Artificial Neural Network based surrogate modelling for multi- objective optimisation of geological CO2 storage operations

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Energy Procedia 63 ( 2014 ) 3483 – 3491

ScienceDirect

Peer-review under responsibility of the Organizing Committee of GHGT-12 doi: 10.1016/j.egypro.2014.11.377

GHGT-12

Artificial Neural Network based surrogate modelling for

multi-objective optimisation of geological CO

2

storage operations

Indranil Pan, Masoud Babaei, Anna Korre, Sevket Durucan*

Department of Earth Science and Engineering, Royal School of Mines, Imperial College London, London SW7 2BP, United Kingdom

Abstract

An Artificial Neural Network surrogate modelling approach was used to optimise CO2 storage into a highly heterogeneous

semi-closed saline aquifer which exhibits considerable pressure increase due to injection. The methodology was implemented to minimise the overall field pressure and well bottom-hole pressures, and to maximise the amount of dissolved and trapped CO2 in

the storage aquifer. Different realisations of permeability and porosity were stochastically generated to represent the uncertainty in the model. Artificial neural networks were used to reduce the computational time of the optimisation procedure by approximating the objective functions for CO2 storage as surrogates to the expensive solutions of flow by the simulator. A

multi-objective evolutionary algorithm was run on these approximators to generate solutions of the multi-multi-objective optimisation’s Pareto front. These solutions were compared with the solutions obtained by the computationally expensive optimisation and they were found to give satisfactory results, illustrating that this methodology can be a viable, and low computational cost alternative for optimisation in CO2 storage design.

Keywords: Neural network; multi-objective optimisation; CO2 storage; surrogate modelling

1.Introduction

The problem of pressure build up in CO2 storage operations is a restricting factor for large-volume CO2 injection, especially for closed or semi-closed aquifers [1]. The storage capacity may be significantly limited due to rapid

* Corresponding author. Tel.: +44-20-7594-7354; fax: +44-20-7594-7444.

E-mail address: [email protected]

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pressure buildup or overpressure during CO2 injection which can cause the injection rate to decline exponentially. For the safe and secure long-term CO2 injection it is imperative to consider the risks of increased pressure which may compromise caprock integrity, induce reactivation of critically stressed faults, drive CO2 and/or brine through conductive features into shallow ground water resources, thereby risking contamination of potable drinking water or affecting existing subsurface activities such as oil and gas production [2]. Therefore, it will be necessary to control the pore pressure to avoid the scenario in which the pressure increase propagates too far from the injection point [3].

In order to control the reservoir pressure, McCoy and Rubin [4] implemented a screening model which assumes that the injection rate is limited only by the fracture pressure of the reservoir and the operator will always choose to inject CO2 at a rate such that the bottomhole pressure (BHP) does not exceed the formation fracture pressure. There are, however, various values for suggested fracture pressure limits in the literature and the allowable maximum pressure increase has to be assessed on a case-by-case basis [5].

Numerical modelling methods, on the other hand, estimate pressure buildup and injection pressure using reservoir simulators [6, 7]. Numerical modelling is more widely applicable than the analytical methods because the various flow dynamics including multi-phase flow, density-dependent flow, heat transfer, ground-water hydrodynamics, CO2 dissolution kinetics and geochemical reactions can be fully described by the finite-element or finite-difference discretisation of the partial differential equations. Most importantly, numerical models take into account heterogeneity of the subsurface model which is an important parameter influencing the flow behaviour and pressure distribution in the model [8].

As the subsurface models may contain a large number of unknowns, numerical computations tend to become intensive and inefficient. For the design and optimisation of CO2 storage operations there will also be a need to perform multiple simulations to account for unavoidable uncertainties in the geological or flow models. This also adds to the computational burden. In such cases, the use of fast running surrogate models of the subsurface system is desirable. The surrogate models can be constructed by sparse sampling of the computationally expensive system and are required to generalise well to the other unsampled locations. The flow solutions by surrogate models can generally be obtained orders of magnitude faster than that by the original flow simulators. The surrogate models can be polynomial or kriging models, neural networks or support vector regressors. A survey of these techniques can be found in Jin [9].

The surrogates are also very helpful in optimisation problems, where an objective function needs to be evaluated multiple times. If each evaluation of the objective function takes a long time, then the optimisation itself becomes intractable. Particularly, black box simulators can be used by population-based evolutionary techniques in parallel. However, in cases where there are hardware constraints (lack of suitable parallel computing facilities), a better approach is to use surrogate models to first approximate the objective function and then to use the surrogate model for optimisation.

One surrogate modelling approach is the application of Artificial Neural Networks (ANNs). ANNs are known to be universal function approximators and generally give good results on unseen data and are relatively computationally inexpensive to evaluate. Neural network surrogate modelling has been used previously in reservoir engineering applications by Mohaghegh et al. [10] for oil production, and by Kalantari-Dahaghi and Mohaghegh [11] to analyse the sensitivity of matrix-fracture systems’ petrophysical and geometrical properties in shale gas reservoirs.

In this study, ANN surrogate modelling has been used to optimise CO2 storage into a highly heterogeneous semi-closed saline aquifer which exhibits considerable pressure increase due to injection. In order to account for pressure buildup in the CO2 injection design, one conventional option is to enforce the limit of pressure increase as a constraint [12]. Another option, implemented in this study, is to include the pressure increase as one objective function of a Multi-Objective Evolutionary Algorithm (MOEA). The advantage of this approach is that it enables the system designer to consider a range of pressure limits, which in turn incorporates the uncertainty of the pressure limit definition, and ensures that the designer does not need to speculate the pressure limit value at the start of the optimisation process.

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2.Methodology

A schematic of the proposed approach is presented in Fig. 1 (a). Initially, a sampling scheme is constructed as shown in step 1 of Fig. 1 (a). Schlumberger’s ECLIPSE [13] solver is sampled at these points to generate the output objectives as indicated by step 2. The inputs are the injection rates of the wells and outputs are either the average pressure of the aquifer, or the bottomhole pressures of the wells, and stored fraction of CO2 is either dissolved or residually trapped. This forms the set of training data with which the ANN models are trained as in step 3 of the Figure. Several realisations of permeability and porosity are considered to account for uncertainty of the subsurface system. The trained ANN essentially functions like a surrogate of the ECLIPSE model itself and generalises to data-points outside the training data-set. By varying the injection rates, it is possible to maximise the fraction of CO2 stored and at the same time minimise either the pressure buildup in the aquifer or the bottomhole pressure of the injection wells. By finding the Pareto fronts between the objectives of this multi-objective optimisation problem, the design engineer would be able to quantify the trade-offs between the different objectives for the system as shown in step 4 of Fig. 1 (a). It is then possible to suggest a suitable compromise solution for the field implementation and use intuitive judgement for a-posteriori decision making. The geometry of the computationally expensive ECLIPSE model is shown in Fig. 1 (b).

(a) (b)

Fig. 1. (a) Schematic for the ANN based optimisation scheme (b) The geometry of the aquifer model and well positions of the ECLIPSE model.

2.1.Governing equations and model description

The compositional simulator ECLIPSE E300 [13] was used to conduct the reservoir simulation. The formulation for this simulator includes the extended Darcy law for multiphase flow of each phase (

D

) and the mass conservation equation for each component (

i

) in present phases including the diffusive flux:

( ), r k p g z D D D D D D U U P K v (1) ( . ) i i i i S x x q t D D D D D D D D M U § · w ¨ _¸ ¨ ¸ w ¨ ¸ ©

¦

¹

¦

v J (2)

where v_D , P_D , krD, pD , and UD are respectively the velocity, viscosity, relative permeability, pressure and

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mass conservation, M is porosity of the porous medium, S_D is the saturation of phase D, _xi

D is the mass fraction

of component i in fluid phase D. Hence, _xi. i

D vD JD is the mass flux of component i, within fluid phase D,

written as the sum of the advective (Darcy) and nonadvective (diffusive) mass fluxes. The diffusive flux of component i in phase D in a porous medium is computed using Fick’s law as:

i _S _Di _xi

D M D DU D D

J I (3)

where _Di

D is the normal diffusion coefficient of component i, within fluid phase D.

The reservoir model used is a synthetic two-dimensional channelised structure 6 km×6 km×30 m in size which represents a very large thin aquifer. The number of grid blocks in x, y and z directions are Nx = 100, Ny = 100 and Nz = 1. Four vertical CO2 injection wells, denoted as W1, W2, W3 and W4 in Fig. 1 (b), are located within the model domain in a regular grid spacing 2 km apart in x and y directions. The depth of the aquifer is set to 1 km and the aquifer temperature is set to 50 °C. The initial molar composition of brine is set to 0.08 NaCl, 0.01 CaCl2 and 0.91 water. The normal diffusion coefficients are set to 0.001 cm2_{/sec for water and CO2 components in the gaseous} phase, and 0.0001 cm2_{/sec for all components present in the aqueous phase. For the relative permeability the} hysteretic curves given in Juanes et al. [14] are used. The pore volumes of the gridblocks situated in the northern and southern boundaries are enlarged so that the total pore volume reach to around 850 million reservoir cubic meters (rm3_{) at an assumed initial reference pressure of 62}_bar_.

The aim here is to study the effect of injection rate choice on the bottomhole pressure for each well, the average reservoir pressure and the overall amount of CO2 stored for the given choice. It is assumed that each well has a fixed volume of 1.8×103_{million standard cubic meters (sm}3_{) of CO2 available to inject within a period of 15 to 100 years.} This equates to approximately 3.4 million tonnes CO2 per well or 13.6 million tonnes CO2 overall (1 tonne CO2 equates to 535 sm3_{for CO2 density of 1.87 kg/m}3_{at standard conditions, 15°C and 1 atm). Therefore, the injection} rate for each well varies between 0.034 and 0.2267 million tonnes CO2 per year. These values are relatively small considering that the reservoir is not thick or multi-layered.

In order to consider the uncertainty in the geological model, six realisations of permeability and porosity in the synthetic channelised model were created using training images and the S-GeMS software [15]. The channel facies are designed to exhibit long range correlation length features in the orientation of channel streaks, whereas the interchannel facies exhibit smoothly varying, but yet heterogeneous porosity and permeability distributions (Fig.2).

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(b)

Fig. 2. Example realisations illustrating (a) porosity and (b) permeability (md) heterogeneity within the channel and interchannel areas.

I 0 0.1 0.2 0.3 0.4 0.5 I 0 0.1 0.2 0.3 0.4 0.5 I 0 0.1 0.2 0.3 0.4 0.5

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3.Evolutionary multi-objective optimisation and surrogate modelling using artificial neural networks

3.1.ev- MOGA algorithm and Objective functions

The ev-MOGA algorithm [16] used in this study is an elitist multi-objective evolutionary algorithm based on the concept of H - dominance. Deb et al. [17] provide a comparison of multi-objective evolutionary algorithms such as NSGA II (Non-dominated Sorting Genetic Algorithm II), PESA (Pareto Envelope based Selection Algorithm), SPEA II (Strength Pareto Evolutionary Algorithm II) etc., where the ev-MOGA has been found to be superior.

The first objective function for the optimisation problem studied here is:

, , 0 1 1 , , , 0 1 ( ) ( ) ( ) n b n b t N i t i t R D t i t N i t i t i t R D F t i m m J m m m

¦¦

x

(4) where i t, R m , i t, D m and i t, M

m are respectively the mass of residually trapped CO2 in the gaseous phase, dissolved CO2 in aqueous phase and free CO2 in gaseous phase of the gridblock

i

at time t. Therefore, J₁ is the fraction of CO2 which is either dissolved or residually trapped and thus stored at a time t_n during simulation.

For the second objective function, the aim is to minimise the maximum bottomhole pressure of the four wells in the model so that:

^

1 2

`

2( ) max Wit , Wit ,..., Witn , {1, , 4}

J x P P P i (5)

where t Wi

P is the well i’s bottomhole pressure (bar) at time t. The max operator, therefore, considers all the wells and over the entire injection and post-injection period.

3.2.Surrogate modelling using artificial neural networks

Feed-forward neural network are used for approximating the components of the objective functions. The dataset is divided into training, validation and testing sets by randomly dividing the input dataset into three groups, based on a pre-specified ratio of 70:15:15. The training dataset is used to modify the weights of the neural network using an error minimisation mechanism. The back-propagation algorithm is used to train the network. In this technique, the errors in the output layer are propagated backwards to the preceding hidden layers and their weights are adjusted in a way so that the error is reduced. The objective function used for training the neural network and updating its weights is the mean squared error (MSE).

4.Simulation and results

4.1.Neural network fits of the objective functions

Six different feedforward neural networks, each with one input layer, one hidden layer and one output layer is trained for each different component of the objective functions

J

₁ and

J

₂. The ANNs represent the residually trapped CO2, the dissolved CO2 and the bottomhole pressures of each of the four injection wells.There are 4 inputs, 10 neurons in the hidden layer with tan-sigmoid type activation function. The output layer contains a single neuron with a linear transfer function. The R values (the squared root of the coefficient of determination, as a measure of the goodness of fit), which reflect the performance of the networks, is above 90% for all the training, validation and testing data-sets. Fig. 3 shows the error histogram with the corresponding regression plots for training, testing and

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validation of one of the trained ANNs. It is observed that the errors are very small and the ANN predicts the output quite well and also generalizes to unseen data.

(a) (b)

Fig. 3. (a) Error histogram for the training, validation and testing data for one of the ANNs (b), the corresponding regression plots for the original data versus the prediction,

4.2.Optimisation Results

With respect to the Pareto front obtained by the ANN-based multi-objective optimisation shown in Fig. 4 (a) for J1 versus J2, the stored amount of CO2 expressed as mass fraction of the injected CO2 varies between 0.6 to 0.7. The maximum bottomhole pressure of the wells varies between 140 to 280 bar.

(a) (b)

Fig. 4 (a) Pareto fronts (including 50 solutions) for J1-J2 obtained by ANN-based MOGA and substitution in ECLIPSE, (b) the injection rates

corresponding to each Pareto solution. 0 10 20 30 40 50 60 70

Error Histogram with 20 Bins

In

st

an

c

es

Errors = Targets - Outputs

-1 2. 4 -11. 04 -9. 676 -8. 312 -6. 947 -5. 583 -4. 219 -2. 855 -1 .4 9 -0 .1 26 1 1. 238 2. 602 3. 967 5. 331 6. 695 8. 059 9. 424 10. 79 12. 15 13. 52 Training Validation Test Zero Error 150 200 250 140 160 180 200 220 240 260 Target O ut p ut ~ = 0 .9 8 *T a rge t + 3 .2 Training: R=0.99011 Data Fit Y = T 150 200 250 140 160 180 200 220 240 260 Target O u tp u t ~ = 1 *T a rg e t + -4 Validation: R=0.97777 Data Fit Y = T 150 200 250 140 160 180 200 220 240 260 Target O u tp u t ~ = 0. 99 *T ar g e t + 1 .1 Test: R=0.97683 Data Fit Y = T 150 200 250 140 160 180 200 220 240 260 Target O u tp u t ~ = 0. 99 *T ar g e t + 1 .8 All: R=0.98622 Data Fit Y = T 0.6 0.62 0.64 0.66 0.68 0.7 140 160 180 200 220 240 260 280

J₁: Mass fraction of stored CO₂ by dissolution and residual entrapment [-] J2 : M a xi mu m P W fo r a ll we lls d u rin g th e s imu la tio n [b a r] ECLIPSE ANN 0 10 20 30 40 50 0 0.05 0.1 0.15 0.2 0.25

Pareto front solution (number)

CO 2 In je ct io n r a te ( m illio n to n n e s p e r y e a r) W1 W2 W3 W4

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Using the same injection rates for all the Pareto optimal solutions, the ECLIPSE simulation results were analysed. Some of the points in ECLIPSE-based Pareto front are dominated, in other words some solutions are not Pareto front solutions (see Fig. 4 (a)). However, the trend of the ANN-based and ECLIPSE-based Pareto fronts agree reasonably with one another with a maximum error of ±12 bar in pressure and ±0.0118 in fraction of CO2 stored. Very few of the ECLIPSE-based Pareto front points are dominating ANN-based solutions, indicating that the ANN-based optimal solutions suggest a higher mass of trapped CO2.

Fig. 4 (b) shows the injection rates corresponding to each well for each of the 50 Pareto solutions. All solutions are in the range 0.034 to 0.2267 million tonnes CO2 per year as dictated by the upper and lower bounds of the variables. These figures provide guidance over what injection rates must be chosen for the wells to maximise the storage, if the aim is not to exceed a given bottomhole pressure in any of the wells. For example, assuming that the site analysis indicated a maximum allowable bottomhole pressure of 225 bar, it would be feasible to optimise the storage fraction nominally up to an average of 0.67 over the realisations considered.

(a) (b)

Fig. 5. (a) Fraction of mobile, dissolved and residually trapped CO2 for the case in which the allowable maximum bottomhole pressure is 225 bar,

(b) the corresponding profiles of the bottomhole pressures over 50 years four wells.

Fig. 6. The CO2 saturation profiles for three of the realisations at the end of simulation after 200 years

For the ANN- and ECLIPSE-based optimisation points within the red circle in Fig. 4 (a), the injection rates of 0.0802, 0.1683, 0.1124 and 0.0784 million tonnes per year for wells W1, W2, W3 and W4 respectively can be derived from Fig. 4 (b) to maximise CO2 fraction stored. For this configuration of injection rates, the actual fraction of mobile, dissolved and residually trapped CO2, obtained by ECLISPE simulation are shown in Fig. 5 (a) for all realisations. At the end of the simulation the average fraction of stored CO2 is around 0.67. In this case, Fig. 5 (b) shows the evolution of the bottomhole pressures of the four wells for all realisations, where the bottomhole pressures of W1 and W3 reach the 225 bar limit.

0 50 100 150 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Simulation time [year]

M ass fr act ion of C O2 fo r 6 r e a lis a tio n s [-] Mobile CO₂ Residually trapped CO₂ Dissolved CO₂ 0 10 20 30 40 50 0 50 100 150 200 250

Simulation time [year] PW fo r w el ls 1 t o 4 and for 6 r ea lisa tio ns [ ba r] W1 W2 W3 W4

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Finally for the optimal case, the profiles of the gas saturation (mobile CO2) for three realisations at the end of simulation are presented in Fig. 6. It is shown that W1 and W3 allow higher shares of mobile CO2 to remain, although all wells have been used to inject equal amounts of cumulative CO2.

For the analysis of CPU time, the time required for parallelised run of the ECLIPSE data files for 6 realisations is around 6 minutes on a Windows-operating system with 3.40GHz CPU and 16.0 GB RAM. In order to obtain the Pareto front directly by ECLIPSE the run-time required for 6 realisations, with parallelisation, is around 530×6 minutes, which is approximately 2.21 days, since there are 530 evaluations of objective functions. In contrast, the ANN-based Pareto front was obtained in a fraction of a minute. However, the time required for the objective functions’ evaluation of 400 sampling points for all 6 realisations in the training period of ANNs, which is around 1.67 days, should also be considered. Nevertheless, the time savings due to ANN based surrogate modelling is more significant when additional processes, such as single objective optimisation or other algorithms of optimisation, are used in comparison to an optimisation using ECLIPSE outputs. In that case, the additional work required is achievable by the ANN surrogate model without the computationally expensive objective functions’ evaluation.

5.Conclusions

The variables used for the CO2 storage optimisation presented in this paper were the injection rates at all four wells that form part of the injection design. As a result, the improvement in the fraction of stored CO2 was limited to a range of less than 10 % increase in all cases. If other assumptions were considered for maximisation of storage, such as multiple periods of injection, brine cycling, new well setting, existing well(s) use etc., the fraction of stored CO2 would have varied more widely. In other words, there would have been more opportunity for optimising the injection design. Using many parameter controls would require more sampling, hence more time for the initial ECLIPSE runs. This would be potentially a more complex ANN architecture and also modestly increase the ANN running time. However, this is not a limitation to the overall performance of the proposed methodology, besides the higher computational costs.

Another important consideration is that the surrogate ANN model relies on a representative sampling scheme and may not generalise well over the whole input variable space. Hence the minima or Pareto front as predicted by the model might be sub-optimal [9]. One of the ways to overcome this is to include the surrogate modelling scheme in each iteration of the optimisation algorithm [9]. In such a case the surrogate model is dynamically updated over the iterations and, in each iteration, the minima pointed out by the surrogate are chosen as the next sampling point by the optimisation algorithm [9]. This generally results in better minima, but requires dynamic coupling of the surrogate modelling methods and the optimisation algorithm with the computationally expensive flow solver.

In summary, the surrogate based optimisation using ANN was found to give good results for the design of CO2 storage systems. This kind of optimisation is important from the engineering design point of view and presents the system designer with a wide array of choices to decide on the best compromise solution for the specific case. Future work can be directed towards comparison with other surrogate based optimisation methods.

Acknowledgements

This research was carried out as part of the UK Research Councils' Energy Programme funded project "Multiscale whole systems modelling and analysis for CO2 capture, transport and storage", Grant Reference: NE/H01392X/1.

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