case with 0.0001 mD, it is seen that, for the initial 5 years, the eﬀect of spacing does not aﬀect the cumulative gas production; however, as time progresses, the closely spaced 50-ft fractures assist in a higher recovery than the widely spaced 300-ft fractures. For the case of 0.001 mD and 0.01 mD, it can be seen that spacing aﬀects the initial gas production rates, which do diﬀer, but the permeability factor takes over, with the diﬀerence steadily

plays an insigniﬁcant role in it, beyond a certain threshold value. When the simulations were run for the linearly decreasing permeability case, as shown in Figure 14, it was seen that, similar to the exponential case, higher conductivity (from 40 mD ft to 400 mD ft) fractures spaced 700 ft apart lead to a signiﬁcantly higher initial gas rates and a better recovery than the lower conductivity fractures. Eﬀect of Primary Fracture Spacing. The previous simulation runs were done with a primary fracture spacing of 700 ft. Primary fracture spacing was reduced to 500 and 600 ft, to observe its eﬀect on the productivity of the well, as shown in Figure 15 for the exponential case, and Figure 16 for the linear case. The secondary fracture spacing is kept constant at 100 ft, the lateral length of the well is the same as in the previous runs, and the SRV is also kept constant. In both cases, the change in the curves with a corresponding increase in the initial gas rate denote that, with regard to the primary fracture spacing, the 0.0001 mD case was not as eﬀective in draining the reservoir as the 0.01 mD case. Also, the relative impact of lesser fracture spacing diminishes with increasing reservoir permeability. This is due to the fact that a lesser pressure drop is required to produce the hydrocarbons, when the higher permeability allows a greater amount of ﬂuid ﬂow to take place. This happens in the 19795

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Figure 18. Eﬀect of secondary fracture spacing (linear).

Figure 19. Eﬀect of permeability anisotropy (exponential).

Figure 20. Eﬀect of permeability anisotropy (linear).

Figure 21. Pressure distribution inside the fractures (linear versus exponential), for a primary fracture conductivity of 400 mD ft.

opposed to the exponential case, where a diﬀerence was seen between the cumulative values for the 0.0001 mD case, there is insigniﬁcant diﬀerence between the values for the same

decreasing toward the end of 30 years. Figure 18 discusses the eﬀect of secondary fracture spacing on the cumulative gas production for a linear permeability degradation case. As 19796

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Figure 22. Pressure distribution inside the fractures (linear versus exponential), for a primary fracture spacing of 700 ft.

the pressure in the linear case for the same time frame is 1000 psi higher. An exponential decline in the permeability leads to a higher pressure drop in the fractures needed for the hydrocarbons to ﬂow. This higher pressure drop leads to a lower recovery, as shown in Figures 13 and 14. Figure 22 discusses the pressure distribution inside the fractures as well as the reservoir, when the primary fracture spacing was 700 ft. This pressure distribution is for a reservoir with a permeability of 0.001 mD, and the bottom hole pressure for both cases was constant at 1000 psi. This ﬁgure shows that, although the spacing is the same, and the cumulative plot trends (as seen in Figures 15 and 16) are similar, a higher pressure drop is observed in the exponential runs, for both the fractures as well as the matrix. This higher pressure drop leads to a loss in the overall production. Eﬀect of Gridding Scheme. As mentioned earlier, the reservoir is coarsely gridded while the hydraulic fractures are ﬁnely gridded. The eﬀect of sensitivity of the gridding scheme on the productivity of the results was analyzed. As shown in Figure 23, LGR has been implemented in the created hydraulic

reservoir for the linear case. Also, in the linear case, higher drainage is observed, because of the steeper initial gas production rate proﬁle. Eﬀect of Permeability Anisotropy. Generally, horizontal permeability plays an important role in the ﬂow of ﬂuids toward the wellbore. In sandstones, vertical permeability is usually ∼10% of the horizontal permeability. Since shale reservoirs are much tighter than sandstones ones, investigations were done in this study to observe the eﬀect of permeability anisotropy (kv/ kh = 0.1, 0.01, 0.001) on the productivity of the well. Primary fracture spacing was maintained at 600 ft, with a fracture halflength of 750 ft. Secondary fracture spacing was kept constant at 100 ft. Figure 19 discusses the results obtained. This ﬁgure shows that the eﬀect of anisotropy in the 0.0001 mD case begins to show only at late times, whereas in the higher permeability cases, their eﬀect on production proﬁles can be observed right from the ﬁrst year. In all three reservoirs, it is observed that the base case curve, with no anisotropy, remains consistently higher than those with anisotropy. However, because of the fact that the loss in production due to anisotropy is less, we can conclude that the impact of anisotropy is not seen to be signiﬁcant in all three cases. Similarly, when simulations were run for a linear case, as shown in Figure 20, it is observed that the trend of the cumulative gas production curve remains the same for all three reservoir cases, conﬁrming that the role that anisotropy plays in the overall recovery is not very signiﬁcant. The pressure drop in the reservoir and inside the created fractures is dependent on the permeability, and, consequently, the production is a function of it. A change in permeability aﬀects the pressure drop considerably. Figure 21 shows the comparison of the pressure drop inside the fractures for the case of change in primary fracture conductivity. The 400 mD ft conductivity case is shown in Figure 21, for both the linear as well as the exponential case. Reservoir permeability is assumed to be 0.0001 mD; the fracture half-length extends 750 ft away from the wellbore, and the secondary fracture spacing is 100 ft. The comparison is performed at three time steps: after 10, 20, and 30 years of production. This ﬁgure shows that the pressure inside the fractures for the exponential case at the 30-year time step (end of simulation) reaches a value close to 1650 psi, while

Figure 23. Gridding scheme in created hydraulic fractures.

fractures. The reﬁnement is higher in the fractures, compared to the reservoir, which is coarsely gridded. To investigate the eﬀect of gridding on the solution (productivity) of a well, we investigated a particular case of a k = 0.0001 mD permeability reservoir, with hydraulic fractures placed 700 ft apart. The ﬁrst case (base case) was run with the fractures gridded as 7 × 7 × 1 in the i, j, and k directions, respectively. This was further 19797

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toward the productivity of the completed well. In the previous runs, the primary fractures were placed 700 ft apart, and the impact of spacing of the secondary fractures was revealed. Figures 18 and 19 show that the impact of the secondary fractures on the productivity of the well is not signiﬁcant. Simulations were run to observe the impact of changing secondary fracture spacing on a much denser placed hydraulic fractures. This was done by changing the primary fracture spacing from 700 ft to 100 ft. Figure 25 shows that, when the

improved by running multiple runs with further reﬁnement of the grid within the fractures. Results of this grid reﬁnement on the productivity are shown in Figure 24.

Figure 24. Eﬀect of gridding scheme sensitivity.

It can be seen that as the grid reﬁnement increases, the run time increases correspondingly, and the results are aﬀected; however, a signiﬁcant diﬀerence is not visible. With higher grid reﬁnement, the pressure is computed much more accurately and, hence, the productivity results are more accurate. Table 3 Figure 25. Scaling study for primary and secondary fractures.

Table 3. Gridding Scheme Sensitivity sample 1 2 3 4 5 a

gridding scheme (i × j × k) 7×7× 15 × 15 27 × 27 39 × 39 51 × 51

1 × × × ×

1 1 1 1

computational time (s)

productivity (bcf)a

23 47.76 120.07 233.83 403.23

5.18 5.12 5.04 5 4.96

spacing was 700 ft, secondary fractures spaced even more densely could not lead to higher production. However, when the primary spacing was changed to 100 ft, it can be seen that the well deliverability increases multifold. This ﬁgure also shows that the impact of a lesser dense secondary fracture leads to a lower productivity, even though the primary fracture spacing remains the same. However, the relative change in the productivity diminishes as the spacing is increased. From these runs, it can be concluded that, for densely placed hydraulic fractures, the density of the secondary fractures can only contribute up to a certain point, after which the law of diminishing returns starts to become applicable. Simulation Study of Quad-Porosity Systems. As mentioned in the Background section of this paper, quadporosity systems have been deﬁned as porosity in an organic matrix, an inorganic matrix, micro (natural), fractures and macro (induced) fractures, thereby providing a deeper insight into understanding these unconventional reservoirs. Furthermore, the eﬀect that ﬂow through these nanopore spaces has on the ultimate recovery is correctly deﬁned by incorporating processes such as diﬀusion, desorption, slippage, and transport through natural and hydraulic fractures. In this work, each of the subsystems is deﬁned as diﬀerent rock types within the created model. These have been assigned diﬀerent properties, with separate relative permeability curves. It has been noted in the literature5 that the wettability of the organic shale is inclined toward being oil wet, because of the adsorption of hydrocarbon on the surface. Literature25 also mention that gas diﬀusion in kerogen is added to rock types within the reservoir model by means of supplying gas from kerogen bodies to surfaces. Under equilibrium conditions, gas concentrations at the kerogen surface and body are identical. Once gas concentration at the surface decreases due to gas desorption, the resulting gas concentration gradient invokes the gas diﬀusion mechanism. In turn, gas diﬀusion in kerogen is a very slow process at the geological framework; nonetheless, a signiﬁcant gas concentration gradient makes gas diﬀusion an important part of modeling gas production in shale-gas

bcf = billions of cubic feet.

discusses the change of gridding scheme on the results, along with an increase in computational time. It can be seen from the grid reﬁnement analysis that a local grid reﬁnement of the grid blocks comprising the hydraulic fracture does reduce the total cumulative gas production value, but the decrease is not considerable. However, note that the values from the reﬁned grid-block runs do give a much more accurate value. It can be seen from the table that, as the grid is reﬁned, the percentage increase in time increases by more than 100%. For large-ﬁeldscale shale gas models, this increase in computational time can become signiﬁcant, thereby leading to a loss of time and eﬃciency. Scaling Study. The role of secondary fractures, accompanied by a relatively denser fracture density, remain sensitive when the fracture spacing (the distance between the centers of two parallel fracture axes) remains relatively small, as the secondary fractures also signiﬁcantly contribute to the resultant enhancement in intrinsic permeability of the considered rock mass, while the inﬂuence of secondary fractures is minimal with a relatively larger fracture spacing accompanied by a smaller fracture density. In essence, upscaling critically depends on the details of fracture spacing and fracture density, as well as the positioning of the generated fractures. The denser spacing and positioning of the generated fractures ensure an eﬀectively larger ﬂow path to the ﬂuids being fed from the reservoir through the secondary fractures to the wellbore. Simulations were run in order to observe the eﬀect of contribution of secondary fractures in combination with primary fractures 19798

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formations. It also conﬁrms that the additional physical transport mechanisms give rise to measurable deviations from conventional models used to forecast gas production. After conducting simulation runs on the model that was constructed,25 the authors conﬁrm and conclude that, at late times, production enhancement due to diﬀusion accounted for a signiﬁcant percentage of the overall recovery. In the created model, hydraulic fractures are placed in accordance with the speciﬁed spacing. The secondary fractures are assumed to be in a direction normal to the created hydraulic fractures, in such a way that maximum connectivity is achieved. The organic matter, as well as the inorganic matter, is randomly and evenly distributed within the reservoir. Figure 26 shows a schematic of

Va =

VP l P + Pl

(2)

where Va is the amount of gas adsorbed by one ton of rock; Vl is the Langmuir volume, which represents the maximum sorption capacity; and Pl represents the Langmuir pressure, which is the pressure at Vl/2, which has a signiﬁcant eﬀect on the curvature of the isotherm. At low pressure or in low permeability formations (0.0001 mD), the Klinkenberg eﬀect is too signiﬁcant to be ignored when modeling gas ﬂow in reservoirs. The eﬀective absolute permeability for gas is assumed to be related to permeability measured for liquids as given by eq 3: ⎛ b⎞ Ka = K i⎜1 + ⎟ ⎝ P⎠

(3)

where Ka is the apparent permeability, Ki the intrinsic permeability, and b the slip coeﬃcient. Multiple empirical models have been developed and proposed to account for this slip ﬂow and Knudsen diﬀusion, in the form of apparent permeability based on the ﬂow mechanism.2 The empirical model used in this work, which accounts for gas diﬀusion and slippage,2 is given by eq 4:

Ka = Figure 26. Subsystems within a shale reservoir. (Reproduced with permission from ref 5. Copyright 2009, Society for Petroleum Engineers, Richardson, TX.)

Dk kμM + FKD RTρ

(4)

where Dk is the Knudsen diﬀusion constant, μ the gas viscosity, M the gas molecular mass, R the universal gas constant, T the temperature, KD the Darcy permeability, and ρ the gas density. F is the gas slippage factor, which is deﬁned by eq 5:

how these subsystems are distributed in a typical reservoir.5 Specifying the location of organic content, in relation to the inorganic content, is not possible within the capabilities of the commercial reservoir simulator used. Furthermore, this work utilizes a fracture-completed horizontal well in diﬀerent conﬁgurations of quad-porosity shale gas reservoir models to assess the eﬀect of gas ﬂow and storage in these systems on production parameters. The completed well is made to produce gas from a reservoir under a constant bottom-hole pressure of 1000 psi for 30 years. Simulations are run to understand the impact of this detailed characterization on well performance. In addition, this work also investigates the dependency of stress on these quad-porosity systems, as stress acting on the matrix and fractures play a signiﬁcant role in porosity and permeability alteration. The results from the quad-porosity shale gas models are compared with the results of the dual-porosity models, which were presented in the earlier section of this paper. The quad-porosity models are simulated using a commercially available simulator. The system is discretized using the gridding tool of the simulator, with the induced hydraulic fractures ﬁnely gridded, to capture the pressure transient accurately. Within the shale gas matrix, the organic matter is assumed to be the main gas-bearing porosity system with the gas ﬂow being governed by both Darcy’s law and diﬀusion. The inorganic porosity is assumed to be saturated with water and the ﬂow in this subsystem is due to gas adsorption, diﬀusion, and slippage. The Langmuir equation, which considers the adsorption mechanism to be monolayer adsorption, is widely used to describe adsorption/desorption in unconventional reservoirs. In this work, the Langmuir isotherm is used for deﬁning the adsorbed gas in the organic part of the matrix. The Langmuir equation is given by eq 2:

⎛ 8ΠRT ⎞0.5 μ ⎛ 2 ⎞ ⎜ ⎟ − 1⎟ F = 1⎜ ⎠ ⎝ M ⎠ pr ⎝ α

(5)

Here, p is the average reservoir pressure (Pa); r is the pore radius, and α is the fraction of molecules striking the pore wall, which are diﬀusely reﬂected. When the quad-porosity models were run, with the modiﬁcations mentioned above, the OGIP was determined to be higher than the gas in place for the dualporosity models. The OGIP for the dual-porosity model was computed to be 1.65 Tcf, while the OGIP for the quad-porosity model was computed to be 3.27 Tcf (slightly less than twice that of the dual-porosity model). [Here, Tcf denotes trillions of cubic feet.] This higher value can be attributed to the gas stored in the additional porosity subsystems that have been deﬁned earlier. Although it is seen that the gas-in-place numbers are signiﬁcantly diﬀerent for the quad-porosity models, the ease with which the stored gas ﬂows toward the wellbore is an entirely diﬀerent phenomenon. Figure 27 discusses the cumulative production of a quad-porosity system versus a dual-porosity system. It can be seen from the plot that, although the OGIP volume for the quad-porosity system is almost double that of the dual-porosity system, the cumulative production from the quad-porosity system is lesser, over a certain period of time. This can be attributed to the fact that the process of desorption and diﬀusion in the quad-porosity system is a relatively slow process, compared to the Darcy ﬂow of gas in a typical dual-porosity system. Also, the ﬂow of gas in these nanopores is a function of the size of the gas molecule, with respect to the pore throat size. This is also evident from Figure 28, which discusses the gas rate from these two systems. It can be seen that the gas rate for the quad-porosity system is less than that for the dual-porosity system. This ﬁgure also shows 19799

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done for both the linear as well as the exponential decreasing permeability cases within the created hydraulic fractures. As seen in Figure 29, the impact of changing half-length in the exponentially decreasing case is much more pronounced than that for the linear case. However, note that, although the diﬀerence in cumulative productions for the exponential cases is large, the cumulative production for the 750-ft half-length case is 12 bcf for the exponential case, whereas, within the same amount of time, the 750-ft half-length linear case produces ∼22 bcf of gas. This indicates the fact that, although more reservoir area is contacted with a higher half-length, the slow production mechanism in the quad-porosity reservoirs overrules the increased contacted area. The linear case also emphasizes the fact that the incremental production added by higher halflengths is not signiﬁcant, although the initial rates are higher than the exponential case. This is in contrast to the dualporosity linear case, as shown in Figure 12, where the incremental production added by progressively increasing the half-length was signiﬁcant. Eﬀect of Primary Fracture Spacing. Fracture spacing aﬀects the productivity of the well in a signiﬁcant manner. Generally, the lower the fracture spacing, the higher the number of fractures, the higher the contact area, and, consequently, the higher the production. In this run, the primary fracture spacing was varied from 500 ft to 700 ft, in the created fractures. As seen in Figure 30 for the exponential case, the higher fracture spacing had an initial higher productivity, but a lesser overall cumulative production of ∼24 bcf, whereas the higher number of fractures that are due to lesser spacing gives a higher cumulative production of ∼26 bcf, albeit with a lesser initial rate. A cost-to-beneﬁt ratio is deemed necessary to increase the number of fractures versus the incremental production obtained from it. In the linear case, it is seen that, although the initial rates for all the spacing had the same slope, the impact of larger reservoir connectivity due to lesser spacing takes over. Eﬀect of Permeability Anisotropy. Along with the very low permeability of 0.0001 mD in the reservoir, the production potential for a well completed in a quad-porosity system becomes complex, if permeability anisotropy is factored in. Simulations were run with a primary fracture spacing of 700 ft, a half-length of 750 ft, and a secondary fracture spacing of 100 ft. As given in Figure 31, the productivity continues to decrease from the base case, as the permeability anisotropy increases. It is seen that, for the linear case, both the initial as well as cumulative production is higher, which signiﬁes that, although

Figure 27. Cumulative production from a quad-porosity system, versus that from a dual-porosity system.

Figure 28. Production rate for a quad-porosity system, versus that for a dual-porosity system.

that, after the initial production (IP) rate, the stabilized rate for the quad-porosity system, which is represented as having achieved a pseudo-steady state, is lesser than that for the dual porosity system. Impact of Half-Length. As mentioned earlier, in the case of very low permeability reservoirs, a longer half-length helps connect with the maximum reservoir area. In the following runs, only a single reservoir with 0.0001 mD was considered. Also, the sensitivity analysis on the quad-orosity systems was

Figure 29. Impact of eﬀective fracture half-length on the quad porosity system (exponential and linear). 19800

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Figure 30. Impact of primary fracture spacing on the quad-porosity system (exponential and linear).

Figure 31. Impact of permeability anisotropy on the quad-porosity system (exponential and linear).

Figure 32. Impact of primary fracture conductivity on the quad-porosity system (exponential and linear).

Figure 33. Impact of secondary fracture spacing on the quad-porosity system (exponential and linear). 19801

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Figure 34. Pressure distribution in the fractures of a quad-porosity simulation model (exponential).

Figure 35. Pressure distribution in the fractures of a quad-porosity simulation model (linear).

drastic drop in ﬂowing potential is unable to overcome the advantage of having a denser network of secondary propped fractures. The phenomenon of adsorption/desorption, as well as diﬀusion, is a function of pressure. The pressure drawdown to which the well is subjected, along with the ﬂow regime that the hydraulic fractures would induce, would ultimately aﬀect the production from such an unconventional type of reservoir. The pressure was studied in the reservoir, as well as in the producing fractures, to investigate its eﬀect on the ultimate productivity of the completed well. While taking into account all the ﬂow and storage processes within the reservoir, the simulation model was run for 30 years, with a primary fracture conductivity of 400 mD ft, a fracture half-length of 750 ft, and primary fracture spacing as 700 ft. Pressure was studied at time steps of 10, 20, and 30 years, for both the cases of linear as well as exponentially decreasing permeability. Figure 34 shows the pressure variation for the exponential case. Because of the exponential decline in permeability in the created fractures, the production potential of the well decreases, because of a smaller drawdown. It is observed that, even after 30 years of production, the pressure within the fractures is considerably higher, and the ﬂow regime is on the verge of becoming elliptical. As seen in Figure 35 for the linear case, the pressure within the fracture was observed to be ∼2200 psi, even after producing for 30 years. The pressure within the fractures is ∼600 psi higher than that when the model was run as a dualporosity model. This diﬀerence in pressures within the fractures can be attributed to the fact that the adsorbed gas on the organic matrix requires a higher drawdown to get desorbed, thereby leading to a lower production rate and lower cumulative production values, the pressure within the fractures have reached a level of 2025 psi, even after 10 years of production. Also, the ﬂow regime at the end of 30 years is

permeability anisotropy plays a role, the lesser pressure drop leads to higher production. Eﬀect of Primary Fracture Conductivity. To investigate the eﬀect of the primary fracture conductivity on the production from quad-porosity shale gas reservoirs, the conductivity values were changed to 4, 40, 100, 200, 300, and 400 mD ft, respectively, to enable comparison with the dual-porosity model runs. Similarly, the fractures were spaced 600 ft apart, with a total stimulated reservoir volume (SRV) of 180 × 106 ft3. The primary fractures length was 750 ft, and the secondary fracture spacing was kept constant at 100 ft. Figure 32 shows that, although the shape of the 4 mD ft curve is the same for both exponential as well as linear, the production is almost double in the linear case. When compared with the dualporosity model results, the cumulative production for the 4 mD ft is lower in the quad-porosity case. Also, along with a higher initial production, the relative impact of increasing the conductivity diminishes with time, which could be attributed to the ﬂow mechanisms, which change as time progresses. Eﬀect of Secondary Fracture Spacing. The denser the network of secondary or natural fractures, the better the induced hydraulic fractures will connect with them, thereby increasing the connectivity to the wellbore. In the case of dualporosity models, it was observed that there was no signiﬁcant diﬀerence when the spacing was reduced, for all three reservoirs considered. However, in the quad-porosity model runs, it is seen that when the spacing is reduced, a higher cumulative production is obtained for the linear case, as shown in Figure 33. A reduction in the spacing means a denser network, where chances of some of the secondary fractures also getting propped are higher. This is due to the combined eﬀect of propped secondary fractures coupled with the gas stored in the additional porosity systems of the reservoir. This is not evident in the case of exponentially decreasing permeability, as the 19802

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(2) Reservoir permeability plays a critical role in the recovery, when it when it is coupled with the created fracture half-length, for a case when permeability in the fracture is exponentially degrading. (3) Increasing the primary fracture conductivity plays a rather insigniﬁcant role beyond a threshold value, from a cumulative production standpoint. This is due to the fact that the incremental increase in cumulative gas production remains constant, even at higher primary conductivity values. This may be due to the fact that, although the ﬂowing capacity of the fractures increases as the conductivity increases, the ability of the matrix to deliver ﬂuids at that speciﬁc drawdown remains the same. (4) The pressure distribution comparison denotes higher pressure drops in the exponentially decreasing permeability case, which leads to a lower overall recovery. This higher pressure drop is to compensate for the rapidly decreasing permeability in the fractures, while allowing the ﬂuids to ﬂow under the imposed drawdown. (5) Fracture complexity or natural fracture spacing plays a critical role in the ultimate productivity of a well. A closer (smaller) secondary fracture spacing, along with a linearly degrading permeability in the created hydraulic fractures, leads to considerably higher initial gas rates and a signiﬁcantly higher recovery. (6) Although it is observed that the productivity from a reservoir with no permeability anisotropy is higher, compared to a reservoir with permeability anisotropy, a higher anisotropy ratio is unlikely to be a critical parameter that aﬀects drainage. (7) Matrix permeability and its interaction with the natural fracture permeability is important for the optimal design of a fracture treatment. If this relation is known, then the impact of a better-designed hydraulic fracture, resulting in higher stimulated reservoir volume (SRV), can be assessed more accurately. (8) Results indicate that a considerable diﬀerence in well productivity and pressure transients is observed when reservoirs are modeled as quad-porosity systems. Also, they emphasize the fact that a correct description of these reservoirs is critical for the assessment of their production potential and, furthermore, for forecasting the corresponding economic scenarios. (9) Because of the multiple subsystems within the quadporosity model of the shale reservoir, the original gas in place (OGIP) volume is signiﬁcantly higher than that in the dualporosity models. (10) When the primary fracture conductivity is varied, it is seen that, for the same case, the cumulative production for the quad-porosity model was lesser than that for the dual-porosity model. This is a result of the ﬂow regimes, which, in turn, impact the pressure drop. (11) The incremental production from a quad-porosity system is higher than the dual-porosity system, when the secondary fracture spacing is reduced. This is due to the better connectivity of the placed hydraulic fractures with the denser natural fractures, thereby leading to more accessibility to gas stored in the kerogen of the organic matter. (12) For densely packed (closely spaced) hydraulic fracture treatments, the productivity of the well is a function of secondary fracture spacing, up to a certain point, beyond which the law of diminishing returns becomes applicable. (13) Both diﬀusion and adsorption have an impact on the well deliverability, as well as the OGIP. However, small changes in both the parameters do not have a signiﬁcant impact.

almost on the verge of becoming pseudo-radial, just before hitting the boundaries. Eﬀect of Diﬀusion and Adsorption Coeﬃcients. The adsorption and diﬀusion processes are incorporated in the model through the diﬀusion coeﬃcients (cm2/s) and Langmuir adsorption coeﬃcient (1/psi). These coeﬃcients were changed, and it was observed that the deliverability of the well, as well as the OGIP, changes. Figure 36 discusses the scenario when the

Figure 36. Eﬀect of diﬀusion coeﬃcient sensitivity.

diﬀusion coeﬃcients were changed by an order of magnitude, and it was observed that the ultimate recovery from a reservoir changes considerably. The change is observed toward the latter part of the recovery cycle, which is consistent with the phenomenon that diﬀusion is a slow process, and the mechanism of gas ﬂow through this process come at the end of the producing cycle. However, note that if the diﬀusion coeﬃcient is not altered by an order of magnitude, the well deliverability changes by a smaller percentage. Table 4 discusses Table 4. Adsorption Coeﬃcient Sensitivity

a

sample

adsorption coeﬃcient (1/psi)

OGIP (Tcf)a

1 2 3 4

0.002 0.004 0.006 0.008

3.2 3.41 3.56 3.62

Tcf = trillions of cubic feet.

how the OGIP values change, when the adsorption coeﬃcient is changed in the model. This table shows that the relative increase in the OGIP plateaus out as the adsorption coeﬃcient is increased, which means there is a limit to the amount of gas that can be adsorbed on the shale surface.

■

CONCLUSIONS After a detailed parametric investigation that was done in this study, the following conclusions are provided. Note that these conclusions are based on the limited number of simulation runs performed on the models. However, the parameters used in this study are representative of the unconventional reservoirs and their common completion practices that are used for fracturetreated horizontal wells. (1) When cumulative gas production is assessed for the case of linear and exponential permeability degradation in the created hydraulic fracture, a signiﬁcant diﬀerence is observed. 19803

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(14) Accuracy of the results increases with an increase in the gridding scheme, but at the cost of computational time, which increases multifold. This increase in computational time will prove detrimental for large-scale shale gas reservoir simulation projects. (15) The linear case of a gradual decrease in permeability within the created fractures leads to the ﬂow regime transitioning toward the pseudo-steady state, consequently increasing the cumulative production. The impact of all critical parameters that can aﬀect the well productivity has been observed in this simulation study. Because of the fact that all shale reservoirs are diﬀerent in their characteristics, a validation of the results of this study against actual production from fracture-treated horizontal wells drilled in shale reservoirs would provide a better insight.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the ﬁnal version of the manuscript. Notes

The authors declare no competing ﬁnancial interest.

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ACKNOWLEDGMENTS The authors would like to thank Dr. Somnath Nandi and Prof. Pramod Ghatage (MIT, Pune) for their assistance and CMG for supporting us in the publication of this work.

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REFERENCES

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Industrial & Engineering Chemistry Research

Article

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dx.doi.org/10.1021/ie501116j | Ind. Eng. Chem. Res. 2014, 53, 19788−19805

Simulation and Mathematical Modeling of Stimulated Shale Gas

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