# 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 non-Newtonian 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 near-wellbore 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 near-wellbore treatment, such as fractures near wells.

### Keywords

Polymer rheology Polymer injectivity Chemical EOR Local grid refinement Non-Newtonian flow## 1 Introduction

^{4}s

^{−1}near the wellbore and decrease sharply to about 1–10 s

^{−1}within a well gridblock. To eliminate the grid effects, several empirical or analytical models were proposed based on effective properties of the well blocks. For instance, Sharma et al. (2011) proposed to use an effective well radius to calculate the shear rate and match the polymer injectivity from very fine-grid simulation results; Li and Delshad (2014) proposed an effective viscosity using mathematical integration of in situ viscosity by assuming a radial velocity distribution within the well block. However, these approaches are not rigorous for other near-well effects apart from polymer rheology, e.g., non-zero skin factor, polymer permeability reduction, and injection induced fractures near the wellbore, which are often encountered during injection of polymer solutions. Therefore, in order to have a more accurate polymer injectivity adaptive to most reservoir conditions, it is necessary to refine simulation grids. However, grid refinement for the whole reservoir model leads to excessive computational costs. It is thus important to develop a local grid refinement (LGR) technique (or similar unstructured gridding approaches), such as shown in Fig. 2, so that the grid refinement is only applied to the regions where it is needed.

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 cell-based and patch-based 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 cell-based 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. Karimi-Fard 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 coarse-grid 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 three-dimensional multi-phase multi-component 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 volume-occupying species; and (3) the energy conservation equation which will not be discussed here.

### 2.1 Mass conservation equations

*κ*,

*ρ*

_{κ}is the density of component

*κ*,

*C*

_{κl}is the concentration of component

*κ*in phase

*l*,

*n*

_{p}is the phase number, and

**u**_{l}is the Darcy flux of phase

*l*which is calculated using Darcy’s law:

*is the intrinsic permeability tensor,*

**k***k*

_{rl}is the relative permeability,

*μ*

_{l}is the viscosity,

*γ*

_{l}is the specific weight of phase

*l*, and

*h*represents the vertical depth.

*κ*in the mobile and stationary phases expressed as

*S*

_{l}is the saturation of phase

*l*,

*n*

_{cv}is the total number of volume-occupying components, and \(\hat{C}_{\kappa }\) is the adsorbed concentration of component

*κ*. In UTCHEM, the liquid phase

*l*includes aqueous (

*l*= 1), oleic (

*l*= 2), and microemulsion (

*l*= 3).

**K**_{κl}is calculated as

*D*

_{κl}is the molecular diffusion,

*τ*is the tortuosity factor of the porous media,

*α*

_{Ll}and

*α*

_{Tl}are phase

*l*longitudinal and transverse dispersivities, and

*δ*

_{ij}is the Kronecker delta function.

*R*

_{κ}is the source term which is a combination of all rate terms for a particular component

*κ*. It may be expressed as

*r*

_{κl}and

*r*

_{κs}are the reaction rates for component

*κ*in phase

*l*and solid phase

*s*, respectively, and

*Q*

_{κ}is the injection/production rate for component

*κ*per bulk volume.

### 2.2 Pressure equation

*P*

_{cl1}is the capillary pressure between phase

*l*and phase 1 (the aqueous phase), and

*λ*

_{rlc}is the relative mobility expressed as

*C*

_{t}represents the total compressibility which is the volume-weighted sum of the rock matrix (

*C*

_{r}) and component compressibilities (

*C*

_{κ}

^{0}):

*P*

_{R}and

*P*

_{R0}are rock and reference rock pressures.

### 2.3 Rheological viscosity of the polymer solution

*μ*

_{app}is the apparent viscosity of the polymer solution; \(\mu_{\infty }\) is the polymer solution viscosity at infinite shear rate which is assumed to be brine viscosity; \(\dot{\gamma }_{1/2}\) is the shear rate at which the apparent viscosity is the average of \(\mu_{\infty }\) and \(\mu_{\text{p}}^{0}\); \(P_{\alpha }\) is a fitting parameter. For the synthetic polymer, e.g., HPAM, polymer solutions show shear-thinning behavior at intermediate shear rates and shear-thickening (dilatant) behavior at high rates. To remediate the deficiency of Meter’s equation, Delshad et al. (2008) developed a comprehensive polymer viscosity model which covers the whole shear-rate regime. The apparent viscosity consists of two parts:

*a*

_{1},

*a*

_{2}, and

*τ*are all fitting model parameters obtained by matching experimental data;

*μ*

_{max}is given as

*AP*

_{11}and

*AP*

_{22}are fitting parameters. When

*AP*

_{11}and

*AP*

_{22}are zero, the comprehensive polymer viscosity model reduces to the Carreau model.

*n*is the slope of the linear portion of bulk polymer viscosity vs. shear rate plotted on a log–log scale (bulk power-law index);

**u**_{w}is the Darcy flux of the aqueous polymer solution; \(\bar{k}\) is the average permeability;

*k*

_{rw}is the aqueous phase relative permeability;

*S*

_{w}is the aqueous phase saturation; \(\phi\) is the porosity;

*C*is a shear correction factor used to explain the deviation of the porous medium from an ideal capillary bundle model (Wreath et al. 1990; Sorbie 1991) and should be a function of permeability, porosity, and polymer molecule properties.

### 2.4 Analytical polymer injectivity model

*Q*

_{inj}and the pressure difference between injector and well block (

*P*

_{inj}−

*P*

_{wb}) can be expressed by

*I*is the well injectivity:

*h*represents the thickness of the well block;

*r*

_{o}represents the Peaceman equivalent radius;

*r*

_{w}is the well radius;

*s*is the skin factor; and

*k*

_{rl,wb}and

*μ*

_{rl,wb}are the relative permeability and viscosity of phase

*l*of well block, respectively.

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.

*μ*

_{app}(

*r*) adopts the form of Eqs. (10) or (11) using the shear rate calculated from the local velocity expressed by Eq. (18). For the detailed derivation, one can refer to Li and Delshad (2014).

## 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 third-order 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

*x*-direction and then the

*y*-direction.

*x*-direction connections (marked in red in Fig. 5) and

*y*-direction connections (marked in blue in Fig. 5). A summary of the block list and connections is given in Table 1.

Mutual indexing of block list and connections

Block No. | Connections ( | Connections ( | Connection No. ( | Block pair | Connection No. ( | 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

*x*and Δ

*y*and the lengths of the fine blocks are half. For the sake of simplicity to describe our approach, we assume in Fig. 6 isotropic permeabilities without a gravity effect and define

*λ*as the total fluid mobility, i.e., \(\lambda = k_{\text{abs}} \sum_{l = 1}^{{n_{\text{p}} }} \frac{{k_{{\text{r}}l}^{\text{ups}} }}{{\mu_{l} }},\) where \(k_{{{\text{r}}l}}^{\text{ups}}\) is the relative permeability of phase

*l*defined on the block interface with an upstream scheme. The upstream scheme to obtain \(k_{{{\text{r}}l}}^{\text{ups}}\) is the same as that to obtain the upstream concentration, \(C_{\kappa }^{\text{ups}} ,\) which we will explain in the next subsection.

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 second-order 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 cell-centered 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 self-consistent under the homogeneous condition or the condition that fine-block 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

*C*represents the component concentration.

*C*′ at the auxiliary points, e.g.,

*i*1 and

*i*2. After that, we extend Leonard’s scheme to this case:

*m*and

*n*.

## 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 | 300 |

Initial water saturation | 0.35 |

Polymer rheology exponent | 1.8 |

Shear rate at half zero-rate viscosity | 10 |

Wells | 1 injector; 1 producer |

Injection rate, ft | 500 |

Producer bottomhole pressure (BHP), psi | 1000 |

Water injection | 0–150 and 270–365 days |

Polymer injection | 150–270 days (0.3 wt%) |

*P*

_{α}and

*γ*

_{1/2}, used in the polymer rheology equation (Eq. 10), are set as 1.8 and 10 s

^{−1}, respectively. These parameters lead to a relatively sharp shear-thinning curve. CMG_STARS uses a different polymer rheology equation, which is a power-law equation:

*n*

_{thin}is the power-law exponent, and

*u*

_{lower}is defined by the point on the power-law curve when \(\mu_{\text{app}}\) is equal to \(\mu_{\text{p}}^{0}.\) To be close to the UTCHEM polymer equation (Eq. 10) for Case 1 using the CMG_STARS equation, we found out that

*n*

_{thin}must be small and it causes numerical stability issues which are also indicated in the manual of CMG_STARS (2012). Therefore, to achieve a relatively similar polymer rheology curves for both simulators, we use

*P*

_{α}= 1.5 and

*γ*

_{1/2}= 3.8 s

^{−1}for UTCHEM and

*n*

_{thin}= 0.5 and

*u*

_{lower}= 0.02 ft/day for CMG_STARS. Figure 11 shows a comparison of the results between UTCHEM and CMG_STARS using the original grid, 4-level LGR, and 4-level FGR, respectively. It is found that the injection pressure curves for the original grid match very well between UTCHEM and CMG_STARS. In addition, the 4-level LGR simulation results of UTCHEM and CMG_STARS are also close, with only a minor difference. This is acceptable because CMG_STARS and UTCHEM use different polymer concentration-dependent viscosity models and shear-thinning models as mentioned in Goudarzi et al. (2013a). Again, for FGR results, we observe both UTCHEM and CMG_STARS match well with LGR results.

CPU times for UTCHEM and CMG_STARS using different grids for Case 1

Grid | Original grid (225 gridblocks) | 4-level LGR (471 gridblocks) | 4-level 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

*x*–

*y*plane.

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 co-solvent 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% co-solvent | |

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) | 3-level LGR (13,185 gridblocks) | 3-level 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 near-wellbore 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 shear-dependent 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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