SPE-173269-MS

A Comprehensive Reservoir Simulator for Unconventional ReservoirsBased on the Fast Marching Method and Diffusive Time of Flight

Yusuke Fujita, Akhil Datta-Gupta, and Michael J. King, Texas A&M University

Copyright 2015, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Reservoir Simulation Symposium held in Houston, Texas, USA, 2325 February 2015.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contentsof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the writtenconsent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations maynot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Modeling of fluid flow in unconventional reservoirs requires accurate characterization of complex flowmechanisms because of the interactions between reservoir rocks, microfractures, and hydraulic fractures.The pore size distribution in shale and tight sand reservoirs typically ranges from nanometers tomicrometers resulting in ultralow permeabilities. In such extremely low permeability reservoirs, desorp-tion and diffusive processes play important roles in addition to heterogeneity-driven convective flows.

For modeling shale and tight oil and gas reservoirs, we can compute the well drainage volumeefficiently using a Fast Marching Method (FMM), and by introducing the concept of Diffusive Time ofFlight (DTOF). Our proposed simulation approach consists of two decoupled steps - drainage volumecalculation and numerical simulation using DTOF as a spatial coordinate. We first calculate the reservoirdrainage volume and the DTOF using the FMM, and then, the numerical simulation is conducted alongthe 1D DTOF coordinate. The approach is analogous to streamline modeling whereby a multidimensionalsimulation is decoupled to a series of 1-D simulation resulting in substantial savings in computation timefor high resolution simulation. However, instead of a convective time of flight, a diffusive time of flightis introduced to model the pressure front propagation.

For modeling physical processes, we propose a triple continua whereby the reservoir is divided intothree different domains: micro-scale pores (hydraulic fractures and microfractures), nano-scale pores(nanoporous networks), and organic matters. The hydraulic fractures/microfractures primarily contributeto the well production, and are affected by rock compaction. The nanoporous networks contain adsorbedgas molecules, and gas flows into fractures by convection and Knudsen diffusion processes. The organicmatters act as the source of gas. Our simulation approach enables high resolution flow characterization ofunconventional reservoirs because of its efficiency and versatility. We demonstrate the power and utilityof our approach using synthetic and field examples

IntroductionPredicting oil and gas production from subsurface permeable media is an important aspect of reservoirdevelopment and management strategy. Subsurface dynamic models often involve several mathematicaland physical assumptions to simplify the description of Earths internal structure and to reduce thedemand of computation time. Analytical solutions (i.e. material balance method, pressure transient

analysis, rate transient analysis) are the most restricted or simplified models that require the reservoir tobe isotropic and homogeneous in most cases. Numerical simulation removes such approximations andlimitations by decomposing a continuous domain into a finite set of discrete counterparts. In the petroleumindustry, reservoir simulation model is traditionally used for constructing a subsurface system associatedwith spatial heterogeneities (i.e. porosity, permeability, water saturation). The simulation outcomes areutilized for the purpose of improving estimation of hydrocarbon reserves, identifying fluid flow andgeological characteristics, and more importantly, optimizing the strategies regarding the field develop-ments. For convection dominated flows, streamline-based flow simulation has been widely recognized asan efficient approach for modeling fluid dynamics in porous media. The principle underlying thestreamline simulation is to decompose the multidimensional transport equations into a series of 1-Dequations along streamlines (Datta-Gupta and King 2007). The evolution of flood fronts and theinteractions between production and injection wells can be easily identified using the concept ofconvective time of flight (CTOF).

A related concept for pressure propagation in porous media has been proposed by Lee (1982) whodefined a radius of investigation as the propagation distance of a peak pressure disturbance for animpulse source or sink. The radius of investigation can be analytically calculated under limitations ofhomogeneous and isotropic reservoirs; however, such analytical solution is not applicable for complexgeometries and heterogeneous media. Datta-Gupta et al. (2011) generalized the concept of radius ofinvestigation to heterogeneous media by introducing a diffusive time of flight (DTOF) which correspondsto the arrival time of a peak pressure response. Using high frequency asymptotic solution to thediffusivity equation, they derived the Eikonal equation for DTOF calculations and pressure frontpropagation in the presence of spatial heterogeneities (Vasco and Datta-Gupta 1999, Vasco et al. 2000,Datta-Gupta and King 2007). This Eikonal equation can be solved very efficiently by using the FastMarching Method (FMM) (Sethian 1996, Sethian 1999). The FMM is a class of front tracking algorithmfor solving the Eikonal equation and is similar to the Dijkstra algorithm (Dijkstra 1959) that finds theshortest path on graphs. The DTOF can be obtained using the FMM without explicit trajectory construc-tion. Well drainage volume at a given time can be calculated by contouring the corresponding DTOF andby summing up the pore volumes inside the contour. Zhang et al (2014) proposed a DTOF-basednumerical simulation associated with the transformation of a fluid transport coordinate from the physical3-D space to the 1-D DTOF space. As in the CTOF applied to the streamline simulation, the DTOFembodies geological heterogeneities and reduces 3-D heterogeneity to a 1-D homogeneous problem alongits coordinate. This dimension reduction results in substantial savings in computational time and allowsfor high resolution simulation of unconventional reservoirs.

Over the past decade, the transport mechanisms in unconventional reservoirs have been widely studiedin order to better understand their characteristics (Kuila et al 2011, Javadpour et al. 2007, Javadpour 2009,Sakhaee-Pour et al. 2012). It has been found that the techniques and mathematical flow models used inconventional reservoirs may not be adequate for unconventional reservoirs (Aguilera 2010, Michel et al.2011, Swami et al. 2012, Arogundade et al. 2012). The fluid flow mechanisms in hydraulically-fracturedshale and tight sand reservoirs are farther complicated by many co-existing physical factors, such as (1)severe geological heterogeneities of the permeable media due to the variation among fractures, inorganicrock matrix, and organic matters, (2) Knudsen diffusion and slippage effects in nano-scale pores, (3)high-velocity turbulent flow in the perforations or hydraulic fractures, (4) adsorption/desorption on thesurface of organic rocks, (5) geomechanical effects in the fractured rocks, and (6) high capillarity inconfined systems. Currently, there is no consensus or standardized approach on the theory and frame-works for modeling the transport behaviors in such complex reservoirs, although there is a growingdemand in the area of unconventional resource evaluation and predictions.

In shale gas reservoirs, the hydrocarbon usually exists in several states in fracture, matrix, and organicmatter. Aguilera et al. (2010) suggest that gas molecules trapped and stored in shale can be divided into

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five different types: (1) gas adsorbed into the Kerogen material, (2) free gas trapped in inorganic matrixporosity, (3) free gas trapped in natural fractures, (4) free gas stored in hydraulic fractures created duringthe stimulation of the shale reservoir, and (5) free gas trapped in a pore network developed within theorganic matter or Kerogen material. Biswas (2011) pointed out that the flow of gas through the fracturenetwork in shale is the consequence of gas desorption and diffusion which transport it through thematrix-fracture interface. Nelson (2009) investigated the pore-throat distributions in sandstones, tightsandstones, and shales using scanning electron microscopy (SEM) and mercury injection. The pore-throatsize (diameter) of conventional sandstones ranges from 2 to 20 m, whereas the pore-throat size of tightsandstones ranges from 20 nm to 1 m. The pore-throat size of shales ranges from 5 to 100 nm, whichis approximately 100 times smaller than that of conventional sandstones.

In such confined situations, nano-scale pores (nanopores) play two important roles for gas flowbehavior (Javadpour et al. 2007). First, for same pore volume, the exposed surface area in nanopores ismuch larger than that in micro-scale pores (micropores). The increase of the exposed surface area resultsin an increase of the volume of adsorbed gases. Suppose that there is a spherical pore covered by organicmaterial. Within this pore, gas molecules are contained in two states which are free gas and adsorbed gas.The volume of free gas compressed in a spherical pore is simply defined by the pore volume, which is4r3/3, where r is the radius of a sphere. In contrast, adsorbed gases are attached to the surface area ofthe pore, which is 4r2. Consequently, the relative importance of the adsorbed gases to free gas is definedby the ratio of the exposed surface area to the volume of free gas, that is 3/r. This implies that the relativeimportance of adsorbed gas is inversely proportional to the size of the pore. Secondly, nano-scale porestructures can cause the violation of the basic assumption behind the usage of the standard Darcys lawbecause of Knudsen diffusion and slippage effects on the pore surface. As described above, the relativeimportance of pore surface area to pore volume is inversely proportional to the pore size. This means thatthe frequency of slippage and collision on the pore surface will increase as the pore size becomes smaller.The gas molecules tend to collide on the pore walls and slip at the wall surface instead of having theno-slip Hagen-Poiseuille flow.

Klinkenberg (1941) recognized gas slippage in porous media and observed that at very low pressure,the actual flow rate significantly deviates from the one predicted by the conventional laminer flowapproximation (Darcys law), a phenomena called Klinkenberg effect. He proposed the followingcorrection to gas permeability accounting for its pressure dependency due to slippage on the pore surface.

(1)

where k denotes the permeability measured with non-slip boundary condition (Darcys permeability)and b represents the correction factor for slippage (slippage factor). Over the decades, many authors havemeasured the apparent permeability and defined the Klinkenberg slippage factor based on the observationsand theoretical works (Jones et al. 1980, Sampath et al. 1982, Ertekin et al. 1986, Florence et al. 2007,Javadpour et al. 2007, Civian 2010, Michel et al. 2011, Swami et al. 2012). Contrary to the conventionalunderstanding that the Klinkenberg effect has an impact on the fluid flow at low pressure only, severalauthors observed that it affects the flow behavior for smaller pore throat size and for low flowingbottom-hole pressures as well.

Transformation of Transport Domain along Diffusive Time of Flight (DTOF)Our proposed simulation approach relies on transforming the 3-D pressure diffusion equation in hetero-geneous media into an equivalent 1-D homogeneous equation using the DTOF as the spatial coordinate.Thus, the approach can be viewed as analogous to the streamline-based simulation which has beensuccessfully used for modeling convective processes. However, unlike the streamline approach, noexplicit trajectory constructions are necessary here. Instead, the DTOF is obtained by solving the Eikonal

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equation on the grid using the Fast Marching Method (FMM). The mathematical details of the derivationare given in Appendix A.

We define a coordinate transformation as follows.

(2)

where is the Diffusive Time of Flight (DTOF) acting as a spatial coordinate and w() represents thederivative of well drainage volume with respect to the DTOF.

(3)

where Vp denotes the well drainage pore volumes. To start with, the DTOF can be calculated efficientlyby using the Fast Marching Method (FMM). The FMM is a class of front tracking methods, whichcomputes the DTOF sequentially from small value to large values using a single-pass algorithm. Thedrainage volumes Vp() are successively calculated as a function of by summing up the pore volumesinside a specific -contour. We then tabulate the Vp() as a function of to compute the w() function inEq. (3).

On the basis of the transformed 1-D coordinate as given in the Appendix A, the pressure equation canbe written as

(4)

Notice that Eq. (4) is a 1-D transport equation that fully embeds the geological heterogeneities (i.e.porosity, permeability) on the -coordinates and can be numerically solved using a finite differencescheme. This simulation approach is quite similar to that of streamline approach and we solve the pressureequation along 1 -D DTOF coordinate instead of solving the saturation equation along the streamlineconvective time of flight.

Dual-Porosity ModelingFractured reservoirs are characterized by the presence of two distinct porous systems fractured porousnetworks and fine grained matrix blocks. In naturally fractured reservoirs or hydraulically fractured wells,the mass exchange between matrix and fracture is an important component due to their geologicalcharacteristics. The fracture network is highly conductive, but can store very little fluid because of its verylow porosity, while the matrix system has low conductivity and large storage capacity relative to thefracture. The concept of dual-porosity single-permeability (DPSP) model is that the two over-lappingcontinua, fracture system and matrix system, coexist and interact with each other (Barenblatt et al. 1960,Warren and Root 1963, Kazemi 1979). The fluid transport equation in the fracture system is given by anordinary porous medium with an additional connection to the matrix block, whereas the matrix blocks actonly as a source to the fracture system. The advantage of the dual-porosity modeling is that this approachis computationally inexpensive and structurally simplified compared to the Discrete Fracture Network(DFN) method which incorporates all fractures in various locations on the basis of complex fracturegeometries. The mass balance equation in the fractured system is written by general mass balance equationwith the addition of a matrix-fracture mass exchange term.

(5)

where f represents the fracture porosity, kf denotes the fracture permeability, and represents thematrix-fracture transfer function. The sink or source term qf is imposed on the inner boundary of the

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fracture domain. In Eq. (5), the transfer function is given by the Darcy equation-like form (Kazemi et al.1976).

(6)

where denotes the shape factor (fracture density) that defines the connectivity between the matrixblock and the surrounding fracture network. It is reasonable to assume that the matrix-fracture transfer isalways governed by the matrix permeability (km) because of its low conductivity. Based on Eqs. (5) and(6), we obtain the following fracture equation.

(7)

The matrix flow equation can be written as follows.

(8)

where m and km represent the matrix porosity and permeability, respectively. On the matrix coordinatesystem, the both inner and outer boundary conditions are imposed as no-flow boundary, thus the well termis absent in Eq. (8). The matrix system only plays as an additional source to the fracture system drivenby the differential pressure between fracture and matrix blocks. Applying the coordinate transformationon the fracture flow equation, we can obtain the DTOF-based fracture transport equation.

(9)

where f,init is the initial fracture porosity. The matrix equation can be written as follows.

(10)

For the application of the dual-porosity modeling using the DTOF-based 1-D flow simulation, we makethe following assumptions.

The FMM and DTOF calculation only involve the fracture coordinate system. This means that theFMM calculates the pressure front propagation on the fracture domain only and takes into accountthe fracture heterogeneities (kf and f) without consideration of the matrix system. Thus, andw() are obtained on the fracture domain only. This is consistent with the dual porosity formalismthat assumes that flow primarily occurs in the fracture system.

The matrix properties (i.e. porosity, permeability, shape factor) are assumed to be spatiallyuniform. The geological heterogeneities of the matrix system are not considered in the DTOFcalculation.

Above assumptions will be reasonably held when there is a high contrast between the fracture andmatrix permeabiities. Thus, the pressure front propagates primarily in the fracture network and the matrixserves as a fluid source to the fracture system.

Triple-Continuum Approach for Shale Gas ModelingBased on the pore size variation in naturally- or hydraulically-fractured shale reservoirs, the gas transportdomain are divided into three distinct systems: (1) natural and hydraulic fracture networks (macro-scaleporous media), (2) nanopores in inorganic matrix and/or in organic matters (nano-scale porous media),and (3) organic bulk or Kerogen (not containing pore space). Fig. 1 illustrates a typical gas flow processand physical mechanisms encountered in fractured shale reservoirs. The reservoir gas is producedprimarily through the fracture networks, and the other two systems, nanopores and Kerogen content act

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as additional gas source to the fracture system.There are three distinct sources of gas compressedin the pore spaces of the three domains: the free gasin fracture and nanopores, adsorbed gas on nano-pore surface, and dissolved gas in the organic matterbulk.

For each coordinate system, the shale gas physicsare incorporated as follows.

Primary coordinate: Fracture network

a. Fracture is the primary coordinate forfluid flow and production.

b. Flow is governed by convective trans-port.

c. Fractures are affected by rock compaction from geomechanical effect.

Secondary coordinate: Nanopores in organic- and/or inorganic-rocks

a. Nanopore contains two types of compressed gases - free and adsorbed gas.b. Flow is governed by convection and Knudsen diffusion.

Tertiary coordinate: Organic bulk (Kerogen)

a. Kerogen is the hydrocarbon source and contains the dissolved gas.b. Dissolved gas diffuses to nanopores driven by concentration gradient.

The triple-continuum approach provides a generalized framework that is able to account for all dominantphysical effects and processes that govern flow in shale gas reservoirs. Our numerical implementationinvolves slippage and Knudsen diffusion effects, rock compaction in fractures, adsorption/diffusion on theshale surface, and gas diffusion from the Kerogen content. Fig. 2 illustrates the gas transport processesbased on the triple-continuum model and the connectivity among the fracture, nanopores, and Kerogensystems. The approach is similar to the DPSP model whereby a tertiary coordinate (organic matter) isincorporated in addition to the matrix-fracture coordinate systems. The inter-coordinate mass transferbetween the fracture and nanopore is governed by the Convection and Knudsen diffusion flow modeledusing an apparent permeability.

The mass transfer between the nanopores and Kerogen is governed by diffusive transport driven by thegas concentration difference between the nanopores (adsorbed gas) and Kerogen bulk (dissolved gas), asgiven by the Ficks law of diffusion.

(11)

where is the shape factor, g,sc is the surface gas density, Dc is the gas diffusion coefficient, Ck is gasconcentration dissolved in the Kerogen, and Cm is gas concentration adsorbed on the surface of nanopores.The adsorbed gas concentration on the nanopore surface Cm is given by the Langmuir isotherm model(Langmuir 1916).

(12)

where VL is Langmuir volume and PL is Langmuir pressure. Notice that the Langmuir volume VL hasthe units of scf/rcf. This is obtained from the bulk rock density b (gm/cc) and the adsorbed gas contentVm (scf/ton).

Figure 1Gas flow process among fracture, nanopore (matrix), andorganic matters.

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(13)

Mengal et al. (2011) suggested that the approximate values of b, Vm, and PL in the Barnett shale are2.38 (gm/cc), 96 (scf/ton), and 650 (psia), respectively. Hence, the Langmuir volume VL is expected to beabout 7.13 (scf/rcf). Fig. 3 illustrates the gas flow process from organic matter bulk to nanopores. At staticcondition, the Kerogen gas concentration is in equilibrium with the adsorbed gas concentration. After thewell starts production and results in pressure depletion in the nanopore system, the equilibrium conditionis disrupted due to desorption of the adsorbed gas molecules. The concentration imbalance causes thedissolved gas in the Kerogen to diffuse through the Kerogen-nanopore interface, and also additional gasmolecules start to be adsorbed on the pore surface.

The mass balance in the fracture system is given in Eq. (7). We now incorporate additional physics suchas rock compaction and Knudsen diffusion. The gas slippage and Knudsen diffusion are simply incor-porated by using an apparent permeability kapp in the matrix-fracture transfer instead of using the ordinarymatrix permeability km. The rock compaction is accounted for by applying porosity and permeabilitymultipliers.

(14)

whereM andMk represent the multipliers for porosity and permeability, respectively, and FM denotesthe shape factor of fracture-nanopore connectivity (fracture density). The mathematical formulation of theapparent permeability is described in Appendix B. In Eq. (14), the multipliers of permeability andporosity, Mk and M are given by the rock compaction table as a function of pressure instead of aconventional exponential rock compressibility function such as

(15)

Applying the coordinate transformation into the 1-D -coordinate, we obtain the 1-D fracture equationalong the -coordinate.

Figure 2Illustration of the triple-continuum approach.

Figure 3Gas diffusion from organic matter to nanopore.

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(16)

For the nanopore system, the mass balance equation is written as

(17)

where g,sc represents the gas density at standard conditions (14.7 psia and 60 F), and MK denotes theshape factor of nanopore-Kerogen connectivity (density of nanopores in the organic matter). Theaccumulation term in Eq. (17) contains the mass of two states of gas which are free gas compressed withinthe pore and the adsorbed gas compressed on the pore surface. The first term in the right hand side of Eq.(17) represents the mass transfer between fracture and nanopore given by the convection-Knudsendiffusion flow, and the second term represents the mass transfer term between nanopore and Kerogengiven by diffusion.

For the Kerogen system, the mass balance equation is written as follows.

(18)

This completes the mathematical formulation of the DTOF-based triple-continuum model for shale gasreservoirs. The fluid flow is governed by Eqs. (16) - (18) and the corresponding primary unknowns to besolved for are the fracture pressure (Pf), matrix pressure (Pm), and Kerogen dissolved gas concentration(Ck).

Simulation of Dual-Porosity Model (Naturally Fractured Gas Well)A dual-porosity modeling is an efficient way to simplify the geological description of the complexfractured reservoirs. In this example, we demonstrate the naturally fractured gas reservoir model with avertically completed well. The reservoir size is 1,990 ft, 1,990 ft, and 500 ft along x, y, and z directions,respectively, and the model consists of a total of 19919910 grids. The first 5 layers represent thematrix system and the other 5 layers are the fracture system. These two distinct systems are connected bythe convective transfer function. The initial reservoir pressure is 5,470 psi at static conditions. Fig. 4

Figure 4Spatial distributions of fracture permeability and porosity. (a) x-permeability, (b) y-permeability, (c) z-permeability, and (d) porosity.

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shows the permeability and porosity distributions in the fracture coordinate. The fracture permeability isranging from 0.32 to 4.93 md in x direction, 0.32 to 4.98 md in y direction, and 0.034 to 0.634 md in zdirection. The fracture porosity ranges from 0.97 to 12 %.

In the matrix system, the permeability and porosity are assumed to be uniform (1104 md and 10 %).The matrix rocks have a low permeability but large storage capacity relative to the fracture system. Thewell is vertically placed in the center of the model and completed through the five fracture layers. Thegeomechanical rock compaction is included to account for the effects of the rock deformation andcompaction in the fractured space. In the conventional stratified sandstone reservoirs, geomechanicaleffects on porosity and permeability are generally small and usually neglected. However, in fracturedreservoirs, the geomechanical effects can be relatively large and may have a significant impact particularlyin near-wellbore region because of the large pressure drawdown. The resulting rock compaction cansignificantly alter the flow conductivity and the fluid storage capacity in the fracture space. In this model,the rock compaction is incorporated as a function of pressure as shown in Fig. 5. For this example, thefracture permeability (multiplier) changes nonlinearly as a function of pressure. The porosity (multiplier)is compressed linearly instead of the conventional exponential function.

Fig. 6 shows the numerical simulation results for two different well constraints cases. Fig. 6 (a) showsthe prediction of the gas production rate with a constant bottom-hole pressure (4,000 psi). In this figure,the case 1 represents the simulation results without the rock compaction effect, and case2 represents

Figure 5Rock compaction table for fracture system. The brown line represents permeability multiplier and the red line shows porosity multiplier.

Figure 6Simulation results of the (a) gas production rate with constant BHP and (b) BHP with constant gas production. The plots are the resultsof commercial simulator and the lines are the results of DTOF-based simulation. The case 1 represents the results of no rock compaction model,and case2 denotes the results of rock compaction model.

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the simulation result with the rock compaction effect. Fig. 6 (b) is the predicted bottom-hole pressure witha constant gas rate constraint (1,000 Mscf/day). It is obvious that the fracture rock compaction has asignificant impact on the behavior of the production rate and the bottom-hole pressure. Under theinfluence of rock compaction, the rate and bottom-hole pressure rapidly decline because the fractureconductivity and storage capacity are dramatically decreased as the fracture pressure decreases. Forcomparison purposes, we have superimposed the results from a commercial finite difference simulator(ECLIPSE). A very close agreement can be observed.

Simulation of Triple-Continuum Model (Shale Gas Well)The triple-continuum approach allows us to simulate unconventional shale gas reservoirs taking intoaccount the dominant physical mechanisms and flow characteristics. The traditional dual-porosity modelis not sufficient to explain the existence of organic matters. In the triple-continuum approach, wedecompose the reservoir domain into three distinct subdomains (1) natural and hydraulic fracturenetworks (micropores), (2) inorganic matrix (nanopores), and (3) organic matter bulk (Kerogen). Itenables us to incorporate various types of physical phenomena in each subdomain, i.e. geomechanicalrock compaction, slippage and Knudsen diffusion effect in nanopores, and gas desorption and diffusionfrom organic matters. In this example, we present a synthetic shale gas well model stimulated bymultistage hydraulic fracturing. The model size is 2,000 ft, 4,000 ft, and 150 ft along x, y, and z directions,respectively, and the model consists of the 20040090 grids. The first 30 layers comprise the fracturedomain, the next 30 layers comprise the nanoporous domain, and the remaining 30 layers represent theKerogen bulk domain. The initial reservoir pressure is 1,500 psi at equilibrium. Fig. 7 shows the fracturepermeability distribution that ranges from 1104 to 0.15 md in the horizontal direction and from1106 to 1.5103 md in the vertical direction. The well is horizontally placed along y direction andcompleted with 12-stage hydraulic fractures.

The horizontal length of the well is approximately 3,300 ft. The average fracture half-length and heightare 200 ft and 100 ft respectively. In the hydraulically fractured grids, the permeability is increased by afactor of 10,000, and ranges from 1.3 to 1,532 md in horizontal direction and from 0.013 to 15.3 md invertical direction. The reservoir properties are summarized in Table 1. The apparent matrix permeabilityis calculated for each grid for each time-step by using pressure, temperature, gas properties, and nanoporesize as shown in Eq. (B.15); thus, the matrix permeability is not explicitly input in the model. In Table1, there are two shape factors listed viz. fracture-matrix shape factor and Kerogen-nanopore shape factor.The latter represents the density of the nanopores in the organic matter.

The adsorption/desorption processes are modeled by the Langmuir isotherm. This isotherm model hasseveral assumptions.

The adsorption equilibrium is instantaneous (only relates to pressure, not to time).

Figure 73D Heterogeneous Horizontal Well Model with 12-stage Hydraulic Fractures.

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The adsorption layer forms only a mono-layer at the maximum.

There are no phase transitions and surfacediffusion in the adsorbed layer.

This isotherm is characterized by two parameters Langmuir pressure and Langmuir volume. Lang-muir pressure represents the pressure at which onehalf of the Langmuir volume can be adsorbed.Langmuir volume is defined as the maximumamount of gas that can be adsorbed to the surface ofnanopores at infinite pressure. In this model, theBarnett shale gas data are used (Mengal et al. 2011)as shown in Fig. 8.

Fig. 9 illustrates the Knudsen number as a function of pressure and pore radius at given temperatureand fluid compositions. The Knudsen number is calculated by Eq. (B.1) as a function of mean-free-pathand pore size. When the pore size is smaller than 100nm, the flow regime falls in the slip or the transitionflows at low pressure conditions. In contrast, the viscous flow regime is dominant in pore sizes ofmicrometers (1,000 nm~).

Fig. 10 shows the changes of apparent gas permeability ((a)) and the permeability ratio ((b)) as afunction of pressure and pore size at given reservoir temperature and fluid compositions.

We conduct the simulation of the triple-continuum model with fracture, nanopore, and Kerogendomains. We first predict 10 years of gas production rate with constant bottom-hole pressure constraint(500 psia) as shown in Fig. 11. The simulation is conducted based on several pore size conditions (100,50, 20, 10, and 5 nm cases). The early-time production behavior significantly varies with the pore sizeselection which controls the matrix-fracture conductivity. The decline rate of the gas production isconsiderably higher in smaller pore condition and less in large pores, because the fracture obtains littleassistance from less-conductive smaller nanopores. For this example, after 10 days from the firstproduction, the pore size appears to make no difference on gas production rate.

Fig 12 shows the volume transfer rate between fracture and matrix ((a)) and between matrix andorganic matter ((b)). The matrix-fracture transfer responds quickly to the pressure depletion due to itsconvective nature. The diffusive flow between nanopore and Kerogen proceeds slowly due to its diffusivenature. Finally, we predict the well bottom-hole pressure under the constant gas rate constraints as shownin Fig 13. Warren and Root (1963) suggested that dual-porosity reservoirs typically exhibit two distinctparallel pressure responses (or pseudo-pressure response). Our model shows similar behavior as the

Figure 8Langmuir isotherm model (Barnett shale gas case).

Table 1Reservoir properties (3D triple-continuum model)

Reservoir properties

Initial pressure (psia) 1,500

Temperature (degF) 250

Matrix properties

Porosity (fraction) 0.1

Rock compressibility (1/psi) 1 106

Fracture-matrix shale factor (1/ft2) 0.15

Langmuir pressure (psi) 650

Langmuir volume (scf/rcf) 7.13

Kerogen properties

Diffusion coefficient (ft/day) 0.02

Kerogen-matrix shape factor (1/ft2) 0.15

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dualporosity model as shown in Fig 13 (a). The firststraight line represents the fracture-dominant behav-ior, and the second straight line represents the be-havior of the dual system (fractures nanopores).In the 100 nm case (blue line), the first straight linedoes not explicitly appear because the matrix instan-taneously responds to the pressure drawdown due tothe high conductivity between the fracture andnanopores. In contrast, the result of the 5 nm (purpleline) shows an extended first straight line because ofits low conductivity and a slow response of thematrix system. The pressure derivative plots areshown in Fig. 13 (b). The responses of the matrixand Kerogen systems are also examined by plottingthe volume transfer rate between the fracture andmatrix and between the matrix and Kerogen (Fig. 14). An instantaneous response can be seen in the 100nm case (blue line), and then the smaller pores are gradually activated as the pressure drawdown proceedsinto the matrix system (Fig. 14 (a)). Notice that the steady-state matrix-fracture transfer rate (the rate after

Figure 11Gas production rates with different pore size conditions (10years of simulation).

Figure 9Knudsen number as a function of pressure and pore size (T 250 F).

Figure 10Slippage and Knudsen diffusion effects. (a) Apparent permeability and (b) Permeability ratio.

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Figure 12Inter-coordinate fluid transfer between (a) matrix and fracture, and (b) Kerogen and matrix.

Figure 13Simulation results of the (a) bottom-hole pressures and (b) its derivatives

Figure 14Inter-coordinate fluid transfer between (a) matrix and fracture, and (b) Kerogen and matrix.

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10 days) is insensitive to pore sizes. The Kerogen system transfer is more pronounced in the late timeperiod and the transfer rate is independent of pore size variations (Fig. 14 (b)).

ConclusionsWe have proposed a novel DTOF-based flow simulation for modeling shale gas reservoirs. Our approachis computationally efficient and is well-suited for high resolution flow simulation because of severalinherent advantages. First, the FMM calculations generally take seconds because each cell needs to bevisited essentially only once (single-pass algorithm) and do not require any global matrix solution. Thus,the computation of the DTOF can be carried out very rapidly starting from the source point to the outerboundary. Second, the multidimensional transport equation is reduced to a 1-D transport equation basedon a coordinate transformation from the physical space to the DTOF coordinate. As a result, duringnumerical computation, the sizes of the Jacobian matrix and unknown vector are dramatically reduced foreach iteration step, for example, from millions to several hundreds elements, especially for high resolutionflow simulation. Third, in the transformed 1-D space, the geological heterogeneities are fully embeddedin the drainage volume and the -coordinate. The complex reservoir geometries (i.e. corner point grid) arealso transformed to a relatively simple 1-D grid coordinate. Furthermore, the drainage volume is asmoothly varying property increasing monotonically from small to large along the 1-D coordinate. Asa result, the grid complexities and spatial heterogeneities are considerably simplified in the DTOFformulation. All these lead to substantial savings in computation time.

We have also extended the DTOF formulation to multi-continuum modeling. Specifically, we proposeda generalized framework for the modeling of shale gas reservoirs using a triple-continuum approach. Theflow characteristics of unconventional gas reservoirs were comprehensively investigated by accountingfor all dominant physical mechanisms including the Knudsen diffusion and slippage effects, adsorption/diffusion in nanopore surfaces, rock compaction in fractures due to geomechanical effects, and gasdiffusion from the Kerogen content. The results of the numerical simulation show that the apparentpermeability, which governs the mass transfer between the fracture and nanopores, can be significant inlow pressure conditions. The apparent permeability also shows high dependency on the matrix poreradius. The matrix-fracture interaction has a significant impact on the early time transient behavior, whilethe Kerogen system is activated slowly and provides sustainable gas production at late times because ofits diffusive nature.

Nomenclatures

A Surface areab Slippage factorB Formation volume factorCk Dissolved gas concentration in KerogenCm Adsorbed gas concentration on matrix surfacect Total compressibilityDc Gas diffusion coefficientDm Effective Knudsen diffusion coefficientDk Knudsen diffusion coefficientF Dimensionless slippage factorJ FluxKn Knudsen numberk Absolute permeabilitykapp Apparent permeabilityk Darcy permeability

14 SPE-173269-MS

Mk Permeability multiplierMw Molecular weightM Porosity MultiplierP PressurePL Langmuir pressureq Production rate per unit volumeR Universal gas constantr Pore radiusT Temperatureu Fluid velocityVL Langmuir volumeVp Drainage volumew() Derivative of drainage pore volume with respect to z Compressibility factordiff Diffusivity Matrix-Fracture transfer function Pore geometric factor Fluid viscosity Fluid density Porosity Diffusive time of flight

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Arogundate, O., and Sohrabi, M. 2012. A Review of Resent Developments and Challenges in ShaleGas Recovery. Paper SPE-160869 presented at the SPE Saudi Arabia Section Technical Sympo-sium and Exhibition, Al-Khobar, Saudi Arabia, 8-11 April.

Barenblatt, G.E., Zheltov, I.P., and Kochina, I.N. 1960. Basic Concepts in the Theory of Seepage ofHomogeneous Liquids in Fissured Rocks. Journal of Applied Mathematics and Mechanics 24(5):12861303.

Biswas, D. 2011. Shale Gas Predictive Model (SGPM) An Alternative Approach to Predict ShaleGas Production. Paper SPE-148491 presented at the Eastern Regional Meeting, Columbus, Ohio,USA, 17-19 August.

Blair, P.M. 1964. Calculation of Oil Displacement by Countercurrent Water Imbibition. Society ofPetroleum Engineers Journal 4(3): 195202.

Brown, G.P., Dinardo, A., Cheng, G.K., and Sherwood, T.K. 1946. The Flow of Gases in Pipes at LowPressures. Journal of Applied Physics 17(10): 802813.

Civian, F. 2010. Effective Correlation of Apparent Gas Permeability in Tight Porous Media. Transportin Porous Media 82(2): 375384.

Civian, F., Rai, C.S., and Sondergeld, C.H. 2011. Shale-Gas Permeability and Diffusivity Inferred byImproved Formulation of Relevant Retention and Transport Mechanisms. Transport in PorousMedia 86(3): 925944.

Datta-Gupta, A., Kulkarni, K.N., Yoon, S., and Vasco, D.W. 2001. Streamlines, Ray Tracing andProduction Tomography: Generalization to Compressible Flow. Petroleum Geoscience 7: 7586.

Datta-Gupta, A., and King, M.J. 2007. Streamline Simulation: Theory and Practice. Society ofPetroleum Engineers, Richardson, Texas.

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Datta-Gupta, A., Xie, J., Gupta, N., King, M.J., and Lee W.J. 2011. Radius of Investigation and itsGeneralization to Unconventional Reservoirs. Journal of Petroleum Technology 63(7): 5255.

Dean, R.H., and Lo, L.L. 1988. Simulations of Naturally Fractured Reservoirs. SPE ReservoirEngineering 3(2): 638648.

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Ertekin, T., King, G.R., and Schwerer, F.C. 1986. Dynamic Gas Slippage: A Unique Dual-MechanismApproach to the Flow of Gas in Tight Formations. SPE Formation Evaluation 1(1): 4352.

Florence, F.A., Rushing, J.A., Newsham, K.E., and Blasingame, T.A. 2007. Improved PermeabilityPrediction Relations for Low-Permeability Sands. Paper SPE-107054 presented at the SPE RockyMountain Oil & Gas Technology Symposium, Denver, Colorado, USA, 16-18 April.

Grathwohl, P. 1998. Diffusion in Natural Porous Media: Contaminant Transport, Sorption/Desorp-tion and Dissolution Kinetics. Springer Publishers, UK.

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SPE-173269-MS 17

Appendix A

Derivation of the 1-D DTOF-based Transport Equation

Zhang et al. (2012) presented the mathematical derivation of the 1-D coordinate transformation using the divergence theory.In this section, we derive the same equation by using a simplified approach starting from the general diffusivity equation.

To start with, a single-phase mass balance equation is given by the following diffusivity equation.

(A.1)

where is the Darcy velocity with an anisotropic permeability .

(A.2)

Suppose the flow domain is given by the closed finite permeable media with a source or sink point (inner boundary). Whenthe fluid flow takes place only by the convective transport, the fluid particle moves along the pressure gradient direction.Furthermore, in the primary depletion stage, the evolution of the fluid flow proceeds outwardly from the sink/source point andis given by the gradient of the series of the non-overlapping contour surfaces (pressure contours).

The direction of the convective fluid transport is identical to the gradient direction of the contour surface, ps, as shownin Fig. A.1. Therefore, the flux coordinate is transformed from the physical space to the series of surface contours.

(A.3)

where A(s) is the surface area of the contour and q is the total flux across the surface contour. Substituting Eq. (A.3) intoEq. (A.1), we obtain the diffusivity equation as follows.

(A.4)

On the contour surface, the total flux is given by

(A.5)

where denotes a normal vector. Consider the drainage pore volume inside the contour surface.

(A.6)

Differentiating Eq. (A.6), we obtain the surface area of the contour.

(A.7)

or, we have

Figure A.1Pressure contour map and fluid path along the pressure difference.

18 SPE-173269-MS

(A.8)

Inserting Eqs. (A.2) and (A.8) into Eq. (A.5), we obtain

(A.9)

Now we approximate the trajectory s by the trajectory of the pressure front propagation, . The gradient direction of thesurface contour, ps is replaced by the gradient direction of the diffusive time of flight, p.

(A.10)

From the Eikonal equation, we have the following relationship.

(A.11)

Notice that the Fast Marching Method is performed to solve for the DTOF on the initial reservoir state (i.e. the porosity,total compressibility, and fluid viscosity at the initial condition). Substituting Eq. (A.11) into Eq. (A.10), we obtain the fluxequation along .

(A.12)

Now, we define the w-function as follows.

(A.13)

Substituting Eqs. (A.7) and (A.12) into Eq. (A.3), we define the coordinate transformation as follows.

(A.14)

Notice that, at inner boundary ( w), the well production rate is given by Eq. (A.12).

(A.15)

The surface rate is obtained by dividing Eq. (A.15) by the formation volume factor. The drainage volume is assumed tobe zero at the wellbore. The fluid viscosity is given by upstream weighting.

SPE-173269-MS 19

Appendix B

Use of Apparent Permeability as Shale Permeability

Knudsen number is a widely-recognized parameter for system identification that determines a flow regime at given flowcondition and fluid properties. This dimensionless parameter is defined by the ratio of a mean-free-path to a physical length(usually pore radius) (Civan et al. 2011).

(B.1)

The mean-free-path is the average distance travelled by a moving molecule between successive collisions on pore wallor with another molecule. Knudsen number, which is given as the collision distance scaled by the pore radius, indicates thefrequency of molecular-molecular and molecular-wall collisions when a molecule travels unit length. For ideal gas, is definedas follows (Civan et al. 2011).

(B.2)

where T is temperature, R is a universal gas constant, and Mw is molecular weight. For real gas situations, themean-free-path is corrected by multiplying by the compressibility factor z (Swami et al. 2012).

(B.3)

Schaaf and Chambre (1961) identified five flow regimes on the basis of Knudsen number as shown in Table B.1.The Darcys law (viscous flow) is valid only in a fairly low Knudsen number range (Kn 0.001), whereas the non-slip

boundary condition is broken as Knudsen number becomes higher (Kn 0.001). In the slip flow regime (0.001 Kn 0.1),the collisions between gas molecule and pore surface become more pronounced and consequently the linear flow approximationis broken down. In the transition flow (0.1 Kn 10), the additional pore surface effect plays an important role on the fluidflow, that is Knudsen diffusion becomes important. Knudsen diffusion represents the diffusive flow driven by the collisionsbetween the molecule and pore surface, which is different from the molecular diffusion driven by the molecule-moleculecollision (Fig.B.1), and it occurs in porous media where the physical length r of the fluid flow path approaches comparable orsmaller than the mean-free-path . The transition flow is identified as the combination of flow contributed by convection,slippage, and Knudsen diffusion. Some authors (Javadpour et al. 2007, Swami et al. 2012) pointed out that most of shales andmany tight gas reservoirs fall in the transition flow regime. At a very high Knudsen number (Kn 10), the fluid flow is mainlydriven by Knudsen diffusion flow, not by the convective drive. This regime is not frequently encountered in shales and tightgas plays. Knudsen flow is usually modeled by using the molecular simulation instead of the continuum flow approach.

Table B.1Flow regime identification based on Knudsen number

Knudsen Number, Kn Flow Regime

Kn 0.001 Viscous flow

0.001 Kn 0.1 Slip flow

0.1 Kn 10 Transition flow

Kn 10 Knudsen flow

Figure B.1Diffusion types in porous media.

20 SPE-173269-MS

On the basis of the above flow regime identification, we focus on the fluid flow modeling for the slip and transition flowregimes. The target Knudsen number is 0.001 Kn10 (Table B.1). During the slip and transition flow regimes, the total massflux JT in nanoporous media is governed by the Convection-Knudsen diffusion equation in addition to the slip surface boundarycondition.

(B.4)

where JC is the convective mass flux, given by the Darcys equation with the correction for slippage effects.

(B.5)

where F is the slippage factor. Brown et al. (1946) proposed a theoretical dimensionless slippage factor for slip velocityin a capillary tube.

(B.6)

where is the tangential momentum accommodation coefficient or, simply, the part of gas molecules reflected diffuselyfrom the pore wall relative to specular reflection. The value of varies theoretically in the range from 0 to 1, depending uponthe pore surface smoothness, gas type, temperature, and pressure (Javadpour et al. 2007). In Eq. (B.4), the Knudsen diffusionmass flux JKn is given by the Ficks first law with the gas concentration difference.

(B.7)

where Dm is the effective Knudsen diffusion coefficient. For convenience, we use the gas density difference pg in Eq.(B.7) instead of using the gas concentration differencepC. Grathwohl (1998) suggested that the Knudsen diffusion coefficientbe scaled based on the matrix porosity and surface tortuosity due to the complexity of the geometry of the porous medianetwork. The approach is to consider the porous network as consisting of a certain percentage of open pores (matrix porosity)and having a degree of interconnection resulting in the actual path of the porous media longer than the straight path (tortuosity).

(B.8)

where m represents the matrix porosity and denotes the tortuosity, and Dk represents the Knudsen diffusion coefficientin a long smooth straight tube given by the function of mean molecular speed (Igwe 1987).

(B.9)

Notice that the Knudsen diffusion coefficient is proportional to the pore radius and temperature. From Eqs. (B.4), (B.5),and (B.7), the Convection-Knudsen diffusion equation for nanoporous media is written as

(B.10)

where cg is gas compressibility. In Eq. (B.10), we define the apparent permeability as follows (Javadpour et al. 2007, Swamiet al. 2012).

(B.11)

Finally, the total mass flux (Eq. (B.4)) is written by the same equation form as the Darcys law on the basis of apparentpermeability.

(B.12)

The Hagen-Poiseuille equation, which assumes laminar flow in a cylindrical pipe with non-slip side boundary, gives atheoretical value for the Darcy permeability k.

(B.13)

If the representative shape of pore is not a straight cylinder, the Darcy permeability is also corrected based on the porosityand pore geometry (tortuosity) as shown in Eq. (B.8) (Grathwohl 1998).

SPE-173269-MS 21

(B.14)

Substituting Eqs. (B.6), (B.8), (B.9) and (B.14) into Eq. (B.11), the apparent permeability is defined as follows.

(B.15)

The apparent permeability is proportional to the pore radius and given by the pressure, temperature, gas properties, and poreradius. The ratio of the apparent permeability to the Darcy permeability (Permeability Ratio) is obtained by dividing Eq.(B.15) by Eq. (B.14).

(B.16)

In the above expression, we see that the apparent permeability is comprised of three dimensionless components. The firstterm in the right hand side represents the relative importance of the viscous flow (convective permeability), that is scaled to1. The second and third terms indicate the importance of the slippage and Knudsen diffusion relative to the viscous flow,respectively. If the permeability ratio is close to 1, the pore surface effects have little significance on the fluid flow (purelyviscous flow). If the permeability ratio is larger than 1, the fluid flow behavior is affected by the slippage and Knudsendiffusion. Notice that the permeability ratio is proportional to pressure and inversely proportional to the pore size. Conse-quently, diffusion effects become considerable at low pressure conditions and in small pores.

22 SPE-173269-MS

A Comprehensive Reservoir Simulator for Unconventional Reservoirs Based on the Fast Marching Met ...IntroductionTransformation of Transport Domain along Diffusive Time of Flight (DTOF)Dual-Porosity ModelingTriple-Continuum Approach for Shale Gas ModelingSimulation of Dual-Porosity Model (Naturally Fractured Gas Well)Simulation of Triple-Continuum Model (Shale Gas Well)Conclusions

References