EP2453106B1 - Method for characterising the network of fractures of a fractured deposit and method for exploiting same - Google Patents

Description

The present invention relates to the field of the exploitation of underground deposits, such as hydrocarbon deposits comprising a network of fractures.

In particular, the invention relates to a method for characterizing the fracture network and thus to construct a representation of the deposit. The invention also relates to a method using this representation to optimize the management of such an operation by means of a prediction of fluid flows likely to occur through this medium, to simulate a production of hydrocarbons according to various production scenarios. .

The oil industry, and more specifically the exploration and exploitation of deposits, especially oil, require the acquisition of the best possible knowledge of the underground geology to efficiently provide an assessment of reserves, a modeling of production , or the management of the operation. In fact, the determination of the location of a production well or an injection well, the constitution of the drilling mud, the completion characteristics, the choice of a hydrocarbon recovery process (such as the injection of water for example) and the parameters necessary for the implementation of this process (such as the injection pressure, the production rate, etc.) require a good knowledge of the deposit. Knowing the deposit means knowing the petrophysical properties of the subsoil at any point in space.

To do this, the oil industry has for a long time been combining on-field measurements (in situ) with experimental (laboratory-generated) and / or digital (software-based) models. Modeling oil fields is therefore a technical step essential to any exploration or exploitation of deposits. These models are intended to provide a description of the deposit.

Cracked reservoirs are an extreme type of heterogeneous reservoirs with two contrasting media, a matrix medium containing most of the oil in place and having a low permeability, and a cracked medium representing less than 1% of the oil in place. and highly conductive. The cracked medium itself can be complex, with different sets of cracks characterized by their respective density, length, orientation, inclination and aperture.

The specialists in charge of the operation of fractured reservoirs, need to know perfectly the role of the fractures. A "fracture" is a plane discontinuity, very thin in relation to its extension, which represents a plane of rupture of a rock in the deposit. On the one hand, the knowledge of the distribution and the behavior of these fractures makes it possible to optimize the localization and the spacing between the wells that one to drill through the oil field. On the other hand, the geometry of the fracture network conditions the displacement of the fluids both at the reservoir scale and at the local scale where it determines elementary matrix blocks in which the oil is trapped. Knowing the distribution of fractures, is therefore very useful, also, at a later stage, for the tank engineer who tries to calibrate the models he builds to simulate the deposits in order to reproduce or predict past production curves. or future. For these purposes, geoscientists have three-dimensional images of the deposits, making it possible to locate a large number of fractures.

Thus, to reproduce or predict (ie "simulate") the production of hydrocarbons during the production of a deposit according to a given production scenario (characterized by the position of the wells, the recovery method, ...) , the tank engineering specialist implements a calculation software, called a "reservoir simulator" (or "flow simulator"), which calculates the flows and the evolution of the pressures within the reservoir represented by the reservoir model. tank. The results of these calculations allow it to predict and optimize the deposit in terms of flow and / or quantity of recovered hydrocarbons. The calculation of the behavior of the reservoir according to a given production scenario constitutes a "reservoir simulation".

A method is known for optimizing the exploitation of a fluid reservoir traversed by a fracture network, in which fluid flows in the reservoir are simulated by means of a simplified but realistic modeling of the deposit. This simplified representation is called "double-medium approach", it is proposed by Warren JE et al. in "The Behavior of Naturally Fractured Reservoirs", SPE Journal (September 1963), 245-255 . This technique consists in considering the fractured medium as two continua exchanging fluids between them: matrix blocks and fractures. This is called the "double medium" or "double porosity" model. Thus, the "double medium" modeling of a fractured deposit consists in discretizing this deposit into two sets of meshes (called grids) superimposed, constituting the grid "crack" and the grid "matrix". Each elementary volume of the fractured deposit is thus conceptually represented by two meshes, one "crack" and the other "matrix", coupled together (that is to say, exchanging fluids). In the reality of the fractured field, these two meshes represent the set of matrix blocks delimited by fractures present in this place of the reservoir. Indeed, most often, meshes have lateral dimensions MF (commonly 100 or 200 m) given the size of the fields and limited possibilities of simulation software in terms of capacity and calculation time. As a result, for most fractured fields, the fracture reservoir elemental volume (mesh) contains innumerable fractures forming a complex network delimiting multiple matrix blocks. of variable dimensions and shapes depending on the geological context. Each of the constituent real blocks exchanges fluids with the fractures that surround it at a rate (flow) that is specific to it because of the size and shape of this particular block.

Faced with such a geometrical complexity of the real environment, the approach consists, for each elementary volume (mesh) of reservoir, to represent the real fractured medium as a set of matricial blocks all identical, parallelepipedic, delimited by an orthogonal and regular network of fractures oriented along the main directions of flow: for each mesh, the so-called "equivalent" permeabilities of this fracture network are thus determined and defines a "representative" matrix block (of the real (geological) distribution of the blocks), unique and of parallelepipedal shape. It is then possible to formulate and calculate the matrix-crack exchange fluxes for this "representative" block, and to multiply the result by the number of such blocks in the elementary volume (mesh) to obtain the flux at the scale of this mesh.

It should however be noted that the calculation of the equivalent permeabilities requires knowing the flow properties (ie. The conductivities) discrete fracture of the geological model.

This is why, prior to the construction of this equivalent reservoir model (called the "double-medium tank model") as described above, it is first necessary to simulate the flow responses of a few wells (test results). transient or pseudo-permanent flow, interferences, flowmetry, etc.) on models extracted from the geological model giving a discrete (realistic) representation of the fractures feeding these wells. Adjusting the simulated pressure / flow responses on the field measurements calibrates the conductivities of the families of fractures. Although only covering a limited area (drainage area) around the well, such a well testing simulation model still has a large number of computational nodes if the fracture network is dense. As a result, the size of the systems to be solved and / or the duration of the calculations often remain prohibitive.

Another method, described in the patent application US 2008/0091396 A1 relates to a method of modeling hydraulic fractures, in which zones are built in the region of the well approaches.

To overcome this difficulty, the invention involves a simplification of the fracture networks at the local scale of the well drainage area, in order to be able to simulate the wells tests of fractured reservoirs and thus to calibrate the conductivities of the families of fractures. This hydraulic calibration of fractures results in a set of parameters characterizing the fracture network (or fracture model). This fracture model is then used to construct a reservoir-scale dual-flow model.

The method according to the invention

1. a) on déduit desdits paramètres statistiques (PSF) un tenseur de perméabilité équivalente et une ouverture moyenne desdites fractures, à partir desquels on construit une image représentative du réseau de fractures délimitant des blocs poreux et des fractures ;
2. b) on déduit dudit tenseur une direction d'écoulement de fluide autour dudit puits ;
3. c) on définit autour dudit puits, une première zone de frontière elliptique centrée sur ledit puits et contenant ledit puits, et au moins une seconde zone de frontière elliptique centrée sur ledit puits et formant une couronne elliptique avec la frontière elliptique de ladite première zone, lesdites zones étant orientées selon ladite direction d'écoulement de fluide ;
4. d) on simplifie ladite image représentative du réseau de fractures de façon différente dans chacune desdites zones ;
5. e) on utilise ladite image simplifiée pour construire ladite représentation du gisement de fluide ;
6. f) on utilise ladite représentation du gisement de fluide et un simulateur d'écoulement pour optimiser l'exploitation dudit gisement de fluide.

Thus, the object of the invention relates to a method for optimizing the exploitation of a fluid reservoir traversed by a fracture network and by at least one well, in which a representation of said fluid reservoir is constructed, said deposit being discretized deposit in a set of meshes, and fractures are characterized by statistical parameters (PSF) from observations of said deposit. The method has the following steps:

1. a) deducing from said statistical parameters (PSF) an equivalent permeability tensor and an average opening of said fractures, from which a representative image of the fracture network delimiting porous blocks and fractures is constructed;
2. b) deducing from said tensor a direction of fluid flow around said well;
3. c) defining around said well, a first elliptical boundary zone centered on said well and containing said well, and at least one second elliptical boundary zone centered on said well and forming an elliptical crown with the elliptical boundary of said first zone, said zones being oriented in said direction of fluid flow;
4. d) simplifying said representative image of the fracture network differently in each of said zones;
5. e) using said simplified image to construct said representation of the fluid reservoir;
6. f) using said representation of the fluid reservoir and a flow simulator to optimize the exploitation of said fluid reservoir.

According to the invention, the statistical parameters (PSF) can be chosen from the following parameters: fracture density, fracture length, fracture orientation, fracture inclination, fracture opening, and distribution of fractures within the deposit.

According to one embodiment, a shape ratio for each zone, defined from the lengths of the axes of the ellipse constituting the zone boundary, is determined so as to reproduce an anisotropy of flow around the well, and the areas so as to respect the form report. This aspect ratio can be determined by the main values of the permeability tensor.

According to one embodiment, a distance between the boundaries between zones is defined, so as to give a weight equal to each zone in terms of the pressure difference observed on each zone in steady-state flow regime. This distance can be defined by fixing the lengths of one of the two axes of two successive ellipses to values in geometric progression of constant reason.

According to another embodiment, three zones are constructed, a first zone (ZNS) containing the well within which no simplification of the image is made, a second zone (ZP) in contact with the first zone within of which a first simplification of the image is performed, and a third zone (ZL) in contact with the second zone in which a second simplification of the image is performed, the second simplification being more important than the first simplification.

• ladite seconde zone est divisée en un nombre de sous-zones égal à un nombre de blocs de mailles présents dans ladite zone, un bloc de mailles désignant un empilement vertical de mailles ;
• ladite troisième zone est divisée en réalisant les étapes suivantes :
  • ∘ on découpe tous les degrés la frontière de ladite troisième zone, définissant 360 arcs ;
  • ∘ on définit une sous-zone en reliant des points-extrémités de chacun desdits arcs au centre de l'ellipse formant la frontière ;
  • ∘ pour chacune desdites sous-zones, on calcule un tenseur de perméabilité de fracture équivalente à partir duquel on détermine une orientation des écoulements dans ladite sous-zone ;
  • ∘ on compare les valeurs de perméabilité de fracture équivalente et l'orientation de l'écoulement entre sous-zones voisines ; et
  • ∘ on regroupe des sous-zones voisines en une seule sous-zone lorsqu'une différence entre les valeurs de perméabilité est inférieure à un premier seuil et lorsqu'une différence entre les orientations des écoulements est inférieure à un second seuil.

Advantageously, the second and third zones can be divided into sub-zones by applying the following steps:

• said second zone is divided into a number of sub-zones equal to a number of mesh blocks present in said zone, a mesh block denoting a vertical stack of meshes;
• said third zone is divided by performing the following steps:
  • ∘ the borders of said third zone, defining 360 arcs, are cut off all the degrees;
  • A sub-area is defined by connecting end-points of each of said arcs to the center of the ellipse forming the boundary;
  • ∘ for each of said sub-zones, an equivalent fracture permeability tensor is calculated from which a direction of the flows in said sub-zone is determined;
  • Comparing the values of equivalent fracture permeability and the orientation of the flow between neighboring sub-zones; and
  • Grouping neighboring sub-zones into a single sub-zone when a difference between the permeability values is lower than a first threshold and when a difference between the orientations of the flows is lower than a second threshold.

• on construit un réseau de fracture équivalent (RFE) à ladite image, au moyen d'une représentation dite de Warren et Root, dans laquelle le réseau est caractérisé par des espacements de fractures (s₁ ^fin, s₂ ^fin) dans deux directions orthogonales de perméabilité principale, par un paramètre d'ouverture de fractures (e^fin), par des conductivités de fractures (C_f1 ^fin et C_f2 ^fin), et une perméabilité (k_m ^fin) d'un milieu matriciel entre fractures ;
• on simplifie ledit réseau de fracture équivalent (RFE) au moyen d'un coefficient d'écartement des fractures (G) du réseau dont la valeur est inférieure à une valeur G_max-zone définie sur chacune desdites zones afin de garantir une connectivité suffisante entre zones simplifiées et zones non simplifiées.

According to the invention, the image can be simplified by carrying out the following steps:

• constructing an equivalent fracture network (RFE) to said image, by means of a so-called Warren and Root representation, in which the network is characterized by fracture spacings (s ₁ ^end , s ₂ ^end ) in two orthogonal directions main permeability, by a fracture opening parameter (e ^fin ), by fracture conductivities (C _f1 ^fin and C _f2 ^end ), and a permeability (k _m ^fin ) of a matrix medium between fractures;
• said equivalent fracture network (EFR) is simplified by means of a network gap fracture coefficient (G) whose value is lower than a maximum G _{-zone value} defined on each of said zones in order to guarantee sufficient connectivity between simplified areas and un-simplified areas.

avec :

• DLM : dimension latérale minimale de la sous-zone donnée ;
• s₁ ^fin, s₂ ^fin : les espacements de fractures dans la représentation dite de Warren et Root.

It is possible, for a given subfield, to define the G _max-zone value as follows: $${\mathrm{BOY\; WUT}}_{\max -\mathit{zoned}}=\frac{\mathit{DLM}}{6.\mathit{Max}\left(s_1^{\mathit{end}},s_2^{\mathit{end}}\right)}$$

with:

• DLM: minimum lateral dimension of the given sub-area;
• s ₁ ^end , s ₂ ^fin : fracture spacings in the representation of Warren and Root.

• on réitère à l'étape a) en modifiant lesdits paramètres statistiques (PSF) de façon à minimiser une différence entre un résultat de test de puits et un résultat de simulation de test de puits à partir de ladite image simplifiée ;
• on associe à chacune desdites mailles au moins une valeur de perméabilité équivalente et une valeur d'ouverture moyenne desdites fractures, lesdites valeurs étant déterminées à partir desdits paramètres statistiques (PSF) modifiés.
• on simule des écoulements de fluides dans ledit gisement au moyen d'un simulateur d'écoulement et des valeurs de perméabilité équivalente et des valeurs d'ouverture moyenne desdites fractures associées à chacune desdites mailles ;
• on sélectionne un scénario de production permettant d'optimiser la production du gisement à l'aide de ladite simulation des écoulements de fluides; et
• on exploite ledit gisement selon ledit scénario permettant d'optimiser la production du gisement.

Finally, the invention also relates to a method for optimizing the management of a deposit. It comprises the following steps:

• repeating said statistical parameters (PSF) in step a) so as to minimize a difference between a well test result and a well test simulation result from said simplified image;
• each of said meshes is associated with at least one equivalent permeability value and an average opening value of said fractures, said values being determined from said modified statistical parameters (PSF).
• fluid flows are simulated in said reservoir by means of a flow simulator and equivalent permeability values and mean opening values of said fractures associated with each of said meshes;
• a production scenario is selected which makes it possible to optimize the production of the deposit by means of said simulation of the flows of fluids; and
• the said deposit is exploited according to the said scenario making it possible to optimize the production of the deposit.

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 illustrates the different steps of the method according to the invention.
• The figure 2 illustrates a realization of a network of fractures / faults at the scale of a reservoir
• The figure 3 illustrates an initial discrete fracture network (DFN)
• The figure 4 illustrates an equivalent fracture network (RFE), said Warren and Root.
• The figure 5 illustrates the creation of zones and sub-areas necessary for the simplification of the equivalent fracture network (RFE).
• The figure 6 illustrates a simplified fracture network (RFES) simplified according to the invention.

Detailed description of the method

1. 1- Discrétisation du gisement en un ensemble de mailles (MR)
2. 2- Modélisation du réseau de fractures (DFN, RFE, RFES)
3. 3- Simulation des écoulements de fluides (SIM) et optimisation des conditions de production du gisement (OPT)
4. 4- Exploitation (globale) optimisée du gisement (EXPLO)

The method according to the invention for optimizing the exploitation of a deposit, using the method according to the invention for characterizing the fracture network, comprises four steps, as illustrated in FIG. figure 1 :

1. 1- Discretization of the deposit into a set of meshes (MR)
2. 2- Modeling the fracture network (DFN, RFE, RFES)
3. 3- Simulation of fluid flows (SIM) and optimization of production conditions of the deposit (OPT)
4. 4- Optimized Exploitation (Global) of the Deposit (EXPLO)

1- Discretization of the deposit into a set of meshes (MR)

For a long time, the oil industry has combined on-field measurements (in situ) with experimental (laboratory-generated) and / or digital (software-based) models. The modeling of the oil fields, therefore constitute a technical step essential to any exploration or exploitation of deposits. These modelizations are intended to provide a description of the deposit, characterized by the structure / geometry and the petrophysical properties of the deposits or geological formations that constitute it.

These modelizations are based on a representation of the deposit, in a set of meshes. Each of these meshes represents a given volume of the deposit, and constitutes an elementary volume of the deposit. The set of meshes is a discrete representation of the deposit, called reservoir model.

Specialists are familiar with many software tools for constructing a reservoir model, using data (DG) and measurements (MG) for the deposit.

The figure 2 illustrates a two-dimensional view of a tank model. Fractures are represented by lines. Meshes are not represented.

2- Modeling the fracture network

To take into account the role of the fracture network in the simulation of the flows within the deposit, it is necessary to associate with each of these elementary volumes (meshes of the reservoir model) a fracture modeling.

1. a. on caractérise les fractures par des paramètres statistiques (PSF) à partir d'observations du gisement,
2. b. on déduit de ces paramètres statistiques (PSF) un tenseur de perméabilité équivalente et une ouverture moyenne des fractures, à partir desquels on construit une image représentative du réseau de fractures délimitant des blocs poreux et des fractures ;
3. c. on déduit de ce tenseur une direction d'écoulement de fluide autour du puits ;
4. d. on définit autour du puits, une première zone de frontière elliptique centrée sur le puits et contenant ce puits, au moins une seconde zone de frontière elliptique centrée sur le puits et de limite intérieure confondue avec la frontière elliptique de la première zone, les zones étant orientées selon la direction d'écoulement de fluide ;
5. e. on simplifie l'image représentative du réseau de fractures dans chaque maille appartenant à au moins une zone ;
6. f. on réitère à l'étape b) en modifiant les paramètres statistiques (PSF) de façon à minimiser la différence entre le résultat de test de puits et le résultat de simulation de test de puits à partir de l'image simplifiée ; et
7. g. on associe à chacune des mailles au moins une valeur de perméabilité équivalente et une valeur d'ouverture moyenne des fractures, ces valeurs étant déterminées à partir des paramètres statistiques (PSF) modifiés.

Thus, an object of the invention relates to a method for constructing a representation of a fluid reservoir traversed by a network of fractures and by at least one well. This method involves the discretization of the deposit into a set of meshes (step 1 previously described). Then the method has the following steps:

1. at. Fractures are characterized by statistical parameters (PSF) from observations of the deposit,
2. b. from these statistical parameters (PSF) is deduced an equivalent permeability tensor and an average fracture opening, from which a representative image of the fracture network delimiting porous blocks and fractures is constructed;
3. vs. a tension direction of fluid around the well is deduced from this tensor;
4. d. a first elliptical boundary zone centered around the well and containing the well, at least a second elliptical boundary zone centered on the well and an inner boundary coinciding with the elliptical boundary of the first zone, the zones being defined around the well; oriented in the direction of fluid flow;
5. e. the representative image of the fracture network is simplified in each mesh belonging to at least one zone;
6. f. reiterating in step b) modifying the statistical parameters (PSF) so as to minimize the difference between the well test result and the well test simulation result from the simplified image; and
7. boy Wut. each mesh is associated with at least one equivalent permeability value and an average opening value of the fractures, these values being determined from the modified statistical parameters (PSF).

These steps are detailed below.

Characterization of fractures

The specialists in charge of the statistical characterization of the reservoir, perform direct and indirect observations (OF) of the reservoir. For this, they have 1) well cores extracted from the reservoir on which a statistical study of the intersected fractures is carried out, 2) characteristic outcrops of the reservoir which have the advantage of providing a large-scale vision of the fracture network 3 ) seismic images allowing them to identify large geological events.

These measurements make it possible to characterize the cracks by statistical parameters (PSF): their density, their length, their orientation, their inclination and their respective opening, and of course, their distribution within the tank.

At the end of this stage of fracture characterization, we have statistical parameters (PSF) describing the fracture networks from which realistic images of the real (geological) networks can be reconstructed (generated) at the scale of each of the meshes (cells) of the reservoir model considered (simulation domain).

The objective of the characterization and modeling of the deposit fracture network is to propose a model of fractures validated on the local flows around the wells. This fracture model is then extended to the reservoir scale in order to perform production simulations. To do this, flow properties are associated with each mesh of the reservoir model (MR) (permeability tensor, porosity) of the two media (fracture and matrix).

These properties can be determined either directly from the statistical parameters (PSF) describing the fracture networks, or from a discrete fracture network (DFN) obtained from the statistical parameters (PSF).

Construction of a Discrete Fracture Network (DFN) - Figure 2 and 3

From a reservoir model of the studied deposit, a detailed representation (DFN) of the internal complexity of the fracture network is associated in each mesh as faithful as possible of the direct and indirect observations of the reservoir. The figure 2 illustrates a realization of a network of fractures / faults at the scale of a reservoir. Each mesh of the reservoir model thus represents a discrete network of fractures delimiting a set of porous matrix blocks of irregular shapes and sizes delimited by fractures. Such an image is represented on the figure 3 . This discrete fracture network constitutes a representative image of the real network of fractures delimiting the matrix blocks.

To build a discrete fracture network in each cell of a reservoir model, we can use modeling software, known in the art, such as FracaFlow ^® software (IFP, France). These programs use the statistical parameters determined in the fracture characterization step.

The next step is to determine the flow properties of the initial fractures (C _f , e) and then calibrate these properties by means of well-test simulations on discrete, realistic image-inherited local flow models. the actual (geological) fracture network at the reservoir scale. Although only covering a limited area (drainage area) around the well, such a well testing simulation model still has a large number of computational nodes if the fracture network is dense. As a result, the size of the systems to be solved and / or the duration of the calculations often remain prohibitive. Hence the need for a procedure to simplify the fracture network.

Simplification of fracture networks - Figure 5

Because of its extreme geometric complexity, the fracture network obtained in the previous step and representative of the actual fractured reservoir can not be used to simulate, i.e. reproduce and / or predict, the local flows around the well.

• une première zone, dans laquelle aucune simplification du réseau de fracture n'est réalisée. Cette zone contient le puits et son centre. Elle est notée ZNS, pour "Zone Non Simplifiée".
• une seconde zone, au contact de la première zone, dans laquelle une simplification modérée du réseau de fracture est réalisée. Cette zone est notée ZP, pour "Zone à simplifier Proche du puits".
• une troisième zone, au contact de la seconde zone, dans laquelle une simplification importante du réseau de fracture est réalisée. Cette zone est notée ZL, pour "Zone Lointaine du puits".

To circumvent this obstacle, the method according to the invention uses a method based on the division of the simulation domain (that is to say the reservoir model) into at least three types of zones around each well ( figure 5 ):

• a first zone, in which no simplification of the fracture network is performed. This area contains the well and its center. It is noted ZNS, for "Not Simplified Zone".
• a second zone, in contact with the first zone, in which a moderate simplification of the fracture network is performed. This zone is marked ZP, for "Zone to be simplified Near the well".
• a third zone, in contact with the second zone, in which a major simplification of the fracture network is performed. This zone is marked ZL, for "Far Zone of the Well".

The invention is not limited to the definition of three zones. It is also possible to divide the domain into n zones, the simplification of the network being increased from the zone 1 (ZNS) to the zone n (the furthest away from the well). It is thus possible to create a zone ZNS, n1 zones of type ZP, and n2 zones of type ZL.

Construction of areas - Figure 5

The purpose of fracture network modeling is to simulate well-flow responses (transient or pseudo-permanent flow tests, interference, flowmetry, etc.). This involves simulating, for example, the production of oil via each well drilled through the reservoir.

• l'orientation de l'ellipse, c'est-à-dire la direction du grand axe de l'ellipse (perpendiculaire à la direction du petit axe) ;
• les dimensions de l'ellipse, c'est-à-dire la longueur des axes.

For each well, each zone is defined according to an outer limit forming an ellipse centered on the well. The three zones are therefore concentric and elliptical boundaries. The two simplified zones ZP and ZL have internal limits corresponding to the outer limit of the respective zones ZNS and ZP. With the exception of the unimplified zone, the zones are therefore elliptical rings centered on the well. To construct each zone, it is therefore necessary to define:

• the orientation of the ellipse, that is to say the direction of the major axis of the ellipse (perpendicular to the direction of the minor axis);
• the dimensions of the ellipse, that is to say the length of the axes.

Their orientation is determined by the flow directions deduced from a calculation of equivalent permeabilities on the ZNS zone. This type of calculation of equivalent permeabilities is well known to those skilled in the art. One can for example use the numerical method for calculating equivalent properties of fractured media, implemented in the FracaFlow software (IFP Energies nouvelles, France) and recalled hereinafter.

• La première méthode, analytique, dite "local analytical upscaling", est basée sur une approche analytique décrite dans les documents suivants :
  • Oda M. (1985): Permeability tensor for discontinuous Rock Masses, Geotechnique Vol 35, 483-495
  • demande de brevet EP 2 037 080

According to this method, a permeability tensor representative of the flow properties of the discretized fracture network (DFN) can be obtained via two scaling methods (so - called upscaling methods).

• The first method, known as "local analytical upscaling", is based on an analytical approach described in the following documents:
  • Oda M. (1985): Permeability tensor for discontinuous Rock Masses, Geotechnical Vol 35, 483-495
  • request EP Patent 2,037,080

• La seconde méthode, numérique, dite "local numerical upscaling", est décrite dans les documents suivants :
  • Bourbiaux, B., et al., 1998, "A Rapid and Efficient Methodology to Convert Fractured Reservoir Images into a Dual-Porosity Model", Oil & Gas Science and Technology, Vol. 53, No. 6, Nov.-Déc. 1998, 785-799 .
  • brevet FR 2.757.947 ( US 6.023.656 ) pour les perméabilités équivalentes, et brevet FR 2.757.957 ( US 6.064.944 ) pour les dimensions de bloc équivalent

It has the advantage of being very fast. Its scope is, however, limited to well-connected fracture networks. Otherwise, significant errors on the permeability tensor can be found.

• The second method, digital, called "local numerical upscaling", is described in the following documents:
  • Bourbiaux, B., et al., 1998, "A Rapid and Efficient Methodology to Convert Fractured Reservoir Images into a Dual-Porosity Model," Oil & Gas Science and Technology, Vol. 53, No. 6, Nov.-Dec. 1998, 785-799 .
  • FR 2,757,947 ( US 6,023,656 ) for equivalent permeabilities, and patent FR 2 757 957 ( US 6,064,944 ) for equivalent block dimensions

It is based on the numerical resolution of the flow equations on a discrete mesh of the fracture network for different boundary conditions of the considered block of computation. The tensor of equivalent permeabilities is obtained by identifying the ratios between flow and pressure drop at the limits of the calculation block. This approach, more expensive than the previous one, has the advantage of well characterizing a given network (even if it is not connected).

According to one embodiment, one of the two preceding methods may be chosen so as to optimize the precision and speed of the calculations, by applying the method described in the application of EP Patent 2,037,080 , based on the calculation of a connectivity index.

This technique makes it possible, as a preliminary step, to determine the permeability tensors of a few cells of the reservoir model surrounding the well, and considered representative of the flow of the ZNS. The diagonalization of these permeability tensors provides the eigenvectors oriented along the main directions of flow sought. We are then able to orient the qualified elliptical domain of ZNS and centered on the well, along the semi-major axis of the ellipse of permeability determined by means of these preliminary calculations.

• rapport de forme des ellipses. En notant L_max la demi longueur du grand axe de l'ellipse, et L_min la demi longueur du petit axe de l'ellipse, ce rapport de forme est défini par L_max /L_min. II convient de ne pas le fixer de manière arbitraire mais au contraire de le choisir de manière à reproduire l'anisotropie d'écoulement. On peut utiliser le tenseur de perméabilité équivalente calculé pour déterminer l'orientation. Si l'on note K_min et K_max les valeurs principales de ce tenseur, alors les ellipses sont orientées suivant les directions principales de perméabilité avec un rapport de forme L_maxlL_min égal à la racine carrée du rapport K_max/K_min : $$\frac{L_{\max }}{L_{\min }}=\sqrt{\frac{K_{\max }}{K_{\min }}}$$
  
• distance entre ellipses concentriques (frontières entre zones) : un critère de dimensionnement conforme à une précision de modélisation uniformément distribuée sur le domaine de simulation consiste à fixer les demi-longueurs du grand axe L_max (ou du petit axe L_min ) de 2 ellipses successives i+1 et i à des valeurs en progression géométrique de raison r (égale à 2 par exemple) constante, soit : $\frac{L_{\max \left(i+1\right)}}{L_{\max \left(i\right)}}=\frac{L_{\min \left(i+1\right)}}{L_{\min \left(i\right)}}=r,$
  
  pour tout i, avec initialisation à la valeur L_max0 de la zone non simplifiée. Cette règle permet de donner un poids égal à chaque zone i (i=1 à n) en termes de différence de pression constatée sur chaque couronne en régime d'écoulement permanent.

Then, to define the dimensions of the ellipse, we define a shape ratio of this ellipse as well as a distance between concentric ellipses (distance separating the inner and outer elliptic limits of a given concentric ring):

• shape ratio of ellipses. Noting L _max the half length of the major axis of the ellipse, and L _min the half length of the minor axis of the ellipse, this aspect ratio is defined by L _max / L _min . It should not be fixed arbitrarily but rather to choose it so as to reproduce the flow anisotropy. The calculated equivalent permeability tensor can be used to determine the orientation. If we denote K _min and K _max the principal values of this tensor, then the ellipses are oriented according to the principal directions of permeability with a form ratio L _max lL _min equal to the square root of the ratio K _max / K _min : $$\frac{{\mathrm{The}}_{\max }}{{\mathrm{The}}_{\min }}=\sqrt{\frac{K_{\max }}{K_{\min }}}$$
  
• distance between concentric ellipses (boundaries between zones): a dimensioning criterion conforming to a uniformly distributed modeling precision on the simulation domain consists in setting the half-lengths of the major axis L _max (or the minor axis L _min ) of 2 ellipses successive i + 1 and i to values in geometric progression of reason r (equal to 2 for example) constant, that is: $\frac{{\mathrm{The}}_{\max \left(i+1\right)}}{{\mathrm{The}}_{\max \left(i\right)}}=\frac{{\mathrm{The}}_{\min \left(i+1\right)}}{{\mathrm{The}}_{\min \left(i\right)}}=r,$
  
  for all i, with initialization to the L _max0 value of the _unimproved area. This rule makes it possible to give equal weight to each zone i (i = 1 to n) in terms of the pressure difference observed on each ring in steady-state flow.

Once this overall dimensioning has been carried out, the delimitation and simplification procedures of the sub-areas are implemented, also relying on the methods for calculating the equivalent properties of the fractured media.

Subzone Construction Within Each Zone - Figure 5

• La zone ZNS est une zone à conserver intacte (non simplifiée); aucune sous-zone n'y est créée.
• Les zones à simplifier les plus proches d'un puits (zones de type ZP), nécessitent une attention particulière. Pour modéliser correctement les variations locales des propriétés d'écoulement dans cette zone, les ZP sont divisées en un nombre de sous-zones égal au nombre de blocs de mailles présents dans les zones ZP. Le terme "bloc de mailles" est employé pour désigner un "empilement" de mailles verticales (de type CPG (Corner Point Grid) par exemple) du modèle de réservoir, délimitées par les mêmes poteaux droits sub verticaux. Il ne s'agit donc pas des blocs équivalents.
• Les zones à simplifier les plus éloignées d'un puits (zone de type ZL), sont des couronnes elliptiques concentriques qui couvrent des surfaces de plus en plus importantes au fur et à mesure qu'on s'éloigne du puits. Étant plus éloignées des puits, il est acceptable d'être moins précis dans la détection des hétérogénéités que dans le cas des zones de type ZP. L'hétérogénéité des propriétés d'écoulement d'une zone ZL reste toutefois le facteur principal contrôlant le découpage en sous-zones. Pour échantillonner une zone ZL, un balayage angulaire, centré sur le puits, est effectué sur la limite elliptique extérieure séparant la zone ZL considérée de sa voisine extérieure. Tous les degrés, un bloc de mailles est sélectionné. Ainsi, la zone ZL est caractérisée par au plus 360 blocs. Pour chacun de ces blocs, un calcul du tenseur de perméabilité de fracture équivalente est effectué. Ce tenseur permet de caractériser les propriétés dynamiques du réseau de fractures du bloc étudié. Les valeurs de perméabilité et l'orientation de l'écoulement obtenues sont alors comparées entre blocs voisins. Si les propriétés de deux blocs voisins sont proches, ces deux blocs seront considérés comme appartenant à une même sous-zone. Dans le cas contraire (différence de 20 % sur les valeurs de perméabilité principale ou de 10 degrés sur les directions principales de perméabilité par exemple), les deux blocs sont affectés à des sous-zones différentes ainsi définies. De cette manière, la limite extérieure de la zone ZL considérée est divisée en arcs de forme elliptique (figure 1). Les sous-zones sont alors obtenues en reliant les points-extrémités de chacun de ces arcs au centre (puits) de l'ellipse (lignes en pointillées de la figure 5) : chaque sous-zone est ainsi définie comme l'aire comprise entre le cloisonnement radial intra-zone (traits pointillés) et les limites elliptiques inter-zones (en trait plein).

Each of the zones, except the zone ZNS, is then divided into sub-zones (ssZP, ssZL1, ssZL2) within which a simplification of the fracture network is carried out. The number of sub-areas per area depends on the type of area (ZNS, ZP or ZL) and the heterogeneity of the area. So :

• The ZNS zone is an area to be kept intact (not simplified); no subzone is created there.
• Areas to be simplified closer to a well (ZP type zones) require special attention. To correctly model the local variations of the flow properties in this area, the PZs are divided into a number of subfields equal to the number of mesh blocks present in the ZP zones. The term "block of meshes" is used to designate a "stack" of vertical meshes (of the CPG type (Corner Point Grid) for example) of the reservoir model, delimited by the same right sub vertical posts. It is not therefore equivalent blocks.
• The zones to be simplified furthest from a well (ZL-type zone) are concentric elliptical crowns that cover increasingly large surfaces as one moves away from the well. Being further away from wells, it is acceptable to be less accurate in detecting heterogeneities than in the case of ZP-type zones. The heterogeneity of the flow properties of a ZL zone, however, remains the main factor controlling the subzone cutting. To sample a zone ZL, an angular sweep, centered on the well, is performed on the outer elliptical boundary separating the zone ZL considered from its outer neighbor. Every degree, a block of meshes is selected. Thus, the ZL area is characterized by at most 360 blocks. For each of these blocks, a calculation of the equivalent fracture permeability tensor is performed. This tensor makes it possible to characterize the dynamic properties of the fracture network of the studied block. The permeability values and the flow orientation obtained are then compared between neighboring blocks. If the properties of two neighboring blocks are close, these two blocks will be considered as belonging to the same sub-zone. In the opposite case (difference of 20% on the principal permeability values or 10 degrees on the principal directions of permeability for example), the two blocks are assigned to different sub-zones thus defined. In this way, the outer limit of the zone ZL considered is divided into elliptic arcs ( figure 1 ). The subfields are then obtained by connecting the endpoints of each of these arcs to the center (well) of the ellipse (dashed lines of the figure 5 ): each subfield is thus defined as the area between the intra-zone radial partitioning (dashed lines) and the inter-zone elliptical boundaries (solid line).

As indicated above, the delimitation of the sub-zones (boundary points of the inter-zone elliptic boundary arcs) is based on the comparison of the equivalent permeabilities calculated on the neighboring "blocks" materializing these arcs. The analytical method for calculating equivalent permeabilities is preferably used because it is in this case to determine if a block has the same dynamic behavior as the neighboring block. Although, for a weakly connected network, the analytical approach provides erroneous results, the errors are systematic, similar from one block to the other, which allows the comparison of the results between blocks which, let us specify it, does not require a great precision considering the simple objective of definition of zone. Thus the analytical approach is quite justified, with the considerable advantage of allowing much faster calculations, guaranteeing a practical feasibility.

Simplification of the fracture network in sub- zones (RFE, RFES)

Once the sub-zones have been cut, the scaling calculations will allow the fracture network of these sub-zones to be replaced by a simplified network having the same flow properties as the original network. In this case, and contrary to the above, the calculation of the equivalent fracture permeability tensor must be as accurate as possible.

• Réseau bien connecté : pour ce cas caractérisé par un indice de connectivité voisin ou dépassant 3 (au moins 3 intersections par fracture du réseau en moyenne), la méthode d'upscaling analytique est choisie car sa précision est garantie compte tenu de la bonne connectivité du réseau, avec l'avantage supplémentaire essentiel de la rapidité.
• Réseau peu/mal connecté : dans ce cas, l'indice de connectivité est compris entre 1 et 3 (ce qui correspond à un nombre d'intersections de fractures compris entre une et trois fois le nombre de fractures) et la méthode d'upscaling numérique est utilisée pour calculer de manière fiable le tenseur de perméabilité.
• Lorsque le réseau est très peu voire non connecté (nombre d'intersections voisin ou inférieur au nombre de fractures), les fractures originelles (peu nombreuses) sont conservées, c'est-à-dire que la sous-zone en question n'est pas simplifiée.

In order to take full advantage of the two scaling methods, the permeability tensor is determined by either of these two methods, for example, following the selection procedure described in the document. EP 2,037,080 , and based on the value of the connectivity index of the fracture network. This index, representative of the ratio between the number of intersections between fractures and the number of fractures, is calculated for each unit of the block considered ( ie to 2D). Its value makes it possible to consider the network as very well connected, little / badly connected or not connected. The choice of the scaling method is then performed as follows ( Delorme, M., Atfeh, B., Allken, V. and Bourbiaux, B. 2008, Upscaling Improvement for Heterogeneous Fractured Reservoir Using a Geostatistical Connectivity Index, edited in Geostatistics 2008, VIII International Geostatistics Congress, Santiago, Chi the. :

• Well connected network: for this case characterized by a connectivity index close to or greater than 3 (at least 3 intersections per network fracture on average), the analytical upscaling method is chosen because its accuracy is guaranteed given the good connectivity of the network. network, with the essential additional benefit of speed.
• Poorly / poorly connected network: in this case, the connectivity index is between 1 and 3 (which corresponds to a number of fracture intersections between one and three times the number of fractures) and the upscaling method digital is used to reliably calculate the permeability tensor.
• When the network is very little or not connected (number of intersections close to or less than the number of fractures), the original fractures (few) are preserved, ie the sub-zone in question is not not simplified.

• calcul d'un premier réseau équivalent (réseau parallélépipédique dit de Warren et Root) dit "fin", issu directement de la méthode du brevet FR 2.757.957 ( US 6.064.944 ). Ce réseau (Figure 4) est caractérisé par des espacements des fractures s₁ ^fin, s₂ ^fin dans les 2 directions orthogonales de perméabilité principale 1 et 2 définissant le réseau de fractures.
  Un paramètre supplémentaire, l'ouverture des fractures (e^fin), caractérise les fractures de ce réseau fin. La valeur des ouvertures de fracture est en pratique quasiment toujours négligeable devant l'espacement entre fractures : il est tenu compte de cette hypothèse dans les formules qui suivent, où on suppose également la même valeur d'ouverture pour les 2 familles de fractures. Compte tenu de l'égalité des porosités du réseau DFN initial (φ_f ) et du réseau équivalent fin, e^fin se déduit du volume de fractures du réseau DFN initial, V_f ^init, et du volume total de roche V_T comme suit : $$\boxed{e^{\mathit{fin}}=\frac{1}{\left(\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_1^{\mathit{fin}}}+\frac{1}{s_2^{\mathit{fin}}}\right)}\frac{V_f^{\mathit{init}}}{V_T}=\frac{1}{\left(\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_1^{\mathit{fin}}}+\frac{1}{s_2^{\mathit{fin}}}\right)}{\phi }_f}$$
  
  
  Pour mémoire, les perméabilités principales de fracture équivalentes, issues des calculs décrits plus haut, valent k₁= k_eq ^Max et k₂= k_eq ^Min suivant les directions principales 1 et 2. On déduit les conductivités des fractures du réseau équivalent fin, C_f1 ^fin et C_f2 ^fin, suivant ces deux directions d'écoulement, en écrivant la conservation des flux par unité d'aire de milieu fracturé : $${{\mathrm{<figure-callout\; id="1"\; label="C\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{f}}}^{\mathrm{fin}}={{\mathrm{s}}_{\mathrm{2}}}^{\mathrm{fin}}\mathrm{.}{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{1}}{{\mathrm{et\; C}}_{\mathrm{f}\mathrm{2}}}^{\mathrm{fin}}={{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_{\mathrm{1}}}^{\mathrm{fin}}\mathrm{.}{\mathrm{k}}_{\mathrm{2}}$$
  
  
  Enfin, le milieu matriciel entre fractures possède une perméabilité k_m ^fin.
• remplacement de ce réseau fin (Figure 4) par le réseau équivalent dit "grossier" (Figure 6), car comportant des fractures plus espacées afin d'accroître le degré de simplification en vue des simulations d'écoulement ultérieures. Les propriétés géométriques et d'écoulement de ce réseau grossier sont les suivantes:
  • ∘ des espacements de fractures s₁ ^gros et s₂ ^gros tels que : $${{\mathrm{s}}_{\mathrm{1}}}^{\mathrm{gros}}=\mathrm{G}\mathrm{.}{{\mathrm{s}}_{\mathrm{1}}}^{\mathrm{fin}}$$
    
    $${{\mathrm{s}}_2}^{\mathrm{gros}}=\mathrm{G}\mathrm{.}{{\mathrm{s}}_2}^{\mathrm{fin}},$$
    
    où G est un coefficient de grossissement (d'écartement des fractures) du réseau dont la valeur est laissée au libre choix de l'utilisateur avec cependant une limite supérieure G_max-zone à ne pas dépasser afin de garantir une connectivité suffisante entre zones simplifiées et non simplifiées :
    • G<G_max-zone où G_max-zone est tel que
    • Max(s₁ ^gros,s₂ ^gros) < (dimension latérale minimale de la sous-zone )/6
    Ainsi : $$G_{\max -\mathit{zone}}=\frac{\mathit{DLM}}{6.\mathit{Max}\left(s_1^{\mathit{fin}},s_2^{\mathit{fin}}\right)}$$
    
    avec :
    • DLM dimension latérale minimale de la sous-zone donnée ;
    • s₁ ^fin, s₂ ^fin : les espacements de fractures dans la représentation dite de Warren et Root.
  • ∘ des conductivités de fractures C_f1 ^gros et C_f2 ^gros, dont les valeurs permettent de conserver les flux par unité d'aire de milieu fracturé, c'est-à-dire également les perméabilités équivalentes, soit : C_f1 ^gros=s₂ ^gros .k₁ et C_f2 ^gros=s₁ ^gros.k₂
    ou encore, sachant que C_f1 ^fin=s₂ ^fin .k₁ et C_f2 ^fin=S₁ ^fin.k₂, $${{\mathrm{<figure-callout\; id="1"\; label="C\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{f}}}^{\mathrm{gros}}=\mathrm{G}\mathrm{.}{{\mathrm{<figure-callout\; id="1"\; label="C\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">C</figure-callout>}}_{\mathrm{f}}}^{\mathrm{fin}}{{\mathrm{et\; C}}_{\mathrm{f}\mathrm{2}}}^{\mathrm{gros}}=\mathrm{G}\mathrm{.}{{\mathrm{C}}_{\mathrm{f}\mathrm{2}}}^{\mathrm{fin}}$$
    
  • ∘ une ouverture de fracture e^gros permettant de conserver à nouveau la porosité de fracture φ_f du réseau initial (égale à celle du réseau équivalent grossier) : $${\mathrm{\phi }}_{\mathrm{f}}={\mathrm{e}}^{\mathrm{gros}}\mathrm{.}\left(\mathrm{1}/{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_{\mathrm{1}}}^{\mathrm{gros}}+\mathrm{1}/{{\mathrm{s}}_{\mathrm{2}}}^{\mathrm{gros}}\right)={\mathrm{e}}^{\mathrm{fin}}\mathrm{.}\left(\mathrm{1}/{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_{\mathrm{1}}}^{\mathrm{fin}}+\mathrm{1}/{{\mathrm{s}}_{\mathrm{2}}}^{\mathrm{fin}}\right)$$
    
    d'où: $$e^{\mathit{gros}}=e^{\mathit{fin}}\frac{\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_1^{\mathit{fin}}}+\frac{1}{s_2^{\mathit{fin}}}}{\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_1^{\mathit{gros}}}+\frac{1}{s_2^{\mathit{gros}}}}=G\mathrm{.}{e}^{\mathit{fin}}$$
    
  • ∘ une perméabilité de matrice k_m ^gros conservant la valeur du paramètre d'échange matrice-fissure : $${\lambda }_{\mathit{fin}}=r_w^2\frac{k_m^{\mathit{fin}}}{k_f^{\mathit{<figure-callout\; id="1"\; label="fin\; \alpha \; S"\; filenames="imgf0004.png"\; state="\{\{state\}\}">fin</figure-callout>}}}\left(\frac{\alpha }{S_{}^{}}\right)\left(\frac{{}_1^{{\mathit{fin}}^2}}{}+\frac{\alpha }{S_2^{{\mathit{fin}}^2}}\right)={\lambda }_{\mathit{grossier}}=r_w^2\frac{k_m^{\mathit{gros}}}{k_f^{\mathit{gros}}}\left(\frac{\alpha }{S_1^{{\mathit{gros}}^2}}+\frac{\alpha }{S_2^{{\mathit{gros}}^2}}\right)$$
    

où α est une constante et où les perméabilités équivalentes de fracture des réseaux fin et grossier, égales, sont ici notées $k_f^{\mathit{fin}}$

et $k_f^{\mathit{gros}}\mathrm{.}$

soit $$k_m^{\mathit{gros}}=k_m^{\mathit{fin}}\frac{\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="imgf0004.png"\; state="\{\{state\}\}">S</figure-callout>}}_1^{{\mathit{fin}}^2}}+\frac{1}{S_2^{{\mathit{fin}}^2}}}{\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="imgf0004.png"\; state="\{\{state\}\}">S</figure-callout>}}_1^{{\mathit{gros}}^2}}+\frac{1}{S_2^{{\mathit{gros}}^2}}}=G^2\mathrm{.}{k}_m^{\mathit{fin}}$$

Once these equivalent permeability calculations are made for each sub-area, the other equivalent flow parameters characterizing the simplified sub-areas are easily determined according to the following methods and equivalents.

• calculation of a first equivalent network (parallelepipedal network called Warren and Root) said "end", directly from the method of FR 2,757,957 ( US 6,064,944 ). This network ( Figure 4 ) is characterized by spacings of fractures s ₁ ^end , s ₂ ^end in the 2 orthogonal directions of main permeability 1 and 2 defining the network of fractures.
  An additional parameter, the opening of fractures (e ^end ), characterizes the fractures of this fine network. In practice, the value of the fracture openings is practically always negligible compared to the spacing between fractures: this assumption is taken into account in the following formulas, where the same opening value is also assumed for the two families of fractures. Given the equal porosity of the initial DFN network ( φ _f ) and the fine equivalent network, e ^end is deduced from the fracture volume of the initial DFN network, V _f ^init , and the total volume of rock V _T as follows: $$\boxed{e^{\mathit{end}}=\frac{1}{\left(\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_1^{\mathit{end}}}+\frac{1}{s_2^{\mathit{end}}}\right)}\frac{V_f^{\mathit{init}}}{V_T}=\frac{1}{\left(\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_1^{\mathit{end}}}+\frac{1}{s_2^{\mathit{end}}}\right)}{\phi }_f}$$
  
  
  For memory, the main permeabilities equivalent fracture, resulting from the calculations described above, are worth k ₁ = k _eq ^Max and k ₂ = k _eq ^Min following principal directions 1 and 2. We deduce the conductivities of the equivalent network end fractures, C _f1 ^end and C _f2 ^end , following these two directions of flow, by writing the conservation of flows per unit area of fractured medium: $${{\mathrm{<figure-callout\; id="1"\; label="VS\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">VS\; </figure-callout>}}_{\mathrm{f}}}^{\mathrm{end}}={{\mathrm{s}}_{\mathrm{2}}}^{\mathrm{end}}\mathrm{.}{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="imgf0004.png"\; state="\{\{state\}\}">k</figure-callout>}}_{\mathrm{1}}{{\mathrm{and\; C}}_{\mathrm{f}\mathrm{2}}}^{\mathrm{end}}={{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_{\mathrm{1}}}^{\mathrm{end}}\mathrm{.}{\mathrm{k}}_{\mathrm{2}}$$
  
  
  Finally, the matrix medium between fractures has a permeability k _m ^fin .
• replacing this fine network ( Figure 4 ) by the so-called "coarse" equivalent network ( Figure 6 ), because with more spaced fractures to increase the degree of simplification for subsequent flow simulations. The geometric and flow properties of this coarse network are as follows:
  • ∘ fracture spacings of ₁ ^large and ₂ ^large such as: $${{\mathrm{s}}_{\mathrm{1}}}^{\mathrm{large}}=\mathrm{BOY\; WUT}\mathrm{.}{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_{\mathrm{1}}}^{\mathrm{end}}$$
    
    $${{\mathrm{s}}_2}^{\mathrm{large}}=\mathrm{BOY\; WUT}\mathrm{.}{{\mathrm{s}}_2}^{\mathrm{end}},$$
    
    where G is a coefficient of magnification (of the separation of the fractures) of the network whose value is left to the free choice of the user with however an upper limit G _max-zone not to be exceeded in order to guarantee a sufficient connectivity between simplified zones and not simplified:
    • G <G _max-zone where G _max-zone is such that
    • Max (s ₁ ^large , s ₂ ^large ) <(minimal lateral dimension of the sub-area) / 6
    So : $${\mathrm{BOY\; WUT}}_{\max -\mathit{zoned}}=\frac{\mathit{DLM}}{6.\mathit{Max}\left(s_1^{\mathit{end}},s_2^{\mathit{end}}\right)}$$
    
    with:
    • DLM minimum lateral dimension of the given subfield;
    • s ₁ ^end , s ₂ ^fin : fracture spacings in the representation of Warren and Root.
  • ∘ fracture conductivities C _f1 and C ^{large large} _f2, whose values help maintain flows through the middle of fractured unit area, that is to say also the equivalent permeabilities, ie: C _f1 = s ^big ₂ ^big .k ₁ and C _f2 ^big = s ₁ ^big .k ₂
    or again, knowing that C _f1 ^end = s ₂ ^end .k ₁ and C _f2 ^fi n = S ₁ ^end .k ₂ , $${{\mathrm{<figure-callout\; id="1"\; label="VS\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">VS\; </figure-callout>}}_{\mathrm{f}}}^{\mathrm{large}}=\mathrm{BOY\; WUT}\mathrm{.}{{\mathrm{<figure-callout\; id="1"\; label="VS\; f"\; filenames="imgf0004.png"\; state="\{\{state\}\}">VS\; </figure-callout>}}_{\mathrm{f}}}^{\mathrm{end}}{{\mathrm{and\; C}}_{\mathrm{f}\mathrm{2}}}^{\mathrm{large}}=\mathrm{BOY\; WUT}\mathrm{.}{{\mathrm{VS}}_{\mathrm{f}\mathrm{2}}}^{\mathrm{end}}$$
    
  • ∘ a ^large fracture opening e allowing retaining the fracture porosity φ _f of the initial network (equal to that of the coarse equivalent network): $${\mathrm{\phi }}_{\mathrm{f}}={\mathrm{e}}^{\mathrm{large}}\mathrm{.}\left(\mathrm{1}/{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_{\mathrm{1}}}^{\mathrm{large}}+\mathrm{1}/{{\mathrm{s}}_{\mathrm{2}}}^{\mathrm{large}}\right)={\mathrm{e}}^{\mathrm{end}}\mathrm{.}\left(\mathrm{1}/{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_{\mathrm{1}}}^{\mathrm{end}}+\mathrm{1}/{{\mathrm{s}}_{\mathrm{2}}}^{\mathrm{end}}\right)$$
    
    from where: $$e^{\mathit{large}}=e^{\mathit{end}}\frac{\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_1^{\mathit{end}}}+\frac{1}{s_2^{\mathit{end}}}}{\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="imgf0004.png"\; state="\{\{state\}\}">s</figure-callout>}}_1^{\mathit{large}}}+\frac{1}{s_2^{\mathit{large}}}}=\mathrm{BOY\; WUT}\mathrm{.}{e}^{\mathit{end}}$$
    
  • ∘ a matrix permeability k _m ^large retaining the value of the matrix-crack exchange parameter: $${\lambda }_{\mathit{end}}=r_w^2\frac{k_m^{\mathit{end}}}{k_f^{\mathit{<figure-callout\; id="1"\; label="end\; \alpha \; S"\; filenames="imgf0004.png"\; state="\{\{state\}\}">end\; </figure-callout>}}}\left(\frac{\alpha }{S_{}^{}}\right)\left(\frac{{}_1^{{\mathit{end}}^2}}{}+\frac{\alpha }{S_2^{{\mathit{end}}^2}}\right)={\lambda }_{\mathit{coarse}}=r_w^2\frac{k_m^{\mathit{large}}}{k_f^{\mathit{<figure-callout\; id="1"\; label="large\; \alpha \; S"\; filenames="imgf0004.png"\; state="\{\{state\}\}">large\; </figure-callout>}}}\left(\frac{\alpha }{S_{}^{}}\right)\left(\frac{{}_1^{{\mathit{large}}^2}}{}+\frac{\alpha }{S_2^{{\mathit{large}}^2}}\right)$$
    

where α is a constant and where the equivalent fracture permeabilities of fine and coarse, equal networks are noted here $k_f^{\mathit{end}}$

and $k_f^{\mathit{large}}\mathrm{.}$

is $$k_m^{\mathit{large}}=k_m^{\mathit{end}}\frac{\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="imgf0004.png"\; state="\{\{state\}\}">S</figure-callout>}}_1^{{\mathit{end}}^2}}+\frac{1}{S_2^{{\mathit{end}}^2}}}{\frac{1}{{\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="imgf0004.png"\; state="\{\{state\}\}">S</figure-callout>}}_1^{{\mathit{large}}^2}}+\frac{1}{S_2^{{\mathit{large}}^2}}}={\mathrm{BOY\; WUT}}^2\mathrm{.}{k}_m^{\mathit{end}}$$

Finally, once this fine-network network equivalence operation performed for each sub-zone (or "block" of the ZP), the fractures of the simplified networks obtained are extended beyond the limits of sub-zones (or "blocks") to ensure sufficient partial "overlap" and thus sufficient horizontal connectivity of the simplified sub-area networks (or "blocks" of ZP) nearby. For this purpose, and according to a procedure tested by tests, the fractures of the simplified network can thus be extended by a length equal to 60% of the maximum spacing (s ₁ ^large , s ₂ ^large ) fractures of this network.

Such an image is represented on the figure 6 .

Calibration of fracture flow properties

The next step is the calibration of the fracture flow properties (conductivity and opening of fractures), locally around the wells. This requires the simulation of well tests. According to the invention, this simulation of well tests is performed on the simplified flow models ( figure 6 ).

This type of calibration is well known to specialists. For example, the method described in FR 2,787,219 . The flow responses of a few wells (transient or pseudo-permanent flow tests, interferences, flowmetry, etc.) are simulated on these models extracted from the geological model giving a discrete (realistic) representation of the fractures feeding these wells. Then, we compare the result of the simulation with the actual measurements made at the wells. If the results diverge, we modify the statistical parameters (PSF) describing the fracture networks, then we redetermine the flow properties of the initial fractures, and we perform a new simulation. The operation is repeated until the simulation results and the measurements converge.

The results of these simulations make it possible to calibrate (estimate) the geometry and the fracture flow properties, such as the conductivities of the fracture networks of the studied reservoir and the openings.

3- Simulation of fluid flows (SIM) and optimization of production conditions of the deposit (OPT)

At this point, the reservoir engineer then has the data required to build the reservoir scale flow model. Indeed, fracture reservoir simulations often adopt the "double porosity" approach, proposed for example by Warren JE et al. in "The Behavior of Naturally Fractured Reservoirs", SPE Journal (September 1963), 245-255 , and according to which any elementary volume (mesh of the reservoir model) of the cracked reservoir is modelized in the form of a set of identical parallelepipedic blocks, called equivalent blocks, delimited by an orthogonal system of continuous uniform fractures oriented along the principal directions of 'flow. Fluid flow, at the reservoir scale, occurs through fractures only, and fluid exchanges occur locally between fractures and matrix blocks. For example, the tank engineer can use the methods described in the following documents, applied to the entire tank this time: FR 2,757,947 ( US 6,023,656 ) FR 2 757 957 ( US 6,064,944 ) and EP 2,037,080 . These methods make it possible to calculate the equivalent fracture permeabilities and the equivalent block dimensions for each of the cells of the reservoir model.

The reservoir engineer chooses a production process, for example the water injection recovery process, the optimal implementation scenario for the field in question. The definition of an optimal water injection scenario will consist, for example, in determining the number and location (position and spacing) of the injection and production wells in order to best take into account the impact of the fractures on the water. progression of fluids within the reservoir.

Depending on the chosen scenario, the double-middle representation of the deposit, and the formula linking the mass and / or energy exchange flux to the matrix-fracture potential difference, one is then able to simulate the production of hydrocarbons, by means of the so-called dual medium flow simulator (software).

At any instant t of the simulated production, from the input data E (t) (fixed or variable data as the simulated time), and from the formula connecting the flow (f) of exchange to the potential difference (ΔΦ), the simulator solves all the equations specific to each mesh and each of the two grids of the model (equations involving the matrix-crack exchange formula described above) and thus delivers the solution values of the unknowns S (t) (saturations, pressures, concentrations, temperature, etc.) at this instant t. From this resolution, comes the knowledge of the quantities of oil produced and the state of the deposit (distribution of pressures, saturations, etc ...) at the moment considered.

4- Optimized exploitation of the deposit (EXPLO)

By selecting various scenarios, characterized for example by various respective locations of the injectors and producers, and by simulating the production of hydrocarbons for each of them according to step 3, one can select the scenario to optimize the production of the deposit. fractured considered according to the selected technical and economic criteria.

The specialists then exploit the deposit according to this scenario making it possible to optimize the production of the deposit.

Claims (11)

1. A method for optimizing the development of a fluid reservoir traversed by a fracture network and by at least one well, wherein a representation of said fluid reservoir is constructed, said reservoir is discretized into a set of cells and the fractures are characterized by statistical parameters (PSF) from observations of said reservoir, characterized in that the method comprises the following stages:

   a) deducing from said statistical parameters (PSF) an equivalent permeability tensor and an average opening for said fractures, from which an image representative of the fracture network delimiting porous blocks and fractures is constructed,

   b) deducing from said tensor a direction of fluid flow around said well,

   c) defining around said well a first elliptical boundary zone centered on said well and containing said well, and at least a second elliptical boundary zone centered on said well and forming an elliptical ring with the elliptical boundary of said first zone, said zones being oriented in said direction of fluid flow,

   d) simplifying said image representative of the fracture network in a different manner in each one of said zones,

   e) using said simplified image to construct said representation of the fluid reservoir,

   f) using said representation of the fluid reservoir and a flow simulator to optimize the development of said fluid reservoir.
2. A method as claimed in claim 1, wherein statistical parameters (PSF) are selected from among the following parameters: fracture density, fracture length, fracture orientation, fracture inclination, fracture opening and fracture distribution within the reservoir.
3. A method as claimed in any one of the previous claims, wherein an aspect ratio is determined for each zone, defined from the lengths of the axes of the ellipse making up the boundary of said zone, so as to reproduce a flow anisotropy around said well, and the zones are constructed so as to respect said aspect ratio.
4. A method as claimed in claim 3, wherein said aspect ratio is determined by means of the principal values of said permeability tensor.
5. A method as claimed in any one of the previous claims, wherein a distance is defined between said boundaries between zones, so as to give an equal weight to each zone in terms of pressure difference recorded in each zone under permanent flow regime conditions.
6. A method as claimed in claim 5, wherein said distance is defined by setting the lengths of one of the two axes of two successive ellipses at values in geometric progression of constant ratio.
7. A method as claimed in any one of the previous claims, wherein three zones are constructed, a first zone (ZNS) containing the well, within which no simplification of said image is performed, a second zone (ZP) in contact with said first zone, within which a first simplification of said image is performed, and a third zone (ZL) in contact with said second zone, within which a second simplification of said image is performed, said second simplification being more significant than said first simplification.
8. A method as claimed in claim 7, wherein said second and third zones are divided into sub-zones by applying the following stages:

   - said second zone is divided into a number of sub-zones equal to a number of blocks of cells present in said zone, a block of cells designating a vertical pile of cells,

   - said third zone is divided by carrying out the following stages:

   ∘ dividing every degree the boundary of said third zone, defining 360 arcs,

   ∘ defining a sub-zone by connecting end points of each of said arcs to the centre of the ellipse forming the boundary,

   ∘ for each one of said sub-zones, calculating an equivalent fracture permeability tensor from which an orientation of the flows in said sub-zone is determined,

   ∘ comparing the equivalent fracture permeability values and the flow orientation between neighbouring sub-zones, and

   ∘ grouping neighbouring sub-zones together into a single sub-zone when a difference between the permeability values is below a first threshold and when a difference between the flow orientations is below a second threshold.
9. A method as claimed in any one of the previous claims, wherein said image is simplified by carrying out the following stages:

   - constructing a fracture network (RFE) equivalent to said image, by means of a so-called Warren and Root representation, wherein the network is characterized by fracture spacings (s₁ ^fin, s₂ ^fin) in two orthogonal directions of principal permeability, by a fracture opening parameter (e^fin), by fracture conductivities (C_f1 ^fin and C_f2 ^fin) and a permeability (k_m ^fin) of a matrix medium between fractures,

   - simplifying said equivalent fracture network (RFE) by means of a network fracture spacing coefficient (G) whose value is less than a value G_max-zone defined on each of said zones in order to guarantee sufficient connectivity between simplified zones and non-simplified zones.
10. A method as claimed in claim 9 wherein, for a given sub-zone, value G_max-zone is equal to: $$G_{\max -\mathit{zone}}=\frac{\mathit{DLM}}{6.\mathit{Max}\left(s_1^{\mathit{fin}},s_2^{\mathit{fin}}\right)}$$

    

    with:

    - DLM : minimum lateral dimension of the given sub-zone;

    - s1^fin, s2^fin : fracture spacings in the so-called Warren and Root representation.
11. A method as claimed in any one of the previous claims, wherein the following stages are carried out:

    - repeating stage a) while modifying said statistical parameters (PSF) so as to minimize a difference between a well test result and a well test simulation result from said simplified image,

    - associating with each one of said cells at least one equivalent permeability value and an average opening value for said fractures, said values being determined from said modified statistical parameters (PSF),

    - simulating fluid flows in said reservoir by means of a flow simulator, of the equivalent permeability values and the average opening values of said fractures associated with each one of said cells,

    - selecting a production scenario allowing the reservoir production to be optimized by means of said fluid flow simulation, and

    - developing said reservoir according to said scenario allowing the reservoir production to be optimized.

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