# Optimization of waterflooding performance by using finite volume-based flow diagnostics simulation

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## Abstract

From a visual point of view, volumetric information about reservoir portioning and communication such as sweep, flow patterns, and drainage zones are longer better interpreted and pictured when presented by an average volumetric flux calculation. To this hand, finite volume discretization can be used to substitute streamline simulation-based finite difference to assess flow diagnostic information. Herein, we use finite volume-based flow diagnostics to optimize waterflooding. In particular, we discretize in finite volume the flow equation from Darcy’s law single-phase incompressible flow steady state combining with two auxiliary flow equations, time of flight and stationary tracers using the two-point flux approximation to describe fluid particles motion and flow lines. In addition, with the estimation of dynamic heterogeneity, we compute the Lorenz coefficient to highlight the reservoir flow and storage capacity characterization. To optimize waterflooding rates, we first, use an objective function the equalized Lorenz coefficient got through the evaluation of average travel time in cells to increase sweep efficiency and decrease the dynamic heterogeneity coefficient. Second, following the same target, we use the flow diagnostic interactive tools to study the volumetric sweep displacement front and harmonize the flooding breakthrough. In this work, our conceptual approach is to see the reservoir initially filled with oil; then, optimizing the Lorenz coefficient leads us to an oil recovery improvement. To be pragmatic, we apply our waterflooding performance optimization model on two case studies, the ninth SPE comparative solution project, a reexamination of black-oil (synthetic case) and ZHNBA Chinese oilfield (real field dataset).

## Keywords

Flow diagnostics Waterflooding Finite volume Optimization Dynamic heterogeneity Volumetric sweep## List of symbols

- \( K \)
Absolute permeability, md

- \( p \)
Reservoir pressure, bar

- \( P \)
Reservoir discrete pressure, bar

- \( p_{\text{bh}} \)
Bottom hole pressure, bar

- \( \rho \)
Fluid density, \( {\text{Kg/m}}^{3} \)

- \( g \)
Gravity acceleration, \( {\text{m/s}}^{2} \)

- \( Z \)
Depth vector function \( (x,y,z) \)

- \( t \)
Time, \( {\text{s}} \)

- \( v \)
Volume, \( {\text{m}}^{3} \)

- \( V \)
Discrete volume, \( {\text{m}}^{3} \)

- \( \mu \)
Fluid viscosity, \( {\text{cP}} \)

- \( \overrightarrow {u} \)
Darcy’s velocity, \( {\text{m/s}} \)

- \( \overrightarrow {n} \)
Normal direction

- \( q \)
Mass flow rate, \( {\text{m}}^{3} / {\text{day}} \)

- \( c_{\text{r}} \)
Rock compressibility

- \( c_{\text{f}} \)
Isothermal compressibility

- \( \alpha_{\text{f}} \)
Thermal expansion

- \( c_{\text{t}} \)
Total compressibility

- \( \lambda \)
Total mobility

- \( T \)
Transmissibility

- \( u \)
Flux

- \( \tau \)
Time of flight (TOF)

- \( \phi \)
Porosity

- \( \varPhi \)
Storage capacity

- \( L_{\text{c}} \)
Lorenz coefficient

- \( F \)
Flow capacity

- TPFA
Two-point flux approximation

- MRST
MATLAB Reservoir simulation toolbox

## Introduction

The road toward strategies for recovery improvement of hydrocarbon resources compulsory goes through reservoir rock characterization and the study of fluids flow in porous media. Herein, reservoir modeling is taken as an investigation of the interactive motion of fluid particles, depicting the physics and mathematical equations involved. This means disassembling the partial differential equations describing the fluid behaviors in the reservoir and then looking for numerical solutions and implementation. Thereby, the flow diagnostics workflow module from MATLAB Reservoir Simulation Toolbox (MRST) is the perfect modeling tool that can be used as a waterflooding project processor to provide a reservoir model to establish connections and quickly procure a qualitative picture of the flow patterns. Flow diagnostics are the result of flow field analysis properties, and they are the study of well responses and fluid displacement front distribution to understand the flow paths and communication patterns in reservoir models. Howsoever, flow diagnostics-based finite volume discretization can be used to measure and optimize the dynamic heterogeneity, and increase well volumetric sweep efficiency. Flow diagnostic with the interactive tools can be used to manage the flooding breakthrough, and forecast an optimal injection well rates configuration.

The study and use of flow diagnostic tools have been the purpose of several types of research. Authors have mostly involved flow diagnostics to develop strategies for hydrocarbon resources optimal recovery. However, we have distinguished two main approaches, the streamline-based finite difference, and finite volume-based flow diagnostics.

Streamline-based flow diagnostics has received attention these last years; it is a simulation method based on the finite-difference discretization. This started with modeling reservoir geostatistical realizations based on the study of time flight equations by Idrobo et al. (2000), and then Thiele and Batycky (2003) with an automatic method for waterflooding and well rates optimization. Then, later the same year, Ates et al. (2003) ranked and up-scaled geostatistical reservoir models, and then Akhil Datta-Gupta and King (2007) stated the features of streamline technology and complement to conventional finite-difference simulation. Batycky et al. (2005) revisited class flood surveillance methods applied to injection production data, and Park and Datta-Gupta (2011) worked on the workflow for waterflood rate optimization using streamline-based flood sufficiency maps. Lastly, Haegland et al. (2019) simulated fluid flow and transport in porous media.

Streamline-finite difference-based method has been efficient for these above purposes, but it has limitations in terms of computational complexity in its extensibility to irregular computational domains like complex geological structures such as faults. However, reservoir drainage and volumetric sweep information can be well-interpreted and pictured when presented by an average volumetric flux calculation.

Recently, finite volume discretization has been explored as a mathematical ground for reservoir simulation methods. In the waterflooding framework, we can quote Shook and Mitchell (2009) for their work on the new method for estimating heterogeneity in the earth model using time of flight and volumetric flow rate information. Then, Shahvali et al. (2011) proposed finite volume method as an alternative to streamline for obtaining flow diagnostic information. Finally, Møyner et al. (2015) proxied flow diagnostics for reservoir optimal management workflows.

Having dynamic heterogeneity volumetric grid cells average values (optimal values)

Harmonizing the flooding breakthrough

Optimizing the water injection well rates

Optimizing the waterflooding performance

## Mathematical model

The mathematical model for the incompressible single-phase flow is derived by the fusion of the conservation of mass, Darcy’s law on steady state and the state equation on a control volume (\( v \)) shown by Lie 2016. The mass conservation implies that the mass accumulated in \( v \) equals the mass flow across the boundary of \( v \) plus the mass injected into \( v \) via wells with \( \phi \), the fraction of volume, \( v \) available for flow, \( \rho \) the fluid density per unit volume, \( \partial v \) the computational domain with normal vector \( \vec{n} \), \( \vec{u} \) the macroscopic Darcy’s velocity and \( q \) the mass flow rate or fluid sources or sinks (inflow, outflow) per volume unit at certain locations.

*T*and

*p*are the indication that the above process takes place at constant pressure and temperature. Also, we can notice that for a fixed number of particles, \( \rho v \) can only be constant, \( {\text{d}}\rho v = \rho {\text{d}}v \); then, the last equation can be rewritten as follows:

*τ*is the time of flight; therefore, using directional derivative, we can write:

The resulting tracer distribution from the last equation defines the reservoir portion which will be influenced by a coming inflow boundary flow. By reversing the flow field sign, we can identify the reservoir portion that will be influenced by an outflow boundary or sink. Going further by stretching this method to all parts of source flow, we can compute the instantaneous flow field.

## Numerical model: discretization of equations using the two-point flux approximation (TPFA)-based finite volume

### TPFA of the single-phase incompressible flow equation steady state

We can see in this last equation the introduction of \( T_{i,c} \) which is one-side transmissibility or half transmissibility that links the flux across a cell precisely between the cell center and the edge. Finally, let us apply the flux continuity through all faces as follows.

### TPFA of time of flight and tracer equations

## Flow diagnostics impact of dynamic heterogeneity on flooding sweep

One of the best ways to characterize a reservoir rock capacity to store and transmit fluids goes through the deep investigation of its intrinsic properties. Heterogeneity developed by Shook and Mitchell 2009 is the variety of constituent particles encountered in porous rock. In other words, it is the property that defines the reservoir rock permeability and porosity. Studies on the measure of heterogeneity have converged on two models. The first static heterogeneity is all about the rock storage and transmissibility distribution of fluid. Therefore, high static heterogeneity may conduct to a rock good capacity to store and transmit fluid. On the other hand, the dynamic heterogeneity focuses on the measurement of interconnected void space. It describes fluid motion and flow-path connections, thus, a high dynamic heterogeneity may cause a large residence time (time a particle takes to travel from an injector well to a producer well) value.

*F*–

*ϕ*diagram (Fig. 1).

The left plot represents the volume \( V_{i} \), plotted as a function of the flow rate \( q_{i} \); the blue curve describes a heterogeneous fluid motion and the green shows a homogeneous displacement. On the other hand, the right plot presents the *F*–*ϕ* diagrams, where the flow rate and volume have been normalized.

## Lorenz coefficient

*F*–

*ϕ*diagram. It is the difference in flow capacity from which a piston-type displacement has been applied, defined as follows (Fig. 2).

Graphically, the Lorenz coefficient is two times the area bounded by the *F* (*ϕ*) curve (blue) and the *F* = *ϕ* line (green) and has a value from (0) to (1). A Lorenz coefficient of 0 falls along the *F *= *ϕ* line; green line describes a homogenous fluid motion with an equal volumetric displacement front, and the breakthroughs occur at the same residence times 100% sweep. A value of (1) describes an infinitely heterogeneous fluid motion blue line on the *F*–*ϕ* diagram, and the breakthroughs occur at different residence times and lead to stagnant regions.

## Flow diagnostics on case studies: SPE9 (Killough 1995) and ZHNBA China oilfield

### SPE9 case study

#### SPE9 reservoir model setup

#### SPE9 dynamic heterogeneity and sweep efficiency

*F*–

*ϕ*quantities and plot the diagram

*F*versus

*ϕ*heterogeneity developed by Shook and Mitchell 2009 to get the dynamic heterogeneity expression. We note that for a homogeneous displacement, the

*F*–

*ϕ*gives a straight line; all flow paths breakthrough at the same time and gives a Lorenz coefficient of value zero. On the other hand, for a heterogeneous flow motion,

*F*–

*ϕ*plot gives a concave curve with the steep initial slope that denotes the high flow regions giving early breakthrough and the flat regions that correspond to the low flow or stagnant regions where there is a breakthrough delay. The Lorenz coefficient takes the values nonzero to one. In the case of SPE9, the Lorenz coefficient is equal to 0.5340. Stepping further, we can define the sweep efficiency diagram which is the measure of the injected fluid volumetric effectiveness. It is the ratio of oil contracted by water at a time (

*t*) and an infinite time (Fig. 5).

#### SPE9 flow model and volumetric connections

### ZHNBA China oilfield case study

#### ZHNBA reservoir model setup

#### ZHNBA measure of dynamic heterogeneity and sweep efficiency

*F*–

*ϕ*diagram for the measure of dynamic heterogeneity, and we get the Lorenz coefficient evaluation. From the Lorenz coefficient computation, we get \( Lc = 0.6623 \); this shows that ZHNBA is a highly heterogeneous reservoir (Fig. 11).

#### ZHNBA flow model and volumetric connections

## Waterflooding optimization models

Generally, the way to optimize waterflooding requires mathematical methods with several forward simulations; these kind of full and rigorous simulations are costly. Here in this work, we will first present a framework that can be used as a waterflooding optimization proxy. Then secondly, we will optimize the wells flow rate by using flow diagnostics interactive tools. The idea behind the first method is to use flow diagnostics to design a simple and lightweight algorithm easy to implement for waterflooding optimization. On the other hand, the second method is based on the instant TOF snapshot studying. Here, we will see how flow diagnostics with low-cost tools can iteratively improve water injection configurations.

### Optimization by using the objective function

In the above equation, we have the time of flight (\( \tau \)), Darcy’s velocity (\( \vec{u} \)) and the pressure (\( p \)) as unknown. The full flow diagnostics is practically one part based on forward and backward time of flight equations; the other part on tracer equations, but in this work, we will precisely examine the forward time of flight equation and simply consider the treatment of the tracer equations as it is analogous.

**p**and

**τ**which with their values determined in the center of each cell, and \( \vec{u} \) is replaced by the vector fluxes across all cell faces

**u**. Then, \( p(q,\vec{u}) = 0 \) becomes \( P_{h} \left( {{\mathbf{q}},{\mathbf{u}}} \right) \, = \, 0 \), with \( P_{h} \) the expression of the discretized system. Since in this work, we will only work with the discretized fluid flow equations; we can drop \( h \) on the \( P_{h} \), and then we can introduce the discrete model equations as representation of an implicit system which has his linearization coming from the Newton linearized system. Our system will not be constructed explicitly because the pressure and time of flight equations are linked and will be solved sequentially; the system is written as follows:

Our approach to obtain an optimal water injection is based on the conception of getting an ideal piston-like displacement that carries us to deeply focus or observe the Lorenz heterogeneity coefficient. The Lorenz coefficient highlights the fluid flow capacity and storage capacity based on piston-like fluid motion; this makes us considering the optimized Lorenz coefficient as our objective function to improve waterflooding performance. Herein, the optimized Lorenz coefficient will seek to equalize the total travel time through the standard two-point flux approximation in all cells and then improve the sweep.

### Solution strategy and rate optimization

Following the same steps, expression (37) can be extended to tracers if required. When we finish with the computation of all forward equations, we then start looking for the objective function gradient, and to carry this investigation, we can use two ways, applying a numerical differentiation or adjoint method.

In this work, we use the adjoint method although its application requires alteration of the model equations; it is more efficient than a numerical differentiation which is simple to implement and non-intrusive but can be computationally costly because all forward equations need to be solved on each perturbation.

Going deeply, our objective function will not depend on the control \( w \) exclusively but on state variable \( x \); in particular, the term \( \frac{\partial G}{\partial w} \) can disappear. We can conclude this solution strategy study by highlighting the most important steps which are first, solving the forward system Eq. (35) using the block-wise approach explained previously to get state variables and upwind directions, and secondly solving the backward problem Eq. (40) with the same approach to get the objective function sensitivities.

### Application on SPE9

*F*–\( \phi \) diagram and the reservoir sweep efficiency (Fig. 14).

SPE9 initial well configurations

‘I1’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | 0.003241 | ‘rate’ |

‘I2’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | 0.003241 | ‘rate’ |

‘P1’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

‘P2’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

‘P3’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

‘P4’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

‘P5’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

‘P6’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

SPE9 optimized well configurations

‘I1’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | 7.96E−09 | ‘rate’ |

‘I2’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | 0.006481 | ‘rate’ |

‘P1’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

‘P2’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

‘P3’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

‘P4’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

‘P5’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

‘P6’ | 15 × 1 double | 15 × 1 double | 15 × 1 double | 15 × 1 char | 0.1 | − 0.00081 | ‘rate’ |

SPE9 initial and optimal Lorenz and rate values

I1 | I2 | P1 | P2 | P3 | P4 | P5 | P6 | Lorenz coefficient | |
---|---|---|---|---|---|---|---|---|---|

Initial | 280 | 280 | 70 | 70 | 70 | 70 | 70 | 70 | 0.5340 |

Optimized | 6.9647e−04 | 568.7500 | 70 | 70 | 70 | 70 | 70 | 70 | 0.3328 |

### Optimization by using the interactive flow diagnostic tools

The interactive flow diagnostic tools are powerful visualization routines provide by MATLAB and MRST (Lie 2016) to make the user interacting with the input parameters and simulate the results. These tools are particularly differenced with the objective function on the control ability, and they give access through the input screen. Using the interactive tools is controlling the flow diagnostics basic quantities, the backward and forward time of flight and tracers, having the option of plotting the Lorenz coefficient, sweep efficient and more. The flow diagnostics quantities which govern the interactive are computed following the standard two-point flux approximation-based finite volume discretization.

### Study of volumetric sweep displacement front snapshot

Optimizing waterflooding using the interactive tool is looking for the harmonization of water breakthrough in production wells. It is searching for a suitable well control configuration to improve the volumetric sweep through manual modifications. The first step in this method is to investigate the fluid displacement front propagation when the flow field control values are kept constant. In the reservoir, the displacement front moves following Darcy’s velocity obtained from the flux field; then, swept region at a time \( t \) is assimilated to grid cells where \( \tau_{\text{f}} \le t \). The idea behind on the one hand is to study every sweep displacement front steps and notifying different breakthrough, and on the other hand, investigating the drainage instant displacement then, modifying the well controls in the aim to decrease the Lorenz coefficient value.

**A**pplication on ZHNBA China oilfield

The producer P6 records the earliest water breakthrough from the injector I1

Following by the producers, P8 and P7 get their breakthrough from I3

The remaining producer water breakthroughs are considered as late; then, they are not recorded including those coming from the injector I2.

ZHNBA flooding rate optimization experiment

I1 | I2 | I3 | P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 | Lorenz-C | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Base case | 175 | 175 | 175 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 0.6623 |

Case 1 | 175 | 250 | 175 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 0.5763 |

Case 2 | 200 | 300 | 200 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 0.4930 |

Case 3 | 210 | 300 | 250 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 0.4590 |

## Conclusion and summary

Waterflooding is one of the best methods used to improve oil recovery and has sown its efficiency in many applications. However, herein, we have presented a pathway to improve its performance using finite volume discretization-based flow diagnostics simulation. By drawing its sweep efficiency improvement scheme based on the optimization of the dynamic heterogeneity Lorenz coefficient, we harmonized the flooding breakthroughs. Designing a waterflooding optimization performance framework using flow diagnostics-based finite volume method is having the reservoir flow quantities fluxes computed on volumetric average portioning. In this work, the finite volume discretization showed its perfect ability to handle both Cartesian and unstructured grids, when flow diagnostic tools deepen our understanding of reservoir flow characterization summarized in two parts: firstly, the solution of the incompressible steady-state flow equation through the standard two-point flux approximation to get the fluid bulk motion. Secondly, the solution of the mixture time of flight and tracer equations to split the reservoir into volumetric flow and describing the flow path regions.

## Notes

### Acknowledgements

We give special thanks to Xi’an Shiyou University for giving us a conducive atmosphere that led us to complete this work. Special thanks to Professor Xianlin Ma and Dr. Landry Biyoghe for guiding us all along this journey.

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