WO2021161057A1

WO2021161057A1 - Well completion in a hydrocarbon reservoir

Info

Publication number: WO2021161057A1
Application number: PCT/IB2020/000091
Authority: WO
Inventors: Finlay BERTRAM
Original assignee: Total Se
Priority date: 2020-02-11
Filing date: 2020-02-11
Publication date: 2021-08-19

Abstract

The invention relates to a process for well completion in a hydrocarbon reservoir. The hydrocarbon reservoir has one or more wellbores. The process includes a computer-implemented method. The method comprises providing a geological simulation grid representing the hydrocarbon reservoir. The method further comprises providing the one or more wellbores. The method further comprises providing a flow simulator. The flow simulator takes as inputs the geological simulation grid and a distribution of inflow devices in the one or more wellbores. The method further comprises providing an objective function of which evaluation includes an application of the flow simulator. The method further comprises optimizing the objective function to determine an optimal distribution of the inflow devices. The optimization is performed according to a covariance matrix adaptation evolution strategy (CMA-ES). This constitutes an improved process for well completion.

Description

WELL COMPLETION IN A HYDROCARBON RESERVOIR

FIELD OF THE INVENTION

The invention relates to the field of computer programs and systems, and more specifically to a process, system and program for well completion in a hydrocarbon reservoir.

In the hydrocarbon (oil and/or gas) production context, well completion designates a process of making a well ready for production and/or injection after drilling operation. Well completion typically includes installing inflow control devices in wellbores.

Existing methods for well completion rely on human intervention and manual problem boundingand lack of accuracy and/or efficiency. Other existing methods rely on reservoir scale brute force methodologies where different combinations and strategies are run. These methods are very time consuming, computationally heavy and often do not lead to the most optimum answer but rather the identification of the best answer from a series of simulations determined by the engineers with consideration of their knowledge.

Therefore, there is a need for an improved method for well completion in a hydrocarbon reservoir.

SUMMARY OF THE INVENTION

It is therefore provided a process for well completion in a hydrocarbon reservoir. The hydrocarbon reservoir has one or more wellbores. The process includes a computer-implemented method. The method comprises providing a geological simulation grid representing the hydrocarbon reservoir. The method further comprises providing the one or more wellbores. The method further comprises providing a flow simulator. The flow simulator takes as inputs the geological simulation grid and a distribution of inflow devices in the one or more wellbores. The method further comprises providing an objective function of which evaluation includes an application of the flow simulator. The method further comprises optimizing the objective function to determine an optimal distribution of the inflow devices. The optimization is performed according to a covariance matrix adaptation evolution strategy (CMA-ES).

The computer-implemented method may present any one or any combination of the following features:

- the distribution of the inflow devices includes inflow device positions each of a respective inflow device on a wellbore, the inflow device positions being free variables of the optimization;

- the geological simulation grid includes one or more series of segments each representing a respective wellbore, the inflow device positions corresponding each to a respective segment;

- the distribution of the inflow devices is represented by a vector, the vector having coordinates each corresponding to a respective segment and each having a respective value indicative of absence or presence of an inflow device;

- the optimization is iterative and comprises, at each iteration, determining one or more search points each representing a respective new value of the vector, and applying the flow simulator with the one or more search points;

- the determining of the one or more search point comprises performing a random sampling based on results of previous iterations;

- the method further comprises, prior to the optimization, a pre- processing of the distribution of the inflow devices;

- the inflow devices comprise several types of inflow devices;

- the inflow devices comprise one or more packers and/or one or more inflow control devices; and/or

- the objective function rewards high value of a hydrocarbon production measure, and/or the objective function penalizes low value of a water production measure.

It is further provided a computer program comprising instructions for performing the method. It is further provided a data storage medium having recorded thereon the computer program.

It is further provided a computer comprising a processor coupled to a memory, the memory having recorded thereon the computer program.

The process may consist of the method, which already forms a well completion stage. Alternatively, the process may comprise one or more steps prior to the method and/or one or more steps after the method. For example, the process may further include performing a physical well completion based on the optimal distribution. This may be performed after the method.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will now be described, by way of non-limiting example, and in reference to the accompanying drawings, where:

FIG.l to 8 illustrate the process; and FIG. 9 shows an example of the system.

DETAILED DESCRIPTION OF THE INVENTION

It is provided a process for well completion in a hydrocarbon reservoir. The hydrocarbon reservoir has one or more wellbores. The process includes a computer- implemented method. The method comprises providing a geological simulation grid representing the hydrocarbon reservoir. The method further comprises providing the one or more wellbores. The method further comprises providing a flow simulator. The flow simulator takes as inputs the geological simulation grid and a distribution of inflow devices in the one or more wellbores. The method further comprises providing an objective function of which evaluation includes an application of the flow simulator. The method further comprises optimizing the objective function to determine an optimal distribution of the inflow devices. The optimization is performed according to a covariance matrix adaptation evolution strategy (CMA-ES).

This constitutes an improved process for well completion in a hydrocarbon reservoir.

Notably, the process comprises a method which determines an optimal distribution of the inflow devices in the one or more wellbores. As known perse, the inflow devices allow to manage hydrocarbon production and/or injection. The determination of their distribution eventually allows to perform a physical well completion based on the determined optimal distribution. The process may in fact comprise performing a physical well completion of the hydrocarbon reservoir based on the optimal distribution. The performing of the physical well completion may for example comprise positioning real physical inflow devices in one or more wellbores of a hydrocarbon reservoir according to the determined optimal distribution. Because the distribution is optimal, the physical well completion is improved as the inflow devices are thereby physically optimally distributed on the one or more wellbores. For examples, they are optimally distributed with respect to maximizing hydrocarbon production, e.g. if the objective function rewards high value of a hydrocarbon production measure, and/or with respect to minimizing water production, e.g. if the objective function rewards low value of a water production measure.

As a matter of fact, the optimal distribution of the inflow devices is notably the result of the optimizing of the objective function. In other words, optimizing the objective function, which amounts to searching for an optimum of the objective function, results in the optimal distribution. This means that the optimal distribution of the inflow devices tends to be, or at least to be close to, an optimum of the objective function. The objective function may be any function that pertains to the management of hydrocarbon flow in the one or more wellbores. In examples, the objective function may reward high value of a hydrocarbon production measure, and/or the objective function may reward low value of a water production measure, as previously discussed.

The method is efficient and fast, as it reduces the use of human interventions and manual operations for determining the optimal distribution. This is notably allowed by the CMA-ES according to which the objective function is optimized. The CMA-ES allows to determine efficiently and accurately the optimal distribution, notably because it aims at optimizing the objective function. The CMA-ES is known to other fields of technology, but, surprisingly, it proves itself efficient for optimizing the distribution of the inflow devices in the present case. This is all the more surprising in that CMA-ES is known to be usually performed for continuous variables. However, in the present context, it has been successfully applied to discrete variables.

That being said, as further discussed hereinafter, the CMA-ES requires evaluation of the objective function, as it pertains to its optimisation. The evaluation of the objective function is carried out at each iteration of the CMA-ES, as further discussed hereinafter. The evaluation includes applying the flow simulator. However, applying the flow simulator may be time-consuming and greedy in terms of computer resources consumption. Nevertheless, the CMA-ES is efficient in that it tends to require an amount of iterations which is relatively small, e.g. as compared to other optimization methods. Therefore, not too many flow evaluations are performed. This makes the method efficient, but nonetheless accurate.

The method is now further discussed.

The geological simulation grid may be any type of geological simulation grid representing the hydrocarbon reservoir and the one or more wellbores, and suitable for being inputted to the flow simulator. The providing of the grid may be carried out by a user e.g. by designing the grid or by retrieving an already designed grid. The grid has cells (also referred to as blocks) and nodes.

The objective function may be a function to maximize by the optimizing, or alternative a function to minimize by the optimizing. As previously said, the objective function may be any function that pertains to the management of hydrocarbon flow in the one or more wellbores. For example, the objective function may capture any hydrocarbon (oil and/or gas) production measure and/or any water production reduction measure, and/or any infrastructure construction cost, e.g. in correlation with hydrocarbon production and/or water production. In examples, the objective function may be to be maximized (respectively, to be minimized) and comprise a term rewarding high value of a hydrocarbon production measure, e.g. the term being equal to a quantity of hydrocarbon production (respectively, minus a quantity of hydrocarbon production), and/or a term rewarding low value of a water production measure, e.g. the term being equal to minus a quantity of water production (respectively, equal to a quantity of water production). In particular examples, the objective function may consist of any of said two terms, or of the addition of both terms.

The providing of the objective function may comprise a user-selecting the objective function among a (e.g. predefined) list of available objective functions, or user-establishing the objective function. Alternatively, the objective function may be provided automatically.

Any inflow device herein is a physical device suitable to be positioned in a wellbore in order to manage hydrocarbon and/or water flow in the wellbore. Any inflow device herein may be an inflow control device, which controls hydrocarbon and/or water flow in the wellbore. Any inflow device herein may be any one of an inflow control device (also referred to as "ICD"), an autonomous inflow control device (also referred to as "AICD"), a flow control valve (also referred to as "FCV"), or an inflow control valve (also referred to as "ICV"). Any inflow device herein may additionally or alternatively be a packer, which is a physical device suitable to be positioned in a wellbore in order to provide a seal between the outside of the production tubing and the inside of the casing, liner, or wellbore wall. In examples, the inflow devices of which distribution is optimized may comprise several types of inflow devices, e.g. one or more packers and/or one or more inflow control devices (e.g. one or more ICDs and/or ICVs).

The flow simulator designates any simulator suitable for simulating fluid (e.g. hydrocarbon and/or water) flow in the reservoir. The flow simulator may implement one or more numerical methods suitable for simulating fluid flow in the reservoir. The flow simulator takes as input the geological simulation grid and the distribution of the inflow devices, and outputs data pertaining to fluid flow in the reservoir. The values of the objection function may be computed directly based on outputs of the flow simulator. The providing of the flow simulator may comprise a user selecting the flow simulator among a (e.g. predefined) list of available simulators. Alternatively, the flow simulator may be provided automatically, e.g. in connection with the objective function, for example as a flow simulator required to evaluate the objective function. It is to be noted that any inflow device herein may be any type of Smart/Intelligent/ Advanced completion device. Such a device has a strong effect on the flow of the reservoir. As such, the distribution of the inflow devices, when inputted to the flow simulator, e.g. for evaluation of the objective function, may lead to extreme simulation times. The CMA-ES used by the method allows to reduce this difficulty, as it allows to determine the optimal distribution of the inflow devices in as few iterations as possible, and thus by applying the flow simulator as few as possible.

The method thus comprises the providing of tools used for the present CMA- ES-based optimization, namely the geological simulation grid, the flow simulator and the objective function. The optimization includes free variables representative of the distribution of inflow devices. The providing may in fact further comprise providing an initial value of the free variables. This initial value may result from a pre-processing step, prior to the optimization, as further discussed hereinafter. The method then operates the optimization of the objective function according to the CMA-ES. The optimization iteratively modifies the values of the free variables of the optimization until a termination criterion is met (e.g. a maximum number of iterations criterion and/or a convergence criterion). The optimization is performed according to the CMA-ES, i.e. the optimization implements steps of a CMA-ES.

The CMA-ES is well-known. In the present context, it is a genetic algorithm based approach that is used to find exact or approximate solutions of the problem defined by the objective function. It determines the most optimal distribution of the inflow devices with respect to constraints defined by the objective function, e.g. constraints pertaining to hydrocarbon and/or water flow in the wellbores and/or to infrastructure construction cost. With each loop of the algorithm, otherwise known as mutation, the algorithm learns and adapts the completion accordingly based on the new information it has received from the latest simulation and calculation of the objective function. For example, water production reduction, oil production maximisation, or any other constraints pertaining to hydrocarbon production and/or injection. The genetic algorithm mutates and creates a new completion design (e.g. a new distribution of the devices) based on the new information or stops mutating if the results are becoming less optimal and if the most desirable outcome has already been achieved. In contrast to most other evolutionary algorithms, the CMA-ES is, from the end user's perspective, quite straight forward. The user may choose an initial solution point (i.e. an initial distribution of the devices), and an initial step-size, and optionally, a number of candidate samples. The latter allows the user to change the characteristic search behavior and termination condition can be adjusted to the problem at hand. It is also to be noted that, besides providing optimal results and to lead to a completion designed with the most optimal outcome, the CMA-ES is user- friendly and simple to use with a graphical user interface, which makes the method ergonomic. A typical genetic algorithm requires two key parts, a genetic representation of the problem and an objective (or fitness) function. With complicated well architectures having a significant effect on simulation times, being able to determine the optimum design in the minimal number of runs is a significant competitive advantage in the reservoir engineering domain. Furthermore, the mechanised design loop removes the need for the currently required knowledge of smart/intelligent/advanced inflow device placement, as such it makes the design process far simpler for the end user engineer. As the process of design is simplified, advanced/smart/intelligent well completions will be more readily considered.

Due to the present context of well completion, the objective function may be non-convex, non-separable, ill-conditioned, multi-modal, and/or noisy, and the search space dimension may range between two and a few hundred. The CMA-ES nevertheless achieves the optimization successfully and with accuracy and efficiency. Furthermore, it does not require gradients evaluation, but only evaluations of the objective function, which makes it particularly efficient and reduces cost of search.

In examples, the distribution of the inflow devices includes inflow device positions each of a respective inflow device on a wellbore. In these examples, the inflow device positions are free variables of the optimization.

The optimization thus varies the inflow device positions. Specifically, at each step of the optimization, the optimization, based on the current inflow device positions, searches, among candidates inflow device positions, the next inflow device positions which would optimize better the objective function. The optimization then selects these next inflow device positions for the next iteration.

In examples, the geological simulation grid includes one or more series of segments each representing a respective wellbore. In these examples, the inflow device positions correspond each to a respective segment. A segment is a representation of a respective portion of a wellbore between two adjacent cells of the grid. A segment may be either a couple of cells of the grid or a couple of nodes along the grid. The segments may be defined by a user beforehand, e.g. at an initial stage of designing the geological simulation grid. Alternatively, the segments may already be defined when the grid is provided. Each inflow device position corresponds to a respective segment in that the inflow device position specifies a position of the inflow device on the respective segment.

In examples, the distribution of the inflow devices is represented by a vector. The vector has coordinates each corresponding to a respective segment and each having a respective value indicative of absence or presence of an inflow device. For examples, the value may be equal to 1 when there is an inflow device at the respective segment and equal to 0 when there is none. The vector may be a matrix or an array. For example, the vector may be a combination (e.g. a concatenation) of a first vector (e.g. array) having coordinates each corresponding to a respective segment and each having a respective value indicative of absence or presence of an inflow control device and of a second vector (e.g. array) having coordinates each corresponding to a respective segment and each having a respective value indicative of absence or presence of a packer. The combination may result in an array or in a matrix (e.g. if there are N segments, an array of size 2 N or a matrix of size 2 x N).

In examples, the optimization is iterative and comprises, at each iteration, determining one or more search points each representing a respective new value of the vector, and applying the flow simulator with the one or more search points. In other words, the current value of the vector is fed as input to the CMA-ES (i.e. at each iteration/search step of it), which, based on the current value of the vector, determines the one or more search points each representing a new value for the vector. The applying of the flow simulator with the one or more search points yields an evaluation of the one or more search points on the objective function, which yields a new value of the vector. The iteration goes on this way, until a termination criterion is met as previously discussed. At the end, the CMA-ES outputs an optimized value for the vector.

The determining of the one or more search points may be carried out as follows. The CMA-ES searches among candidate search points, which each represent a candidate value of the vector, one or more search points each representing a new value of the vector that optimizes better the objective function. The candidate search points may represent candidate vectors each being a candidate value of the vector. The candidate vector may in an implementation each be a row or a line of a same matrix comprising all possible values of the vector. In such an implementation, the determining explores this matrix.

In examples, the determining of the one or more search point comprises performing a random sampling based on results of previous iterations (e.g. based on all the search points of all the previous iterations). In such an example, the sampling takes as input the current value of the vector. As known per se, this current value, resulting from a previous iteration of the CMA-ES, may depend on all previous iterations/search steps and results of these iterations. Indeed, as known per se the CMA-ES is based on the principle to increase the probability of successful candidate solutions and search steps. The mean of the distribution may be updated such that the likelihood of previously successful candidate solutions is maximized. The covariance matrix of the distribution may be updated (incrementally) such that the likelihood of previously successful search steps is increased. Therefore, at a given iteration, the current value of the vector results from the previous searches/iterations and their results. Based on this value, the determining of the one or more search points may comprise performing a sampling. The sampling results in new one or more search points. The sampling is carried out randomly but may be conditioned to the current value of the vector, so that the new one or more search points may correspond better to the current value of the vector than the previous one or more search points. It is also to be noted that the CMA-ES approach herein may apply a second order of learning to the model, so it can calculate the gradients between a variable, for example the number of inflow devices and how this gradient relate to the final output with respect to the objective function. In such a way, the next guess is further along that gradient and closer to the optimal outcome. This is done for all different parameters defined, and their independent gradients between one another are also calculated to improve the guess of the next step.

In examples, the process further comprising, prior to the optimization, a pre- processing of the distribution of the inflow devices. The pre-processing is any type of pre-processing that allows to render the distribution, before the optimization, conform to one or more well completion criteria. For example, if the inflow devices comprise ICDs/ICVs and Packers, the inflow devices could a priori be freely placed along the well by the optimization. However, there may be some geological/hydrocarbon production logic that dictates placement. For example, this may comprise avoiding a packer to be placed in such a way that it would collide with an ICD/ICV. In such a case, the pre-processing, e.g. comprising a normalisation, allows the first search of the CMA-ES to be reasonable and not arbitrary.

An implementation of the method is now discussed.

This implementation is based on the following conservation equation:

where:

• ΔM_mI is mass of component I in grid block m, (change in mass)

• Δt represents time step

• Q_Ijm represents the inter block flow rate of component I from neighbor block j (inlet) to block m (Flow rate in grid blocks remain constant).

• represents well inflow rate from any grid block in reservoir.

• N represents number of grid blocks

This implementation comprises a segmentation of the grid into the previously discussed segments. FIG. 1 illustrates segments for horizontal wells.

In this implementation, the flow rate equation for a well model is of the type:

where:

• T_j is the transmissibility connection factor (well index).

• is the mobility of the component I at the grid block connection j.

• P_j is the pressure in the grid block j with the segment.

• P_n is the pressure at the node of segment n.

• H_cj represents hydrostatic head correction between the center of the grid block and the depth of the completion (c represents grid block connection), i.e. the depth of the wellbore in the middle of the completed length within the grid block.

• H_nc represents hydrostatic head between the segment's node depth and the depth of the completion, which depends on the density of the fluid mixture within the segment.

The present implementation uses combinations. Specifically, all possible combinations of inflow devices [i.e., in this implementation, packers and inflow control device (ICD/ICV, also referred to as ICD/V)) are calculated with binomial coefficients:

The number of packers and ICD/V depends on segments that are defined along the wellbore and it can be reduced by analyzing well production or another parameter such as permeability or other engineering software. This combination may be a matrix which is such that each row has a combination of packers and ICD/V. In this implementation row numbers are inputs for the optimization.

As known per se CMA-ES is based on a defined number for "population size" and number of wells in each new search input, which are sampled and such that each element is between 1 and the maximum row number of combination matrix previously discussed. These elements can be both integer and floating numbers. After generating an input vector/matrix, a simulator software (i.e. the flow simulator) is used to calculate the objective function (e.g. oil production or water injection) based on the combination which is chosen by a normal random function. This means that the well model (2) is based on each combination and has different grid connections. Therefore, flow into wells varies and is affected by packers and ICD/V.

FIG. 2 shows a flow path when ICD/Vs and packers are defined on a segmented wellbore. There is no connection from the reservoir grid to the place where a packer is defined, therefore, the well model (2) does not calculate any flow rate. This is true wherever a packer is defined. However, if an ICD/V is defined as shown in FIG. 1 flow goes through grid block into well. Therefore, the well model (2) calculates the fluid that comes from a neighboring block and also considers hydrostatic head and pressure.

Segment numbers are fixed throughout the optimization. However, ICD/V and packers' placements are varying in each optimization iteration. In each iteration, CMA-ES is based on results from simulator software (e.g. oil production), to refine step size and search space based on step-size control and covariance matrix that are implemented in the algorithm. Using these two functions (step control and covariance matrix adaptation) reduces the number of necessary iterations and the iteration continues until CMA-ES minimizes the objective function.

The CMA-ES of the implementation is such that new search points are sampled normally distributed, as shown in FIG. 3. The sampling is in fact specifically implemented according the following:

where B, D result from an eigen decomposition of the covariance matrix C with C = BD²B^T = BDDB^T , and where:

• ~ denotes the same distribution on the left and right side.

• D scales the spherical distribution within the coordinate axes. • B defines a new orientation for the ellipsoid, where the new principal axes of the ellipsoid

• correspond to the columns of B.

• N(0, C^g) is a multivariate normal distribution with zero mean and covariance matrix C^g: m^g + σ^gN(0, C^g)~N(m^g, (σ^g)²C^g)

•

k-th offspring (individual, search point) from generation g + 1.

•

represents the mean value of search distribution at generation g.

• the so-called step-size (standard deviation) controls the step

length.

• the covariance matrix at generation g, determines the

shape of the distribution ellipsoid.

• λ ≥ 2, population size, sample size, number of offspring.

Here, all new points are sampled with the same parameters.

The CMA-ES of the implementation proceeds as follows:

Initialize distribution parameters θ⁰ For generation g = 0, 1, 2, ...

• Sample λ independent points from distribution P(X I θ^g) → X₁, ...,X_λ

• Evaluate the sample X₁, ... ,X_λ on ƒ

• Update parameters: θ^g+1 = F_θ (θ^g, (X₁,ƒ(X₁)), ..., (X_λ,ƒ(X_λ)))

• Break, if termination criterion met.

As to the objective function, noted ƒ: it is, in this

implementation, non-linear, non-separable and has at least moderate dimensionality, for example n < 10. ƒ can additionally also be non-convex, multimodal, non-smooth, discontinuous, ill-conditioned and/or noisy. This implementation copes with any of these function properties and is thus efficient. In this implementation, the sample of one offspring from parent m is performed according to the following equation.

X = m + σN(0 , C)

If X is better than, m then this implementation selects X: m ← X.

The implementation may perform here a non-elitist and intermediate recombination (μ/ μ ,λ ): which proceeds as follows:

Given the i-th solution point

Let X_i:λ the i-th ranked solution point, such thatƒ(X_i:λ) ≤ ··· ≤ ƒ(X_λ:λ )·

The new mean reads :

The best m points are selected from the new solutions (non-elitistic) and weighted intermediate recombination is applied.

The final equation rewrites (*) as an update of m:

This implementation uses step-size control, which is illustrated on FIG. 4, illustrating step-size control by (Cumulative Step-size Adaptation, CSA). The step-size control proceeds as follows:

• 1/5 -t h success rule, often applied with "+"-selection:

^■ increase step-size if more than 20% of the new solutions are successful, decrease otherwise

^■ P_s: of successful offspring / offspring (per generation):

_■

^■ IF offspring better parent

^■ ELSE

• σ self-adaptation, applied with ","-selection:

^■ mutation is applied to the step-size and the better search points, according to the objective function value, are selected, this implementation of the method thereby implementing simplified "global" self-adaptation

• path length control (Cumulative Step-size Adaptation, CSA):

^■ self-adaptation derandomized and non-localized

^■ Initialize evolution path P_σ = 0, set C_σ ≃ 4/n , d_σ ≃

1 :

This implementation may perform covariance matrix adaptation according to the likelihood of successful steps, y_w, to appear again. Alternatively, the adaptation may follow a natural gradient approximation of the expected fitness.

The rank-one update proceeds as follows in the present implementation:

• Initialize and C = 1, set s = 1, learning rate

not terminate:

• Learns all pairwise dependencies between variables: Off-diagonal entries in the covariance matrix reflect the dependencies.

• Conducts a principle component analysis (PCA) of steps μ_w , sequentially in time and space. Eigenvectors of the covariance matrix C are the principle components / the principle axes of the mutation ellipsoid, rotational invariant.

• Learns a new, rotated problem representation and a new metric (Mahalanobis). Components are independent (only) in the new representation rotational invariant.

• Approximates the inverse Hessian on quadratic functions.

• Is entirely independent of the given coordinate system, for μ = 1: natural gradient ascent on N.

FIG. 5 illustrates how the CMA increases the likelihood of successful step by changing the search domain.

The implementation performs an evolution path. Conceptually, the evolution path is the search path the strategy takes over a number of generation steps. It can be expressed as a sum of consecutive steps of the mean m, based on the following:

"Cumulation" is a widely used technique also known as:

• Exponential smoothing in time series, forecasting

• Exponentially weighted moving average

• Iterate averaging in stochastic approximation

• Momentum in the back-propagation algorithm for ANNs

The rank- m update proceeds as follows. The rank- m update extends the update rule for large population sizes λ using μ > 1 vectors to update C at each generation step. The matrix

computes a weighted mean of the outer products of the best m steps and has rank min (μ, n) with probability one. The rank- m update then reads:

where C_cov ≃ μ/n² and C_cov ≤ 1.

FIG. 6 illustrates a sampling of λ = 150 solutions where C = I and σ = 1. FIG.

7 illustrates the calculating of C where and C_cov = 1.

FIG. 8 illustrates the new distributions.

The update:

• increases the possible learning rate in large populations, roughly from 2/n²to μ_w/n²

• can reduce the number of necessary generations roughly from

given μ_w λ n. Therefore, the rank-m update is the primary

mechanism whenever a large population size is used, for example λ ≥ 3n + 10. The rank-one update uses the evolution path and reduces the number of necessary function evaluations to learn straight ridges from

• can be combined with the rank-one update.

A pseudo-code of the algorithm implemented by the implementation is now presented.

Set parameters:

Selection and Recombination:

Step-Size control:

Covariance matrix adaptation:

Initialization:

• Set evolution paths P_σ = 0, covariance matrix C = I and g = 0.

• Choose distribution mean and step size

dependent.

Until termination criterion met, g ← g + 1:

• Sample new population of search points, for k = 1,

o Selection and recombination:

Nomenclature Matrix Adaptation :

all the steps) of the method are executed by at least one computer, or any system alike. Thus, steps of the method are performed by the computer, possibly fully automatically, or, semi-automatically. In examples, the triggering of at least some of the steps of the method may be performed through user-computer interaction. The level of user-computer interaction required may depend on the level of automatism foreseen and put in balance with the need to implement user's wishes. In examples, this level may be user-defined and/or pre-defined.

A typical example of computer-implementation of a method is to perform the method with a system adapted for this purpose. The system may comprise a processor coupled to a memory and a graphical user interface (GUI), the memory having recorded thereon a computer program comprising instructions for performing the method. The memory may also store a database. The memory is any hardware adapted for such storage, possibly comprising several physical distinct parts (e.g. one for the program, and possibly one for the database). FIG. 9 shows an example of the system, wherein the system is a client computer system, e.g. a workstation of a user.

The client computer of the example comprises a central processing unit (CPU) 1010 connected to an internal communication BUS 1000, a random access memory (RAM) 1070 also connected to the BUS. The client computer is further provided with a graphical processing unit (GPU) 1110 which is associated with a video random access memory 1100 connected to the BUS. Video RAM 1100 is also known in the art as frame buffer. A mass storage device controller 1020 manages accesses to a mass memory device, such as hard drive 1030. Mass memory devices suitable for tangibly embodying computer program instructions and data include all forms of nonvolatile memory, including by way of example semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks and removable disks; magneto-optical disks; and CD-ROM disks 1040. Any of the foregoing may be supplemented by, or incorporated in, specially designed ASICs (application-specific integrated circuits). A network adapter 1050 manages accesses to a network 1060. The client computer may also include a haptic device 1090 such as cursor control device, a keyboard or the like. A cursor control device is used in the client computer to permit the user to selectively position a cursor at any desired location on display 1080. In addition, the cursor control device allows the user to select various commands, and input control signals. The cursor control device includes a number of signal generation devices for input control signals to system. Typically, a cursor control device may be a mouse, the button of the mouse being used to generate the signals. Alternatively or additionally, the client computer system may comprise a sensitive pad, and/or a sensitive screen.

The computer program may comprise instructions executable by a computer, the instructions comprising means for causing the above system to perform the method. The program may be recordable on any data storage medium, including the memory of the system. The program may for example be implemented in digital electronic circuitry, or in computer hardware, firmware, software, or in combinations of them. The program may be implemented as an apparatus, for example a product tangibly embodied in a machine-readable storage device for execution by a programmable processor. Method steps may be performed by a programmable processor executing a program of instructions to perform functions of the method by operating on input data and generating output. The processor may thus be programmable and coupled to receive data and instructions from, and to transmit data and instructions to, a data storage system, at least one input device, and at least one output device. The application program may be implemented in a high-level procedural or object-oriented programming language, or in assembly or machine language if desired. In any case, the language may be a compiled or interpreted language. The program may be a full installation program or an update program. Application of the program on the system results in any case in instructions for performing the method.

Claims

1. A process for well completion in a hydrocarbon reservoir having one or more wellbores, the process including: a computer-implemented method comprising: providing:

^■ a geological simulation grid representing the hydrocarbon reservoir and the one or more wellbores,

^■ a flow simulator that takes as inputs the geological simulation grid and a distribution of inflow devices in the one or more wellbores, and

^■ an objective function of which evaluation includes an application of the flow simulator; optimizing the objective function to determine an optimal distribution of the inflow devices, the optimization being performed according to a covariance matrix adaptation evolution strategy (CMA-ES).

2. The process of claim 1, wherein the distribution of the inflow devices includes inflow device positions each of a respective inflow device on a wellbore, the inflow device positions being free variables of the optimization.

3. The process of claim 2, wherein the geological simulation grid includes one or more series of segments each representing a respective wellbore, the inflow device positions corresponding each to a respective segment.

4. The process of claim 3, wherein the distribution of the inflow devices is represented by a vector, the vector having coordinates each corresponding to a respective segment and each having a respective value indicative of absence or presence of an inflow device.

5. The process of claim 4, wherein the optimization is iterative and comprises, at each iteration, determining one or more search points each representing a respective new value of the vector, and applying the flow simulator with the one or more search points.

6. The process of claim 5, wherein the determining of the one or more search point comprises performing a random sampling based on results of previous iterations.

7. The process of any one of claims 1 to 6, wherein the process further comprising, prior to the optimization, a pre-processing of the distribution of the inflow devices.

8. The process of any one of claims 1 to 7, wherein the inflow devices comprise several types of inflow devices.

9. The process of claim 8, wherein the inflow devices comprise one or more packers and/or one or more inflow control devices.

10. The process of any one of claims 1 to 9, wherein the objective function rewards high value of a hydrocarbon production measure, and/or the objective function penalizes low value of a water production measure.

11. The process of any one of claims 1 to 10, wherein the process further includes performing a physical well completion based on the optimal distribution.

12. A computer program comprising instructions that, when executed by a computer, cause the computer to perform the computer-implemented method of the process of any one of claims 1 to 11.

13. A data storage medium having recorded thereon the computer program of claim 12.

14. A computer system comprising a processor coupled to a memory, the memory having recorded thereon the computer program of claim 12.