Water Resources Research

Dynamic coupling of pore-scale and reservoir-scale models for multiphase flow

Authors

  • Qiang Sheng,

    1. Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana, USA
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  • Karsten Thompson

    Corresponding author
    1. Department of Petroleum Engineering, Louisiana State University, Baton Rouge, Louisiana, USA
    • Corresponding author: K. Thompson, Department of Petroleum Engineering, Louisiana State University, 2107 Patrick F. Taylor Hall, Baton Rouge, LA 70803, USA. (karsten@lsu.edu)

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Abstract

[1] The concept of coupling pore-scale and continuum-scale models for subsurface flow has long been viewed as beneficial, but implementation has been slow. In this paper, we present an algorithm for direct coupling of a dynamic pore-network model for multiphase flow with a traditional continuum-scale simulator. The ability to run the two models concurrently (exchanging parameters and boundary conditions in real numerical time) is made possible by a new dynamic pore-network model that allows simultaneous injection of immiscible fluids under either transient-state or steady-state conditions. Allowing the pore-scale model to evolve to steady state during each time step provides a unique method for reconciling the dramatically different time and length scales across the coupled models. The model is implemented by embedding networks in selected gridblocks in the reservoir model. The network model predicts continuum-scale parameters such as relative permeability or average capillary pressure from first principles, which are used in the continuum model. In turn, the continuum reservoir simulator provides boundary conditions from the current time step back to the network model to complete the coupling process. The model is tested for variable-rate immiscible displacements under conditions in which relative permeability depends on flow rate, thus demonstrating a situation that cannot be modeled using a traditional approach. The paper discusses numerical challenges with this approach, including the fact that there is not a way to explicitly force pore-scale phase saturation to equal the continuum saturation in the host gridblock without an artificial constraint. Hurdles to implementing this type of modeling in practice are also discussed.

1. Introduction

[2] Pore-scale modeling of subsurface flow applications has the potential to provide links between fundamental pore-scale physics and macroscopic phenomena that are observed at the continuum scale. Qualitative modeling techniques have been used for years for phenomenological research. In recent years, emphasis has shifted toward quantitative and even predictive modeling. This change is driven largely by improvements in high-resolution 3-D imaging, which have allowed rapid advances in image-based modeling techniques (i.e., algorithms that operate directly on digital images of real materials).

[3] The evolution toward predictive pore-scale modeling brings questions about how to integrate pore-level results into reservoir-scale models using upscaling and/or numerical coupling. The most direct approach is simple numerical upscaling: pore-scale simulations are performed on a sufficiently large domain to extract continuum-scale parameters; the predicted parameters are then used in larger-scale models. This approach mirrors what is done with physical laboratory experiments, but does not use multiscale modeling to its full potential. A more integrated approach is to allow communication in both directions: continuum-scale parameters are obtained from fundamental pore-scale models and passed up to the continuum scale; simultaneously, the continuum scale model passes relevant conditions back down to the pore-scale models so that these simulations are performed using updated conditions.

[4] This latter approach is more complicated, but it is difficult to overstate its potential impact if it can be made practical. Consider reservoir-scale modeling of multiphase flow, which is vital to both subsurface hydrology and oil and gas production. Relative permeability is the main continuum-scale parameter used to characterize the dynamic component of continuum-scale multiphase models. For practical reasons, it is usually tabulated as a nonlinear function of phase saturation. However, this basic parameterization significantly oversimplifies the reality, which is that relative permeability is usually hysteretic and depends on numerous other parameters including wettability, capillary number (Ca =µv/σ), viscosity ratio, absolute permeability, and saturation history. True multiscale modeling represents a fundamental change to the traditional approach to reservoir-scale modeling of multiphase flow and has a number of practical implications. First, communication in both directions would allow the continuum model to operate using current relative permeability values from the pore-scale model, thus eliminating the need to tabulate or fit a relative permeability curve. (Here the important benefit is not convenience as much as it is the fact that relative permeability no longer needs to operate as a function of saturation only.) Second, provided that the pore-scale model captures relevant first-principles behavior, relative permeability would respond to unanticipated changes to the system that either are revealed by the continuum-scale model or are forced via changes in applied boundary conditions at the continuum scale. Examples include changes in injection rate, wettability alteration, formation damage, stimulation, or subsidence. Third, a pore-scale model that is coupled in simulated time with the continuum-scale model provides a mechanism to implicitly store saturation history (or other relevant history such as wettability change). This is not possible using traditional techniques.

[5] The general concept of exchanging information between a practical continuum simulation and a first-principles pore-scale simulation is not new. However, the hurdles between concept and implementation are large enough to have prevented a fully coupled approach from being developed until now. This paper describes the first multiscale algorithm that allows a pore-scale model to be embedded inside continuum gridblocks such that relative permeability is passed from the pore-scale model to the continuum models as a current gridblock parameter, while fractional flows are passed from the continuum model to the pore-scale model as current boundary conditions. The two models march forward in time together such that saturation evolves naturally, which in turn allows saturation history to be retained implicitly via the pore-scale model.

2. Background

[6] Pore-scale modeling has seen a resurgence in the last 15 years because microtomography imaging has become commonplace and techniques such as the Lattice Boltzmann Method (LBM) and network modeling can operate directly on these 3-D data [Rassenfoss, 2011]. If modeling is performed on sufficiently large domains, continuum parameters such as permeability can be obtained from the flow simulations, just as with a laboratory flow experiment. Generally, this approach is not viewed as a replacement for experimental tests, but rather a complementary tool. A significant constraint for all pore-scale modeling techniques is the relatively small domain sizes that can be used (due to imaging and/or computational limitations). This makes the general concepts of upscaling and multiscale modeling very appealing.

[7] Upscaling in the continuum domain has been studied extensively, resulting in techniques such as the multigrid method [Durlofsky et al., 2007; Aarnes et al., 2007]. This and similar approaches are designed to allow models to operate over multiple continuum scales, and aid in simulating multiscale physical and chemical phenomena such as water coning near a gas well or hydrodynamic dispersion of a chemical species in groundwater. While certain upscaling issues are similar, pore-to-continuum-scale multiscale modeling has unique challenges [Scheibe et al., 2007; E et al., 2007].

2.1. Boundary Coupling of Pore-Scale and Continuum-Scale Models

[8] For the sake of discussion, we divide pore-to-continuum multiscale techniques into two categories: boundary coupling and hierarchical coupling. Boundary coupling links pore-scale models with adjacent continuum regions at an interface between the two. Balhoff et al. [2007] coupled a pore-network model to a lower-permeability continuum region by matching the flux at a shared interface. They demonstrated that both pore-scale heterogeneity and the resistance in the low-permeability region affect the near-interface flow distribution in the continuum region, making it essential to match flux and pressure at the coupling boundary. Subsequently, Balhoff et al. [2008] developed a more general approach that couples pore-scale network models to other adjacent pore-scale or continuum-scale models using a mortar method. This approach demands a weaker flux match at the interface than the earlier technique but has a more flexible formulation. Boundary coupling techniques work well for joining adjacent regions, but they are not designed to upscale information across multiple length scales at the same location.

2.2. Sequential Coupling of Pore-Scale and Continuum-Scale Models

[9] A second class of multiscale modeling is hierarchical coupling, which links pore-scale and continuum-scale models in the same region (e.g., from an embedded network model to the continuum gridblock in which it resides). This technique must address large disparities in both length and time scales [e.g., Patzek, 2001], and can require both numerical coupling and upscaling. A simple but limited hierarchical approach is sequential coupling, where rock properties or transport parameters such as permeability, relative permeability, or capillary pressure are simulated in a preprocessing step and then used in a reservoir simulation in much the same way as if they had been obtained from a physical laboratory experiment. Blunt et al. [2002] showed that the assignment of relative permeability based on network modeling gives significantly different predictions of oil recovery compared to using traditional empirical relative permeability models, which implies sequential coupling could improve the predictive power of reservoir models if pore-scale prediction of relative permeability can be made more accurate than general empirical methods. Jackson et al. [2003] used the same coupled model to predict oil recovery for different wettabilities. White et al. [2006] developed a multiscale model for simulating single-phase flow through porous media. The pore space was discretized into smaller domains. LBM was used to calculate microscale velocity distributions and permeability at the pore scale for each subdomain, and the results were then transmitted to a continuum model via tabular or empirical permeability relationships as one would do with laboratory data. Lichtner and Kang [2007] developed a multiscale model to study reactive flow. Pore-scale transport equations were solved using LBM at the pore scale, and the results were upscaled through simple volume averaging at slow reaction rate and fit into a multiscale continuum model. Rhodes et al. [2008] studied particle transport at four different scales: pore, core, gridblock, and field. At the pore scale, transport was simulated using a network model. Transport properties were then upscaled to the larger scales using transit time distribution functions. They argued that pore-scale or gridblock-scale heterogeneity delayed transport and increased breakthrough times by up to an order of magnitude compared to those predicted using a traditional advection and dispersion model. Chen et al. [2010] developed a multiscale LBM simulation: pore-scale LBM was used to estimate permeability after colloid deposition, and this estimated that permeability distribution was used in a second LBM model to simulate flow at the continuum scale. Tsakiroglou [2012] developed a multiscale network type algorithm. They estimated capillary pressure and relative permeability functions using a quasi-static network model and fed the information as the input to a macroscopic simulator as part of an analysis of the axial distribution of water saturation and pressure across a column-scale soil sample. Sun et al. [2011] studied the effect of subdomain size on the accuracy of a multiscale method, which couples LBM with a continuum model.

2.3. Concurrent Coupling of Pore-Scale and Continuum-Scale Models

[10] The aforementioned sequential coupling techniques offer the potential to replace or augment coreflood experiments with numerical simulations. However, they do not exploit the most powerful aspect of pore-scale to continuum-scale coupling, which is the ability of the pore-scale model to march forward in time with the reservoir simulator, providing “just-in-time” continuum-scale parameters needed at the larger scale. This approach, which we will refer to as concurrent coupling, has the advantages described earlier.

[11] To date, few studies have been published in the area of concurrent coupling. Heiba et al. [1986] coupled a statistical network model for two-phase flow with a reservoir-scale finite element solver. Relative permeability was calculated using percolation theory. They argued that this approach provides more accurate and reliable prediction of relative permeability and capillary pressure in two-phase displacements compared to empirical correlations, but with a factor of 100 increase in computational cost compared to conventional reservoir simulation because of the capillary pressure and relative permeability simulations. Celia et al. [1993] proposed a concurrent multiscale simulation framework to couple a quasi-static network model with a continuum model. The idea consisted of a two-way exchange process: network simulators located in the center of selected blocks in the continuum model providing bulk properties to these blocks. In turn, the continuum simulator provides boundary conditions (in the form of fluid pressures) for the network model. The focus of their approach was to allow material heterogeneities to be properly accounted for in the continuum model. A concurrent multiscale simulation strategy using LBM was also proposed by Van den Akker [2010]. Neither of these latter two models was implemented at the time that they were proposed because of numerical limitations. Battiato et al. [2011] developed a multiscale approach that allows a pore-scale model to operate at locations where flow or transport violates assumptions inherent in the continuum equations. The averaging process in the pore-scale region produces integral source terms in the continuum model, and multiscale coupling allows these terms to be evaluated. Chu et al. [2012] developed a single-phase multiscale algorithm that couples network models to continuum-scale models. Network properties such as throat conductances are nonlinear functions of the macroscopic properties at a much larger scale, which is designed to allow multiscale simulation of non-Darcy flow. Chu et al. [2013] integrated a dynamic pore-scale model for immiscible displacement with a reservoir model for multiphase flow. They decoupled the pressure and saturation computations by using two separate sets of networks: networks placed at gridblock boundaries were used for pressure calculation, while separate networks were placed along the displacement front (where the saturation gradient is larger) to be used for saturation updates.

[12] For the type of multiscale simulation we are interested in (i.e., embedding a network model into a much larger gridblock for the purpose of predicting continuum parameters in that gridblock), a number of simplifications can be made to allow multiscale coupling and/or to improve numerical efficiency. However, these simplifications have corresponding restrictions associated with the physics of flow, including the following:

[13] 1. Quasi-static models are limited to zero-capillary-number behavior, meaning not only that viscous effects are negligible but also that time is not a parameter in the model. Giving up the ability to model viscous behavior or any time-dependent effects eliminates many reservoir processes that would benefit most from multiscale, multiphase modeling.

[14] 2. Most dynamic network algorithms model immiscible displacement (injection of only one phase) rather than more general multiphase flow processes. These simulations force a large saturation change to occur over very small (order-mm) linear dimensions, which translates to an essentially infinite saturation gradient in the direction of flow at the reservoir scale. Consequently, both the magnitude of the saturation gradient and the spatial distribution of phases (which dictates relative permeability) are inconsistent with what would be expected in a reservoir.

[15] 3. For finite-difference or finite-volume algorithms (at the continuum scale), two possible mappings of pore-scale models onto the continuum grid are to (i) associate a pore-scale model with the interior of a gridblock or (ii) place pore-scale models at gridblock boundaries. The latter allows for a simpler numerical approach if it is run such that a pore-scale model on an yz face of a gridblock (for instance) is operated using only a dP/dx pressure gradient. However, this approach implies that the continuum grid is aligned with the principal directions for the permeability tensor, which restricts generality of the algorithm.

[16] 4. As discussed below, no explicit equation links phase saturation in a pore-scale model with the saturation in its home gridblock, at least in a natural formulation derived directly from the conservation equations. If these values are forced to be equal (for instance, by a forced update to the pore-scale saturation at each time step so that it matches the current gridblock saturation), this procedure in effect erases saturation history.

[17] The issue of viscous versus quasi-static multiphase modeling is not unique to the multiscale framework, and it has a fairly long history in network modeling. Multiphase network models that account for viscous effects (termed dynamic algorithms) remain tricky to implement even after 25 years of research, and research remains active in this area. Lenormand et al. [1988] developed a dynamic two-phase network model that couples viscous force and capillary force together using nonlinear Poiseuille equations. Constantinides and Payatakes [1996] developed a dynamic two-phase model by incorporating Washburn-type equations (for immiscible displacement in a tube) into the material balance. Using their model, they simulated steady state flow in a cubic lattice network with geometrically identical inlet and outlet zones and calculated steady state relative permeability by averaging transient values over a sufficiently long time. Mogensen and Stenby [1998] developed a dynamic two-phase network model and studied the competition between frontal pore-filling and snap-off. Thompson [2002] developed a model in which the two-phase conservation equations were solved simultaneously within each pore, helping to reduce rule-based decisions in the algorithm. Nguyen et al. [2006] developed a two-phase network model for dynamic imbibition; the thickness of wetting films was allowed to vary as a function of local capillary pressure, and pore invasion was controlled by the competition between frontal displacement and snap-off. Joekar-Niasar et al. [2010] used the aforementioned approach in which the phase conservation equations were solved simultaneously, and studied nonequilibrium capillarity effects and the dynamics of two-phase flow. This latter model was also used for the coupled model by Chu et al. [2013] mentioned above.

[18] Using the lattice Boltzmann modeling for multiphase flow simulation automatically addresses the quasi-static problem. However, while LBM models are more rigorous than network models, they are less appealing for this type of multiscale simulation because of their additional computational overhead. Additionally, most LBM models employ periodic or mirror boundary conditions, which prevent them from being used to monitor the natural evolution of saturation history (because overall saturation cannot change due to differences in inflow/outflow).

[19] As with the quasi-static issue, the problem of length scales is not unique to coupled models. In fact, it is a significant concern for nearly all modern image-based modeling applications because of the small size of the samples that can be imaged at high resolutions. However, in the context of multiscale modeling, the problem is particularly striking because pore-scale models derived from high-resolution imaging are usually order-mm in size, whereas continuum-scale gridblocks are typically three orders of magnitude greater. Similarly, timescales for immiscible displacement in a mm-sized pore-scale model are order seconds or even less than one second, while numerical time steps at the continuum scale are orders of magnitude larger. At this point, we should clarify two issues which are not the focus of work in this paper. The first is the traditional role of upscaling over a hierarchy of scales. While we do not discount the importance of this problem for modeling heterogeneous, multiscale geologic media, the focus in this work is on a single transition in scales: from pore-scale (where fundamental equations of motion can be applied) to the continuum scale (where Darcy-type equations are used, and which in turn requires spatially averaged parameters such as permeability). The second issue not addressed here is the data-poor conditions that exist in the context of pore-to-continuum modeling, which will always prevent us from having sufficient pore-scale data to accurately populate a reservoir model. We again acknowledge the significance of this problem, but argue that it should not slow the development of these powerful multiscale techniques. In fact, we point out the similarity to the situation that the subsurface simulation community has dealt with for decades: experimental tests are performed on a limited set of cm-scale core samples or sand columns, and these results are used to populate much larger continuum grids, which in turn describe heterogeneous subsurface structures.

[20] Discussion in the remainder of the paper assumes the following. First, separate tests have been performed to ensure that the length scales used in the pore-scale modeling are sufficient to extract spatially averaged continuum scale parameters such as permeability, relative permeability, and capillary pressure. Second, continuum properties within a gridblock are reasonably uniform and flow conditions are such that the original continuum-scale equation remains valid everywhere.

[21] In addition to the issues of scale discussed in the previous two paragraphs, there are time-scale and length-scale considerations that are more critical to the coupled, multiscale problem than to general pore-scale modeling. Consider first that the most common methods for pore-scale modeling of multiphase flow use immiscible displacement algorithms. In these simulations, both saturation and fractional flow exhibit order-unity changes over very small length scales, thus creating essentially infinite gradients in these parameters at the continuum scale. This behavior is not consistent with the magnitude of continuum-scale saturation gradients, nor is it consistent with the physics of flow at either scale. The problem is of particular concern if relative permeabilities are extracted from these unrealistically sharp saturation fronts because the pore-scale distribution of fluid phases is the dominant factor in dictating relative permeability.

[22] The problems associated with time scales are analogous. The flat front that passes through a pore-scale region during an immiscible displacement simulation represents an order-unity saturation change in approximately one pore volume, which in turn implies a very small time scale for the saturation change to occur (order milliseconds or seconds depending on the capillary number). In contrast, an Eulerian view of a mm-scale section of a reservoir would exhibit a much slower saturation change during most reservoir-scale flow scenarios, during which time an enormous number of pore volumes would pass through the small domain.

[23] Given the above considerations, we argue that the more proper pore-scale simulation to couple to a continuum-scale time step is steady-state multiphase flow over a short period of time. Generally, this would be at a new saturation that has evolved to match the current continuum-scale time step.

3. Materials and Methods

3.1. Porous Materials

[24] Two types of porous materials were used in the current study: (1) unconsolidated sand with grain sizes between 210 and 297 µm [Bhattad et al., 2011]; (2) a consolidated sandstone sample from a subsurface reservoir, described in more detail in Sheng et al. [2011]. The samples were imaged using X-ray microtomography (microCT) to obtain three-dimensional images. Although significantly larger data sets from both samples are available, internal sections of 2003 voxels were used in the current work because of the large computational demands associated with running the steady state algorithm, especially with nonperiodic conditions [Sheng et al., 2011]. These voxel dimensions correspond to 1320.6 µm on a side for the unconsolidated sand and 704 µm on a side for the sandstone. Networks were generated from the microCT images using a network extraction algorithm described by Thompson et al. [2008]. Figure 1 illustrates the network generation process by showing the 2003 voxel section of the reservoir sandstone image with the corresponding network embedded in it. Table 1 contains network specifications. (Throat numbers include external connections on all six faces, which in the case of these small networks gives the appearance of larger pore coordination numbers than what is observed in the interior of the domain.) For simplicity, we assume uniform and strongly wetting surfaces (zero degree contact angle) in all simulations although this is likely not the case for the reservoir sandstone. Table 2 gives the fluid properties used in the simulations. Density differences for the fluids were ignored in this study because the reservoir simulations described below are one-dimensional in the horizontal direction.

Figure 1.

Image of the sandstone data set with the pore-network model superimposed (pores shown as inscribed spheres and throats shown as line connections).

Table 1. Network Specifications
 Number of PoresNumber of ThroatsDimension (m)Pore Radius × 106 (m)Permeability × 1012 (m2)
Sandstone153275400.000704[4.25–38.7]2.90
Sand88248620.00132[7.97–89.7]8.82
Table 2. Fluid Parameters in the Network Model
 SymbolValue
Contact angle (deg)Θ0.0
Interfacial tension (kg/s2)σnw0.0535
Wetting phase viscosity (kg/m·s)μw0.00042
Nonwetting phase viscosity (kg/m·s)μnw0.0155

3.2. Dynamic Two-Phase Network Model

[25] Following Thompson [2002], two-phase flow is modeled by solving mass balance equations for the two phases in each pore, which gives the following coupled nonlinear algebraic equations:

display math(1)

[26] Vi is the volume of pore i and Si is the phase saturation (nondimensional volume fraction) in pore i. The superscripts l and l + 1 denote the current and future times in the simulation as dictated by the time step Δt. The time indexes are intentionally left off the right-hand-side terms to allow the equations to remain general for either explicit or implicit solution. The subscripts w and nw are for the wetting and nonwetting phases. gij is the absolute hydraulic throat conductance, gr,w and gr,nw are nondimensional relative throat conductances for each phase, which account for the loss of conductivity in the throat when a second immiscible phase is present. µ is the viscosity. The dependent variables are related by auxiliary equations and functions. The pressure difference between the nonwetting phase and the wetting phase is the pore capillary pressure:

display math(2)

[27] Pore capillary pressure (not to be confused with the spatially averaged continuum-scale capillary pressure function) is a nonunique function of phase saturation in a pore. It has an unstable branch within the saturation range of 0–1, which is associated with nonequilibium invasions or retractions of nonwetting phase. A second auxilliary equation is the definition of phase saturation:

display math(3)

[28] For the simulations shown here, pressure is solved implicitly and the saturation is updated explicitly, similar to the implicit pressure explicit saturation (IMPES) approach used in reservoir scale modeling. Stability is maintained by limiting time step size [Chen, 2007]. (Alternatively, our algorithm is set up to be switched to an implicit method if necessary.) Other methods to improve stabiltiy have also been proposed [Lux and Anguy, 2012]. The resulting pressure equation for the semiexplicit method is:

display math(4)

[29] In order to solve the governing equations, it is necessary to specify the functionality for relative conductances and local capillary pressures. In the current algorithm, local capillary pressure curves are approximated a priori for each pore using local geometric parameters measured for each pore from CT images. During a simulation, these functions (one for each pore) are retrieved using interpolation tables to improve performance. For simplicity, absolute conductance gij is obtained by the modified Hagen-Poiseuille equation [Bryant and Blunt, 1992]:

display math(5)

[30] The effective radius reff is the arithmetic average of the inscribed radius and the equivalent area radius, while lt is a throat length. In real materials, pore-throats are rarely distinct entities as they are in lattice models, so lt is the pore-to-pore distance of an approximation based on this distance minus some distances that account for the throat expanding into the adjacent pores. A more rigorous computation for hydraulic conductance can be performed if there is a reason to do so [Thompson and Fogler, 1997].

[31] Relative conductances are similar to the macroscopic relative permeabilities, but significant differences arise because of interface dynamics unique to the pore scale [e.g., Morrow, 1971]. For instance, continuum-scale relative permeability is nonlinear but smooth. In contrast, nonequilibrium events at the pore scale lead to a jump discontinuity on the relative conductance curve. We use the following expessions for relative conductances:

display math(6)

where β is a dimensionless resistance factor [Ransohoff and Radke, 1988], At is the throat area, and St is the throat saturation (which is determined based on the saturations of the bounding pores using upstream weighting). Equation (6) is applied to calculate relative conductance if throat saturation is smaller than the maximum stable saturation Smax. The value of Smax is found by subtracting the minimum stable nonwetting-phase volume (which we assume to be the volume of the maximal inscribed sphere in the pore) from the total pore volume and nondimensionalizing. When St > Smax, the wetting-phase relative conductance jumps to unity and the nonwetting-phase relative conductance is assumed to be zero. Zero relative conductance would occur in a variety of conditions in which a pore throat contains no nonwetting phase fluid: for instance, prior to a Haines jump, a snap-off situation, or when a throat is bypassed by the nonwetting phase.

[32] The original implementation of this pore-scale algorithm was for imbibition [Thompson, 2002]. We have extended the algorithm to allow the simultaneous injection of two fluids as an upstream boundary condition (for instance, total flow rate and fractional flow ratio are specified). This nontrivial development, which will be described in a different paper, is essential for the coupling process described below, which requires that the inlet volumetric flow rates of wetting and nonwetting phases can be specified independently. In general, the specified inlet flow rates may not be consistent with the current internal phase distribution or total saturation in the pore-scale model, in which case a transient period is evident prior to reaching steady state. At steady state, the saturation in every pore in the network is unchanging with time (at least when integrated over a few time steps to account for small numerical oscillations). The boundary equations for a nonperiodic network are defined as follows:

display math(7)

[33] This set of boundary conditions leads to two additional unknowns: inlet wetting-phase pressure Pw,inlet and inlet nonwetting-phase pressure Pnw,inlet. Wetting-phase pressure and nonwetting-phase pressure are assumed to be uniform for all inlet pores at any given time step. The model can be run with only one fluid injected (drainage or imbibition) or with two fluids injected simultaneously. In the latter case, total volumetric flow rate qtol,in and fractional flow ratio (qw,in/qtol,in) are specified (or equivalently, the two separate volumetric flow rates qw,in and qnw,in are specified). The outlet boundary condition is fixed pressure in the wetting phase and a saturation-based pore capillary pressure difference that dictates nonwetting phase pressure. The simulation is not highly sensitive to the outlet boundary condition because upstream weighting is used for pore-throat conductances. The result is a unified model for traditional drainage, traditional imbibition, and two-phase flow (the latter including both unsteady and steady processes).

3.3. Relative Permeability Simulation

[34] Various methods for pore-scale prediction of relative permeability have been developed [Sheng et al., 2011]. The method most directly tied to the multiphase Darcy's law (which defines relative permeability) is to perform simulations that mimic steady-state coreflood tests in the laboratory. Total flow rate and fractional flow are specified, and both phases are injected from the inlet until a steady state is reached (determined by a stationary saturation and pressure distribution). Relative permeabilities of both phases are then calculated using the multiphase Darcy's law. The complete relative permeability curve is created by altering the injected fractional flow ratio, waiting for the next steady state to evolve in a physically realistic way from the previous state, and computing the next point on the curve.

[35] This approach to determining relative permeabily alleviates problems associated with other methods. Specifically, quasi-static simulations decouple the capillary and viscous forces and therefore cannot show any rate dependence on relative permeabilty. Simulations performed with periodic boundary conditions (with periodic or mirrored structures) are not capable of simulating physically based hysteresis in relative permeability (because total saturation cannot change, and relatively permeability must therefore be viewed as a state function of saturation). Similarly, the use of periodic boundary conditions does not allow the effects of saturation history to be retained implicitly, which is one of the major potential advantages of concurrent multiscale simulation.

3.4. Macroscopic Reservoir Model

[36] The contimuum-scale model used in this work consists of standard coupled mass balance equations for two-phase flow.

display math
display math(8)

[37]  inline image and inline image are phase saturations in each gridblock (a finite volume or fintie element), inline image and inline image are phase pressures, Qw and Qnw are phase mass flow rates, ρw and ρnw are phase densities, z is vertical coordinate based on a global datum level, μw and μnw are phase viscosities, ϕ is porosity, k is permeability, and krw and krnw are relative permeabilities. The constraint on phase saturation is

display math(9)

[38] The pressure difference between the two phases is given by the macroscopic capillary pressure:

display math(10)

[39] Capillary pressure and relative permeability are functions of saturation and computed at the current saturation by empirical formulas or interpolation tables for gridblocks without embedded networks. Source or sink terms Qw and Qnw are specified in order to close the system of equations [Chen, 2007]. Pressure and saturation are solved here using an IMPES method.

[40] Fractional flow must be computed for the coupled algorithm (described in the next section), so that it can be passed down from the reservoir simulator to the pore-scale model. These computations are made for flow rate Q and fractional flow ratio F at the upstream boundary of a simulation gridblock. FL and FR (fractional flow ratios at both boundaries of gridblock i for a 1-D reservoir) can be derived using relative permeability values if capillary pressure and gravity are ignored [Bear, 1988]. The result is the expressions in equation (11), in which krw and krnw are wetting and nonwetting relative permeabilities (which are functions of saturation at the upstream or downstream gridblock faces).

display math(11)

[41] Alternatively, related formulas can be used if capillary pressure is deemed to be important.

[42] For simplicity in the current paper, the reservoir-scale simulation is performed using a 1-D grid in which gravity is ignored. We have performed initial 2-D and 3-D simulations; the changes to the algorithm are straightforward but nontrivial because of the application of multidimensional (multiphase) boundary conditions on the network model.

3.5. Coupled Multiscale Algorithm

[43] The coupled multiscale algorithm can be viewed as a pore-scale network model embedded into one or more continuum-scale gridblocks. The network model provides relative permeability to the reservoir model and the reservoir model returns fractional flows that are imposed as boundary conditions on the network model.

[44] The main assumptions associated with the coupled algorithm are the following: (1) each gridblock is seen as a homogeneous porous medium, and network(s) are located inside one or more gridblocks that will obtain information from the coupling process. The network is representative of the porous medium in its home gridblock and can be used to assign two-phase properties such as relative permeability and macroscopic capillary pressure. (2) Because of the very small time and length scales associated with the pore-scale model (compared with the continuum model), two-phase flow at the pore scale can be viewed as a steady state process during a given reservoir-scale time step. Relative permeability is determined and upscaled from the steady state computation.

[45] In the current paper, flow through two- or three-dimensional networks is approximated by applying pressure on only two opposing faces, corresponding to the alignment with the 1-D reservoir model. (This assumption can be relaxed in 2-D and 3-D models. We have successfully tested 2-D and 3-D implementations for single-phase flow, and believe the same approach can be extended to multiphase flow.) Phase flow rates, qa, for a network model are calculated by downscaling the upstream flow rate Qa at the host gridblock boundary. The value for fractional flow ratio imposed on the network model Fnet is calculated by linear interpolation between fractional flows at both boundaries of the gridblock. For a network model placed in the center of a gridblock, this would be

display math(12)

[46] In the current study, network models are embedded into at least one continuum-scale gridblock. We assume that initial conditions in the reservoir are known, and further assume that the initial pore-scale saturation is known. In practice, the initial pore-scale phase distribution would come from either the end of a separate simulation (for instance, at the end of a nonaqueous phase liquid (NAPL) spill simulation but prior to a remediation simulation, or at the end of primary oil production but prior to water flooding) or inferred from a known condition (such as equilibrium water saturation in a Vadose zone).

[47] A flowchart of the algorithm is shown in Figure 2. Given the initial conditions at both the reservoir and the pore scale, two-phase properties for the first time step can be determined. For each subsequent continuum-scale iteration, the reservoir model provides fractional flows that are imposed as boundary conditions on the pore network models, while the networks return “real-time” relative permeability to their host gridblocks. Pore-scale saturation and pressure distributions are stored locally for each coupled gridblock. The initial conditions for subsequent pore-scale simulations are the local saturation and pressure distributions from the previous time step, which is the process that generates the implicit saturation history discussed earlier.

Figure 2.

Flowchart of the coupled multiscale algorithm.

[48] Because of the large number of runs that were made in the current work, simulations were performed with network coupling in only a few gridblocks. The remaining gridblocks were assigned relative permeability and capillary pressure values from tabulated data. To maintain as much consistency as possible in values, the tabulated data were also obtained from a network model simulation, but with the network running as a stand-alone tool prior to the multiscale simulation.

4. Results

4.1. Steady State Relative Permeability Test

[49] The two materials described in section 'Porous Materials' were used for modeling relative permeability. Absolute permeabilities were calculated using a single-phase flow algorithm applied to the network models.

[50] For noncoupled gridblocks, relative permeability curves were calculated using steady state simulations with nonperiodic conditions in a single extended run. An example is shown for the sandstone network: the system was started with an overall saturation Sw = 0.002. The first fractional flow (qw/qtol) selected was 0.005. Fluids were injected until a stationary saturation distribution and pressure field were reached. Each steady state solution provided a single relative permeability data point for each phase, after which the fractional flow ratio was increased and the process was repeated until a complete relative permeability curve was acquired. A plot of saturation versus pore volumes injected is shown in Figure 3a. Fractional flow and relative permeability curves are shown in Figures 3b and 3c. As with physical experiments, it takes longer time to reach steady state for a high fractional flow ratio because the corresponding nonwetting phase flow rate is small and must drain to the equilibrium state through pathways with low hydraulic conductance.

Figure 3.

Plots from the steady state relative permeability test on the sandstone sample: (a) phase saturation versus pore-volumes injected; each plateau represents a steady state relative permeability data point; (b) fractional flow ratio versus saturation; (c) relative permeability curves.

4.2. Rate Effect on Relative Permeability

[51] An obvious benefit of the multiscale modeling proposed here is the ability of relative permeability values to respond to changes in Darcy velocity at the reservoir scale. To help quantify the impact of this effect prior to multiscale modeling, relative permeability curves were generated using different flow rates for both the sandstone and the sand networks as shown in Figures 4a and 4b. Results show that the nonwetting-phase relative permeability is an increasing function of flow rate. This trend is consistent with experimental results by Avraam and Payatakes [1999] and numerical results by Li et al. [2005]. Results also show that rate effects are more significant at intermediate flow rates, where two-phase flow undergoes a transition from viscous-dominated regime to capillary-dominated regime.

Figure 4.

(a) Relative permeability curves for the sandstone network at different flow rates; (b) relative permeability curves for the unconsolidated sand network at different flow rates.

[52] Visualizations were used to help interpret differences in relative permeability predictions by mapping the network model saturations onto voxels as described by Sheng et al. [2011]. By applying similar visualization methods at different flow rates, we observed that pore-scale saturation distributions are consistent with the relative permeability results: higher flow rates create more connected nonwetting phase paths across the sample, thus leading to larger nonwetting relative permeabilities. (Likewise, some nonwetting phase connections are lost at low flow rate because of strong capillary forces.)

4.3. Concurrent Model With a Constant Injection Rate

[53] The concurrent multiscale model is run using a water injection in a 1-D reservoir (more on the application is described below). The reservoir is divided into 100 blocks along the flow direction. Injection occurs in the first block and fluids flow out of the last block. Networks are embedded into selected gridblocks, and saturations in the networks and associated gridblocks at these locations are used for most of the results shown below. The remaining gridblocks obtain multiphase flow parameters from tabulated functions that are calculated a priori using network simulations. Parameters used in the reservoir model are listed in Table 3.

Table 3. Simulation Specification in the Reservoir Model
 SymbolValue
Wetting phase viscosity (kg/m·s)µw0.00042
Nonwetting phase viscosity (kg/m·s)µnw0.0155
Gridblock cross-section area (m2)A937.5
Gridblock length (m)L6.09600

[54] Figure 5a shows network saturation and gridblock saturation versus time in block #10. Saturations in both the network and the gridblock start at inline image but after the first time step the two saturation values have already come to different values as fluid is reconfigured from the spatially uniform saturation under an applied pressure gradient (see below). The more important observation is that the network and gridblock saturations are not in good agreement. This discrepancy presents a problem, but it is not entirely unexpected since there is no explicit equation in the algorithm to force saturations in the network and associated gridblock to be equal. It is tempting to add a saturation constraint to the algorithm to maintain equality. However, we have avoided this approach because the constraint does not arise naturally from the governing equations and because it would require either an explicit correction at the continuum scale (which would conflict with the underlying conservation equation), or an ad hoc redistribution of phases in the pore-scale model (which would, in effect, erase all implicit history dependence).

Figure 5.

(a) Saturation comparison between the network and its associated gridblock for β = 8 (cylindrical throat shape). The sandstone network was embedded in gridblock 10. (b) comparison of the network and reservoir fractional flow curves for β = 8. (c) saturation comparison between the network and its associated gridblock β = 65.02 (for equilateral triangular throat shape). (d) comparison of the network and reservoir fractional flow curves for β = 65.02.

[55] Numerous tests were performed to track the divergent saturations, both at the outset of the simulations and in cases where the two saturation curves crossed, so were essentially equal for a time step, but then diverged again. These numerical tests showed that the problem can be traced back to relatively small differences in the fractional flow curve used at the continuum scale (based on equation (11)) versus the fractional flow functionality predicted by the pore-scale model. The connection is evident from the fact that fractional flows are sent continuously from the reservoir model to the network model (equation (11)). The value communicated to the pore-scale model, along with any differences between the network fractional flow behavior and the macroscopic curves, governs whether the pore-scale saturation agrees with saturation in its host gridblock. In the current case, these two functionalities are slightly different over an intermediate saturation range as shown in Figure 5b, the network saturation being higher than what is predicted by equation (11).

[56] In order to correct the discrepancy between the two fractional flow curves (network and reservoir), we examined the sensitivity of fractional flows to one of the critical parameters in the network model: the resistance factor β, which is a function of corner geometry, surface shear viscosity, and contact angle. This particular parameter was chosen because it has a somewhat tenuous connection to pore-throat geometries in the real porous media and thus may need to be adjusted. Figures 5a and 5b are the saturation versus time plots and the corresponding fractional flow curve for the case β = 8, which corresponds to cylindrical pore throats. In contrast, Figures 5c and 5d are the two equivalent curves but for β = 65.02, corresponding to equilateral triangular throat geometry. The latter fractional flow curves show much better agreement (network versus reservoir). Additionally, the recomputed saturation comparison between the pore-scale model and its associated gridblock is likewise improved, showing that the network model tracks the reservoir reasonably well even through the two saturations evolve independently of one another. While both network throat geometries are idealized compared to real porous media, the triangular throat is clearly a better choice for the current multiphase flow model, almost certainly because it allows for wetting phase flow in the corners.

[57] Other notes include the following. First, saturations in both the network and its host gridblock begin at inline image, but the two saturation values decrease slightly at the beginning of the simulation (in a location the front has not reached yet). This behavior is caused by the differences between the network and the reservoir fractional flow functions at low saturation, as observed in Figure 5d. Second, step-like patterns in the raw data used to plot Figure 5c suggest that larger networks should be used. We ran the same test with a sand network and obtained a better agreement: the two saturations are almost the same as their initial values before the water front reaches coupled gridblock 10, as shown in Figure 6a. Correspondingly, the two fractional flows predicted directly from the network and predicted by equation (11) match well even at low saturation, as shown in Figure 6b. The two saturation values drift apart slightly at the end of the simulation because of accumulated numerical errors. Relative permeability curves obtained from the network model as a stand-alone tool compared to the network relative permeability values that were passed to the simulator during the coupled simulation are essentially identical, as shown in Figure 7. This agreement is to be expected since all conditions (total flow rate, viscosity, wettability, etc.) were identical in both the relative permeability simulation and the multiscale simulation. Finally, we note (in agreement with a reviewer suggestion) that direct comparison of a large pore-scale simulation with a shrunk coupled model may provide valuable insight to the issues discussed above. These tests were begun; however, at the present time, the largest domain that can be run with a full dynamic simulation is not large enough to run the coupled model. To perform the comparison, the model would require at least five gridblocks (i.e., a minimum of three interior gridblocks along the flow direction), and these gridblocks would have to be sufficiently large that they can contain a significantly smaller embedded network that is still large enough to generate continuum relative permeability.

Figure 6.

(a) Comparison of network versus gridblock saturations for the unconsolidated sand; (b) comparison of network versus reservoir fractional flow curves for the unconsolidated sand network.

Figure 7.

Relative permeability comparisons: the black curve shows relative permeability predicted using the network model as a stand-alone tool; the red curve shows relative permeability determined from a coupled model with constant-rate displacement at Ca = 10−4.

4.4. Concurrent Model With a Variable Injection Rate

[58] To test the two-way communication in the multiscale algorithm, we simulate 1-D water injection with variable rate. Changes in injection rate affect the capillary number, which in turn has the potential to affect relative permeability. Two scenarios are modeled, both of which are simple prototypes of more complicated applied problems. The first case is water injection into a uniformly saturated consolidated sandstone at a decreasing rate (reduced by 0.5% every iteration of the reservoir model). This scenario is a simplistic model of water-drive oil recovery with declining production rate. The second case shown below is viscous-dominated displacement of nonwetting phase liquid from an unconsolidated sand. This scenario is a simplistic model of groundwater flow over a NAPL spill (for instance, a low natural rate initially followed by a higher rate associated with pumping or cleanup). In both cases, rate is dictated by boundary conditions in the reservoir-scale model, but the immiscible displacement regime is governed by a model of the pore-scale physics (in the coupled gridblocks).

4.4.1. Waterflood With Decreasing Rate

[59] The first simulation is performed with an initial saturation inline image in all 100 gridblocks in the 1-D domain. A network model is embedded in block 10 and given the same initial saturation. Water is injected into gridblock 1 at an initial rate of 0.3 m3/s, which corresponds to an initial Darcy velocity of 3.2 × 10−4 m/s. Figure 8a shows saturation versus time in gridblock 10 for both the network and the gridblock. In contrast to the improved agreement shown in Figure 5c, the saturation profiles drift apart over time. The reason is that the surrounding gridblocks (which do not benefit from the embedded network) are operating with a relative permeability curve from a fixed rate (associated with Ca = 10−4, the condition at the outset of the waterflood). Figure 8b helps illustrate this effect with plots of two fixed-rate relative permeability curves, determined from steady state simulations at early and late flow rates that are reached during the simulation. The plot also shows the actual relatively permeability tracked by the network model, which moves from the Ca = 10−4 curve initially toward the lower flow rate curve as the waterflood progresses and the displacement rate decreases. This simulation demonstrates the network model responding to changes in flow rate, even if they had not been anticipated prior to the simulation and therefore were not contained in the a priori relative permeability data. However, the saturation discrepancy also illustrates practical issues that must be overcome before this technique can be effective, namely the ability to embed a network in every gridblock or a technique to transmit information from a single network to multiple gridblocks that are following a similar saturation path, or to communicate information from the gridblocks with embedded networks to neighboring gridblocks that are exposed to similar dynamics and conditions.

Figure 8.

Behavior of the concurrent multiscale model in a variable rate application using the sandstone network: (a) comparison of network saturation with host gridblock saturation; (b) steady state relative permeability for two different rates (black) compared to actual relative permeability values tracked by the coupled network during displacement (red).

[60] As the first step toward addressing these latter issues, we performed a fully coupled simulation with three sandstone networks embedded in multiple gridblocks: 10, 15, and 20, respectively. This setup might be used in a scenario where pore-scale models are positioned at critical locations in the domain, for instance near the wellbore if transient behavior or rate effects might have a more significant impact. This simulation is naturally amenable to parallel computation. In the current setup, each pore-scale simulation was performed on an individual compute node. Pore-scale saturation distributions were stored locally and only the relative permeability values were communicated back to the processor running the continuum simulation so that in principle coupling with multiple networks is as efficient as coupling with a single network. We compare saturation and relative permeability curves for two cases. First, the injection rate is held constant (Ca = 10−4); Second, the injection rate is reduced by 0.5% every reservoir iteration.

[61] Figure 9a shows gridblock saturation profiles at the three different locations for the constant-flow rate simulation. A plot of relative permeability for this run confirms that all three network relative permeability histories track the generic relative permeability curve obtained for Ca = 10−4, as shown in Figure 9b. Figure 10 contains results from the same setup (three embedded networks) but with decreasing injection rate. Because different gridblocks will reach any given saturation at different times and rates, there is the possibility of having different relative permeabilities even for the same saturation, which can be accommodated by the coupling. The consequence of this behavior is the differently shaped trajectories as shown in Figure 10 (compared to Figure 9a where they all exhibited the same behavior, just offset in time). Figures 11a and 11b show the relative permeability curves for the continuum gridblocks as well as relative permeability values tracked by all three networks during the simulation. (Figure 11b is an enlarged view of the critical saturation range.) The plots show that all three networks start with equal relative permeabilities, close to the relative permeability curve at Ca = 10−4. However, permeability values shift toward the lower-rate curve during the displacement process. Clearly, each multiscale gridblock is able to exhibit an independent relative permeability profile depending on the flow rate and saturation history that it was exposed to.

Figure 9.

Concurrent multiscale model with constant rate displacement (Ca = 10−4). Sandstone networks were embedded in gridblocks 10, 15, and 20. (a) Saturation and flow rate profiles; (b) relative permeability for two flow rates. Actual relative permeability values track the Ca = 10−4 curve.

Figure 10.

Same plot as shown in Figure 9a but for the variable rate simulation. Flow rate was reduced by 0.5% for every reservoir iteration.

Figure 11.

Concurrent multiscale model with variable rate displacement. (a) Constant-rate relative permeability profiles compared to the actual relative permeabilities tracked by the three embedded networks; (b) an enlarged view of the critical saturation range.

[62] Figure 12 shows saturation profiles along the flow direction for sequential and concurrent coupling after 220 days. Oil recovery ratios (fraction produced out of original in place) are 10.02% for sequential coupling and 8.46% for concurrent coupling. The value is less for concurrent coupling because nonwetting phase relative permeability shifts toward the relative permeability curve for a smaller flow rate in concurrent coupling, as shown in Figure 11a. The difference in recovery ratio is relatively small for two reasons. First, rate has less of an impact on relative permeability than other factors that the coupled model could account for (e.g., wettability alteration). Second, the model was run with only 3 out of 100 coupled gridblocks and without a mechanism to transmit this information to the other gridblocks. If the model could be run with all of the gridblocks coupled, we would expect the entire saturation front in Figure 12 to track more closely with the three lower data points from the coupled simulation. A more general comment to emphasize at this point is that the important result here is not the magnitude of the difference in the coupled versus noncoupled model (or even which is more correct at this stage), but that the coupled model has the ability to follow a unique trajectory of relative permeability values that would not be found in a data set, and that this has an impact on continuum-scale behavior.

Figure 12.

Saturation versus distance after 220 days in the waterflood simulation: comparison of the sequential versus fully coupled results.

4.4.2. NAPL Displacement With Step Up in Rate

[63] A second illustration for the coupled simulation is a simple prototype for groundwater flow near a nonaqueous phase liquid (NAPL) spill. The porous medium used in this simulation is the unconsolidated sand described above. The NAPL concentration is higher in a central region of gridblocks (initial saturation inline image in gridblocks 31–60), and lower outside this region ( inline image in gridblocks 1–30, 61–100). The central gridblocks represent a region where the spill is most concentrated. The outer gridblocks might represent either a region where concentrated NAPL existed previously but was then brought to residual saturation via natural groundwater flow, or a region that NAPL migrated to from the central zone. In the simulation, water flows slowly (3.56 × 10−3 m3/s or a corresponding Darcy velocity of 3.8 × 10−6 m/s) for the first 800 days, and then jumps to a value 10 times higher at day 800. Two gridblocks contain embedded networks: gridblock 50 inside the concentrated zone and gridblock 70 outside of it (downstream).

[64] Figure 13 shows continuum-scale saturation profiles in gridblocks 50 and 70 for concurrent coupling. It also shows the flow rate, which increases tenfold on day 800. Gridblock 50 shows little change for the first 200 days, as the water injection front has not yet reached this location. The decrease in NAPL saturation is steady but slow between 200 and 800 days. It becomes dramatically more pronounced at day 800. The change is associated largely with the increased displacement rate but the coupling means that it will also capture any improvement in displacement efficiency that would occur because of the shift to a higher capillary number regime for immiscible displacement. Gridblock 70 is downstream of the concentrated NAPL zone. Hence, water saturation decreases dramatically as the NAPL is pushed through this region. It then begins to increase with similar behavior as the upstream gridblock.

Figure 13.

Saturation and flow rate profiles for the NAPL remediation simulation. Saturation plots are for a gridblock in the concentrated spill zone (gridblock 50) and a gridblock downstream of this zone (gridblock 70). Flow rate is lower for the first 800 days, then increased by a factor of 10 at day 800.

[65] Figure 14a is again used to address the impact of the full coupling. It provides a comparison of two purely continuum scale simulations run with exactly the same conditions, but with two different relative permeability curves: one associated with the higher displacement rate, one with the lower rate. A distinct difference exists between the two simulations, cased solely by the rate dependence of the relative permeability curves (neither of which is correct for the entire simulation). In principle, one could of course switch between relative permeability curves at the flow transition point. However, this approach will work only if relative permeability curves exist for all potential operating conditions (and can be parameterized in terms of the conditions that may vary in the simulation). Figure 14a also shows saturation in the network model that is embedded in gridblock 70. It tracks the low flow rate simulation more closely at early times and the high flow rate simulation more closely at later times, as might be expected. The unsmooth behavior in the range 500–800 days is caused by numerical instabilities in the pore-scale model, which are most pronounced in a dynamic multiphase model when operating at low wetting-phase saturations and low capillary number conditions. For completeness, we include Figure 14b, which tracks relative permeability values in gridblock 70 during the displacement. These plots illustrate the jump from one curve to the other at day 800.

Figure 14.

(a) Saturation versus time for gridblock 70 (downstream of the concentrated zone). Comparison of two sequential coupling simulations with one concurrent simulation. The sequential simulations were performed using different relative permeability curves corresponding to the low and high injection rates. The concurrent simulation accounts for the different injection rates automatically. Saturation for the concurrent plot is from the embedded network. (b) The two relative permeability curves (high and low rates) and relative permeability tracked by the network model.

[66] Figure 15 shows saturation profiles along the flow direction for sequential (low flow rate simulation) and concurrent coupling after 1000 days. As with the waterflood example, the sequential versus concurrent coupling lead to different amounts of NAPL recovered at the extraction well: 1.95% for sequential coupling and 3.68% for concurrent coupling at day 1000. The value is higher for the concurrent simulation because the nonwetting phase relative permeabilities shift to higher values once the flow rate is increased.

Figure 15.

Saturation versus distance along the injection direction at day 1000 for the NAPL example. Comparison of sequential versus concurrent coupling.

5. Conclusions

[67] The potential benefits of integrating first-principles pore-scale models into macroscopic simulators have been understood for years. However, the algorithmic challenges and computational demands associated with this type of multiscale modeling have made it slow to be put into practice, even at the research scale. This work represents the first time that multiphase pore-scale models have been embedded into gridblocks in a reservoir simulation so that they operate concurrently: the network provides just-in-time relative permeability values to the host gridblock in the reservoir simulator so that the relative permeability value reflects current properties and dynamics as well as saturation history.

[68] One recent development that allowed this concurrent multiscale algorithm to be implemented is the ability to perform pore-scale simulation of the simultaneous injection of two fluids under transient or steady-state conditions for nonperiodic domains. Typical boundary conditions for these pore-scale simulations are the total volumetric flow rate and the fractional flow of the two immiscible fluids. The ability to apply these boundary conditions to the pore-scale model allows a pseudo-steady-state assumption to be used to reconcile the large difference in time and length scales between the reservoir and the pore-scale models. Furthermore, the ability to apply these boundary conditions to nonperiodic domains allows pore-scale phase distributions to evolve naturally during the simulated timescale. This means the network model(s) contain implicit information about saturation history and can account for behavior associated with both hysteresis and saturation history that otherwise could not be modeled. In operation, the communication of current conditions from the reservoir to the pore scale is via current fractional flow; communication from the pore to continuum scale is via continuum-scale parameters predicted from the pore-scale model, such as relative permeability.

[69] From an algorithmic standpoint, the most important issue to emerge from the research is the challenge in maintaining consistent saturations between the pore scale and the continuum models. This problem occurs because no explicit constraint exists in the governing equations to ensure that gridblock phase saturation is equal to pore-scale phase saturation. Phase saturation in a gridblock is dictated by the governing mass conservation equations; it evolves during a simulation based on the initial condition in the gridblock combined with the subsequent inlet and outlet fractional flows. Phase saturation in an embedded pore-scale model is dictated by current fractional flow values obtained from the gridblock and then applied as boundary conditions in the pore-scale model. In principle, the pore-scale model and its host gridblock should have the same saturation at any point in the simulation. However, the two models can have different fractional flow versus saturation functionality, which can lead to differences in pore-scale versus continuum-scale saturation in the multiscale model. This problem can be corrected by taking steps to reconcile differences between the two curves and by reducing gridblock size (to reduce the difference in fractional flow across a gridblock).

[70] The concurrent multiscale approach was demonstrated using simulations of rate-dependent behavior. However, the advantages of this approach are not limited to this application. The ability for a reservoir-scale model to obtain continuum parameters directly from a first-principles model enables it to respond to any number of phenomena that are not captured by simple empirical relationships or tabulated parameters. Examples include permeability loss due to deep bed filtration, flow instabilities causes by chemical injection, or changes to the porous media itself due to subsidence, fracturing, or acidizing. A more subtle advantage of the concurrent multiscale modeling shown here is the ability for the pore-scale model to serve as an implicit record of the saturation history. Hence, parameters that exhibit hysteresis or history dependence respond to the underlying physics that cause these effects.

[71] Practicality remains a significant problem because of the computational burden added by operating the dynamic pore-scale models concurrently with the reservoir model time steps. The simulations run in this work used small network models, and they were placed in only a few gridblocks. If more fundamental computational models were used (for instance Lattice Boltzmann or traditional Computational Fluid Dynamics (CFD) flow models), the problem would be exacerbated. However, there are still reasons to be positive. First, the steady-state dynamic multiphase flow simulations used in this work are the most computationally demanding of the many network modeling algorithms that we have tested (especially at low wetting-phase saturations or flow rates). Other network modeling algorithms would not require the same computational load on the pore-scale side. It is also important to consider that the pore-scale code used here is a research code, and the main efforts to date have been on reproducing physics rather than optimizing computational performance. Additionally, we have run selected tests in which communication with the embedded networks occurs less often (i.e., not at every reservoir time step). Preliminary results show that considerable speed-up can occur while maintaining good accuracy.

Acknowledgments

[72] The microCT data of the reservoir sandstone were provided by Joanne Fredrich (BP). The unconsolidated sand was imaged at LSU's CAMD facility, which has a synchrotron microCT beamline. We thank Pradeep Bhattad, Kyungmin Ham, and Clinton Willson for obtaining this data set. This work was funded by BP and Schlumberger as part of the PoreSim consortium.