EP2253797B1 - Method of exploitation of a porous medium by modelling of fluid flows - Google Patents

Description

The present invention relates to the field of the exploitation of underground environments. The invention makes it possible in particular to improve the injectivity and the productivity of wells drilled through a porous medium, such as a hydrocarbon deposit or a geological CO ₂ storage tank.

Presentation of the prior art

Local phenomena that can occur around a well, such as damage, have a huge impact on the injectivity or productivity of a well. In the petroleum industry, it is very important to predict injectivity or productivity, especially when there are formation alterations in the vicinity of a well, which change the injection or production capacity of the well.

For years, substantial efforts have been made using experimental techniques, in the laboratory, or numerical modeling methods, in order to take into account these local phenomena around wells, and their impacts on injectivity or productivity.

The numerical methods which make it possible to model the flow of fluids within a well (injectivity and productivity of a well), involve the construction of two distinct models: the model of reservoir (“ reservoir model ”) and the model d ' near-wellbore model'.

• un maillage, dit « maillage de réservoir », constitué d'un ensemble de mailles discrétisant spatialement le réservoir.
• un simulateur d'écoulement. On appelle simulateur d'écoulement, un logiciel permettant de modéliser les écoulements de fluides au sein d'un milieu poreux, grâce au maillage de réservoir. Ce logiciel permet ainsi de simuler des données/propriétés dynamiques des fluides (eau, huile, gaz) : pression, flux (quantité de matière traversant une surface), saturation, débits, concentrations. Par exemple, un simulateur permet d'estimer, pour un scénario d'exploitation de puits donné (scénario de production ou scénario d'injection), et pour un intervalle de temps donné : les saturations en eau, huile et gaz, les débits d'huile, de gaz et d'eau, le water-cut (fraction d'eau dans la production liquide), le GOR (rapport de gaz et huile dans la production), les concentrations en polymère absorbé sur la roche du milieu poreux, les débits d'injection de polymère, si une solution de polymère est injectée dans le réservoir par l'intermédiaire de puits d'injection, ...

A tank model has two elements:

• a mesh, called "reservoir mesh", consisting of a set of meshes spatially discretizing the reservoir.
• a flow simulator. We call flow simulator, a software allowing to model the flows of fluids within a porous medium, thanks to the tank mesh. This software thus makes it possible to simulate dynamic data / properties of fluids (water, oil, gas): pressure, flow (amount of material crossing a surface), saturation, flow rates, concentrations. For example, a simulator makes it possible to estimate, for a given well operating scenario (production scenario or injection scenario), and for a given time interval: water, oil and gas saturations, flow rates d oil, gas and water, water cut (fraction of water in liquid production), GOR (ratio of gas and oil in production), concentrations of polymer absorbed on the rock of the porous medium, the injection rates of polymer, if a polymer solution is injected into the tank via injection wells, ...

• un maillage, dit « maillage d'abords de puits », constitué d'un ensemble de mailles discrétisant spatialement le puits et ses abords. Ses abords appartiennent donc au milieu poreux dans lequel le puits est foré.
• un simulateur d'écoulement, permettant de simuler, grâce au maillage d'abords de puits, des données/propriétés dynamiques des fluides (eau, huile, gaz).

A well approach model has two elements:

• a mesh, called “well access grid”, consisting of a set of meshes spatially discretizing the well and its surroundings. Its surroundings therefore belong to the porous medium in which the well is drilled.
• a flow simulator, allowing the simulation of dynamic data / properties of fluids (water, oil, gas) thanks to the mesh around the wells.

These two types of models, reservoir and well approach, are generally autonomous and decoupled. As local phenomena are generally limited to the immediate vicinity of the well (from a few centimeters to a few meters), very small meshes are necessary for first meshing of wells, while larger meshes are used for reservoir meshes in order to speed up the calculations.

Techniques are known which make it possible to use a single reservoir flow simulator for these two meshes. One can for example use the technique of so-called “hybrid” meshes, combining in a single mesh, meshes for the mesh of the reservoir and meshes for a locally refined mesh around the well. One associates with this type of mesh a single flow simulator to better take into account the behaviors of flows in the vicinity of the well in a simulation of the field.

But the simultaneous simulations of the flows in the reservoir, which require a very large number of meshes, and in the regions close to the well with small meshes, which require small steps of time to ensure the stability of computation, pose numerical problems, in particular, the problem of computation time (CPU time).

Thus, domain decomposition techniques have been developed, described for example in GAIFFE, S. "Hybrid Meshes and Domain Decomposition for the Modeling of Petroleum Reservoirs", Doctoral Thesis, University of Paris 6, 2000 And windowing techniques ( "windowing") as described for example in the following document: MLACNIK, MJ and HEINEMANN, ZE "Using well windows in full field reservoir simulation ", paper SPE 66371 presented at the SPE Reservoir Simulation Symposium, Houston, TX, USA, February 2001 .

However, certain delicate points such as convergence, stability or computation time pose problems during industrial applications. In addition, the domain decomposition method is not always "conservative" (deterioration of the mass balance in the model as a function of time), which is not suitable for practical use of the method. In addition, all these techniques require reformulating the mathematical equations and the boundary conditions developed in the flow simulators, and new developments are necessary to integrate the solutions near and far from the well in a single model, the realization of which is a long and difficult work.

The method according to the invention

• a- on simule les écoulements de fluides au sein du milieu au moyen du premier simulateur sur un intervalle de temps défini entre des temps T₀ et T₁, et on déduit des conditions aux limites mises à jour pour le second simulateur par une interpolation linéaire des résultats du premier simulateur entre les temps T₀ et T₁ ;
• b- on simule les écoulements de fluides aux abords du puits au moyen du second simulateur sur le même intervalle de temps, en utilisant les conditions aux limites mises à jour, et on en déduit des indices de productivité numériques mis à jour pour le premier simulateur en comparant des débits calculés par le premier simulateur et des débits calculés par le second simulateur ; et
• c- on modélise les écoulements de fluides au sein du milieu poreux pendant une période de temps entre T₀ et T_n, où T_n >T₁, en réitérant les étapes a et b, pour des intervalles de temps successifs compris entre T₀ et T_n.

An object of the invention relates to a method implemented by computer for modeling fluid flows within an underground porous medium traversed by at least one well, with a view to the exploitation of said medium produced by injecting one of said fluids into said medium via at least one of said wells, and / or by producing one of said fluids present in said medium by means of at least one of said wells. The method includes the use of a first flow simulator making it possible to simulate the flow of fluids within the porous medium from digital productivity indices relating fluid pressures to fluid flow rates, and the use a second flow simulator to simulate the flow of fluids around the well from boundary conditions. The method includes the following steps:

• a- one simulates the flows of fluids within the medium by means of the first simulator on a time interval defined between times T ₀ and T ₁ , and one deduces from the boundary conditions updated for the second simulator by a linear interpolation results of the first simulator between times T ₀ and T ₁ ;
• b- we simulate the flow of fluids around the well by means of the second simulator over the same time interval, using the updated boundary conditions, and we deduce numerical productivity indices up to date for the first simulator by comparing flows calculated by the first simulator and flows calculated by the second simulator; and
• c- the fluid flows within the porous medium are modeled for a period of time between T ₀ and T _n , where T _n > T ₁ , by repeating steps a and b, for successive time intervals between T ₀ and T _n .

According to the invention, each successive time interval can have a length which is a function of a time step of calculation of the first flow simulator and of a time step of the second flow simulator. For example, each successive time interval can have a length equal to a time step of the first flow simulator.

According to one embodiment, the fluid flows within the medium are simulated by means of the first simulator on a first mesh discretizing the porous medium into a set of meshes, and the fluid flows around the well are simulated by means of the second simulator on a second mesh discretizing the well and its surroundings into a set of meshes. This second mesh is generated by constraining meshes situated on the edge of the second mesh, so that their interfaces coincide with the interfaces of the meshes of the first mesh.

In the case where multiphase flows are modeled, digital productivity index multipliers are updated, instead of the digital productivity indices themselves, for each phase, by comparing flow rates per phase calculated by the first simulator and flow rates per phase calculated by the second simulator.

1. a- on choisit un scénario d'exploitation du réservoir poreux ;
2. b- on associe au maillage de réservoir un premier simulateur d'écoulement permettant de simuler l'écoulement des fluides au sein du réservoir, à partir d'au moins les données suivantes: le scénario de production, des données d'entrées relatives au fluide et au réservoir, des indices de productivité numériques permettant de relier des pressions à des débits, des conditions aux limites ;
3. c- on associe au maillage d'abord de puits un second simulateur d'écoulement pour simuler l'écoulement des fluides aux abords du puits, à partir d'au moins les données suivantes : des données d'entrées relatives au fluide et au réservoir, des conditions aux limites ;
4. d- on modélise les écoulements de fluides au sein du milieu poreux et aux abords du puits, au moyen de la méthode selon l'une des revendications 1 à 7 ; et
5. e- on modifie le scénario d'exploitation et l'on répète l'étape d, jusqu'à obtenir un scénario d'exploitation optimal.

The invention also relates to a method for exploiting an underground porous reservoir by means of at least one well passing through it, at least one fluid circulating between the reservoir and the well. According to this method, we acquire data relating to the geometry of the porous reservoir, from which we construct a discretization of the reservoir in a set of meshes, called reservoir mesh, and we construct a discretization of the well and its surroundings in a set of mesh, called first well mesh. This method also includes the following steps:

1. a- a porous reservoir operating scenario is chosen;
2. b- a first flow simulator is associated with the reservoir mesh, making it possible to simulate the flow of fluids within the reservoir, from at least minus the following data: the production scenario, input data relating to the fluid and the reservoir, numerical productivity indices making it possible to link pressures to flow rates, boundary conditions;
3. c- the second well simulator is associated with a second flow simulator to simulate the flow of fluids around the well, from at least the following data: input data relating to the fluid and to the reservoir , boundary conditions;
4. d- the fluid flows within the porous medium and around the well are modeled, by means of the method according to one of claims 1 to 7; and
5. e- the operating scenario is modified and step d is repeated, until an optimal operating scenario is obtained.

According to this operating method, damage to the well can be taken into account by a drilling fluid by modeling an invasion of the porous reservoir by the drilling fluid in steps d and e.

The operating scenario can include an injection of a polymer solution through the well, and we can then model the flows to prevent water coming in. The operating scenario can also include an injection of an acid solution into the well, and we can then model the flows to assess the impact of acid stimulation.

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

Summary presentation of the figures

• The figure 1 illustrates the main steps of the method according to the invention.
• The figure 2 shows the coupling diagram between the reservoir model and the well approach model. The T axis corresponds to time.
• The figure 3 shows the coarse mesh used for field simulation in a reservoir model.
• The figure 4 shows the fine mesh used to simulate the detailed phenomena of flows around wells in an approaching well model.
• The figure 5 shows the two meshes used in the coupling. The figure on the left represents the reservoir mesh for the field simulation, and the figure on the right represents the mesh in the vicinity of wells in the approach well model. The meshes at the edges (in gray) in the well approach model coincide with the meshes of the same color in the reservoir mesh.
• The figure 6 shows the coarse mesh for the simulation of the field in the case of damage by the drilling fluid. Γ _x- and Γ _{x +} correspond to two borders of this mesh in the direction x , and Γ _y- and Γ _{y +} correspond to two borders of this mesh in the direction y .
• The figure 7 shows the locally refined mesh around the well to simulate the reference solution in the case of damage by drilling fluid.
• The Figures 8A and 8B show the coupling meshes to simulate the damage by the drilling fluid. The left mesh ( figure 8A ) corresponds to the mesh for the field simulation, and the right mesh ( figure 8B ) corresponds to the mesh in the well approach model.
• The figure 9 shows relative permeabilities during drilling and production. The X axis is saturation without units. The Y axis is relative permeability. There is no unity. The "krw drilling" curve is the relative water permeability curve during drilling. The "kro drilling" curve is the relative oil permeability curve during drilling. The "krw production" and "kro production" curves are relative permeability curves of water and oil respectively, during production.
• The Figures 10A and 10B compare the invasion volume of the drilling fluid simulated by the coupling method with that of the reference solution. The figure 10A shows the invasion rate during drilling. The X axis is the time expressed in days. The Y axis is the flow expressed in m ³ / day. The curve R is the reference solution. The CM curve is the simulation with the coupling method. The figure 10B shows the invasion volume as a function of time. The X axis is time in days. The Y axis is the flow in m ³ / d. The curve R is the reference solution. The CM curve is the simulation with the coupling method.
• The figure 11 compare oil production rates. The X axis is time in days. The Y axis is the flow in m ³ / d. The curve R is the reference solution. The CM curve is the simulation with the coupling method. Curve S is the simulation without damaging the well. The CK curve is the simulation with damage only by drilling mud (“cakes”).
• The figure 12 shows the permeabilities in layer 3 on the coarse mesh of the reservoir model in the application to the prevention of water inflow. There is an injector and a producer.
• The figure 13 shows the refined mesh around the producer to simulate the reference solution.
• The Figures 14A and 14B show the meshes of the coupling. The figure 14A is the mesh of the reservoir model, and the figure 14B is the mesh of the well approach model.
• The figure 15 shows the polymer injection rate in the treated well. The X axis is time in days. The Y axis is the flow in m ³ / d. The curve R is the reference solution. The CM curve is the simulation of the reservoir model with the coupling method. Curve S is the direct simulation with the reservoir model without coupling. The NW curve is the simulation of the well approach model with coupling.
• The Figures 16A to 16E show the polymer injection rate in the layers. The figure 16A shows the rate of polymer injection into layer 1. The figure 16B shows the rate of polymer injection into layer 2. The figure 16C shows the rate of polymer injection into layer 3. The figure 16D shows the rate of polymer injection into layer 4. The figure 16E shows the rate of polymer injection into layer 5. The X axis is time in days. The Y axis is the flow in m ³ / d. The curve R is the reference solution. The CM curve is the simulation of the reservoir model with the coupling method. Curve S is the direct simulation with the reservoir model without coupling. The NW curve is the simulation of the well approach model with (obviously) the coupling.
• The figure 17 shows the producer's oil flow. The X axis is time in days. The Y axis is the oil flow in m ³ / d. The curve R is the reference solution. The CM curve is the simulation of the reservoir model with the coupling method. Curve S is the direct simulation with the reservoir model without coupling.
• The figure 18 shows the producer's water flow. The X axis is time in days. The Y axis is the water flow in m ³ / d. The curve R is the reference solution. The CM curve is the simulation of the reservoir model with the coupling method. Curve S is the direct simulation with the reservoir model without coupling.

• The figure 19 shows the producer's water-cut curve. The X axis is the time expressed in days. The Y axis is water-cut, without unit. The curve R is the reference solution. The CM curve is the simulation of the reservoir model with the coupling method. Curve S is the direct simulation with the reservoir model without coupling.
• The figures 20A to 20D show a water saturation map at 1100 days. The figure 20A corresponds to the reference solution in the field. The figure 20B corresponds to the map obtained with the reservoir model in coarse mesh with coupling. The figure 20C shows the reference solution in the vicinity of the well. The figure 20D shows the water saturation in the vicinity of a well simulated with the near well model.
• The Figures 21A to 21D show a pressure card at 1100 days. The figure 21A shows the reference solution in the field. The figure 21B shows the solution obtained with the reservoir model in coarse mesh with coupling. The figure 21C shows the reference solution in the vicinity of the well. The figure 21D shows the solution with the first well model.

Detailed description of the method

The invention relates to a method for exploiting an underground porous medium, by injecting a fluid into the medium via at least one well, and / or by producing a fluid present in the medium by means of at least one well. The method includes a modeling of the flows of fluids in the system constituted by the porous medium (reservoir and surroundings of the wells) and the well. It is therefore, in particular, to model the injectivity or the productivity of wells passing through a porous medium.

1. 1. Choix d'un scénario d'exploitation du milieu poreux, scénario de production et/ou scénario d'injection (SCE) ;
2. 2. Choix d'un simulateur d'écoulement (RSIM) compatible avec un maillage de réservoir donné, et choix d'un simulateur d'écoulement (NWSIM) compatible avec un maillage d'abord de puits donné ;
3. 3. Au moyen d'un couplage entre les deux simulateurs (EST_CAL, et figure 2), estimation des écoulements de fluides, c'est-à-dire, par exemple, du volume injecté ou du volume produit, sur un intervalle de temps donné, et
4. 4. Détermination du scénario d'exploitation optimal par modification du scénario d'exploitation et répétition de l'étape 3 (OPT).

The figure 1 illustrates the main steps of the method:

1. 1. Choice of a porous environment exploitation scenario, production scenario and / or injection scenario ( SCE );
2. 2. Choice of a flow simulator ( RSIM ) compatible with a given reservoir mesh, and choice of a flow simulator ( NWSIM ) compatible with a given well first mesh;
3. 3. By means of a coupling between the two simulators ( EST_CAL, and figure 2 ), estimation of fluid flows, that is to say, for example, of the volume injected or of the volume produced, over a given time interval, and
4. 4. Determination of the optimal operating scenario by modifying the operating scenario and repeating step 3 ( OPT ).

1. Choice of a porous medium exploitation scenario

It can be a scenario of production of hydrocarbons contained in the porous medium (reservoir), or a scenario of injection of acid gas, such as CO2, in an underground reservoir for the storage of acid gas . A scenario is characterized by the position of the wells, the recovery or injection method, the injection and / or production flow rates and duration, the operating conditions in these wells such as the flow rate or the bottom pressure.

In the context of production, the reservoir engineer chooses a production process, for example the recovery process by water injection, of which it then remains to specify the optimal implementation scenario for the reservoir considered. The definition of an optimal scenario consists, for example, in fixing the number and the location (position and spacing) of the injector and producer wells in order to best take into account the impact of heterogeneities within the reservoir, for example permeability channels, fractures, etc., on the progression of fluids within the reservoir. Depending on the scenario chosen, and the geometric representation of the reservoir, we are then able to simulate the expected production of hydrocarbons, using the tool well known to specialists: the flow simulator.

The choice of a scenario, by the definition of multiple technical characteristics, is a step well known to specialists.

2. Choice of flow simulators

To choose a flow simulator, it is necessary to know the type of mesh on which the simulator must operate.

Construction of reservoir (RM) and first well (NWM) meshes

The “reservoir mesh” consists of a set of meshes spatially discretizing the reservoir (porous medium + well). An example of a tank mesh is illustrated on the figure 3 , this mesh is coarse. Some meshes correspond to the “porous medium” part, others correspond to the part where the well is drilled. One speaks for these last of meshs of well of the mesh of reservoir.

The “well access grid” consists of a set of meshes spatially discretizing the well and its surroundings. An example of first well mesh is illustrated on the figure 4 , this mesh is fine to simulate the detailed phenomena around the well. Its surroundings therefore belong to the porous medium in which the well is drilled. Some meshes correspond to the “porous medium” part, others correspond to the “well” part. One speaks for these last of mesh of wells of the mesh of approaches of wells.

The generation of these meshes, whether it is the reservoir mesh or the well mesh first, is a step well known to the specialist, who knows many methods for building them. For example, techniques for building a mesh of wells first are described in the following documents:
Boe, O., Flynn, J. and Reiso, E., "On Near Wellbore Modeling and Real Time Reservoir Management", SPE 66369, Houston, Texas, USA, 11-14 Feb. 2001 .

Methods are also known for constructing reservoir meshes, from data relating to the geometry of the medium (seismic, logging, etc.), described for example in the following documents:
Flandrin, N., Bennis, C. and Borouchaki, H., "3D Hybrid Mesh Génération for Reservoir Simulation", ECMOR, Canne, France, August 30 - September 2, 2004 .

Definition of reservoir and well area models

To define a reservoir model, it is necessary to associate a flow simulator with the reservoir mesh. Similarly, to define a first well model, it is necessary to associate a flow simulator with the first well mesh.

• caractéristiques géométriques du réservoir, caractéristiques de la roche, caractéristiques des fluides en place et des fluides injectés (masse volumique, viscosité), courbes de perméabilités relatives, courbes de pression capillaire, saturations initiales en fluides,...
• conditions aux limites du domaine simulé et aux puits où sont injectés ou produits des fluides. Les conditions aux limites, sont les valeurs de données dynamiques, telles que la pression, le débit ou le flux, les saturations en fluides, sur les bords du maillage ou dans les mailles qui forment les bords du maillage de réservoir ou d'abord de puits. Un exemple de conditions aux limites peut être : un flux nul sur tous les bords du maillage, ou des saturations et pressions imposées sur les mailles aux bords du maillage.
• éventuellement des Indices de Productivité numériques (IP). La connexion entre la pression dans les mailles traversées par un puits et les pressions dans le puits lui-même est réalisée à l'aide d'un Indice de Productivité numérique (IP). L'IP numérique peut être calculé par une formule analytique dans le code ou donné par l'utilisateur du logiciel (le simulateur). En général, le simulateur calcule un IP numérique par une formule analytique au début de simulation. Mais, si l'utilisateur donne un IP numérique dans le jeu de donnée d'entrée, c'est l'IP numérique de l'utilisateur qui est prise en compte dans la simulation.

As is known to those skilled in the art, in order to function, a flow simulator needs certain data, called input data:

• geometric characteristics of the reservoir, characteristics of the rock, characteristics of the fluids in place and the fluids injected (density, viscosity), relative permeability curves, capillary pressure curves, initial saturations in fluids, ...
• boundary conditions of the simulated domain and wells where fluids are injected or produced. The boundary conditions, are the values of dynamic data, such as the pressure, the flow or the flow, the saturations in fluids, on the edges of the mesh or in the meshes which form the edges of the mesh of reservoir or first of well. An example of boundary conditions can be: a zero flux on all the edges of the mesh, or saturations and pressures imposed on the meshes at the edges of the mesh.
• possibly digital Productivity Indices (IP). The connection between the pressure in the cells crossed by a well and the pressures in the well itself is made using a Digital Productivity Index (IP). The digital IP can be calculated by an analytical formula in the code or given by the user of the software (the simulator). In general, the simulator calculates a numerical IP by an analytical formula at the start of the simulation. However, if the user gives a digital IP in the input data set, the user's digital IP is taken into account in the simulation.

According to the invention, it is possible to use any type of flow simulator, whether for the reservoir model, or the well model first. Indeed, an object of the invention relates to a coupling method, which makes it possible to couple in a very simple way a reservoir model, for the simulation of the reservoir, and a well model first, which is an autonomous model for simulating the phenomena detailed around the well.

Regarding the tank model simulator, it can be the Puma ^Flow ® software (IFP, France) for example.

Concerning the simulator of the first well model, we can use the one described in the following document: DING, Y., RENARD, G.: "Evaluation of Horizontal Well Performance after Drilling Induced Formation Damage", J. of Energy Resources Technology, Vol. 127, Sept., 2005 .

3. Estimation of the volume of fluid displaced over a given time interval

It is a question here of modeling the injectivity or the productivity of a well crossing the porous medium, and allowing the exploitation of this medium. This modeling is done over a given time interval, D = [ T ₀ ; T _n ] . For example, we model the behavior of the medium + well system over 20 years, considering the previously chosen operating scenario.

The technique used here consists in coupling between the two flow simulators.

A coarse mesh is often used for the reservoir model, and a fine mesh is usually necessary to simulate the detailed phenomena around the well. The figure 5 shows the two meshes used in the coupling. The figure on the left represents the reservoir mesh for the field simulation, and the figure on the right represents the mesh in the vicinity of wells in the approach well model. The meshes at the edges (in gray) in the well approach model coincide with the meshes of the same color in the reservoir mesh. The cross indicates the location of the well.

The time steps used in the first well model are generally much smaller than those of the reservoir model. The reservoir model is mainly used to simulate flows in the entire reservoir.

• 3a. On initialise les modèles.
  • on initialise le modèle de réservoir (RINIT) en affectant aux mailles du maillage de réservoir des valeurs de porosité, perméabilité, pression et saturations en fluide. L'initialisation comporte également la définition de conditions aux limites du modèle de réservoir. Ces conditions peuvent être définies par un flux nul (pas d'échange vers l'extérieur du domaine) ou par un flux ou une pression imposé sur les bords extérieurs des mailles de bord du maillage du modèle de réservoir (échange avec l'extérieur). Les conditions opératoires dans ces puits, telles que le débit ou la pression de fond, sont imposées sous forme d'un historique d'injection pour les injecteurs, et d'un historique de production dans les producteurs.
  • on initialise le modèle d'abord de puits (NWINIT) en affectant aux mailles du maillage d'abord de puits des valeurs de porosité, perméabilité, pression et saturations en fluide. Cette affectation est réalisée en utilisant des techniques de mise à l'échelle des résultats du modèle de réservoir. Ces techniques étant connues des spécialistes. L'initialisation comporte également la définition de conditions aux limites du modèle d'abord de puits. Ces conditions peuvent également être définies au moyen des résultats du modèle de réservoir.
• 3b. On définit au moins un pas de temps, noté ΔT, pour échanger des données dynamiques entre le modèle de réservoir et le modèle d'abord de puits, pendant la modélisation sur l'intervalle de temps D.
  Ce pas de temps ΔT peut être choisi en fonction du pas de temps ΔTR du simulateur d'écoulement du modèle de réservoir, et le pas de temps ΔTNW du simulateur d'écoulement du modèle d'abord de puits (ΔTR>ΔTNW).
  Théoriquement, ΔT doit être le plus petit possible pour assurer la convergence des solutions dans les deux modèles. Mais généralement l'utilisation du pas de temps utilisé pour la simulation du modèle de réservoir est suffisante. Mais, du point de vue pratique, on a parfois besoin de réaliser une simulation d'abord de puits de façon autonome plus longtemps. Ceci se traduit par une réduction de la fréquence de couplage. C'est pourquoi, selon la méthode, le pas de temps ΔT pour les échanges de données entre le modèle de réservoir et le modèle d'abord de puits est un paramètre ajustable.
  Selon un mode de réalisation, le pas de temps ΔT peut varier sur l'intervalle de temps D. On peut par exemple utiliser un premier pas de temps entre T₀ et T_i , et un second pas de temps entre T_i et T_n. Un exemple d'une telle application est illustré ci-après. Sur la figure 2, on note RSIM(T₁ ) une simulation du simulateur de réservoir effectuée entre T₀ et T₁, et NWSIM(T₁ ) une simulation du simulateur d'abord de puits effectuée entre T₀ et T₁ .
• 3c. On réalise une simulation d'écoulements avec le modèle de réservoir entre le temps T₀ et le temps T₁ = T₀ + ΔT.
  Les résultats de cette simulation sont :
  • la pression et les saturations en fluides à la fin du pas de temps dans chaque maille du maillage de réservoir, en particulier dans les mailles qui sont communes avec les mailles du maillage d'abord de puits et qui vont servir de conditions aux limites du modèle d'abord de puits ;
  • les débits en fluide (eau, huile, gaz) et les pressions dans les puits d'injection et de production.
• 3d. On met à jour les conditions aux limites (MAJCL) du modèle d'abord de puits en utilisant les résultats de la simulation d'écoulements réalisée avec le modèle de réservoir entre T₀ et T₁ (étape 3c).
  Les conditions aux limites, sont les valeurs de données dynamiques, telles que la pression ou le flux, les saturations, dans les mailles qui forment les limites du maillage de réservoir ou d'abord de puits. Selon un exemple, les conditions aux limites sont définies par un flux nul sur tous les bords du maillage des abords de puits, et une porosité très grande (1000000, par exemple) dans toutes ces mailles.
  Ainsi, au cours de cette étape, on utilise les résultats du simulateur d'écoulement du modèle de réservoir, pour déterminer des valeurs que l'on impose comme conditions aux limites pour le simulateur d'écoulement du modèle d'abord de puits au temps T₀.
  Les conditions aux limites peuvent être calculées à chaque pas de temps du modèle d'abord de puits en interpolant linéairement les résultats de simulation du modèle de réservoir entre T₀ et T₁.
• 3e. On réalise une simulation d'écoulements au voisinage du puits avec le modèle d'abords de puits entre le temps T₀ et le temps T₁, avec les conditions aux limites mises à jour à l'étape 3d.
  Les résultats de cette simulation sont, au moins :
  • la pression et les saturations en fluides à la fin du pas de temps dans chaque maille du modèle d'abords de puits;
  • les débits en fluide (eau, huile, gaz) et les pressions dans le puits d'injection ou de production selon du type de puits modélisé dans le modèle d'abords de puits.
  
  Ces résultats permettent de déterminer un Indice de Productivité numérique (IP).
• 3f. La connexion entre la pression dans les mailles traversées par un puits et les pressions dans le puits lui-même est réalisée à l'aide d'un Indice de Productivité numérique (IP). Les formules de Peaceman sont en général utilisées pour calculer cet indice. On met alors à jours les indices de productivité numériques (MAJIP) du modèle de réservoir, en utilisant les résultats de la simulation d'écoulements réalisée avec le modèle d'abord de puits entre T₀ et T₁ . En fait, si à la fin de la simulation, au temps T₁, les résultats simulés au puits avec le modèle d'abord de puits et avec le modèle de réservoir ne sont pas les mêmes, les indices de productivité numériques dans le modèle de réservoir sont modifiés pour ajuster les résultats de la simulation du modèle de réservoir à ceux du modèle d'abord de puits.
• 3g. On répète les étapes 3c (éventuellement 3b) à 3f, avec un nouvel intervalle de temps (de T₁ à T₂ , puis de T₂ à T₃, ..., puis de T_n-1 à T_n )

The time T ₀ is the time at which the coupling begins. The coupling algorithm in a general framework comprises the following steps, illustrated on the figure 2 :

• 3a. We initialize the models.
  • the reservoir model ( RINIT ) is initialized by assigning to the mesh of the reservoir mesh values of porosity, permeability, pressure and saturations in fluid. The initialization also includes the definition of boundary conditions of the reservoir model. These conditions can be defined by a null flow (no exchange towards the outside of the field) or by a flow or a pressure imposed on the external edges of the meshs of edge of the mesh of the tank model (exchange with the outside) . The operating conditions in these wells, such as the flow rate or the bottom pressure, are imposed in the form of an injection history for the injectors, and a production history in the producers.
  • one initializes the model of well first ( NWINIT ) by assigning to the meshes of the mesh first of well values of porosity, permeability, pressure and saturations in fluid. This assignment is made using techniques for scaling the results of the reservoir model. These techniques are known to specialists. The initialization also includes the definition of boundary conditions of the first well model. These conditions can also be defined using the results of the reservoir model.
• 3b. At least one time step, denoted ΔT, is defined to exchange dynamic data between the reservoir model and the well model first, during the modeling over the time interval D.
  This time step ΔT can be chosen as a function of the time step ΔTR of the flow simulator of the reservoir model, and the time step ΔTNW of the flow simulator of the first well model ( ΔTR > ΔTNW ) .
  Theoretically, ΔT must be as small as possible to ensure the convergence of the solutions in the two models. But generally the use of the time step used for the simulation of the reservoir model is sufficient. But, from a practical point of view, there is sometimes a need to perform a well simulation first of all independently for longer. This results in a reduction in the coupling frequency. This is why, according to the method, the time step ΔT for the data exchanges between the reservoir model and the first well model is an adjustable parameter.
  According to one embodiment, the time step ΔT can vary over the time interval D. One can for example use a first time step between T ₀ and T _i , and a second time step between T _i and T _n . An example of such an application is illustrated below. On the figure 2 , we denote RSIM ( T ₁ ) a simulation of the reservoir simulator performed between T ₀ and T ₁ , and NWSIM ( T ₁ ) a simulation of the well simulator first performed between T ₀ and T ₁ .
• 3c. A simulation of flows is carried out with the reservoir model between the time T ₀ and the time T ₁ = T ₀ + ΔT.
  The results of this simulation are:
  • the pressure and the saturations in fluids at the end of the time step in each mesh of the tank mesh, in particular in the meshes which are common with the meshes of the first well mesh and which will serve as boundary conditions of the first well model;
  • the fluid flows (water, oil, gas) and the pressures in the injection and production wells.
• 3d. The boundary conditions ( MAJCL ) of the first well model are updated using the results of the flow simulation performed with the reservoir model between T ₀ and T ₁ (step 3c).
  The boundary conditions, are the values of dynamic data, such as the pressure or the flow, the saturations, in the meshes which form the limits of the mesh of reservoir or initially of well. According to one example, the boundary conditions are defined by a zero flux on all the edges of the mesh around the wells, and a very large porosity (1000000, for example) in all these meshes.
  Thus, during this step, we use the results of the flow simulator of the reservoir model, to determine values that we impose as boundary conditions for the flow simulator of the first well model at time T ₀ .
  The boundary conditions can be calculated at each time step of the first well model by linearly interpolating the results of simulation of the reservoir model between T ₀ and T ₁ .
• 3rd. A flow simulation is carried out in the vicinity of the well with the well approach model between time T ₀ and time T ₁ , with the boundary conditions updated in step 3d.
  The results of this simulation are, at least:
  • the pressure and the saturations in fluids at the end of the time step in each mesh of the model of approaches of well;
  • fluid flow rates (water, oil, gas) and pressures in the injection or production well depending on the type of well modeled in the well approach model.
  
  These results make it possible to determine a Digital Productivity Index ( IP ).
• 3f. The connection between the pressure in the meshes crossed by a well and the pressures in the well itself is carried out using a Digital Productivity Index ( IP ) . Peaceman's formulas are generally used to calculate this index. We then update the digital productivity indices ( MAJIP ) of the reservoir model, using the results of the simulation of flows carried out with the first well model between T ₀ and T ₁ . In fact, if at the end of the simulation, at time T ₁ , the results simulated at the well with the first well model and with the reservoir model are not the same, the numerical productivity indices in the reservoir are modified to adjust the results of the simulation of the reservoir model to those of the first well model.
• 3g. We repeat steps 3c (possibly 3b) to 3f, with a new time interval (from T ₁ to T ₂ , then from T ₂ to T ₃ , ..., then from T _n-1 to T _n )

avec :

i

: numéro de maille de puits dans le maillage (de réservoir ou d'abord de puits)

p

: phase du fluide. Les phases p peuvent être l'eau, l'huile ou le gaz

Q_p,i

: débit de la phase p dans la maille de puits i du maillage (de réservoir ou d'abord de puits)

λ_p,i

: mobilité de la phase p dans la maille de puits i du maillage (de réservoir ou d'abord de puits); λ_p,i dépend essentiellement de la perméabilité relative et de la viscosité de la phase p

IP_i

: indice de productivité numériques dans la maille de puits i du maillage (de réservoir ou d'abord de puits)

P_p,i

: pression de la phase p dans la maille de puits i du maillage (de réservoir ou d'abord de puits)

P_wf,i

: pression dans le puits, au fond, au niveau du réservoir dans la maille de puits i du maillage (de réservoir ou d'abord de puits)

The digital productivity index is noted IP. It is generally used in the models of flows to connect the pressures to the flow in a mesh of wells of the mesh of reservoir or approaches to wells. $$Q_{p,i}={\lambda }_{p,i}.{\mathit{IP}}_i\left(P_{p,i}-P_{\mathit{wf},i}\right)\mathrm{is}{\mathit{IP}}_i=\frac{Q_{p,i}}{{\lambda }_{p,i}\cdot \left(P_{p,i}-P_{\mathit{wf},i}\right)}$$

with:

i

: well mesh number in the mesh (reservoir or first well)

p

: fluid phase. The phases p can be water, oil or gas

Q _{p, i}

: flow rate of phase p in the well mesh i of the mesh (reservoir or first of well)

λ _{p, i}

: mobility of phase p in the well mesh i of the mesh (reservoir or first of well); λ _{p, i} essentially depends on the relative permeability and the viscosity of the phase p

IP _i

: digital productivity index in the well mesh i of the mesh (reservoir or first of well)

P _{p, i}

: pressure of phase p in the well mesh i of the mesh (reservoir or first well)

P _{wf, i}

: pressure in the well, at the bottom, at the level of the reservoir in the well mesh i of the mesh (of reservoir or first of well)

The digital productivity index IP takes into account: the geometric effect of the well mesh i of the mesh, the permeability of the porous medium in the well mesh and a skin coefficient. A skin coefficient is a coefficient, well known to those skilled in the art, used to represent the damage of a well in a mesh.

avec :

i

: numéro de maille de puits dans le maillage de réservoir

j

: numéro de maille de puits dans le maillage d'abord de puits

W_i

: ensemble des mailles de puits du maillage d'abord de puits correspondant à un raffinement de la maille de puits i du maillage de réservoir

p

: phase du fluide. Les phases p peuvent être l'eau (w), l'huile (o) ou le gaz (g)

IP_r,i

: indice de productivité numériques dans la maille de puits i du maillage de réservoir, et utilisé dans le modèle de réservoir

P_nw,p,j

: pression de la phase p dans la maille de puits j du maillage d'abord de puits, calculée avec le modèle abord de puits

P_r,p,i

: pression de la phase p dans la maille de puits i du maillage de réservoir, calculée avec le modèle de réservoir

P_wf,j

: pression dans le puits au niveau du réservoir dans la maille de puits j du maillage d'abord de puits

IP_nw,j

: indice de productivité numériques dans la maille de puits j du maillage d'abord de puits, et utilisé dans le modèle d'abord de puits

Updating a digital productivity index IP at time T ₁ , can be done by comparing the simulated flows with the first well model and the reservoir model by the following formula: $${\mathit{IP}}_{r,i}\left({\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">T</figure-callout>}}_1\right)=\frac{{\displaystyle \underset{j\in W_i}{\Sigma }{\displaystyle \underset{p=w,o,g}{\Sigma }\left(P_{\mathit{nw},p,j}\left(T_1\right)-P_{\mathit{wf},\mathrm{<figure-callout\; id="1"\; label="j\; T"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">j\; </figure-callout>}}\left(T_{}\right)\left({}_1\right)\right){\mathit{IP}}_{\mathit{nw},j}}}}{{\displaystyle \underset{p=w,o,g}{\Sigma }\left(P_{r,p,i}\left(T_1\right)-P_{\mathit{wf},i}\left({\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">T</figure-callout>}}_1\right)\right)}}$$

with:

i

: well mesh number in the reservoir mesh

j

: well mesh number in the well mesh first

W _i

: set of well meshes of the first well mesh corresponding to a refinement of the well mesh i of the reservoir mesh

p

: fluid phase. The phases p can be water ( w ), oil ( o ) or gas ( g )

IP _{r, i}

: numerical productivity index in the well mesh i of the reservoir mesh, and used in the reservoir model

P _{nw, p, j}

: pressure of phase p in the well mesh j of the first well mesh, calculated with the well approach model

P _{r, p, i}

: pressure of phase p in the well mesh i of the reservoir mesh, calculated with the reservoir model

P _{wf, j}

: pressure in the well at the reservoir in the well mesh j of the first well mesh

IP _{nw, j}

: numerical productivity index in the well mesh j of the first well mesh, and used in the first well model

The variables IP _i , P _{nw, p, j} , P _{r, p, i} and P _{wf, j} are a function of time T.

avec :

• Q_nw,i : débit du fluide (unique phase) calculé avec le modèle d'abord de puits dans la section correspondant à la partie du puits dans la maille de puits i du maillage de réservoir
• Q_r,i : débit du fluide (unique phase) calculé avec le modèle de réservoir dans la même section, correspondant à la partie du puits dans la maille de puits i du maillage de réservoir
• IP_r,i (T ₁) et IP_r,i (T ₀) sont les indices de productivité numérique aux temps T₁ et T₀ respectivement, c'est-à-dire, avant et après la mise à jour.

For a problem of pressure P _wf imposed on the well, and in the monophasic case (we can remove the index p ), the above formula is equivalent to the following expression: $${\mathit{IP}}_{r,i}\left({\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">T</figure-callout>}}_1\right)=\frac{Q_{\mathit{nw},i}\left(T_1\right)}{Q_{r,i}\left({\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">T</figure-callout>}}_1\right)}{\mathit{IP}}_{r,i}\left({\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="imgf0006.png,imgf0007.png"\; state="\{\{state\}\}">T</figure-callout>}}_0\right)$$

with:

• Q _{nw, i} : flow rate of the fluid (single phase) calculated with the well model first in the section corresponding to the part of the well in the well mesh i of the reservoir mesh
• Q _{r, i} : fluid flow rate (single phase) calculated with the reservoir model in the same section, corresponding to the part of the well in the well mesh i of the reservoir mesh
• IP _{r, i} ( T ₁ ) and IP _{r, i} ( T ₀ ) are the digital productivity indices at times T ₁ and T ₀ respectively, that is, before and after the update.

et donc IP_r,i (T ₁) = IP _r,i (T ₀). This formula clearly shows that the updating of the numerical productivity index corresponds to the correction of the fluid flow rate of the reservoir model compared to the fluid flow rate of the well approach model: if the two models give the same result in terms debit then $\frac{Q_{\mathit{nw},i}\left(T_1\right)}{Q_{r,i}\left({\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">T</figure-callout>}}_1\right)}=1$

and therefore IP _{r, i} ( T ₁ ) = IP _{r , i} ( T ₀ ) .

4. Determination of the optimal exploitation scenario

By testing various scenarios, characterized for example by various respective locations of the injector and producer wells, and by simulating the production of hydrocarbons for each of them according to step 3, the optimal scenario can be selected. The optimal scenario is the scenario allowing to obtain an optimal production of the deposit within the framework of the production of a reservoir, or the scenario allowing to obtain the optimal injectivity in the deposit within the framework of injection of fluid in the tank (injection of water for improved production, or injection of acid gases).

To test various exploitation scenarios, the scenario selected in step 1 is modified (Δ SCE ), for example by modifying the location of a well.

We then optimize the exploitation of the deposit, by implementing, on the field, the production scenario thus selected.

According to the invention, it is entirely possible to couple a reservoir model with several models of wells first.

variants

According to a particular embodiment of the invention, step 2, during which the meshes are constructed, is modified.

The simulation, carried out using the reservoir model in step 3c, provides dynamic properties of fluids such as pressure or saturation in the period from T ₀ to T ₁ on all coarse meshes. However, the determination of the boundary conditions in step 3b requires the interpolation of the pressure or the flow on the edges of the well approach model. To reduce the errors in the interpolation, we can constrain, during the generation of mesh, the meshs of edge of the model approach of well so that they coincide with the interfaces of the meshs of the model of reservoir. In addition, the edge meshes in the approach well model are also constrained so that they coincide with meshes of the reservoir model ( figure 3 ). In this way, the transfer of dynamic data from the reservoir model to the approach well model is direct on these meshes. In the well approach model itself, the boundary conditions are of zero flux. In order to maintain the dynamic properties at the edges of the model, very high porosities

(1000000, for example) are assigned to the edge meshes. This type of boundary condition is consistent with most flow models, and its implementation is easy.

For certain problems, the changes in flow around the well are linked to multiphase flows. In this case, we can also update the digital productivity indices by phase. To do this, we reformulate the pressure / flow relationship, by introducing a coefficient, called the productivity index multiplier: $$Q_{p,i}={\lambda }_{p,i}.M_{p,i}.{\mathit{IP}}_i.\left(P_{p,i}-P_{\mathit{wf},i}\right)$$

M _{p, i} is the multiplier of the productivity index for phase p in the mesh of wells i .

avec :

Q_r,p,i (T ₁)

: débit de la phase p calculé par le modèle de réservoir dans la maille de puits i du maillage de réservoir au temps T₁

Q_nw,p,i (T ₁)

: débit de la phase p calculé par le modèle d'abord de puits dans le même secteur du puits (voir l'ensemble W_i ) au temps T₁

M_p,i (T ₀)

: multiplicateur d'indice de productivité numérique pour la phase p dans le modèle de réservoir aux temps T₀ (avant mise à jour du modèle)

M_p,i (T ₁)

: multiplicateur d'indice de productivité numérique pour la phase p dans le modèle de réservoir aux temps T₁ (après mise à jour du modèle)

If the physics around the well are related to multiphase flows, we can update the multiplier of the IP instead of the IP itself, using the following formula: $$M_{p,i}\left({\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="imgf0001.png,imgf0006.png"\; state="\{\{state\}\}">T</figure-callout>}}_1\right)=\frac{Q_{\mathit{nw},p,i}\left(T_1\right)}{Q_{r,p,i}\left(T_1\right)}{M}_{p,i}\left({\mathrm{<figure-callout\; id="0"\; label="T"\; filenames="imgf0006.png,imgf0007.png"\; state="\{\{state\}\}">T</figure-callout>}}_0\right)$$

with:

Q _{r, p, i} ( T ₁ )

: flow rate of phase p calculated by the reservoir model in the well mesh i of the reservoir mesh at time T ₁

Q _{nw, p, i} ( T ₁ )

: flow rate of phase p calculated by the first well model in the same sector of the well (see the set W _i ) at time T ₁

M _{p, i} ( T ₀ )

: numerical productivity index multiplier for phase p in the reservoir model at times T ₀ (before updating the model)

M _{p, i} ( T ₁ )

: numerical productivity index multiplier for phase p in the reservoir model at times T ₁ (after updating the model)

Application examples

The coupling method according to the invention can be used to model various detailed phenomena around the well, such as, damage by drilling or completion fluid, acid stimulation, non-Darcean flow around the well, the problem of condensate gas, asphaltene deposition, damage by injection of CO ₂ , prevention of water or gas, sand, mineral deposits, the impact of completions, etc. Here, we present in particular an example of application for the damage of the oil formation by the drilling fluid during the drilling of the well, and an example of application for the prevention of water coming when a well in production produces a significant amount of water, and we are trying to reduce this production of water.

To further simplify the coupling method, we update the data using the values at time T _n , instead of the linear time interpolation between T _n and T _{n + 1} , for the simulation of the first model of wells in the period from T _n to T _{n + 1} . This choice is interesting because it allows simulations in parallel on different machines for the reservoir model and the well approach model.

1 - Application to damage to the oil formation by the drilling fluid

A standard reservoir model is used for the field simulation. The well approach model developed by DING, Y. and RENARD, G.: "Evaluation of Horizontal Well Performance after Drilling Induced Formation Damage" J. of Energy Resources Technology, Vol. 127, Sept., 2005 , is used to simulate damage to the formation by drilling. It should be noted that the advanced physics of the damage is not modeled in the simulation of the field with the model of reservoir.

A 1000m x 1000m x 10m tank is considered. A Cartesian mesh with 20 meshes in the x direction, 20 meshes in the y direction and 1 mesh in the z direction is used for the simulation of the field ( figure 6 ). The mesh sizes are therefore 50m x 50m x 10m. The initial tank pressure is 200 bar. A producing well must be drilled in the block (15, 15, 1). It is represented by a black circle on the figure 6 . The damage to this well by the drilling fluid is studied with the method according to the invention.

The tank is homogeneous with permeability 200 mD and porosity 0.15. The boundary conditions of this reservoir are zero flows, except on the edge Γ _x- ( figure 6 ), where the pressure is constant (200 bars).

To obtain the reference solution, the mesh is refined around the well ( figure 7 ). A specific model, which takes into account the advanced physics of the damage, is used on this mesh to simulate the reference solution. Since damage by drilling fluid is generally limited to a few centimeters or a few tens of centimeters around wells, we need very small meshes in the refined zone (Table 1). The diameter of the well is 21.6 cm. For the well to be included in a mesh, the size of the well mesh is 22 cm. The other meshes around the well are much smaller with a size of 2 cm. The meshes used for the coupling are illustrated on the Figures 8A and 8B . The mesh of the well approach model ( figure 8B ) corresponds to the refined zone and to the meshes around in the reference mesh. The meshes at the edges of the approaching well model coincide with meshes of the reservoir model. Table 1-Sizes of the meshes around the well Mesh size in x direction (m) Mesh size in y direction (m) 50 42.7 30 20 16 8 4 2 1 0.51 0.3 50 42.7 30 20 16 8 4 2 1 0.51 0.3 0.16 0.08 0.04 0.02 0.02 0.02 0.02
0.02 0.16 0.08 0.04 0.02 0.02 0.02 0.02
0.02 0.22 0.22 0.02 0.02 0.02 0.02 0.02 0.04 0.08
0.16 0.02 0.02 0.02 0.02 0.02 0.04 0.08
0.16 0.3 0.51 1 2 4 8 16 20 30 42.7 50 0.3 0.51 1 2 4 8 16 20 30 42.7 50

It is assumed that the tank is thick, and that this model only corresponds to the first layer of the tank. The contact time between the drilling fluid and the reservoir is 2 days. The pressure during drilling at the bottom of the well is 250 bars. The permeability and the thickness of the external “cake” formed by the drilling mud are equal to 0.001 mD and 0.2 cm. The thickness of the internal cake is 2 cm with an average permeability reduced to 20 mD during the drilling period and 40 mD in the production period. The viscosity of the drilling fluid is 30 cPo. The hysteresis of the relative permeability between the drilling and production periods is presented in the figure 9 . An irreducible water saturation of 30% linked to the filtrate (drilling fluid) which will invade the formation during the drilling phase will remain blocked in the porous medium when the well is returned to production.

The drilling fluid invasion volumes are compared to the figure 10 for the simulation with the coupling method and the reference solution obtained using the mesh with local refinement ( figure 7 ). The time steps for updating the data in the coupling are presented in Table 2. The figure 10 shows that the volume of fluid invasion is correctly simulated with the coupling method. The small difference between the coupling solution and the reference solution in the period between 0.1 and 0.3 days can be improved by using small iteration steps in time to exchange the data in the coupling. Table 2-No time for updating the data in the linkage Period (day) No time (day) 0 - 0.01 0001 0.01 - 0.1 0.01 0.1 - 3 0.1 3 - 10 1 10 - 200 10

After the 2 days of drilling, the well is closed for 1 day for the establishment of its completion, then it is put into production. The coupling is effected until 10 ^th day. Beyond 10 days, the effect of damage around wells becomes stable and the digital IPs in the reservoir model hardly change any more. We no longer need coupling to continue simulating the field with the reservoir model. The oil production curve simulated by the reservoir model, which is coupled with the first well model for the first 10 days, is presented in the figure 11 . This curve is very close to the reference solution.

If the damage is not taken into account or if only the presence of "cakes" is considered in the simulation, the results are very imprecise with errors of more than 20% ( figure 11 ). Taking into account phenomena around wells such as damage by the drilling fluid is important for reservoir management, and the proposed coupling method is perfectly suited to simulating this type of problem.

2 - Application to the prevention of water inflow

In the water prevention procedure, a polymer solution is injected into a producing well for a short time in order to reduce the large amount of water produced together with the oil. Part of the polymer is absorbed on the rock, and another part is dispersed in water. The injected polymer has the effect of reducing the mobility of the water phase by increasing its viscosity and by reducing the relative permeability of this phase. Therefore, in the coupling method, the most suitable approach is to update the digital IP multiplier for the water phase.

A 1000m x 1000m x 25m tank is considered as an example. A Cartesian mesh with 20 meshes in the x direction, 20 meshes in the y direction and 5 meshes in the z direction is used for the simulation of the field. The mesh size is 50m x 50m x 5m. The reservoir is heterogeneous. Permeability is presented to the figure 12 . The ratio of permeabilities in the vertical and horizontal directions is 0.1. The initial tank pressure is 200 bar.

There is an injector well ( INJ ) and a producer well ( PROD ) as shown in the figure 12 . The pressure at the injector well is imposed at 300 bars, and the pressure at the producing well is constrained to 150 bars during production. After production for 1000 days, the water-cut (water flow compared to the total flow) of the producing well reaches 85%. The water ingress prevention procedure is then applied to reduce the amount of water produced. A polymer solution with a concentration of 2500 ppm is injected into the producer with a bottom pressure of 300 bars for 2 days. Then, the well is put back into production. This water prevention procedure is simulated with the method according to the invention.

In order to have a reference solution, a local refinement around the producing well is used ( figure 13 ). The mesh size around the well is 0.617 m in the x direction. The mesh for coupling is presented in the figure 14 . Meshes at the edges of the first well model coincide with meshes in the reservoir model. The physics of the polymer can be considered in both models (first well model and reservoir model). Table 3 - No time during coupling Period (day) No time (day) 0 - 950 - 950-970 2 970 - 1000 28 1000 - 1000.1 0.01 1000.1 - 1005 0.1 1005 - 1030 1 1030 - 1100 2 1100 - 3000 -

The coupling begins at 950 days and ends at 1100 days, a period of 150 days in total. The time steps for exchanging data in the coupling method are presented in Table 3. In the first 50 days (from 950 to 1000 days) of coupling, there is no polymer injected. This period is only used to ensure proper initialization of the well approach model. The global digital IPs are updated at the start of the coupling (from 950 to 970 days) to take into account the effects of the meshes between the reservoir model and the approaching well model. During the polymer injection period (between 1000 and 1002 days), the global digital IPs are further recalculated to integrate the effect induced by the injected polymer (we could also update the digital IP multipliers for the water phase ). However, when the well is returned to production (at 1003 days), the digital IP multipliers for the water phase are updated.

The figure 15 compare the injection rates of polymer in the well for the different simulations: the reference solution, the simulation on the reservoir mesh with coupling, the direct simulation on the reservoir mesh without coupling and the simulation with the well approach model (with coupling). The Figures 16A to 16E show the same layer-by-layer comparisons. For the direct simulation with the reservoir mesh without coupling, the volume of polymer injected is greatly overestimated. When the coarse mesh simulation is coupled with the well approach model, the results are significantly improved. At the start of the coupling, the injection rate is high, but it is quickly corrected by the update of the IP due to the coupling. If we want to have more precision on the polymer injection rate, it is enough to refer to the simulation results with the first model of well. With this model, the volume injected and the distribution of the polymer around the well are both correctly simulated.

The figures 17, 18 and 19 show the oil, water and water-cut flow curves for the tank model with coupling, the tank model without coupling and the reference solution. The results of the coupled reservoir model are generally satisfactory. The figure 20 presents the water saturation card at the end of the coupling (1100 days), and the figure 21 shows the pressure card at 1100 days. Compared to the reference solutions, the coupling gives globally satisfactory results.

Claims (9)

1. Method implemented by computer for modelling flows of fluids within a subterranean porous medium traversed by at least one well with a view to the exploitation of the said medium, carried out by injecting one of the said fluids into the said medium via at least one of the said wells, and/or by producing one of the said fluids present in the said medium by means of at least one of the said wells, in which method a first flow simulator is used making it possible to simulate the flow of the fluids within the porous medium on the basis of numerical productivity indices linking pressures of fluids to flowrates of fluids, and a second flow simulator is used to simulate the flow of the fluids in the surroundings of the well on the basis of boundary conditions, characterized in that:

   a- the flows of fluids within the medium are simulated by means of the first simulator over a time interval defined between times T₀ and T₁, and updated boundary conditions for the second simulator are deduced through a linear interpolation of the results of the first simulator between the times T₀ and T₁ ;

   b- the flows of fluids in the surroundings of the well are simulated by means of the second simulator over the same time interval, using the updated boundary conditions, and updated numerical productivity indices for the first simulator are deduced by comparing flowrates calculated by the first simulator and flowrates calculated by the second simulator; and

   c- the flows of fluids within the porous medium during a time period between T₀ and T_n, where T_n >T₁, are modelled by repeating steps a and b, for successive time intervals lying between T₀ and T_n.
2. Method according to Claim 1, in which each successive time interval has a length which is dependent on a calculation timestep of the first flow simulator and on a timestep of the second flow simulator.
3. Method according to Claim 1, in which each successive time interval has a length equal to a timestep of the first flow simulator.
4. Method according to one of the preceding claims, in which the flows of fluids within the medium are simulated by means of the first simulator on a first mesh discretizing the porous medium into a set of mesh cells, and the flows of fluids in the surroundings of the well are simulated by means of the second simulator on a second mesh discretizing the well and its surroundings into a set of mesh cells, the said second mesh being generated by constraining mesh cells situated on the edge of the said second mesh, in such a way that their interfaces coincide with the interfaces of the mesh cells of the said first mesh.
5. Method according to one of the preceding claims, in which multiphase flows are modelled, and, for each phase, multipliers of numerical productivity indices, instead of the numerical productivity indices themselves, are updated by comparing per-phase flowrates calculated by the first simulator and per-phase flowrates calculated by the second simulator.
6. Method, according to one of Claims 1 to 5, for exploiting a subterranean porous reservoir by means of at least one well traversing the reservoir, at least one fluid circulating between the reservoir and the well, in which data relating to the geometry of the said porous reservoir are acquired, on the basis of which data a discretization of the reservoir into a set of mesh cells, called reservoir mesh, is constructed, and a discretization of the well and of its surroundings into a set of mesh cells, called well surroundings mesh, is constructed, characterized in that the following steps are carried out:

   a- an exploitation scenario for the porous reservoir is chosen;

   b- a first flow simulator is associated with the reservoir mesh making it possible to simulate the flow of the fluids within the reservoir, on the basis of at least the following data: the production scenario, data of inputs relating to the fluid and to the reservoir, numerical productivity indices making it possible to link pressures to flowrates, boundary conditions;

   c- a second flow simulator is associated with the well surroundings mesh for simulating the flow of the fluids in the surroundings of the well, on the basis of at least the following data: data of inputs relating to the fluid and to the reservoir, boundary conditions;

   d- the flows of fluids within the reservoir and in the surroundings of the well are modelled by means of the method according to one of Claims 1 to 5; and

   e- the exploitation scenario is modified and step d is repeated, until an optimal exploitation scenario is obtained.
7. Method according to Claim 6, in which damage to the well by a drilling fluid is taken into account by modelling an invasion of the porous reservoir by the said drilling fluid in steps d and e.
8. Method according to Claim 6, in which the exploitation scenario comprises an injection of a polymer solution by the well, and the flows are modelled so as to forestall an ingress of water.
9. Method according to Claim 6, in which the exploitation scenario comprises an injection of an acid solution into the well, and the flows are modelled so as to evaluate the impact of an acid stimulation.

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