Numerical simulation of the impact of polymer rheology on polymer injectivity using a multilevel local grid refinement method
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Abstract
Polymer injectivity is an important factor for evaluating the project economics of chemical flood, which is highly related to the polymer viscosity. Because the flow rate varies rapidly near injectors and significantly changes the polymer viscosity due to the nonNewtonian rheological behavior, the polymer viscosity near the wellbore is difficult to estimate accurately with the practical gridblock size in reservoir simulation. To reduce the impact of polymer rheology upon chemical EOR simulations, we used an efficient multilevel local grid refinement (LGR) method that provides a higher resolution of the flows in the nearwellbore region. An efficient numerical scheme was proposed to accurately solve the pressure equation and concentration equations on the multilevel grid for both homogeneous and heterogeneous reservoir cases. The block list and connections of the multilevel grid are generated via an efficient and extensible algorithm. Field case simulation results indicate that the proposed LGR is consistent with the analytical injectivity model and achieves the closest results to the full grid refinement, which considerably improves the accuracy of solutions compared with the original grid. In addition, the method was validated by comparing it with the LGR module of CMG_STARS. Besides polymer injectivity calculations, the LGR method is applicable for other problems in need of nearwellbore treatment, such as fractures near wells.
Keywords
Polymer rheology Polymer injectivity Chemical EOR Local grid refinement NonNewtonian flow1 Introduction
LGR and similar unstructured gridding approaches have continuously played an important role in reservoir simulations. Successful applications can be found in water flood (Oliveira and Reynolds 2014), miscible gas flood (Suicmez et al. 2011), steam flood (Christensen et al. 2004; Nilsson et al. 2005), etc. LGR methods are classified into cellbased and patchbased approaches (Berger and Oliger 1984), while the former is more frequently used in simulations of flow in porous media. Therefore, in the scope of this paper, we only discuss the cellbased LGR approach. Forsyth and Sammon (1986) developed an LGR algorithm with a rigorous analysis of discretization of flow equations upon the composite grid geometry. However, the accuracy of their numerical scheme is reported to be low because a direct subtraction of pressures of two adjacent blocks is used to calculate the Darcy velocity across the block interface (Rasaei and Sahimi 2009). Nacul et al. (1990) proposed an LGR technique using a domain decomposition method, in which overlapping boundaries are used for the subdomains. KarimiFard and Durlofsky (2012) presented an unstructured LGR method, and the well block is fully refined and solved at a fine scale to determine the effective properties that can be used for coarsegrid simulations over the reservoir domain.
 (a)
An efficient numerical scheme developed to calculate the velocity and the mass flux across the block interface between different grid levels of the composite grid, which is also applied in the heterogeneous cases.
 (b)
An algorithm on how to index the gridblock list and gridblock connections under the LGR composite grid presented in detail. The numerical computations under the LGR grid structure can benefit from this data management, which may also be extended to the classical unstructured grid and provide a good basis for the successive simulator development.
This paper is organized as follows: In the next two sections, we will give the mass balance equations and the chemical flood simulation models. The subsequent section presents the details of the proposed efficient LGR algorithm. We will then test several examples simulated with different levels of refinement to demonstrate the improvement in numerical results. The LGR simulations are also compared to those using the analytical injectivity model proposed by Li and Delshad (2014).
2 Mathematical model
In this section, we briefly present the mathematical framework of the University of Texas Chemical Flooding Simulator, UTCHEM (Delshad et al. 1996) and formulations for modeling polymer rheology and injectivity. UTCHEM is a threedimensional multiphase multicomponent compositional simulator with the capability of modeling geochemical reactions, complex phase behavior, etc. The governing balance equations include (1) the mass conservation equation for each species; (2) the pressure equation obtained by summing up all mass conservation equations for all volumeoccupying species; and (3) the energy conservation equation which will not be discussed here.
2.1 Mass conservation equations
2.2 Pressure equation
2.3 Rheological viscosity of the polymer solution
2.4 Analytical polymer injectivity model
In traditional simulation models, the polymer solution viscosity (μ _{w,wb}) of the well block is directly calculated from Eqs. (10) or (11), using the averaged shear rate of the block. Thus, the shear rate is smeared and consequently gives significant error in well injectivity depending on the flow rate and the size of the gridblocks.
3 UTCHEM flowchart
In each time step, the simulator first solves the pressure equation (Eq. 7) implicitly and then solves concentration equations for each component (Eq. 1) explicitly using a thirdorder scheme with a flux limiter. After that, phase behavior calculations will be performed if a surfactant is present. In the last step, properties are updated by taking into account water reactions and polymer adsorption, as well as other chemical and physical changes. All the newly updated variables and properties will be provided for the initial values of the next time step. This continues until it reaches the final time.
4 Local grid refinement algorithm
The current form of the UTCHEM simulator is developed based on a structured grid, and the use of LGR will transform the grid from structured to unstructured as the connections between blocks are no longer regular. This makes it necessary to change the original data structure and computational model for solving the pressure equation and concentration equations.
4.1 Block list and connections
Mutual indexing of block list and connections
Block No.  Connections (xdirection)  Connections (ydirection)  Connection No. (xdirection)  Block pair  Connection No. (ydirection)  Block pair 

1  1  1  1  1, 2  1  1, 4 
2  1, 2  2  2  2, 3  2  2, 5 
3  2, 5, 3  3  3  3, 8  3  3, 7 
4  4  1, 4  4  4, 5  4  4, 6 
5  4, 5  2, 5  5  5, 3  5  5, 6 
6  6  4, 5  6  6, 7  
7  6, 7  3  7  7, 8  
8  3, 7  – 
The indexing of a block list and connections facilitates the search for neighboring blocks and the assignment of properties evaluated at the block interfaces, such as transmissibility, velocity, and mass flux using the list of connections.
4.2 Coupling of governing equations
As the IMPEC scheme is used, the pressure equation and concentration equations are solved separately during computations.
4.2.1 Coupling of the pressure equation
To calculate the fluxes across the block interfaces, such as \(u_{(m)}\) and \(u_{(n)}\), an early approach (Forsyth and Sammon 1986) used the pressures at the block centers to obtain the pressure difference in Darcy’s law. However, it was pointed out that it generated high truncations (Rasaei and Sahimi 2009). Gerritsen and Lambers (2008) proposed in their anisotropic grid adaptivity method to use bilinear interpolation to obtain pressures of the auxiliary points (such as \(P_{{\left( {i1} \right)}}\) and \(P_{{\left( {i2} \right)}}\) in Fig. 6) for calculating the interfacial velocity using Darcy’s law. This method proves to be secondorder accurate when solving the pressure equation for homogeneous cases. However, the accuracy of bilinear interpolation is insufficient for heterogeneous cases because the discontinuity of the pressure gradient across the block interface is not taken into account. Actually, handling heterogeneity is an important factor to weigh up the reliability of the numerical scheme. As far as we know, there has not been a rigorous numerical scheme in the scope of the cellcentered finite volume method for accurately coupling the pressure equations with the LGR composite grid.

It has a simple form as it does not require any additional information from other blocks except for the current three connected blocks.

It is easy to use as it does not need any interpolation/extrapolation.

It is based on the continuity of mass flux across the interfaces and it is rigorously selfconsistent under the homogeneous condition or the condition that fineblock permeabilities are identical. The latter condition is often met for most LGR applications when the permeabilities of the refined blocks are directly from the coarse block permeability.
4.2.2 Coupling of mass conservation equations
5 Case study
To validate the LGR method proposed in this paper, we tested four simulation examples. These examples show comparisons of simulation results using the LGR method with those using the analytical polymer well model, and full grid refinement (FGR) where the whole model has the smallest grid size of the LGR.
5.1 Case 1: Polymer flooding in a 2D homogeneous reservoir
Reservoir and well descriptions (Case 1)
Model description  Values 

Reservoir size  450 ft × 450 ft × 10 ft 
No. of gridblocks  15 × 15 × 1 
Simulation time, day  365 
Number of components  3 
Permeability in the x or y directions, mD  300 
Initial water saturation  0.35 
Polymer rheology exponent P _{ α }  1.8 
Shear rate at half zerorate viscosity γ _{hf}, s^{−1}  10 
Wells  1 injector; 1 producer 
Injection rate, ft^{3}/day  500 
Producer bottomhole pressure (BHP), psi  1000 
Water injection  0–150 and 270–365 days 
Polymer injection  150–270 days (0.3 wt%) 
CPU times for UTCHEM and CMG_STARS using different grids for Case 1
Grid  Original grid (225 gridblocks)  4level LGR (471 gridblocks)  4level FGR (14,400 gridblocks) 

CPU time  
UTCHEM  1 min 26 s  3 min 40 s  141 min 39 s 
CMG_STARS  4 min 14 s  9 min 12 s  318 min 45 s 
5.2 Case 2: Polymer flooding in a 2D reservoir with a fracture near the injector
5.3 Case 3: Polymer flooding in a 3D heterogeneous reservoir
Reservoir and well descriptions (Case 3)
Model description  Values 

No. of gridblocks  17 × 21 × 25 
No. of components  6 
Total injection volume for simulation, PV  0.32 
Polymer injection volume, PV  0–0.16 (0.2 wt%) 
Water injection volume, PV  0.16–0.32 
BHP, psi  Injectors 4500; Producers 700 
5.4 Case 4: A pilot of alkaline cosolvent polymer (ACP) flood
Reservoir and well descriptions (Case 4)
Model description  Values 

Reservoir dimension  5512 ft × 4856 ft × 98 ft 
No. of gridblocks  42 × 37 × 5 
No. of components  12 
Total simulation time, day  7300 
Optimum salinity, meq/mL  0.26 
Wells  6 injectors; 22 producers 
ACP injection  0–3650 days 
1.5 wt% cosolvent  
0.275 wt% polymer  
Polymer injection  3650–7300 days 
0.225 wt% polymer 
CPU times for Case 4 using original grid, LGR, and FGR
Grid  Original grid (7770 gridblocks)  3level LGR (13,185 gridblocks)  3level FGR (124,320 gridblocks) 

CPU time, h  0.2  1.9  23 
6 Summary and conclusions
We have used an efficient LGR algorithm to improve the accuracy of numerically estimating the nearwellbore solutions when dealing with complex rheology of polymer or emulsion solutions. We present an algorithm to generate the block list and connections and propose an efficient numerical scheme to couple the pressure and mass conservation equations using the LGR composite grid and with consideration of heterogeneous reservoir properties. Several numerical examples are carried out, focusing on polymer flooding, reservoir with fractures near injection wells, and ACP flooding. Simulation results reveal that the LGR is able to obtain more accurate polymer injectivity compared to using the coarse grid and the analytical injectivity model. The LGR can deal with more complex and realistic reservoir conditions such as fractures and skin. CPU time is significantly reduced using LGR compared to FGR. This offers a reliable and efficient solution to handle the general concern of reservoir simulations for the sheardependent polymer rheology in chemical flooding projects.
Notes
Acknowledgments
The authors would like to acknowledge the sponsors of the Chemical EOR Industrial Affiliates Project at The University of Texas at Austin.
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