EP1630348A1 - Method for modelling the production of an oil-bearing formation - Google Patents

Abstract

The method has five stages: Stage 1 consists of constructing a flow simulator based on physical data measured at the deposit; Stage 2 consists of determining a first analytical model linking the production of the deposit to a time function, taking into account parameters that influence production. In Stage 3 the first model is adjusted to a finite number of production values obtained from the simulator. In Stage 4 a new production value is selected from the simulator, and in Stage 5 a second model is produced by adjusting the first model in such a way that the second model interpolates the new productin value.

Description

The present invention relates to the study and optimization of the production patterns of oil deposits. It aims at modeling the behavior of an oil field in order to be able to compare several production schemes, and to define an optimal scheme considering a given production criterion (oil recovery, water inflow, production flow rate). ..).

The study of a deposit has two main phases.

The reservoir characterization phase consists in determining a numerical flow model or flow simulator that is compatible with the real data collected in the field. Engineers only have access to a small part of the field they are studying (measurements on cores, logs, well tests, etc.). They must extrapolate these point data over the entire oil field to build the digital simulation model.

The production forecasting phase uses the numerical simulation model to estimate future reserves and productions or to improve the production scheme in place. This phase is carried out thanks to the numerical model of simulation constructed from numerous and varied data, but coming from only a tiny part of the deposit. Consequently, the notion of uncertainty must be constantly taken into account.

In order to fully characterize the impact of each uncertainty on oil production, the largest number of production scenarios must be tested and, therefore, a significant number of reservoir simulations are needed. Given the significant time required to perform a flow simulation, it can clearly not be considered to test all possible scenarios via the numerical flow model. In this context, the use of the experimental design method can allow the construction of a simplified model of the flow simulator based on a reduced number of parameters. Experiment plans make it possible to determine the number and spatial location of the parameters of the simulations to be carried out in order to obtain the maximum of relevant information at the lowest possible cost. This simple model translates the behavior of a given response (for example the cumulated oil produced at 10 years) according to some parameters. Its construction requires a reduced number of simulations, defined beforehand thanks to a plan of experiments.

During the production forecasting phase, the simplified model is used because it is simple and analytical and, therefore, every simulation 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 wishes, regardless of the time required to perform a numerical flow simulation.

The methods presented in the French documents FR 2 855 631 and FR 2 855 633 use simplified models to optimize the production of a petroleum deposit or to assist in the decision making for the management of a petroleum deposit, in the presence of uncertainties.

The simplified model obtained by experimental design assumes that the response obtained by the model is a linear function of the parameters taken into account. However, in the majority of cases, this is not true. When the interval in which a parameter can evolve (permeability, porosity, etc.) is relatively small and its contribution is reasonable, we can assume that its behavior is linear. But when this interval becomes too wide or when the contribution of the parameter is no longer linear, the linearity assumption biases the knowledge of the oil field.

It is therefore necessary to establish a criterion for detecting nonlinearities and to set up an effective and rapid methodology for effectively predicting non-linear response behaviors.

The present invention proposes to model a petroleum field by making iterative adjustments in order to reproduce the behavior of the oil field, while controlling the number of simulations.

• a) on construit un simulateur d'écoulement à partir de données physiques mesurées sur le gisement pétrolier,
• b) on détermine un premier modèle analytique exprimant la production du gisement en fonction du temps en tenant compte de paramètres ayant une influence sur la production du gisement, le premier modèle s'ajustant au mieux sur un nombre fini de valeurs de production obtenues par le simulateur d'écoulement,
• c) on sélectionne au moins une nouvelle valeur de production associée à un point situé dans un domaine du gisement choisi en fonction de la non linéarité de la production du gisement dans ce domaine, la nouvelle valeur étant obtenue par le simulateur d'écoulement,
• d) on détermine un deuxième modèle en ajustant le premier modèle de manière à ce que la réponse du deuxième modèle audit point corresponde à la nouvelle valeur de production.

In general, the present invention relates to a method for simulating the production of a petroleum deposit in which the steps are carried out:

• a) a flow simulator is constructed from physical data measured on the oil reservoir,
• b) determining a first analytical model expressing the production of the deposit as a function of time taking into account parameters having an influence on the production of the deposit, the first model fitting best on a finite number of production values obtained by the flow simulator,
• c) selecting at least one new production value associated with a point situated in a field of the deposit chosen as a function of the non-linearity of the production of the deposit in this area, the new value being obtained by the flow simulator,
• d) determining a second model by adjusting the first model so that the response of the second model at said point corresponds to the new production value.

• on détermine un sous-modèle qui s'ajuste au mieux sur ledit nombre fini de valeurs de production, à l'exception d'une valeur test choisie parmi ledit nombre fini de valeurs de production,
• on calcule un résidu de prédiction associé à ladite valeur test en effectuant la différence entre la réponse du sous-modèle et ladite valeur test,
• on calcule le résidu de prédiction associé à chacune desdites valeurs de prédiction en répétant les deux étapes précédentes en attribuant successivement à la valeur test chacune des valeurs comprises dans ledit nombre fini de valeurs de production,
• on sélectionne la nouvelle valeur de production dans un domaine du gisement voisin du point associé à la valeur de production ayant le plus grand résidu de prédiction.

According to the invention, in step c), the following steps can be performed:

• determining a submodel that best fits over said finite number of production values, except for a test value selected from said finite number of production values,
• calculating a prediction residue associated with said test value by making the difference between the submodel response and said test value,
• the prediction residue associated with each of said prediction values is calculated by repeating the two preceding steps by successively assigning to the test value each of the values included in said finite number of production values,
• the new production value is selected in a field of the deposit adjacent to the point associated with the production value having the largest prediction residue.

The new production value can be selected by taking into account the production gradient at the point associated with the production value with the largest prediction residual.

In addition, a new value can be selected in step c) and step d) can be performed provided that the largest prediction residue is greater than a previously fixed value.

• on détermine une première variance de krigeage du premier modèle pour ledit nombre fini de valeurs de production obtenues par le simulateur d'écoulement,
• on choisit un premier point pilote dans le gisement à l'endroit où la première variance de krigeage est maximale,
• on détermine une deuxième variance de krigeage du premier modèle pour ledit nombre fini de valeurs de production obtenues par le simulateur d'écoulement et le premier point pilote,
• on choisit un deuxième point pilote dans le gisement à l'endroit où la deuxième variance de krigeage est maximale,
• on attribue une valeur à chacun desdits points pilotes en effectuant les cinq opérations suivantes pour chacun des points pilotes :
  • on détermine un sous-modèle qui s'ajuste au mieux sur le nombre fini de valeurs de production et sur la valeur associée à un des points pilotes, à l'exception d'une valeur test choisie parmi le nombre fini de valeurs de production et la valeur associée au point pilote,
  • on calcule un résidu de prédiction associé à la valeur test en effectuant la différence entre la réponse du sous-modèle et la valeur test,
  • on calcul le résidu de prédiction associé à chacune des réponses du sous-modèle en répétant les deux opérations précédentes en attribuant successivement à la valeur test chacune des valeurs comprises dans l'ensemble consistant en ledit nombre fini de valeurs de production et la valeur associée au point pilote,
  • on calcule la somme des valeurs absolues des résidus de prédiction calculé pour chacune des valeurs tests,
  • on attribue audit point pilote la valeur qui minimise cette somme
• on détermine un deuxième sous-modèle qui s'ajuste au mieux sur ledit nombre fini de valeurs de production et sur les valeurs desdits points pilotes,
• pour chacun des points pilotes, on effectue la différence entre la réponse du deuxième sous-modèle et la réponse du premier modèle,
• on associe ladite nouvelle valeur de production de l'étape c) au point pilote pour lequel ladite différence est la plus grande,

De plus, à l'étape d), on peut déterminer le deuxième modèle en ajustant le premier modèle de manière à ce que la réponse du deuxième modèle audit point pilote sélectionné corresponde à la nouvelle valeur de production et, en outre, aux valeurs attribués aux autres points pilotes.According to a variant of the invention, in step c), the following steps can be performed:

• a first kriging variance of the first model is determined for said finite number of production values obtained by the flow simulator,
• a first pilot point is chosen in the field where the first variance of kriging is maximal,
• determining a second kriging variance of the first model for said finite number of production values obtained by the flow simulator and the first pilot point,
• a second pilot point is chosen in the field where the second maximum kriging variance is
• a value is assigned to each of said pilot points by performing the following five operations for each of the pilot points:
  • a submodel is determined which best fits the finite number of production values and the value associated with one of the pilot points, except for a test value selected from the finite number of production values and the value associated with the pilot point,
  • a prediction residue associated with the test value is calculated by making the difference between the submodel response and the test value,
  • calculating the prediction residual associated with each of the replies of the submodel by repeating the two preceding operations, successively assigning to the test value each of the values comprised in the set consisting of the said finite number of production values and the value associated with the pilot point,
  • the sum of the absolute values of the prediction residuals calculated for each of the test values is calculated,
  • the pilot point is assigned the value that minimizes this sum
• determining a second sub-model which best fits said finite number of production values and the values of said pilot points,
• for each of the pilot points, the difference is made between the response of the second sub-model and the response of the first model,
• said new production value of step c) is associated with the pilot point for which said difference is the largest,

Moreover, in step d), the second model can be determined by adjusting the first model so that the response of the second model to said selected pilot point corresponds to the new production value and, in addition, to the assigned values. other pilot points.

• on détermine un modèle analytique exprimant la dérivée de la production du gisement en fonction du temps, le modèle s'ajustant au mieux sur les dérivées aux points associés auxdites valeurs de production utilisées à l'étape b),
• à partir du modèle exprimant la dérivée, on sélectionne au moins une nouvelle valeur de production associée à un point dont la réponse du modèle exprimant la dérivée est nulle.

According to another variant of the invention, in step c), the following steps can be performed:

• an analytical model expressing the derivative of the production of the deposit as a function of time is determined, the model adjusting at best on the derivatives at the points associated with said production values used in step b),
• from the model expressing the derivative, selecting at least one new production value associated with a point whose response of the model expressing the derivative is zero.

A new value can be selected in step c) and step d) can be performed provided that the prediction residual of the new selected value is greater than a previously fixed value.

• on détermine un troisième modèle analytique exprimant la dérivée de la production du gisement en fonction du temps, le troisième modèle s'ajustant au mieux aux dérivées aux points associés audit nombre fini de valeur de production et à la valeurs de production sélectionnées à l'étape c),
• si la réponse du troisième modèle analytique au point sélectionné à l'étape c) est supérieure à zéro, on détermine un point associé à la valeur maximum de la réponse du deuxième modèle au voisinage du point sélectionné à l'étape c),
• si la réponse du troisième modèle analytique au point sélectionné à l'étape c) est inférieure à zéro, on détermine un point associé à la valeur minimum de la réponse du deuxième modèle au voisinage du point sélectionné à l'étape c),
• on détermine une nouvelle valeur de production par le simulateur d'écoulement au point associé à la valeur minimum ou maximum précédemment déterminée,
• on détermine un quatrième modèle en ajustant le deuxième modèle de manière à ce que la réponse du quatrième modèle corresponde à la nouvelle valeur déterminée à l'étape précédente.

According to the invention, after step d), the following steps are carried out:

• a third analytical model expressing the derivative of the production of the deposit as a function of time is determined, the third model fitting best to the derivatives at the points associated with said finite number of production value and with the production values selected at the step vs),
• if the response of the third analytical model to the point selected in step c) is greater than zero, determining a point associated with the maximum value of the response of the second model in the vicinity of the point selected in step c),
• if the response of the third analytical model to the point selected in step c) is less than zero, determining a point associated with the minimum value of the response of the second model in the vicinity of the point selected in step c),
• a new production value is determined by the flow simulator at the point associated with the previously determined minimum or maximum value,
• a fourth model is determined by adjusting the second model so that the response of the fourth model corresponds to the new value determined in the previous step.

According to the invention, steps c) and d) can be repeated.

In step b), one can choose said production values using a plan of experiments.

In step b), the first model can be adjusted using one of the following approximation methods: approximation by polynomials, neural networks, vector support machines.

In step d), one of the following interpolation methods can be used: kriging method and spline method.

Thus, the method according to the invention provides the reservoir engineer with a simple and inexpensive formalism in terms of numerical simulations for the management of scenarios and the optimization of production patterns, to help him in his decision-making in order to to minimize the risks.

• la figure 1 schématise la méthode selon l'invention,
• la figure 2 schématise une fonction « chameau » et l'approximation de cette fonction par des modèles obtenus par plans d'expériences,
• la figure 3 schématise l'amélioration de l'approximation de la fonction « chameau », en mettant en oeuvre l'invention.

Other features and advantages of the invention will be better understood and will become clear from reading the description given below with reference to the drawings among which:

• FIG. 1 schematizes the method according to the invention,
• FIG. 2 schematizes a "camel" function and the approximation of this function by models obtained by experimental design,
• FIG. 3 schematizes the improvement of the approximation of the "camel" function, by implementing the invention.

The method according to the invention is shown schematically by the diagram of FIG.

Step 1: Build the tank flow simulator

The oil field is modeled using a digital tank simulator. The reservoir simulator or flow simulator makes it possible to calculate the production of hydrocarbons or water over time according to technical parameters such as the number of layers of the reservoir, the permeability of the layers, the strength of the aquifer , the position of oil wells, etc. In addition, the flow simulator calculates the derivative of the production value at the point considered.

The numerical simulator is built from data characteristic of the oil field. For example, the data is obtained through laboratory measurements of cores and fluids taken from the oil field, by well logs, well tests, and so on.

Step 2: Approximation of the flow simulator

Since the flow simulator is complex and time-consuming, a simplified model of the oil reservoir's behavior is constructed.

Parameters influencing the hydrocarbon or water production profiles of the reservoir are selected. 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: a permeability multiplier for some layers of the reservoir, the force of the aquifer, the residual oil saturation after a water scan.

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

Points are selected in the experimental domain for which numerical flow simulations will be performed. These points are used to construct a simplified model that best replicates the deposit flow simulator. These points are chosen by the experimental design method, which makes it possible to determine the number and the location of the simulations to be carried out in order to obtain the maximum information at the lowest possible price and thus to determine a reliable model reflecting at best the production profile. It should be noted that the choice of this experimental device is very important: the initial experimental plan has a primordial role in the development of the modeling of the first model, the results depend strongly on the disposition of the experiments.

• 1. Dejean, J.P. and Blanc, G., "Managing uncertainties on production predictions using integrated statistical methods", SPE 56696, SPE Annual Technical Conference and Exhibition, Houston, USA, oct. 3-6, 1999.
• 2. Box, G.E.P. and Hunter, J.S., "The 2k-p fractional factorial designs", Part I, Technometrics, 2, 311-352, 1961a
• 3. Box, G.E.P. and Hunter, J.S., "The 2k-p fractional factorial designs", Part II, Technometrics, 3, 449-458, 1961b
• 4. Box, G.E.P and Wilson, K.B., "On the experimental attainment of optimum conditions", Journal of the Royal Statistical Society, Series B, 13, 1-45
• 5. Draper, N. R., "Small composite designs", Technometrics, 27, 173-180, 1985
• 6. Atkinson, A.C. and Donev, A.N., "Optimum experimental designs", Oxford University press, 1992

The choice of simulation points can be made through different types of experimental designs, for example, factorial designs, composite planes, Latin Hypercubes, maximin distance plans, and so on. It is possible to use the experimental plans described by the following documents:

• 1. Dejean, JP and White, G., SPE Annual Technical Conference and Exhibition, SPE 56696, Houston, USA, Oct. 3-6, 1999.
• 2. Box, GEP and Hunter, JS, "The 2k-p fractional factorial designs", Part I, Technometrics, 2, 311-352, 1961a
• 3. Box, GEP and Hunter, JS, "The 2k-p fractional factorial designs", Part II, Technometrics, 3, 449-458, 1961b
• 4. Box, GEP and Wilson, KB, "On the Experimental Achievement of Optimal Conditions," Journal of the Royal Statistical Society, Series B, 13, 1-45
• 5. Draper, NR, "Small composite designs", Technometrics, 27, 173-180, 1985
• 6. Atkinson, AC and Donev, AN, Optimum Experimental Designs, Oxford University Press, 1992

After the construction of this first experimental design and when the numerical simulations are realized, an approximation method is used to determine a first model which gives a trend of the behavior of the response function, that is to say which approaches the flow simulator.

The first model expresses a criterion of production studied over time, this criterion being expressed as a function of the selected parameters. The production criterion can be oil recovery, water inflow, production flow. The first analytical model is constructed using the previously selected values of this criterion and obtained by means of the flow simulator.

By approximation method, we intend to consider polynomials of the first or second order, neural networks, vector-support machines or possibly polynomials of order greater than two. The choice of this model depends on the one hand on the number of maximum simulations envisaged by the user and secondly, the initial experimental plan used.

Step 3: Adjust the first model

There may be a difference between the production value given by the first analytical model obtained in step 2 and the simulated production values used to construct this first model.

In this case, the residues are determined at the various simulation points. The residuals correspond to the difference between the response of the first model and the value obtained by the reservoir flow simulator. Then the residues are interpolated. Any n-dimensional interpolation method may be suitable. In particular, the method of kriging or splines can be envisaged. These methods are explained in the book "Statistics for spatial data" of Cressie, N., Wiley, New York 1991.

The residual interpolation structure lends itself well to this sequential approach because it is broken down into two parts: a linear model, which corresponds to the first model determined in step 2, and a term "corrector" which makes it possible to bridge the gap. difference between the prediction of the first model and the simulation point. In the case where the analytical model is satisfactory, it is not necessary to add the term "corrector". In the opposite case, it makes it possible to interpolate the responses and, thus, to take into account the detected non-linearities of the surface.

Thus, a second adjusted model is determined by adding the results of the interpolation of the residues to the first model determined in step 2.

Step 4: Model predictivity test and choice of additional simulation points

At this stage of modeling, the second model exactly interpolates the simulations, so the adjustment of the response function is optimal. Since the interpolation method is correct, the "classical" residues are zero. Thus, according to the invention, we are interested in the prediction residues. As a result, the predictivity of the model for out-of-plane points is examined. Predictions should be as precise as possible. Therefore, a model predictivity test is then performed to assess the quality of the approximation to judge the need for improvement by adding new points to the initial plan.

• le calcul de la prédictivité a priori avec le calcul des résidus de prédiction
• le calcul de la prédictivité a posteriori avec l'utilisation de points de confirmation.

By predictivity test, we mean two criteria:

• the calculation of predictivity a priori with the calculation of the prediction residuals
• the calculation of the posterior predictivity with the use of confirmation points.

Prediction a priori

The prediction residuals are the residues obtained at a point of the plane by adjusting the first model without this point. Deleting a point and redoing the model estimate will determine if this point (or the area near the point) provides decisive information or not. The calculation of these prediction residues is done for each point of the initial experimental plan. In the neighborhood of the points considered the least predictive of the current plane, that is to say the points having the greatest prediction residue, new points are simulated. To do this, a subsampling zone is defined in the vicinity of the points. The addition of these points may be conditioned by the fact that the residues are greater than a value set by the user.

The size of this subsampling area can be defined by using the gradient information of the production at the points and / or the value of the prediction residuals. Indeed, a high gradient value reflects a strong variation of the response. It may therefore be informative to add a new point close to the existing one. On the other hand, a low gradient value in a given direction indicates that there are no irregularities in that direction. Therefore, it is not necessary to investigate a large range of variation in this direction. On the other hand, the variation range for one of the parameters is greater the greater the value of the gradient in this direction. This approach makes it possible to eliminate certain directions (those where the value of the gradient is not significant) and, therefore, to reduce the number of simulations to be performed. This sub-sampling may, for example, result from the construction of a new experimental plan defined on this zone. The choice of this experimental design (factorial plane, composite plane, Latin Hypercube) results from the necessary compromise between the cost and the quality of modeling.

Alternatively, the pilot point method can be implemented to improve the second model.

For a given number of experiments, there are a large number of estimators (exact interpolators) passing through all the experiments and respecting the spatial structure (expectation and covariance) of the process. In this class of estimators respecting the data, we look for the estimator that maximizes the predictivity a priori. In order to browse this class of estimators, we proceed to the addition of fictitious information, that is to say that we add pilot points to the simulated experiments. These pilot points are then considered as data although no simulation has been carried out and will make it possible to browse all the estimators passing through all the experiments. The objective is to select the interpolator that maximizes the predictive coefficient a priori of the model, that is to say that the pilot points are positioned so as to achieve maximum predictivity.

• la capacité du point pilote à réduire l'écart entre les observations et les résultats de simulation numériques d'écoulement
• la contribution du point pilote dans la réduction des incertitudes sur le modèle d'approximation courant.

The location of a pilot point is determined taking into account the following two criteria:

• the ability of the pilot point to narrow the gap between observations and numerical flow simulation results
• the contribution of the pilot point in reducing uncertainties on the current approximation model.

For this choice to be made optimally, it is necessary to be able to quantify the impact of a possible pilot point on each of the two criteria.

In order to remove the uncertainty of the prediction which reigns on the poorly represented places, it is interesting to apply local disturbances on the areas where the variance of kriging is important (absence of observations). A pilot point is therefore placed where the variance of kriging is maximal. Methods for determining kriging variance are explained in the book "Statistics for spatial data" by Cressie, N., Wiley, New York, 1991.

• on détermine la variance de krigeage sur le domaine incertain du deuxième modèle déterminé à l'étape 3 pour le nombre fini de valeurs de production obtenues par le simulateur d'écoulement,
• on place un premier point pilote là où la variance de krigeage est maximale

To determine the location of a pilot point, perform the following operations:

• the kriging variance over the uncertain domain of the second model determined in step 3 is determined for the finite number of production values obtained by the flow simulator,
• a first pilot point is placed where the variance of kriging is maximal

Suppose that in addition to the production values obtained by the flow simulator, a certain number of pilot points have already been positioned in the uncertain domain and that we are trying to place new pilot points to improve the predictivity of the model. We then assimilate the existing pilot points to local data of zero variance. It is to take into account the location of already existing points that we optimize the location of the pilot points sequentially.

• on détermine la variance de krigeage du premier modèle pour le nombre fini de valeurs de production obtenues par le simulateur d'écoulement et le premier point pilote,
• on détermine l'emplacement d'un deuxième point pilote là où la variance de krigeage est maximale.

Thus, to determine the location of a second pilot point, the following operations are performed:

• the kriging variance of the first model is determined for the finite number of production values obtained by the flow simulator and the first pilot point,
• the location of a second pilot point is determined where the kriging variance is maximum.

Several pilot points can be added by repeating the two previous operations.

Preferably, it is chosen to add a number of pilot points less than or equal to the number of actual experiments present, so as not to disturb the model too much. Once the optimal location of the pilot points is determined, a "fictitious" answer value must be assigned to these points.

The addition of pilot points to improve the prior predictivity of the model, we must therefore define the value of pilot points from an objective function that measures this predictivity. Since kriging is an exact interpolation method, "classical" residues are null. They therefore do not provide any information on predictivity, and therefore, the prediction residuals are considered. By a priori predictivity, we mean the calculation of the prediction residuals in each of the points of the initial experimental plan. The prediction residuals are the residues obtained at a point of the initial experimental plane by adjusting the first model without this point.

• on détermine un sous-modèle qui s'ajuste sur le nombre fini de valeurs de production et sur la valeur associée au point pilote, à l'exception d'une valeur test choisie parmi le nombre fini de valeurs de production et la valeur associée au point pilote,
• on calcule un résidu de prédiction associé à la valeur test en effectuant la différence entre la réponse du sous-modèle et cette valeur test,
• on calcule le résidu de prédiction associé à chacune des réponses du sous-modèle de prédiction en répétant les deux étapes précédentes en attribuant successivement à la valeur test chacune des valeurs comprises dans le nombre fini de valeurs de production et la valeur associé au point pilote,
• on calcule la somme des valeurs absolues ou des carrés des résidus de prédiction déterminé pour chacune des valeurs test
• on attribue au point pilote la valeur qui minimise cette somme.

To determine the production value associated with one of the pilot points whose location has been previously determined, the following steps can be performed:

• a submodel is determined that adjusts to the finite number of production values and the value associated with the pilot point, except for a test value selected from the finite number of production values and the value associated with the pilot point,
• a prediction residue associated with the test value is calculated by making the difference between the response of the submodel and this test value,
• the prediction residual associated with each of the replies of the prediction model is calculated by repeating the two preceding steps by successively assigning to the test value each of the values included in the finite number of production values and the value associated with the pilot point,
• the sum of the absolute values or squares of the prediction residuals determined for each of the test values is calculated
• the pilot point is assigned the value that minimizes this sum.

Deleting a point and re-estimating the model makes it possible to determine whether the point or zone of the experimental domain close to this point provides decisive information or not. The calculation of the prediction residuals is performed in a neighborhood of the pilot point to be optimized. We set initial values for the pilot points and then we consider these data as real and we vary the value of the pilot point to obtain a model that is as predictive as possible, that is, we want to minimize the error mean prediction of the model.

The determination of the optimal value of the pilot point is thus performed to minimize the average prediction error of the model over the entire uncertain domain. Likewise, this determination of the optimum value of the pilot point can be carried out so as to minimize the error of local prediction of the model (i.e., in a vicinity of the pilot point, independent of other prediction errors).

Once the value and position of the pilot points are determined, we test the sensitivity of the model to the newly added points, then we perform simulations at points that seem very sensitive in the approximation. For this, we compare the estimator obtained without pilot points to the estimator obtained by kriging with pilot points (i.e. the achievement of maximum predictivity).

The points for which the greatest disagreement is found, ie where the difference is the most important, reflect a strong instability of the approximation. Therefore, it is essential to improve the quality of the approximation in these places. Thus, the simulations corresponding to the points of strongest disagreements are made in order to stabilize the approximation.

• on détermine un sous-modèle à partir des points pilotes et du nombre fini de valeurs de production,
• pour chacun des points pilotes, on effectue la différence entre la réponse de ce sous-modèle et la réponse du deuxième modèle déterminé à l'étape 3,

Selon une première variante :
on sélectionne le point pilote pour lequel la différence entre la réponse du sous-modèle et la réponse du deuxième modèle est la plus grande. C'est le point choisi pour améliorer le premier modèle, les autres points pilotes sont alors ignorés dans la suite du processus.
Selon une deuxième variante :
on sélectionne un ou plusieurs points pilotes pour lesquels la prédictivité est la plus mauvaise (inférieure à un seuil plus petit que 1), puisque cette faible prédictivité traduit une forte sensibilité du point. Dans la suite du processus, on prend en compte, d'une part, les valeurs de productions associées aux points pilotes sélectionnés, ces valeurs de production étant obtenues par le simulateur d'écoulement et, d'autre part, les valeurs de productions associées aux autres points pilotes dont la prédictivité est meilleure, ces valeurs de production correspondant aux valeurs estimées selon la prédictivité a priori mentionnée ci-dessus.
Selon la deuxième variante, si on réitère le processus, il convient ensuite de réévaluer la prédictivité locale aux points pilotes non simulés, pour s'assurer que cette valeur correspond toujours à une stabilisation satisfaisante. Si tel n'est pas le cas, le point pilote non simulé n'est plus considéré dans la nouvelle estimation.To select the pilot points for which we will perform a simulation, we can perform the following steps:

• a sub-model is determined from the pilot points and the finite number of production values,
• for each of the pilot points, the difference between the response of this submodel and the response of the second model determined in step 3 is carried out,

According to a first variant:
the pilot point is selected for which the difference between the response of the submodel and the response of the second model is the largest. This is the point chosen to improve the first model, the other pilot points are then ignored in the rest of the process.
According to a second variant:
one or more pilot points are selected for which the predictivity is the worst (less than a threshold smaller than 1), since this low predictivity reflects a strong point sensitivity. In the following process, the production values associated with the selected pilot points are taken into account, these production values being obtained by the flow simulator and, on the other hand, the associated production values. at other pilot points with better predictivity, these production values corresponding to the values estimated according to the a priori predictivity mentioned above.
According to the second variant, if the process is repeated, it is then necessary to re-evaluate the local predictivity at the non-simulated pilot points, to ensure that this value always corresponds to a satisfactory stabilization. If this is not the case, the non-simulated pilot point is no longer considered in the new estimate.

The addition of these new simulations then allows the realization of a study of residues. By residues here, we mean, for each pilot point, the difference between the simulated value and the value obtained during the optimization of the pilot points:

As before, if the residues are large, there is a discrepancy between the current approximation with the pilot points and the simulations; this reflects a lack of predictivity of the model. In this case, an improvement of the current model is necessary, this again passes by the realization of new simulations. It is therefore necessary to proceed to one or more new iterations.

On the other hand, if the residues are weak, the prediction at these points is good and therefore the model seems predictive in the considered domains. But, the overall predictivity of the model requires confirmation, for this we propose to add confirmation points. These new simulations make it possible to determine whether or not to continue the iteration process.

Predictivity a posteriori

Confirmation points, ie production values obtained by the flow simulator constructed in step 1, can be added to the experimental design by examining the derivative of the production values. In fact, a criterion for adding simulations can be based on: the value of the derivative of the production values obtained by the flow simulator, the direct identification of points whose maximum production value or identification points whose production value is minimal.

A model that approximates the values of the derivatives at the points chosen by the experimental design in step 2 is determined. Then, a new simulation point is added at the point where the response of the derivative model is canceled. provided that this point is sufficiently distant from the simulations already carried out. These confirmation points make it possible to test the predictivity of the second model, in this new investigated zone. If the prediction residuals calculated at the newly selected points exceed a user-defined value, these new points are used to perform a new interpolation phase.

The addition of simulations to the current device, whether it is the consequence of a lack of prior or posterior predictivity, makes it possible to increase the quality and the quantity of the information on the response function in order to obtain a more representative sampling. .

Step 5: Build and fit a third model

From the second model determined in step 2, the residues are determined at the new simulation points selected in step 4. The residuals correspond to the difference between the response of the first model and the simulation value obtained by the simulator. flow of the tank. Then the residues are interpolated. Any n-dimensional interpolation method may be suitable. For example, kriging or splines can be used.

The interpolation structure of the residues is divided into two parts: the first model determined in step 2, and a "corrector" term which makes it possible to bridge the gap between the prediction of the first model and the new simulation (s) selected at Step 4. The new simulation makes it possible to interpolate the responses and, thus, to take into account the detected non-linearities of the surface.

A second adjusted model is determined by adding the results of the residual interpolation to the first model determined in step 2.

Iteration

In addition, according to the invention, it is possible to improve the model iteratively by repeating steps 4 and 5.

In this case, during the new step 4, we add simulation points relative to the model determined in the previous step 5. And in the new step 5, we build and adjust a new model from the points of simulations selected in the new step 4 and adjusting the first model determined in step 2.

Step 6: Finding inflection points

• on détermine un modèle analytique exprimant la dérivée de la production du gisement en fonction du temps, le modèle s'ajustant au mieux sur les dérivées aux points associés aux valeurs de production sélectionnées aux étapes 2 et 4,
• on vérifie qu'au point ajouté à l'étape 4, la réponse du modèle analytique exprimant la dérivée de la production du gisement est nulle si cette réponse est supérieure à 0, on détermine le maximum du troisième modèle déterminé à l'étape 5 situé au voisinage du point ajouté à l'étape 4,
  si cette réponse est inférieure à 0, on détermine le minimum du troisième modèle déterminé à l'étape 5 situé au voisinage du point ajouté à l'étape 4
• on détermine la valeur du minimum ou du maximum par le simulateur d'écoulement
• on détermine un nouveau modèle en ajustant le troisième modèle de manière à ce que la réponse du nouveau modèle corresponde à la nouvelle valeur minimum ou maximum obtenue par le simulateur d'écoulement.

In the case where the posterior method was used in step 4, the model can be improved in step 5 by adding simulation points by performing the following steps:

• an analytical model expressing the derivative of the production of the deposit as a function of time is determined, the model adjusting at best on the derivatives at the points associated with the production values selected in steps 2 and 4,
• it is verified that at the point added in step 4, the response of the analytical model expressing the derivative of the production of the deposit is zero if this response is greater than 0, the maximum of the third model determined in step 5 in the vicinity of the point added in step 4,
  if this answer is less than 0, the minimum of the third model determined in step 5 located in the vicinity of the point added in step 4 is determined.
• the value of the minimum or maximum is determined by the flow simulator
• a new model is determined by adjusting the third model so that the response of the new model corresponds to the new minimum or maximum value obtained by the flow simulator.

The interest of the method according to the invention is illustrated hereinafter with reference to FIGS. 2 and 3.

The strongly nonlinear analytic function studied has two x and y parameters in order to better visualize the results. This is the "camel" function, which is characterized by its strong non-linearity. The expression of this function is as follows: $$F(x,\mathrm{there})=4x^4-\frac{21}{10}{x}^4+\frac{1}{3}{x}^6+\mathit{xy}-4{\mathrm{there}}^{\mathrm{two}}+4{\mathrm{there}}^4$$

It is represented graphically in the unit cube [-1,1] ² referenced A in FIG.

Reference B in Figure 2 presents the graph of the estimation of the "camel" function by a linear model obtained from a factorial plane with 4 simulations. Reference C in FIG. 2 represents the graph of the estimation of the "camel" function by a second-order polynomial model obtained from a composite plane centered on 9 simulations.

The disparity of the results between the function to be modeled (cube A) and the models (cubes B and C) confirm the limits of the theory of classical experimental designs for modeling nonlinear functions .

FIG. 3 illustrates the optimization, according to our invention, of the model approaching the "camel" function. The function represented in the unit cube [-1,1] ² referenced D is obtained by performing steps 2) and 3), from a Latin Hypercube of initial maximin distance containing nine tests. Then, the functions represented in the unit cube [-1,1] ² referenced E, F and G are obtained by adjusting this function obtained from a Latin Hypercube and adding seven points of simulations. Steps 4) and 5) are repeated three times.

By comparing the function referenced G in FIG. 3 with respect to the "camel" function referenced A in FIG. 2, it can be seen that the curves are relatively close, the non-linearities have indeed been detected. The evolutionary method according to the invention is well adapted and the results are very satisfactory.

Claims (13)

Method for simulating the production of a petroleum deposit, in which the steps are carried out: a) a flow simulator is constructed from physical data measured on the oil reservoir, b) determining a first analytical model expressing the production of the deposit as a function of time taking into account parameters having an influence on the production of the deposit, the first model fitting best on a finite number of production values obtained by the flow simulator, c) selecting at least one new production value associated with a point situated in a field of the deposit chosen as a function of the non-linearity of the production of the deposit in this area, the new value being obtained by the flow simulator, d) determining a second model by adjusting the first model so that the response of the second model at said point corresponds to the new production value. The method of claim 1, wherein in step c) the following steps are performed: a sub-model is determined which best fits over said finite number of production values, except for a test value selected from said finite number of production values, a prediction residue associated with said test value is calculated by making the difference between the response of the submodel and the said test value, the prediction residue associated with each of said prediction values is calculated by repeating the two preceding steps by successively assigning to the test value each of the values included in said finite number of production values, the new production value is selected in a field of the deposit adjacent to the point associated with the production value having the largest prediction residue. The method of claim 2, wherein the new production value is selected by taking into account the production gradient at the point associated with the production value having the largest prediction residue. Method according to one of Claims 2 and 3, in which a new value is selected in step c) and step d) is carried out, provided that the largest prediction residue is greater than a previously fixed value. The method of claim 1, wherein in step c) the following steps are performed: a first kriging variance of the first model is determined for said finite number of production values obtained by the flow simulator, a first pilot point is chosen in the deposit where the first variance of kriging is maximum, a second kriging variance of the first model is determined for said finite number of production values obtained by the flow simulator and the first pilot point, a second pilot point is chosen in the field where the second maximum kriging variance is a value is assigned to each of said pilot points by performing the following five operations for each of the pilot points: • a submodel is determined which best fits the finite number of production values and the value associated with one of the pilot points, except for a test value chosen from the finite number of production values and the value associated with the pilot point, A prediction residue associated with the test value is calculated by making the difference between the submodel response and the test value, The prediction residual associated with each of the replies of the submodel is calculated by repeating the two preceding operations, successively assigning to the test value each of the values comprised in the set consisting of the said finite number of production values and the associated value at the pilot point, • the sum of the absolute values of the prediction residuals calculated for each of the test values is calculated, • the pilot point is assigned the value that minimizes this sum a second sub-model is determined which best fits on said finite number of production values and on the values of said pilot points, for each of the pilot points, the difference between the response of the second submodel and the response of the first model is carried out, said new production value of step c) is associated with the pilot point for which said difference is the largest, A method according to claim 5, wherein in step d), determining the second model by adjusting the first model so that the response of the second model to said selected pilot point corresponds to the new production value and further , to the values assigned to the other pilot points. Method according to one of claims 1 to 6, wherein in step c), the following steps are carried out: an analytical model expressing the derivative of the production of the deposit as a function of time is determined, the model fitting at best on the derivatives at the points associated with said production values used in step b), from the model expressing the derivative, selecting at least one new production value associated with a point whose response of the model expressing the derivative is zero. A method according to claim 7, wherein a new value is selected in step c) and step d) is performed provided that the prediction residual of the new selected value is greater than a previously fixed value. Method according to one of claims 7 and 8, wherein after step d), the following steps are carried out: a third analytical model expressing the derivative of the production of the deposit as a function of time is determined, the third model fitting best to the derivatives at the points associated with said finite number of production value and with the production values selected at the time; step c), if the response of the third analytical model at the point selected in step c) is greater than zero, determining a point associated with the maximum value of the response of the second model in the vicinity of the point selected in step c), if the response of the third analytical model at the point selected in step c) is less than zero, determining a point associated with the minimum value of the response of the second model in the vicinity of the point selected in step c), a new production value is determined by the flow simulator at the point associated with the minimum or maximum value previously determined, a fourth model is determined by adjusting the second model so that the response of the fourth model corresponds to the new value determined in the previous step. Method according to one of the preceding claims, in which steps c) and d) are repeated. Method according to one of the preceding claims, wherein, in step b), said production values are selected using a plan of experiments. Method according to one of the preceding claims, in which, in step b), the first model is adjusted using one of the following approximation methods: approximation by polynomials, neural networks, vector support machines. Method according to one of the preceding claims, wherein in step d) one of the following interpolation methods is used: kriging method and spline method.

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