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by A Afanasyev · 2021 · Cited by 11 — An optimal volume of injected CO2 exists at which the rev- enue from the additional oil recovery still exceeds the ex- penses of CO2 injection ( ...

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Adv. Geosci., 56, 19–31, 2021

https://doi.org/10.5194/adgeo-56-19-2021

© Author(s) 2021. This work is distributed under

the Creative Commons Attribution 4.0 License.

Numerical optimisation of CO2 flooding using a hierarchy of

reservoir models

Andrey Afanasyev, Anna Andreeva, and Anna Chernova

Institute of Mechanics, Moscow State University, Moscow 119192, Russia

Correspondence: Andrey Afanasyev ([email protected])

Received: 29 June 2021 – Revised: 19 August 2021 – Accepted: 24 August 2021 – Published: 16 September 2021

Abstract. We present a method for accelerated optimisation

of CO2 injection into petroleum reservoirs. The optimisation

assumes maximisation of the net present value by coupling

reservoir models with the calculation of cash flows. The proposed

method is based on the construction of a hierarchy of

compositional reservoir models of increasing complexity.We

show that in dimensionless volumes, the optimal water and

gas slugs are very close for the 1-D and 2-D areal reservoir

models of the water-alternating-gas (WAG) process. Therefore,

the solution to the 1-D optimisation problem gives a

good approximation of the solution to the 2-D problem. The

proposed method is designed by using this observation. It

employs a larger number of less computationally expensive

1-D compositional simulations to obtain a good initial guess

for the injection volumes in much more expensive 2-D simulations.

We suggest using the non-gradient optimisation algorithms

for the coarse models on low levels of the hierarchy

to guarantee convergence to the global maximum of the

net present value. Then, we switch to the gradient methods

only on the upper levels. We give examples of the algorithm

application for optimisation of different WAG strategies and

discuss its performance. We propose that 1-D compositional

simulations can be efficient for optimising areal CO2 flooding

patterns.

1 Introduction

Gas flooding is a well-established method of enhanced oil recovery

(EOR). The injection of gas, particularly CO2 or the

associated petroleum gas, into oil reservoirs allows for a significant

increase in the microscopic displacement efficiency,

ED, caused by the compositional exchanges between oil and

gas, oil swelling and viscosity reduction (Lake, 1989; Blunt

et al., 1993; Christensen et al., 2001; Thomas, 2008). As

discussed by Pritchard and Nieman (1992), Sanchez (1999),

Jensen et al. (2012), and Johns and Dindoruk (2013), gas injection

is often organised in the water-alternating-gas (WAG)

process to also improve the volumetric sweep efficiency,

EV. The total efficiency of oil recovery can be estimated as

E D EDEV (Ghedan, 2009; Verma, 2015).

Optimisation of gas flooding often requires determining

the well patterns, the distances between injection and producing

wells, and the volumes of water and gas slugs in theWAG

injection (Sanchez, 1999; Kovscek and Cakici, 2005; Chen

and Pawar, 2018). It is acknowledged that a larger volume

of injected CO2 results in a better microscopic displacement

efficiency caused by the miscibility. However, the economic

considerations must always be taken into account when optimising

CO2 flooding. Since the cost of CO2 is higher than

that of water, the expenses of CO2 injection will at some

point outweigh the gain from the increase in oil extraction.

An optimal volume of injected CO2 exists at which the revenue

from the additional oil recovery still exceeds the expenses

of CO2 injection (Fig. 1). Therefore, the efficiency of

a CO2 flood should be evaluated by coupling the reservoir

model with the equations for the cash flows. Typically, the

net present value (NPV) is maximised by considering simplified

economic processes (Salem and Moawad, 2013; Rodrigues

et al., 2019), whereas engineering studies can involve

more elaborate models for asset management decisions (Ettehadtavakkol

et al., 2014). Thereby, the optimisation assumes

finding an injection strategy, e.g. the volumes of gas and water

slugs, that maximises NPV (Fig. 1). The moment in time

at which the maximum is reached corresponds to the production

life of the oil field (Afanasyev et al., 2021).

Numerical modelling of CO2 flooding is often conducted

by employing compositional reservoir simulations, which al-

Published by Copernicus Publications on behalf of the European Geosciences Union.

20 A. Afanasyev et al.: Numerical optimisation of CO2 flooding

Figure 1. Sketch of the areal study (a quarter of the five-spot pattern).

The colour shows the distribution of oil saturation in the optimized

strategy 2(WG)W at PVI D 0:25,  D 2 and the net revenue

for oil exporting of ro DUSD 12.5 per one barrel of oil.

low for detailed estimation of compositional exchanges between

the gas and oil phases (Coats, 1980; Orr et al., 1995).

This complicates the optimisation, because compositional

modelling is computationally expensive. Consequently, the

optimal injection strategy should be determined with a rather

limited number of compositional runs. Additional efforts

to accelerate the numerical optimisation should be undertaken

to allow for an accurate solution to be reached within

a reasonable computational time. The improvement can be

achieved by either acceleration of forward simulations, e.g.

by using proxy modelling (Amini and Mohaghegh, 2019;

Wang et al., 2019), or implementing a modified optimisation

method specially designed for such studies (Chen and

Reynolds, 2016).

In this paper, we aim to present our approaches for the numerical

optimisation of gas flooding scenarios. Here, we constrain

the investigation to determining the optimal volumes

of gas and oil slugs for the case of areal CO2 injection. Finding

optimal well patterns and distances between injection and

producing wells is beyond the scope of this work. The idea

behind our approach is to construct a hierarchy of reservoir

models of increasing complexity (Fig. 2). The top level corresponds

to the reservoir model that needs to be optimised,

whereas the lower levels correspond to coarser approximations

of the top-level model. We intend to begin optimisation

with the most coarse model on the first level, which allows

for many computationally-cheap simulation runs. Thus, we

can explore a large region of the parameter space and find a

good approximation of the injection schedule near the global

maximum of NPV. When going up the hierarchy, the initial

guess for the optimal volumes of the water and gas slugs

(i.e. pore volumes injected (PVI)) on the next level is equal

to the volumes obtained on the previous level of the hierarchy.

The determined PVI are then used as the initial guess on

Figure 2. Sketch of the constructed hierarchy of compositional

reservoir models.

the next level, where a more refined reservoir model is employed.

Since the refined model requires lager computational

resources, we aim to reduce the number of simulation runs

for this model. This is achieved by ensuring that the coarser

model from a previous level provides a good estimate for the

solution to the optimisation study on the next level.

This article is organised as follows. In Sect. 2, we describe

a hierarchy of synthetic models ofWAG and briefly overview

all governing equations for both the fluid transport and economic

processes. In Sect. 3, we discuss the objective function

and its noisy behaviour at large timesteps. In Sect. 4, we

present the details of the proposed optimisation method. In

Sect. 5, we discuss the result of the numerical experiments

related to the optimisation of different gas flooding strategies.

We end the article with the conclusions in Sect. 6.

2 Mathematical model and optimisation criterion

2.1 Overview of the reservoir models

For estimating NPV of CO2 flooding, we use the mathematical

model presented in our previous work (Afanasyev

et al., 2021). Generally, all parameters of the reservoir model

(e.g. the fluid composition) and the economic calculations are

equal to those in the noted paper. Here, we give only a brief

overview of the involved modelling approaches emphasising

several new developments.

A new feature of the present work is that we consider 2-D

reservoir models corresponding to a quarter of the five-spot

pattern (an injection pattern in which four injection wells are

located at the corners of a square and the production well

sits in the centre), in addition to the 1-D simulations corresponding

to the slim-tube experiments. We use a Cartesian

reservoir model of equal lateral extensions L and height H

(Fig. 1). We denote the number of grid blocks along axes x

and y by nx and ny . The number of grid blocks along the

vertical axis z is always 1. Thus, we consider an areal study.

Adv. Geosci., 56, 19–31, 2021 https://doi.org/10.5194/adgeo-56-19-2021

A. Afanasyev et al.: Numerical optimisation of CO2 flooding 21

Two wells are placed in the opposite corners of the sector,

i.e. in the grid blocks .1;1/ and .nx ;ny /. Water and CO2 are

injected into the reservoir through the first well (Injector),

whereas oil, water and gas are recovered through the other

well (Producer).We search for the injection schedule, i.e. the

optimal periods of water and CO2 injection, that maximises

NPV.

Thereafter only two cases of nx and ny are possible:

– 1-D reservoir model (Fig. 3a). nx > 1 and ny D 1, i.e.

the grid dimension is nx1. This one-dimensional study

was investigated in detail by Afanasyev et al. (2021).

The flow occurs only in the direction of axis x, and all

grid blocks in a sequence from the Injector to Producer

are contacting the injected water and CO2. Thus, this

case corresponds to 100% efficient volumetric sweep,

EV D 1.

– 2-D reservoir model (Fig. 3b and c) with nx D ny > 1.

This reservoir model corresponds to a quarter of the

five-spot pattern with a regular rectilinear grid. The volumetric

sweep is less efficient in this case because some

oil near the distant corners from the wells is bypassed

by the injection fluids.

Certainly, the 2-D model is more complicated than the 1-

D model, because its dimension is higher. Therefore, it is not

obvious how the conclusions of Afanasyev et al. (2021) for

the 1-D model can be applied to this simulation setup. In part,

this study concerns comparison of the optimal strategies in

the 1-D and 2-D studies.

2.2 Governing equations

We employ compositional simulations to estimate the efficiency

of oil recovery. We use the three-component mixture

CH4–C6–C16 as a proxy for oil. The initial molar composition

of fluid is 20% CH4, 40% C6 and 40% C16 (Orr, 2007).

The reservoir pressure is 139 bar (the minimum miscibility

pressure (MMP) is  205 bar) and reservoir temperature is

93 C. The phase equilibria are simulated using the Soave-

Redlich-Kwong equation of state with volume shift (Redlich

and Kwong, 1949; Soave, 1972). The viscosity of hydrocarbon

phases is predicted using the LBC-correlation (Lohrenz

et al., 1964). The water viscosity is 0.35 cP. The connate water

saturation is swc D 0:16. The residual saturation of oil

is 0.24. The numerical modelling and optimisation is conducted

in our MUFITS simulator (Afanasyev, 2015, 2020;

Afanasyev and Vedeneeva, 2021).

We investigate immiscible gas injection because the reservoir

pressure is much lower than MMP (Johns and Dindoruk,

2013). We assume that during the injection, the reservoir

pressure does not significantly deviate from its initial value.

This can correspond to the case of a high-permeability reservoir.

Thus, the changes in pressure do not influence the phase

equilibria and miscible displacement is considered in the approximation

often employed in the method of characteristics

(Pope, 1980; Orr et al., 1995; LaForce and Jessen, 2010).

Thus, our study does not concern injection strategies implementing

pressurisation to reach miscibility (Langston et al.,

1988).

To estimate the economic efficiency of the injection, we

couple the reservoir model with a simple model of cash flow.

The economic estimates are based on the calculation of NPV,

where the cash flow includes the expenses for the water injection

and disposal (USD 2 and 1.5 per one barrel of oil) and

CO2 injection and separation (USD 2.55 and 1.33 per one

million standard cubic feet), and the net revenue ro for oil

exporting, typically ro DUSD 12.5 per one barrel of oil (Tzimas

et al., 2005; Salem and Moawad, 2013; Rodrigues et al.,

2019). These and other parameters of the coupled reservoir

and economic model are identical to those in Afanasyev et al.

(2021).

2.3 Dimensionless variables

The hydrocarbon pore