EP2022934A2 - Method of evaluating a production scheme of an underground reservoir, taking into account uncertainties - Google Patents

Abstract

Method to evaluate a production scheme of an underground deposit taking into account uncertainties. Input parameters of a flow simulator are selected, characterizing the deposit and the production scheme. An approximate analytical model is constructed to predict reservoir responses. One defines a degree of precision D p that one wishes to obtain, this degree of precision D p measuring the difference between the answers of the model and those of the simulator. The degree of precision D p (M) of the predictions of the model is calculated. We build a plan of experiments to select simulations to perform, relevant to adjust the model. The simulations selected by the experimental design are carried out, then, for each of the simulated responses by the simulator, the analytical model is adjusted using an approximation method. It is repeated until the desired degree of accuracy D p is reached. Finally, the production scheme is evaluated by analyzing the deposit responses predicted by the approximate analytical model. Application particularly to the exploitation of oil deposits.

Description

The present invention relates to the field of exploration and exploitation of oil deposits. More particularly, the invention relates to the evaluation of such deposits, by studying and optimizing production patterns of such oil deposits.

A production scheme is an option for developing a deposit. It groups all the parameters necessary for the production of a deposit. These parameters can be the position of a well, the level of completion, the drilling technique ...

The study of a deposit has two main phases: a reservoir characterization phase and a production forecasting phase.

The characterization phase of the reservoir consists of constructing a reservoir model. A reservoir model is a model describing the spatial structure of the deposit, in the form of a discretization of space. This discretization is materialized by a set of meshes. At each of these meshes, we associate values of properties characterizing the deposit: porosity, permeability, lithology, pressure, nature of the fluids, ... The engineers have access only to a small part of the deposit they study ( measurements on cores, logs, well tests, ...). They need to extrapolate these point data across the entire oil field to build a reliable reservoir model. Consequently, the notion of uncertainty must be constantly taken into account.

For the production forecast phase at a given moment, to improve this production, or, in general, to increase the economic yield of the field, the specialist has a tool, called "flow simulator". A flow simulator is a software allowing, among other things, to model the production of a deposit as a function of time, from measurements describing the deposit, that is to say, from the reservoir model.

A flow simulator works by accepting input parameters, and solving physical fluid mechanics equations in porous media, to deliver information called responses. The set of input parameters is contained in the reservoir model. The properties associated with the meshes of this model are then called settings. These parameters are associated with the geology of the deposit, the petro-physical properties, the development of the deposit and the numerical options of the simulator. The responses (outputs) provided by the simulator are, for example, the production of oil, water or gas tank and each well for different times. Generally, for each of the values of the different input parameters, the flow simulator returns a single value for each response (output). The flow simulator is then qualified as deterministic.

However, the majority of input parameters are uncertain. These uncertainties result in the fact that one can not assign a single value, which is sure of value, to a parameter of the reservoir model. For example, we can not ensure that the porosity at one point of the deposit is 20%. It can best be considered that the porosity is between 15% and 25% at this point. This is because the input parameters are determined using a limited number of measurements and information. The possible responses of the flow simulator are therefore multiple, given the inherent uncertainty of the reservoir model. In our example, there will be a response from the simulator if the porosity is 15%, a different response if the porosity is 20.5% ... It is thus essential to be able to quantify the uncertainty on the outputs of the simulator. Similarly, a correct characterization of the uncertainty of the input parameters is essential. It is also important to determine input parameters that have a significant effect on interest responses.

The specialist in the exploitation of an oil field must therefore integrate these notions of uncertainty into the evaluation of a deposit, so as to determine, for example, optimal production conditions.

State of the art

In order to properly characterize the impact of each uncertainty on oil production, many production scenarios need to be tested, and as a result, a significant number of reservoir simulations are needed.

However, in the oil industry, to be more reliable and predictive, the trend is to use increasingly complex flow simulators, which require a more and more detailed reservoir model (several million mesh). But given the significant time required to perform a flow simulation, it can not be considered to test all possible scenarios via a flow simulator.

To avoid carrying out a large number of simulations, one knows a technique, described in the patent FR 2,874,706 , and based on the experimental plans. This method makes it possible to manage uncertainties via the construction of approximate models, called "response surfaces", obtained for example by kriging. These surfaces provide approximate responses to those from the flow simulator.

However, any response surface makes a more or less important prediction error, depending on the response it tries to approximate. In general, the addition of information (i.e. simulations) allows the construction of an increasingly predictive response surface.

The object of the invention is an alternative method for evaluating underground deposit production patterns by estimating the production of such deposits using an approximate model, and iteratively adjusted to reproduce at better simulator responses, while controlling the number of simulations required for its construction.

The method according to the invention

• on ajuste ledit modèle analytique approché à l'aide d'un processus itératif comportant les étapes suivantes :
  1. a)- on définit, pour chacune desdites réponses, un degré de précision D_p que l'on souhaite obtenir, ledit degré de précision D_p mesurant l'écart entre les réponses prédites par le modèle et celles simulées par le simulateur ;
  2. b)- on calcule un degré de précision D_p(M) des prédictions du modèle analytique approché ;
  3. c)- si cette valeur D_p(M) est inférieure au degré de prédiction souhaité D_p , le processus itératif s'arrête, sinon on poursuit par les étapes suivantes :
  4. d)- on construit un plan d'expériences de façon à sélectionner des simulations à réaliser, pertinentes pour ajuster ledit modèle ;
  5. e)- on réalise les simulations sélectionnées par le plan d'expérience à l'aide du simulateur d'écoulement, puis, pour chacune des réponses simulées par le simulateur, on ajuste ledit modèle analytique à l'aide d'une méthode d'approximation, de façon à ajuster les réponses prédites par le modèle à celles simulées par le simulateur ;
  6. f)- on recommence à l'étape b), jusqu'à ce que le degré de précision souhaité D_p soit atteint ; et
• on évalue ledit schéma de production, en analysant lesdites réponses dudit gisement prédites par ledit modèle analytique approché.

The invention relates to a method for evaluating a production scheme of an underground deposit. According to the method, physical properties characterizing said deposit and said production scheme are selected. These properties are input parameters of a flow simulator to simulate reservoir responses, such as production. An approximate analytical model is constructed to predict said deposit responses. The method also includes the following steps:

• said approximate analytical model is adjusted using an iterative process comprising the following steps:
  1. a) - one defines, for each of said responses, a degree of precision D _p that one wishes to obtain, said degree of precision D _p measuring the difference between the responses predicted by the model and those simulated by the simulator;
  2. b) - a degree of precision D _p (M) of the predictions of the approximate analytical model is calculated;
  3. c) - if this value D _p (M) is less than the desired degree of prediction D _p , the iterative process stops, otherwise the following steps are continued:
  4. d) - one builds a plan of experiments so as to select simulations to realize, relevant to adjust said model;
  5. e) - the simulations selected by the experimental plane are carried out using the flow simulator, then, for each of the simulated responses simulated by the simulator, said analytical model is adjusted using a method of approximation, so as to adjust the responses predicted by the model to those simulated by the simulator;
  6. f) - is repeated in step b) until the desired degree of accuracy D _p is reached; and
• said production scheme is evaluated by analyzing said responses of said deposit predicted by said approximate analytical model.

According to the invention, it is possible to modify the desired degree of precision D _p at each of the iterations. The input parameters may be uncertain, that is, the values of these input parameters are uncertain.

The deposit responses, predicted by the approximate analytical model, can be analyzed by quantifying an influence of each of the input parameters on each of the responses, using an overall sensitivity analysis, in which sensitivity using the analytical model. With the help of this global sensitivity analysis, we can use the most influential parameters on the deposit's responses, and thus define the measures to be taken to reduce uncertainty in the deposit's responses.

According to the invention, if the input parameters comprise at least one stochastic field, it is possible to decompose this stochastic field into a number n of components via a Karhunen-Loeve decomposition. The components of the stochastic field that have an impact on the responses are then selected using the global sensitivity analysis.

Other characteristics and advantages of the method according to the invention will appear on reading the following description of nonlimiting examples of embodiments, with reference to the appended figures and described below.

Brief presentation of the figures

• the figure 1 represents a framework of the uncertainty management method according to the invention.
• the figure 2 shows an example of evolution of the estimated prediction error (in%) by a response surface (approximate model).

Detailed description of the method

1. 1- Sélection et caractérisation des incertitudes des paramètres d'entrée du simulateur
2. 2- Construction d'un modèle analytique approché du simulateur
3. 3- Ajustement du modèle analytique approché
4. 4- Optimisation du schéma de production du gisement.

The method according to the invention makes it possible to optimize the production scheme of a petroleum deposit. The method is schematized by the diagram of the figure 1 . After choosing a flow simulator, the method includes the following steps:

1. 1- Selection and characterization of the uncertainties of the input parameters of the simulator
2. 2- Construction of an approximate analytical model of the simulator
3. 3- Adjustment of the approximate analytical model
4. 4- Optimization of the deposit production scheme.

Step 1: Selection and characterization of the uncertainties of the simulator input parameters

Any flow simulator makes it possible to calculate the production of hydrocarbons or water as a function of time, based on physical parameters characteristic of the oil reservoir, such as the number of layers of the reservoir, the permeability of the layers, the force of the the aquifer, the position of the oil wells, etc.

These physical parameters constitute the inputs of the flow simulator. They are obtained by laboratory measurements on cores and fluids taken from the oil field, by logging (measurements along a well), by well tests, etc.

Among the physical parameters characteristic of the oil field, it is preferable to select input parameters having an influence on the profiles of production of hydrocarbons or water by the deposit. The selection of the parameters can be done either with respect to the physical knowledge of the oil field, or by a sensitivity study. For example, it is possible to implement a Student or Fischer statistical test.

Parameters may be intrinsic to the oil reservoir. For example, the following parameters may be considered: permeability of certain layers of the reservoir, aquifer strength, residual oil saturation after sweeping with water ...

Parameters may correspond to deposit development options. These parameters can be the position of a well, the level of completion, the drilling technique.

After selecting these input parameters, the uncertainties associated with these parameters are characterized. For example, a value of a parameter can be replaced by a variation interval of this parameter.

Step 2: Construction of an approximate analytical model of the simulator

Since the flow simulator is a complex and time-consuming tool, it can not be used to test all scenarios taking into account all the uncertainties of the parameters. An approximate analytical model of the behavior of the oil field is then constructed. This approximate model is also called "response surface". It consists of a set of analytical formulas, each of which translates the behavior of a given response of the flow simulator. These analytic formulas are based on a small number of parameters, and they are constructed from a limited number of simulations.

This approximate model reflects the behavior of given responses, for example the cumulative oil produced at 10 years, according to some input parameters. Thus, for each response (output) of the flow simulator, necessary for the optimization of production or the evaluation of the deposit, we associate an analytical formula for approximating this response from input parameters.

To construct this approximate model of the flow simulator, two techniques are combined: an approximation method and a sample plan method.

Experimental designs allow to determine the number and location, in the space of the input parameters, of a reduced number of simulations to realize to have the maximum of relevant information, at the lowest possible cost.

The technique of experimental designs is described for example in Droesbeke JJ, et al., 1997; "Plans of experiences, Applications to the company", Technip Editions .

A plan indicates different sets of values for the uncertain parameters. Each set of uncertain parameter values is used to perform a flow simulation. In the input parameter space, each simulation represents a point. Each point corresponds to values for the uncertain parameters and therefore to a possible reservoir model. The choice of these points, thanks to the experimental designs, can involve many types of criteria, such as orthogonality or space-filling.

For this "exploratory" stage, the choice of simulation points can be made thanks to different types of experimental designs, for example, factorial plans, composite plans, maximum distance plans, etc. One can also use a plan of experiment of the type Hypercube Latin Maximin or Sobol LP-τ ( A. Saltelli, K. Chan and M. Scott: "Sensitivity Analysis", New York, Wiley, 2000 ).

After the construction of this experimental design, and when the flow simulations are performed, an approximation method is used to determine an approximate model. This model approaches the responses of the flow simulator. In a very simplified way, we can imagine that by performing four simulations, we obtain four couples (input parameter, response). We then estimate a relationship that best respects these couples.

In practice, the parameters and the outputs being multiple, it is possible to use, as approximation method, first or second order polynomials, neural networks, vector support machines or possibly higher order polynomials than two . Numerous other techniques are known to those skilled in the art, such as wavelet, SVM, self reproducing Hilbert methods, or non-parametric regression based on a Gaussian or kriging process ( Kennedy M., O'Hagan A .: "Bayesian calibration of computer models (with discussion)". J. Statist. Soc. Ser. B Stat. METHODOL. 68, 425-464, 2001 ). The choice of the method depends on the one hand on the number of simulations that can be envisaged by the user, and on the other hand, on the initial experimental design used.

• on construit un plan d'expériences de façon à sélectionner un nombre restreint de simulations ;
• on réalise les simulations sélectionnées par le plan d'expérience à l'aide du simulateur d'écoulement, à partir de paramètres d'entrée sélectionnés ;
• pour chacune des réponses du simulateur, on définit une formule analytique reliant les paramètres d'entrée sélectionnés à la réponse (issue des simulations), à l'aide d'une méthode d'approximation.

Thus, to build the approximate model, we proceed as follows:

• an experimental design is constructed to select a limited number of simulations;
• the simulations selected by the experimental plane are carried out using the flow simulator from selected input parameters;
• for each of the responses of the simulator, we define an analytical formula connecting the selected input parameters to the response (resulting from the simulations), using an approximation method.

Step 3- Adjust the approximate analytical model

The approximate model, thus determined, makes it possible to predict the outputs of the flow simulator with a certain precision. According to the invention, the method comprises a measurement of the prediction accuracy of this model so as to define an evaluation criterion associated with the precision of the constructed approximate model. The figure 2 illustrates an example of the evolution of the estimated prediction error ( Err ) by a response surface (approximate model), as a function of the number of simulations ( Nsim ) used to construct the response surface. In this example, the response surface approximates the output of the flow simulator corresponding to the oil flow of the tank after 10 years of production.

This criterion allows a user to decide whether to add simulations to improve the predictive reliability of the model.

1. a)- on définit un degré de précision D_p de la prédiction du modèle approché que l'on souhaite obtenir pour chaque réponse du simulateur que l'on veut analyser.
2. b)- on estime le degré de prédiction D_p(M) du modèle analytique approché. Cette estimation peut se faire en utilisant des méthodes de type validation croisée ou bootstrap.
3. c)- Si cette valeur D_p(M) est inférieure au degré de prédiction souhaité D_p , le processus itératif automatique s'arrête, sinon on poursuit par les étapes suivantes :
4. d)- on sélectionne p nouvelles combinaisons de paramètres d'entrée dans l'espace des paramètres d'entrée, au moyen d'une méthode adaptative. Une méthode adaptative consiste à ajouter de l'information aux endroits où il en manque, et où le modèle approché n'est pas suffisamment prédictif. De telles méthodes sont bien connues des spécialistes.
5. e)- on réalise les p simulations correspondantes, et l'on modifie le modèle approché en conséquences.
6. f)- puis l'on recommence à l'étape b), jusqu'à ce que le degré de précision soit atteint. On peut également recommencer à l'étape a), de façon à définir un nouveau degrés de précision.

On peut également arrêter "manuellement" le processus.The required degree of prediction is obtained iteratively. This step breaks down as follows:

1. a) - one defines a degree of precision D _p of the prediction of the approximate model that one wishes to obtain for each answer of the simulator that one wants to analyze.
2. b) - we estimate the degree of prediction D _p (M) of the approximate analytical model. This estimate can be done using cross validation or bootstrap methods.
3. c) - If this value D _p (M) is lower than the desired degree of prediction D _p , the automatic iterative process stops, otherwise we continue with the following steps:
4. d) - selecting p new combinations of input parameters in the space of input parameters by means of an adaptive method. An adaptive method is to add information where it is missing, and where the approximate model is not predictive enough. Such methods are well known to those skilled in the art.
5. e) - p corresponding simulations are carried out, and it changes the approximate model accordingly.
6. f) - then one starts again in step b), until the degree of precision is reached. We can also start again at step a), so as to define a new degree of precision.

One can also stop "manually" the process.

The number p of simulations carried out at each iteration can be controlled by the user, depending on the number of machines available to perform simulations for example.

The approximate model thus obtained makes it possible to predict the responses almost instantaneously (in computing time), and thus makes it possible to replace the expensive flow simulator in computing time. We can therefore test a large number of production scenarios, while taking into account the uncertainty of each input parameter.

• Scheidt C., Zabalza-Mezghani I., Feraille M., Collombier D.: "Adaptive Evolutive Experimental Designs for Uncertainty Assessment - An innovative exploitation of geostatistical techniques", IAMG, Toronto, 21-26 August, Canada, 2005 .
• Busby D., Farmer C.L., Iske A.: "Hierarchical Nonlinear Approximation for Experimental Design and Statistical Data Fitting". SIAM J. Sci. Comput. 29, 1, 49-69, 2007

Dans Busby et al., on effectue d'abord une partition de l'espace en différentes zones de taille équivalente (méthode connue des spécialistes sous le nom « adaptive gridding »). Les nouveaux points sont ensuite ajoutés dans les zones où la prédiction du modèle approché n'est pas bonne (i.e. au dessous du degré de précision D_p fixé par l'utilisateur). La prédiction du modèle est calculée indépendamment dans chaque zone. Cette erreur de prédiction est calculée en prenant la moyenne des erreurs obtenue par validation croisée (« leave-one-out »).The methods used to select new points in the parameter space in step d) can be various. For example, one can use a method described in the following documents:

• Scheidt C., Zabalza-Mezghani I., Feraille M., Collombier D .: "Adaptive Evolutive Experimental Designs for Uncertainty Assessment - An Innovative Exploitation of Geostatistical Techniques", IAMG, Toronto, 21-26 August, Canada, 2005 .
• Busby D., Farmer CL, Iske A .: "Hierarchical Nonlinear Approximation for Experimental Design and Statistical Data Fitting". SIAM J. Sci. Comput. 29, 1, 49-69, 2007

In Busby et al., A partition of the space into different zones of equivalent size is first performed (a method known to specialists under the name "adaptive gridding"). The new points are then added in the areas where the prediction of the approximated model is not good (ie below the degree of precision D _p fixed by the user). The prediction of the model is calculated independently in each zone. This prediction error is calculated by taking the average of the errors obtained by cross validation ("leave-one-out").

The addition of simulations in step e) is repeated, automatically, to satisfy a stopping criterion which is related to the degree of prediction desired by the user, defined in step a), by example average prediction of 5% of the response studied. An example of estimation of the prediction is obtained from the average of the cross-validation errors in each zone.

• uniquement à la production cumulée de l'huile (gaz, eau) du réservoir au temps final de production,
• à la production cumulée de l'huile (gaz, eau) du réservoir pour différents temps,
• à l'ajout de la production d'huile et de la production d'eau,
• à la production d'huile pour des valeurs fixées de water cut (ou de production d'eau),
• à la durée du plateau du profil de production,...

The selected interest responses may be direct outputs from the flow simulator or output combinations and interpolations. For example, may be interested:

• only to the cumulative production of the oil (gas, water) of the tank at the final time of production,
• the cumulative production of the oil (gas, water) of the tank for different times,
• the addition of oil production and water production,
• the production of oil for fixed values of water cut (or water production),
• to the shelf life of the production profile, ...

Moreover one can easily add economic uncertainties and combine them with the technical uncertainties to define answers associated with the economic value of the deposit like for example the Net Present Value (NPV) and not to be limited to technical answers (of production of oil, gas, water, ...). Such a method is described in the application EP 1 484 704 .

4- Optimization of the production scheme and evaluation of the deposit

The principle of optimization of the production scheme consists of defining different production scenarios, and for each of them, predicting production. This technique also makes it possible, in the same way, to economically evaluate a petroleum deposit.

During this production forecasting phase, the approximate model is used because it is simple and analytical and, therefore, every estimate obtained by this model is immediate. This saves a lot of time. The use of this model allows the reservoir engineer to test as many scenarios as he wants, without worrying about the time required to perform a numerical flow simulation, and above all it allows him to take into account the uncertainties by testing different values of input parameters.

The approximate analytical model is used with Monte Carlo or Quasi Monte Carlo direct sampling techniques (MCMC, Hypercube Latin, ...) in order to propagate the uncertainties of the input parameters on the selected simulator response (s).

This gives the probability distributions associated with the outputs of the simulator. These distributions are useful for making decisions about the exploitation of the deposit in question, taking into account the potential production or economic value and the associated uncertainty.

According to a particular embodiment, the approximate model is used to perform an overall sensitivity analysis, so as to select the parameters influencing the production of the deposit, in order to carry out the measurements necessary for a better evaluation of the deposit.

For example, it is interesting to know that the activity of the aquifer or the permeability of a particular geological layer plays a major role in the future production results of the deposit.

• Saltelli, K. Chan and M. Scott: "Sensitivity Analysis", New York, Wiley, 2000
• Oakley and A. O'Hagan: "Probabilistic sensitivity analysis of complex models: A Bayesian approach", J. Roy. Statist. Soc. Ser. B, 16, pp. 751-769, 2004 .

L'ASG est basée sur une décomposition de Sobol. Cette décomposition est décrite dans le document suivant : I.M Sobol: "Sensitivity estimates for nonlinear mathematical models". Mathematical Modelling and Computational Experiments, 1 :407-414, 1993 .The Global Sensitivity Analysis (ASG) of the uncertain parameters on the simulator responses allows to analyze in detail the impact of the uncertainty of each uncertain parameter or parameter group, on the uncertainty of the simulator responses. Such a technique is described in:

• Saltelli, K. Chan and M. Scott: "Sensitivity Analysis", New York, Wiley, 2000
• Oakley and A. O'Hagan: "Probabilistic sensitivity analysis of complex models: A Bayesian approach", J. Roy. Statist. Soc. Ser. B, 16, pp. 751-769, 2004 .

ASG is based on a decomposition of Sobol. This decomposition is described in the following document: IM Sobol: "Sensitivity estimates for nonlinear mathematical models". Mathematical Modeling and Computational Experiments, 1: 407-414, 1993 .

To describe the method, we consider a mathematical model described by a function f (x), x = ( x _l , ..., x _p ), and defined in p-space p Ω ^p = {x | 0 ≤ x _i ≤ 1; i = 1, ... p }.

avec f₀ une constante, et $\underset{0}{\overset{1}{\int }}{f}_{i1,\dots ,\mathit{is}}\left(x_{i1}\dots x_{\mathit{is}}\right){\mathit{dx}}_{\mathit{ik}}=0,$

où 1 ≤ i1 < .... < is ≤ p, s = 1,...,p et 1 ≤ k ≤ s.The main idea of the Sobol decomposition is to decompose f (x _l , ..., x _p ) as follows: $$f\left(x_1...x_p\right)=f_0+{\displaystyle \underset{i=1}{\overset{p}{\Sigma }}}{f}_i\left(x_i\right)+{\displaystyle \underset{1\le i\le j\le p}{\Sigma }}{f}_{\mathit{ij}}\left(x_i{x}_j\right)+...+f_{1,2...p}\left(x_1...x_p\right)$$

with f ₀ a constant, and $\underset{0}{\overset{1}{\int }}{f}_{i1,...,\mathit{is}}\left(x_{i1}...x_{\mathit{is}}\right){\mathit{dx}}_{\mathit{ik}}=0,$

where 1 ≤ i 1 <.... < is ≤ p, s = 1, ..., p and 1 ≤ k ≤ s .

et si (il,...,is)≠(j1,...,jl), alors $f_{{\mathrm{\Omega }}^p}{f}_{i1},\dots ,{\mathit{isf}}_{j1,\dots ,\mathit{jl}}\mathit{dx}$

According to this definition, one can write: $$f_0={\int }_{{\mathrm{\Omega }}^p}f\left(x\right)\mathit{dx}$$

and if ( he , ..., is ) ≠ ( j1 , ..., jl ), then $f_{{\mathrm{\Omega }}^p}{f}_{i1},...,{\mathit{isf}}_{j1,...,\mathit{jl}}\mathit{dx}$

$$f_{i,j}\left(x_i{x}_j\right)=-f_0-f_i\left(x_i\right)-f_{\mathit{j}}\left(x_j\right)+f_{{\mathrm{\Omega }}^{p-2}}f\left(x\right){\mathit{dx}}^{\mathit{ij}}$$

avec dxⁱ et dx^ij le produit dx₁...dx_p sans dx_i et dx_i dx_j , respectivement.
La variance totale V de ƒ(x) peut alors s'écrire : $$V={\displaystyle \sum _{i=1}^k}{V}_i+{\displaystyle \sum _{1\le i<j\le p}}{V}_{\mathit{ij}}+\dots +V_{1,2,\dots ,p}$$

ou encore: $V={\int }_{{\mathrm{\Omega }}^p}{f}^2\left(x\right)\mathit{dx}-f_0^2$

Sobol showed that the decomposition of f (x _{1, ...,} x _p ) is unique and that all terms can be evaluated via multidimensional integrals: $$f_i\left(x_i\right)=-f_0+f_{{\mathrm{\Omega }}^{p-1}}f\left(x\right){\mathit{dx}}^{\mathit{i}}$$

$$f_{i,j}\left(x_i{x}_j\right)=-f_0-f_i\left(x_i\right)-f_{\mathit{j}}\left(x_j\right)+f_{{\mathrm{\Omega }}^{p-2}}f\left(x\right){\mathit{dx}}^{\mathit{ij}}$$

with dx ⁱ and dx ^ij the product dx ₁ ... dx _p without dx _i and dx _i dx _j , respectively.
The total variance V of ƒ (x) can then be written: $$V={\displaystyle \underset{i=1}{\overset{k}{\Sigma }}}{V}_i+{\displaystyle \underset{1\le i<j\le p}{\Sigma }}{V}_{\mathit{ij}}+...+V_{1,2,...,p}$$

or: $V={\int }_{{\mathrm{\Omega }}^p}{f}^2\left(x\right)\mathit{dx}-f_0^2$

Then, to explain the part of the response variance due to the input parameters, the following sensitivity index can be defined: $$S_{i1,...,\mathit{is}}=\frac{V_{i1,...,\mathit{is}}}{V}\mathrm{for}1\le i1<...<\mathit{is}\le p$$

S _i is called the first order sensitivity index for the factor x _i . This index measures the part of the variance of the response explained by the effect of x _i .

S _{i, j} , for i ≠ j, is called the second order sensitivity index. This index measures the proportion of response variance due to interactions between the effects of x _i and x _j .

The total sensitivity index, S _Ti for a particular parameter x _i , defined as the sum of all the sensitivity indices involving the parameters, can also be very useful for measure the part of the variance of the response explained by all the effects in which x _i plays a role.

où #i représente tous les termes S _il,....is qui inclus l'indice i. $S_{\mathit{Ti}}={\displaystyle \underset{\mathrm{k\; \#\; i}}{\Sigma }}{S}_k$

where #i represents all the terms S _{il, .... is that} which includes the index i.

The global sensitivity analysis makes it possible to explain the variability of the responses as a function of the input parameters, through the definition of total or partial sensitivity indices. These indices can be estimated by Monte Carlo or Quasi Monte Carlo techniques to approximate the different multidimensional integrals, requiring a large sampling.

Thus, the overall sensitivity analysis can not be used directly using a flow simulator. According to the invention, the sensitivity index calculations are performed using analytical models for each response. These analytical models are constructed as previously described.

The Global Sensitivity Analysis (ASG) used in the invention does not have the usual limitations related to the hypotheses that other methods can be used that allow the calculation of standard sensitivity indices Spearman, Pearson, SRC, rank index, ... The only hypothesis is that the uncertain parameters are independent, which greatly expands the use of ASG using Sobol decomposition. This assumption is generally respected in reservoir engineering problems, since the links between parameters are known a priori.

During this analysis, we determine the contribution of the uncertainty of each parameter to the total variance of the (or the) response (s). The principle consists in calculating several sensitivity indices (first, second, ... Nth order and total indices) allowing to know the precise influence of each parameter or group of parameters on the responses of interest. These indices are calculated by formulas requiring the calculation of multiple integrals that can be performed approximately by Monte Carlo or Quasi Monte Carlo techniques.

The Global Sensitivity Analysis (ASG) of the uncertain parameters on the simulator responses also makes it possible to evaluate the average effect of a parameter on a given response. This average effect can be used for example for controllable parameters, eg position a well, injection flow etc ... and is therefore a simple tool of behavior parameters.

The use of the approximate model to make the ASG, allows to determine the influential parameters, and how they are influential. It is thus possible to know the total impact of a parameter, as well as its combined impact with one or more other parameters on the production or economic response of the deposit. The ASG clearly allows a better understanding of the deposit's behavior. In addition, the determination of the average effects of the parameters is also a tool to characterize the average influence of a parameter, given the uncertainty of the other parameters on the responses in production or economic reservoir.

Finally, additional measures can be determined to better characterize the deposit and thus reduce uncertainty about future production. Quantifying the influence of uncertain parameters on the production of the deposit allows the most influential parameters to be determined. Thus, to limit uncertainty about the production or future economy of the deposit, the most influential parameters are first characterized. The use of the described methodology thus gives the means to the reservoir engineer to determine the parameters which must be better defined, and thus, gives a guide in the choice of the new measurements to realize (logging, coring, SCAL, ... ). Once influencing parameters better characterize by measurements, it is then possible to use again the described methodology in order to propagate the uncertainty to quantify the new uncertainty on the production or economic responses of the reservoir.

Propagation, global sensitivity analysis and average effects calculation require several thousand evaluations of the associated response (s). This makes these methods unusable directly with large numerical codes (as is the case for flow simulators), hence the interest of constructing predictive approximate models, allowing to use these very interesting techniques for their answers to business questions.

According to another embodiment, the input parameters comprise stochastic fields, eg permeability, porosity, facies, etc. The uncertainty coming from the geostatistical maps is often neglected in the methods of uncertainty analysis based on plans. experience.

In the case of stochastic field type parameters, the stochastic field is decomposed into a number n of components via the Karhunen-Loeve decomposition (M. Loève Probability Theory, Princeton University Press, 1955). The geostatistical techniques used in reservoir engineering to model the permeability and porosity quantities of rocks are mostly based on random Gaussian functions, discretized on a mesh covering the physical space of the reservoir. The Karhunen-Loève decomposition of a geostatistical model consists of representing it in the base formed of the eigenvectors of its covariance operator. This gives a functional representation of the random field. By keeping only a limited number of components in this representation, we obtain an approximation of the random field which represents a quantifiable part of the variance of the process. Indeed, at each term of the decomposition is allocated a part of the global variance which is equal to the eigenvalue associated with the corresponding eigenvector. It is thus possible to quantify the approximation error in terms of variance. The number of components needed to reproduce the geostatistical model is often quite high. Numerical tests have shown that a hundred components may be needed in some cases. However, in many cases, only the variation of a limited number of these components will impact the simulated production responses of the reservoir model, for example cumulative oil after 10 years of production. According to the invention, the components of the stochastic field having an impact on the simulated interest responses are selected, using an overall sensitivity analysis with an approximate model as indicated in the previous steps.

Advantages

The method according to the invention is a tool for the analysis of the uncertainties of a flow simulator, and to help an engineer to reduce this uncertainty, focusing on the characterization of the parameters whose uncertainty contributes mainly to the poor characterization of the outputs.

This method provides a robust and cost-effective tool (in terms of the number of simulations) for the global sensitivity analysis and the propagation of uncertainties. It allows the engineer to control the degree of approximation of his results by analyzing in real time the advantages in terms of prediction compared to the number of simulations carried out.

The global sensitivity analysis and the average effect of the parameters make it possible to see the impact of the uncertainty of a parameter on the overall uncertainty of a response, and thus gives a guide in the choice of new measurements to be made. in order to better characterize the parameters having a central role on the production or economic results.

Finally, the method allows the uncertainties of the geostatistical model (permeability, porosity, facies, etc.) to be taken into account by using surface response techniques and global sensitivity analysis.

Claims (6)

A method for evaluating an underground reservoir production scheme, wherein physical properties characterizing said reservoir and production scheme are selected, said properties constituting input parameters of a flow simulator for simulating deposit, such as production, and an approximate analytical model is constructed for predicting said deposit responses, characterized in that the following steps are performed: said approximate analytical model is adjusted using an iterative process comprising the following steps: a) - one defines, for each of said responses, a degree of precision D _p that one wishes to obtain, said degree of precision D _p measuring the difference between the responses predicted by the model and those simulated by the simulator; b) - a degree of precision D _p (M) of the predictions of the approximate analytical model is calculated; c) - if this value D _p (M) is less than the desired degree of prediction D _p , the iterative process stops, otherwise the following steps are continued: d) - one builds a plan of experiments so as to select simulations to realize, relevant to adjust said model; e) - the simulations selected by the experimental plane are carried out using the flow simulator, then, for each of the simulated responses simulated by the simulator, said analytical model is adjusted using a method of approximation, so as to adjust the responses predicted by the model to those simulated by the simulator; f) - is repeated in step b) until the desired degree of accuracy D _p is reached; and said production scheme is evaluated by analyzing said responses of said deposit predicted by said approximate analytical model. The method of claim 1, wherein the desired degree of precision D _p is modified at each iteration. Method according to one of the preceding claims, wherein the values of the input parameters are uncertain. Method according to one of the preceding claims, in which said responses of said deposit predicted by said approximate analytical model are analyzed, by quantifying an influence of each of said input parameters on each of said responses, using a sensitivity analysis overall, in which sensitivity indices are calculated using said analytical model. Method according to claim 4, in which the most influential parameters are selected on the responses of said deposit, using the overall sensitivity analysis, and measures are defined to be made to decrease an uncertainty on the responses of said deposit. . A method according to claim 4, wherein the input parameters comprise at least one stochastic field, the stochastic field is decomposed into a number n of components via a Karhunen-Loeve decomposition, and the components of the stochastic field having an impact are selected. on the answers using the overall sensitivity analysis.

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