1 Introduction
Unconventional reservoirs have received great attention as a primary energy resource in the past decade worldwide. Economic production from these reservoirs depends on effective stimulation by means of hydraulic fracturing. Microseismic measurements and other evidence suggest the creation of complex fracture networks that connect huge reservoir surface areas to the wellbore (Maxwell et al. 2002; Fisher et al. 2002; Mayerhofer et al. 2010). In terms of reservoir development and management, numerical simulation continues to play a critical role in evaluating and optimizing the stimulation and production processes (Cipolla et al. 2010b; Weng et al. 2014).
Because of the ultralow permeability of matrix, a long period of transient flow occurs in unconventional formation. The extreme contrast in conductivity between matrix and fracture also results in steep potential gradients that are difficult to capture. Therefore, detailed nearwell/nearfracture models are necessary to provide sufficient resolution for the matrixfracture interactions (Mayerhofer et al. 2010; Cipolla et al. 2010a; Cipolla et al. 2010b). However, a fine grid simulation requires too much CPU time and it is impractical to perform for the entire domain in field cases with multiple hydraulicfracture stages (Ding et al. 2014; Weng et al. 2014; Panfili et al. 2015).
Several modeling approaches for fracturedwell have been proposed in the literature, attempting to improve the solution efficiency while maintaining the accuracy. One simple approach is applying local grid refinement (LGR) on a coarse background grid (Mallison et al. 2010; Cipolla et al. 2010b; Artus and Fructus 2012; Panfili et al. 2015; Jiang and Younis 2015b; Jiang and Younis 2016; Xue et al. 2019). A number of meshing algorithms are available to generate adaptive and optimized mesh with good quality around discrete fractures. Pruess and Narasimhan (1985) developed a subgridding method called multiple interacting continua (MINC) by subdividing each matrix cell according to the distance from fractures. The MINC method is intended to improve the classical dualporosity model (Warren and Root 1963; Kazemi et al. 1976), and subsequently better characterize the transient effects with large solution gradients. In recent years, MINC is widely applied and extended for the simulations of unconventional reservoirs (Hui et al. 2013; Wu et al. 2014; Jiang and Younis 2015a; Jiang and Younis 2016; Farah et al. 2015; Ren et al. 2016; Ding et al. 2018). In addition, Ding et al. (2014) proposed a coupled modeling method that combines a coarsegrid reservoir model with detailed nearfracture models. The two models are solved, and the associated boundary conditions are updated in an alternate mode. The coupled method can be viewed as dynamic upscaling that computes the timedependent fracture index for the coarse domain. Although some promising results were presented, the detailed models may still take a large fraction of the overall computational expense.
In reservoir simulation, the Fully Implicit Method (FIM) is often the method of choice for the temporal discretization of the conservation equations (Aziz and Settari 1979; Coats 1980). FIM offers unconditional stability, but it requires the solution of large coupled nonlinear systems. For a target timestep, a sequence of Newton iterations is performed until convergence. This iterative process is expensive and can account for a significant fraction of the total cost. Here define
as the total number of degrees of freedom in a system. Considering the costs of computing the residual vector, Jacobian matrix, and thermodynamic properties, the overall complexity of a nonlinear iteration is generally superlinear in
(Younis et al. 2010).Previous works in the literature of unconventional simulations mainly focus on the gridding level that adapts to wells and fractures. Limited research has been conducted on solver techniques that exploit locality across timesteps and nonlinear iterations. The benefit of this type of methods is evident: the solution accuracy is maintained, because neither discretization scheme nor spatial mesh will be modified (Younis et al. 2010; Sheth and Younis 2017). Simulation studies have shown that flow dynamics evolve quite slowly within the ultratight formation. Pressure drop may remain in the vicinity of fractures even after years of production (Cipolla et al. 2010b; Ding et al. 2014; Jiang and Younis 2015b). Conceivably, a significant speedup can be achieved if performing adaptive computations only for the locales that are undergoing changes. In addition to the timestep level, a large degree of locality also presents on the nonlinear (Newton) level. It was reported that an individual Newton update is typically sparse and the nonlinear convergence is constrained by a small portion of the model (Younis et al. 2010; Lu and Beckner 2011; Sheth and Younis 2017). To exploit the locality at each iteration, an apriori
strategy is essential to first identify the active subset of simulation cells. The active set flags the cells that will be updated, and then the corresponding localized linear system is solved. Lu and Beckner (2011) observed that over the course of several iterations, the sparsity pattern of the Newton updates was related to that of the discrete residual vector. They proposed to use nonzero entries in the residual vector as an estimate of the active set for the subsequent iteration. It should be mentioned that their heuristic strategy may suffer from an efficiency issue due to overly conservative estimate. Sheth and Younis (2017) have shown that missing any nonzero update during the localization process may lead to worse nonlinear convergence, comparing to the standard Newton method. A theoretical framework was then developed to predict the sparsity pattern of Newton updates. Analytical derivations were made to ensure a conservative estimate of the active sets. The results in Sheth and Younis (2017) demonstrated that their localization method performs quite well for several challenging models.
In this work we do not intend to rely on an analytical derivation or conservative estimate for the active sets. The objective is to develop localization methods that readily accommodate to complex fracture networks and flow physics for the simulations of unconventional reservoirs. Through aggressive localization, the computational speedup is expected to be greatly improved. We recently reveal that the Newton updates for pressuredriven problems exhibit diffusive and global behaviors, because of its parabolic nature (Jiang and Tchelepi 2018). By utilizing the nature of pressure updates, an adaptive algorithm is proposed to make adequate estimates for the active domains. In addition, we further develop a localized solver based on nonlinear domain decomposition (DD). Comparing to a standard DD method, domain partitions are dynamically constructed from the previous iterations. During the nonlinear DD process, the subproblems of an iteration are solved sequentially, and thus the localization can be naturally achieved. This leads to a reliable strategy that exploits the locality while preserving the convergence behavior of the standard Newton process. Note that the two methods developed involve different complexities and efforts for implementation. Subsequently, their applications depend on specific efficiency and implementation considerations.
We evaluate the localization methods using several complex problems with discrete fracture networks. The test problems consider multiphase and compositional fluid systems with phase changes. The results show that large degrees of solution locality present across timesteps and iterations. Comparing to a standard Newton solver, the new solvers exhibit superior computational performance. Moreover, the Newton convergence behavior is preserved, without any impact on the solution accuracy.
2 Isothermal compositional model
We consider compressible gasoil flow in porous media without capillarity. We ignore water that does not exchange mass with the hydrocarbon phases.
The conservation equations for the isothermal compositional problem containing components are written as,
(1) 
where . and are molar fractions of component in the oil and gas phases, respectively. is rock porosity and is time. is phase molar density. is phase saturation. is well flow rate.
Phase velocity is expressed as a function of phase potential gradient using the extended Darcy’s law,
(2) 
where is rock permeability. is pressure. Capillarity is assumed to be negligible. is gravitational acceleration and is height. Phase mobility is given as . and are relative permeability and viscosity, respectively.
In order to close the nonlinear system, additional equations are needed. These include the thermodynamic equilibrium constraints,
(3) 
where , , and denote pressure, temperature, and overall molar fraction, respectively. is the fugacity of component in phase .
We now write the phase constraints,
(4) 
and the saturation constraint as,
(5) 
The above system of equations provide a complete mathematical statement for twophase multicomponent flow. The local equilibrium constraints are enforced only when both phases are present.
3 Naturalvariables formulation
An important aspect of any compositional formulation is the choice of dividing the equations and unknowns into primary and secondary sets. Here we employ the popular naturalvariables set (Coats 1980; Voskov and Tchelepi 2012). The primary unknowns include pressure, saturations, and molar fractions,
(1) pressure [1],
(2) phase saturations [2],
(3) , phase compositions of each component [2].
The size of each variable is given in square bracket.
The various coefficients can be obtained as functions of the base variables. For a twophase cell, the molar phase fraction is related to saturation as follows,
(6) 
and overall molar fraction of component is written as,
(7) 
Note that for singlephase () mixture, , and .
3.1 Variable substitution
An essential ingredient of the naturalvariables formulation is the ‘variable substitution’ process (Aziz and Wong 1989; Cao 2002; Voskov and Tchelepi 2012). A common strategy for variableswitching between Newton iterations during a timestep is,
1. For any cell whose status in the previous iteration is singlephase, run the phase stability test (Michelsen 1982a) to check if the mixture becomes twophase. For the mixture that splits into two phases, perform the flash to compute the phase compositions (Michelsen 1982b).
2. If a cell is already in the twophase state, the thermodynamic constraints are included in the nonlinear system as part of the global Jacobian.
3. If a phase saturation, or phase fraction, becomes negative between two successive iterations, the phase disappears, and appropriate variableswitching is performed.
The system of conservation equations is solved for singlephase regimes, and the combination of conservation equations and thermodynamic constraints is solved for the twophase regime.
3.2 Phase behavior
Phase behavior computation is usually a standalone procedure for detecting phase changes. For a mixture of components and two phases, the mathematical model describing the thermodynamic equilibrium is (Voskov and Tchelepi 2012),
(8) 
(9) 
(10) 
where is molar fraction of phase . We assume that , , and are known. The objective is to find all the , and .
Phase behavior of a hydrocarbon mixture is commonly described using an Equation of State (EoS) model.
4 Nonlinear solution strategies
The spatial and temporal discretization schemes used for the compositional flow model are summarized in Appendix A.
4.1 Newton method
At each timestep of a FIM simulation, given the current state , and a fixed timestep size , we seek to obtain the new state .
The nonlinear residual system is solved by the Newton method,
(11) 
The Newton method generates a sequence of iterates, , , each involving the construction of a Jacobian matrix and solution of the resulting linear system,
(12) 
where
(13) 
and denotes the Jacobian matrix of with respect to .
Here we assume that entries of Newton update such that are essentially negligible. Given a Newton iteration, the support set is defined for the indices of cells that exhibit nonzero update,
(14) 
4.2 Locality within solution processes
In this work we focus on the solution process for the pressuredriven production problem with multiphase multicomponent fluid. We aim to exploit two levels of locality for improving computational efficiency. The first is on the timestep level. Because of the ultralow matrix permeability in unconventional formation, transient flow within matrix may last a long period. As a result, flow dynamics (e.g. pressure propagation) evolve slowly and locally. During early stage of production, only a small portion of domain undergoes considerable variable changes.
For a timestep, the solution update is the sum of all the corresponding Newton updates. Conceivably, the locality also presents on the nonlinear (Newton) level, even if most of the domain is affected over the timestep. Previous works showed that an individual update computed for flow and transport problems is typically sparse and constrained by a small subset of cells (Younis et al. 2010; Lu and Beckner 2011; Sheth and Younis 2017).
Here we show an example with twofracture to demonstrate the solution behavior. The details of the model will be given in the result section. The flagging profiles for the Newton iterations of a timestep are plotted in Fig. 1. The cells that exhibit nonzero pressure updates are flagged in color blue. As we can see, the first iteration reaches the maximum area, and the region gets smaller as the iterations proceed. In the last iteration, the support set of the updates localizes to just a few cells.
Lu and Beckner (2011) proposed an adaptive Newton strategy that solves localized systems. Their method identifies unconverged cells and their neighbors as the active subset to be updated. The unconverged set is inflated to a heuristic extent (e.g. plus one layer of neighbor cells), as a safety measure.
From the simulation studies, we also observe that under a convergent Newton sequence, the support set of updates always shrink after each iteration, such that,
(15) 
Therefore, the support set from a current Newton iteration becomes an adequate estimate for the subsequent iteration.
The adaptive algorithm by Lu and Beckner (2011) is simple to implement. However, the algorithm starts with the entire domain at the first iteration of a timestep, to ensure conservative estimates for the active set. This will considerably contribute to the overall computational cost.
Our recent studies revealed that Newton iterations are closely tied to the underlying physics problem (Jiang and Tchelepi 2018). The updates for the hyperbolic transport problem may have local support that propagates through the domain as the Newton process goes forward. By comparison, the support of the pressure (flow) problem shows diffusive and global behaviors, due to its parabolic nature. To exploit this mechanism, here we propose a localized Newton strategy, which is aggressive in the way that it does not require an initial conservative estimate for the active domain. We observe that the support of pressure updates tends to reach the solution front at the first iteration. Therefore, the localized algorithm can start with a moderate active domain , and expand it if necessary using the outermost (boundary) layer of .
4.3 Localized Newton algorithm
We describe the algorithm based on the proposed localized Newton strategy. Consider a nonlinear system of equations with unknowns . Let be an index set; i.e., there is one integer for each and unknown . Let be the index set that contains the active cells, and be the dimension of . We define as the cell set for the outermost (boundary) layer of the active domain. Further define the set for the neighbors of cell , and is the number of layers that are incorporated. The illustration for with and is plotted in Fig. 2. The neighbor cells are flagged in yellow.
Let be a Boolean matrix of dimension . corresponds to the restriction operator from to . The transpose matrix is an extension operator from to . Then the nonlinear function and the local unknowns of the active set can be expressed as and , respectively. The Jacobian of is,
(16) 
Note that the submatrix can be directly constructed, and thus the full matrix is never needed.
The localization method takes as input a state at time level and outputs the updated state. The algorithmic details can be described as,

Set the iteration counter to zero, and initialize to the current state . Construct the initial active set and the associated boundary set .

Perform localization: solve the reduced linear system over the active domain, and update the solution.

For each cell in , if the update is larger than the cutoff value, inflate with the neighbor subset .

If there is any nonnegligible update in , the localization status stays in the ‘expand’ mode; otherwise, switch to the ‘shrink’ mode, and is specified as the support set of .

Check convergence criteria of Newton iterations. If not converged, go back to Step 2. Otherwise, the timestep is finished.
The proposed method is also outlined in Algorithm 1. We can see that the localization process determines the active set that needs to be solved for the subsequent iteration. The process consists of two stages. In the first stage, does not fully cover the actual support of the timestep. Based on the diffusive nature of pressure updates, is used to detect and expand the active domain. The localized algorithm is selfadaptive in a sense that Newton update already provides an adequate estimate.
The initial active set for the algorithm can be constructed to comprise the cells in the vicinity of wells or fractures. Subsequently, computational speedup will be greatly increased. On the other hand, the iterations taken during the ‘expand’ mode may degrade the nonlinear convergence, comparing to the standard Newton method. In practice, a balance needs to be achieved between more aggressive localization and increased number of iterations.
It is worth to note that the iterative process for expanding active domain can be viewed as a nonlinear domain decomposition (DD) problem. Each subproblem is solved with Dirichlet boundary (constant pressure) conditions. Similarly to nonlinear preconditioning, the subdomain solutions can provide better initial guesses for Newton iterations. As a result, the localized solver shows satisfying convergence performance from the simulation cases.
4.4 Adaptive Nonlinear Domain Decomposition
For a localized Newton algorithm, more iterations may be required, if the estimate for the support set is not conservative. On the other hand, an excessively conservative estimate will reduce the speedup gained from the localized computations.
In this work we further develop an adaptive method that provides aggressive localization, while preserving the convergence behavior of the standard Newton process. To present the method, consider first a nonlinear domain decomposition (DD) with nonoverlapping partitions,
(17) 
Let be the total number of unknowns and be the total number of unknowns associated with the subset . The restrictions of and to are and , respectively. The Jacobian of the subproblem is given as,
(18) 
with . The boundary conditions for a subproblem are Dirichlettype and taken from the neighboring subdomains.
A global solution of the nonlinear DD method is obtained by solving first subproblems and then gluing them together (Dolean et al. 2016),
(19) 
It is common to apply the additive (Jacobi) form of the Schwarz methods. For a standard DD, static partitions of simulation grid are performed in a preprocessing step (Cai et al. 1998; Skogestad et al. 2013; Dolean et al. 2016).
To exploit the localized behaviors of flow problems, we propose instead an adaptive DD solver based on dynamic partitions. Utilizing the diffusive nature of pressure updates, subdomains are constructed from the previous iterations. To achieve localized computations, the subproblems of a nonlinear DD iteration are solved sequentially. This leads to a multiplicative (GaussSeidel) Schwarz method.
The algorithmic details of the adaptive DD method for a timestep are given as,

Set the iteration counters and to zero, and initialize to the current state . Construct the initial active set , and the associated boundary sets and . Define as the outer layer adjacent to , such that .

Start expanding the active domain, and perform localized computations. Construct the subdomains, and the associated cell sets, using the Newton updates. Specifically, for each cell in , if the update is larger than the cutoff value, inflate and the total active set with the neighbor subset .

If there is any nonnegligible update in , remain in the ‘expand’ mode, and obtain the boundary sets of . Otherwise, switch to the ‘shrink’ mode, indicating that the maximum support set over the timestep is reached.

Set as the number of the constructed subdomains. Start the nonlinear DD loop, with the counter denoting the outer iteration. For each , first collect over , from the latest Newton updates.
If there is any nonnegligible element in or , perform localized computation and update the solution.

Check convergence criteria. Repeat the outer iteration until all the subdomains are converged.
The new adaptive solver is also outlined in Algorithm 2. Note that a subdomain needs to be solved only when the solution is not yet converged or the boundary values change. Therefore the localization is naturally achieved during the nonlinear DD process. This leads to a reliable strategy to exploit the locality prior to each outer iteration.
An outer iteration of the DD method can be written in a fixedpoint form,
(20) 
where the solution operator for a subproblem is,
(21) 
As can be seen, evaluation of the function involves the solution of all the subproblems . Despite its simple form, the fixedpoint method may suffer from slow convergence, or even divergence (Skogestad et al. 2013; Dolean et al. 2016). Recently we proposed several ways of accelerating the nonlinear DD process (Jiang and Tchelepi 2019). The nonlinear acceleration techniques greatly improve the outer convergence behavior, while requiring little additional cost. The investigation on the outer convergence of the adaptive DD solver is subject to a future work.
5 Results
We evaluate the efficacy of the localization methods using several test problems with discrete fracture networks. The problems include an oilwater system and a twophase compositional system with phase changes. A 2D synthetic model is generated to contain a singlestage hydraulicallyfractured horizontal well at the center of a reservoir. The fractures are assumed to fully penetrate the formation.
An embedded discrete fracture model (EDFM) is employed to explicitly describe the discrete fractures. Lee et al. (2001), Li and Lee (2008), Hajibeygi et al. (2011) and Moinfar et al. (2014) introduced and extended EDFM, which does not require simulation grid to conform to fracture geometry. Recent works on the implementations of EDFM for various types of problems include Panfili et al. (2015), Jiang and Younis (2016, 2017), Ren et al. (2018), Hui et al. (2019), Xue et al. (2019), and Rey et al. (2019).
A simple timestepping strategy is employed: starting with a small initial value, timestep sizes gradually increase to the maximum value. Newton convergence is based on the following criterion: solution (pressure) delta (increment) between iterations. The specification of the base model is given in Table 1.
Parameter  Value  Unit 

Initial pressure  2500  psi 
Matrix porosity  0.05  
Rock compressibility  3.4e4  1/psi 
Matrix permeability  1e19  
Fracture permeability  1e10  
Fracture aperture  1e3  m 
Production BHP  1000  psi 
Total simulation time  1500  day 
Max timestep size  100  day 
5.1 Grid sensitivity
We first test a model with two fractures to demonstrate the effect of transient flow. Initial water saturation is set as the connate saturation, so that water is immobile during simulations. Newton tolerance has the value of . Simulations are run for three different levels of grid resolution.
Pressure profiles at the end of simulation are shown in Fig. 3. Oil rates are plotted in Fig. 4. As we can see, oil productions are largely underestimated by the coarse grid systems. The large cell sizes of coarse grid are not adequate for the sharp pressure variations in the vicinity of the fractures. On the other hand, the finegrid case involves a large number of cells and thus requires significant computational efforts.
5.2 Localized Newton method
5.2.1 Case 1
We test the model with grid level. At every iteration, the localized Newton algorithm 1 provides the active set to be updated, and then solves the reduced linear system. Convergence tolerance of is employed as the cutoff value for the active and boundary sets. The neighbor set with is used to expand active domain. The support set of the last timestep is taken as the initial active set for the current timestep.
Pressure profile is shown in Fig. 5. We can observe that only a small fraction of the cells around the fractures undergoes significant changes in pressure. Oil rates are plotted in Fig. 6. As expected, the solution from the localization method exactly matches the reference solution. This is because the iterative processes converge under the same convergence tolerance.
We plot the ratios of active domain (per timestep) versus simulation time in Fig. 7. Note that 24 iterations are taken at each timestep. Computational performance of Case 1 is summarized in Table 2. is defined as the ratio of the active to full sets. For the standard Newton, the total ratio is equal to the total iteration number, with the average ratio as 1.
The results show that the localization method exhibits good convergence performance. As we noted before, the iterations for expanding active domain can be viewed as nonlinear preconditioning which provides better initial guesses. The localized solver achieves an at least 13fold reduction in computations, comparing to the standard solver. The actual simulation speedup depends on the scaling of computational complexity .
Timesteps 

Total ratio 



Localized Newton  54  159  11.4  0.072  
Standard Newton  54  156  156  1 
We rerun the case using the localized Newton for one timestep with the size of 50 days. The timestep size is equal to the total simulation time. The profiles for the flagging of cells over the 4 iterations are plotted in Fig. 9. Four types of cell sets are specified: 1. active set (in color yellow); 2. boundary set (black); 3. the active cells with nonnegligible update (the support set), (blue); and 4. the boundary cells with nonnegligible update (red). A color illustration for the cell types is given in Fig. 8.
As can be seen, the initial active set is small and not conservative, resulting in aggressive localization and thus high computational speedup. As the iterations proceed, the active domain expands until the maximum area is reached.
5.2.2 Case 2
We test a model with grid and a more complex fracture network. The model contains 4 secondary fractures with permeability of 1e13 . Pressure profile is shown in Fig. 10. We plot the ratios of active domain (per timestep) versus simulation time in Fig. 11.
Computational performance of Case 2 is summarized in Table 3. From the results we see that the localization method enables a significant reduction in computations, while preserving the original convergence behavior.
Timesteps 

Total ratio 



Localized Newton  54  155  10.1  0.065  
Standard Newton  54  153  153  1 
We rerun the case for one timestep with the size of 50 days. The profiles for the flagging of cells over the 4 iterations are plotted in Fig. 12. The color illustration of cell sets is the same as specified in the previous section. As can be seen, the sparsity patterns of the updates vary largely from one iteration to the next. After two iterations, the nonlinear convergence is constrained to just several cells.
5.2.3 Case 3
We again consider the model from Case 2, but with grid. The model uses a twocomponent fluid system where the initial oil is made of . Phase density and viscosity depend on pressure and compositions. Phase molar density is evaluated based on the compressibility (Z) factor from the PR EoS. Phase viscosity is computed by the correlation of Lohrenz et al. (1964). Simple relative permeabilities given by quadratic function are used. Initial pressure is 2900 psi and temperature is 340 K. The total simulation time is 500 days, with the maximum timestep size as 50 days. The other parameters in the previous case remain unchanged.
The profiles for the phase status and saturation of gas are shown in Fig. 13 and Fig. 14, respectively. As the pressure drops below the bubblepoint, gas starts to appear around the fractures. The transient effects of saturation and phase dynamics can be difficult to capture using a coarse grid. From the results we observe that the saturation updates exhibit considerable locality. By comparison, the pressure updates affect a much larger region.
We plot the ratios of active domain (per timestep) versus simulation time in Fig. 15. Computational performance of Case 3 is summarized in Table 4. The localized Newton method shows superior performance, with a small increase of iterations.
Timesteps 

Total ratio 



Localized Newton  40  153  16.1  0.1  
Standard Newton  40  148  148  1 
5.3 Adaptive Nonlinear DD method
We study the adaptive nonlinear DD (Algorithm 2) which is based on dynamic partitions to perform localized computations. The developed solver can make adequate estimates of the active set for each inner iteration.
5.3.1 Case 2.1
We use the same model as specified in Case 1. Computational performance of the case is summarized in Table 5. The adaptive DD solver greatly reduces the computational cost, while taking much more iterations. This is because each outer iteration consists of multiple inner iterations in the nonlinear DD process. We note that the size of a subdomain system is relatively small and the total number of outer iterations is comparable to the standard Newton method.
Timesteps 

Total ratio 



Adaptive Nonlinear DD  54  804  20.8  0.026  
Standard Newton  54  156  156  1 
We plot the ratios of active domain (per timestep) versus simulation time in Fig. 16.
We rerun the case for one timestep with the size of 50 days. The neighbor set is specified with . Flagging profiles of cells over the 4 iterations are plotted in Fig. 17. As we can see, the timestep converges with 2 outer iterations. The algorithm constructs the subdomains that adapt to the flow dynamics and pressure updates. During the nonlinear DD process, the subproblems are solved sequentially, to achieve localization. The results confirm that the support set of the timestep is contained in the union of all the flagged subsets.
5.3.2 Case 2.2
The model from Case 2 is used. Computational performance of the case is summarized in Table 6. We plot the ratios of active domain (per timestep) versus simulation time in Fig. 18. We can see that the adaptive solver obtains an at least 11fold reduction in solution effort, without degrading the original Newton convergence.
Timesteps 

Total ratio 



Adaptive Nonlinear DD  54  255  13.6  0.053  
Standard Newton  54  153  153  1 
This work develops a prototype algorithm for the adaptive DD solver, which can be further improved and optimized, e.g. exploiting the solution locality within each subdomain. The algorithm will be expected to prevent any overly conservative estimate. Subsequently, higher simulation speedup can be achieved.
6 Summary
We develop the localized solution strategies for efficient simulations of unconventional reservoirs. By utilizing the diffusive nature of pressure updates, an adaptive algorithm is proposed to make adequate estimates for the active sets to be solved. We also develop a localized solver based on nonlinear domain decomposition (DD). The solver provides effective partitioning that adapts to flow dynamics and Newton updates.
We test several complex problems with discrete fracture networks. The results show that large degrees of solution locality present across timesteps and iterations. Comparing to a standard Newton solver, the developed solvers enable superior computational performance. Moreover, the original Newton convergence is preserved, without any impact on the solution accuracy.
The new solvers can be extended to account for more complex fracture networks and flow physics. The incorporation of models with strong heterogeneity and fieldscale fracture networks is a subject of our ongoing research.
Acknowledgements
This work was supported by the Chevron/Schlumberger/Total INTERSECT Alliance Technology. The author also thanks Chevron for permission to publish this paper.
Appendix A. Discretization methods
A standard finitevolume scheme is applied as the spatial discretization for the mass conservation equations. A twopoint flux approximation (TFPA) is used to approximate the flux across a cell interface. The method of choice for the time discretization is the fullyimplicit scheme. The discrete form of conservation equation is given as,
(22) 
where superscripts denote timesteps, and is the timestep size. is the cell volume. All indices related to the cell numeration are neglected. The accumulation term involves the total density,
(23) 
and,
(24) 
where indicates the density computed at a composition x, and is the density computed in the singlephase regime at a composition z.
The discrete phase flux across the interface between two cells is written as,
(25) 
where subscript denotes quantities defined at the cell interface. is the interface transmissibility. is the phase potential difference with the discrete weights . The phase and compositional coefficients associated with the flux terms are evaluated using the PhasePotential Upwinding (PPU) scheme.
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