EP1832711A1 - Method for modelling and simulation on a larger scale of the stimulation of the hydrocarbon wells - Google Patents

Abstract

The method involves constructing a double medium model by considering two sub-mediums suitable for the formation of respective solution holes, by conserving the mass of one sub-medium and mass of an acid and by depicting a transfer of the acid from one sub-medium to the other sub-medium. The model is initialized from experimental settings, and the acidification is modelized by determining physical parameters representing a porous medium e.g. carbonated reservoir traversed by wells, and physical parameters relative to the acid.

Description

The present invention relates to a method for modeling acidification in a porous medium following the injection of a chemical product, such as an acid.

In particular, the invention makes it possible to optimize acid injection parameters, such as the flow rate and the zones to be treated, in the context of acid well stimulation in a carbonated context.

In the oil industry, the production of a well can be greatly reduced due to damage in the vicinity of the well. The damage results in an alteration of the permeability and the nature of the rock around the well. The operations damaging the well are multiple: drilling, casing, cementing, exploitation, completion and treatment. The consequence of these damages is the clogging of the formation, and therefore the reduction or the cessation of the production of hydrocarbon. It is therefore very important for the oil industry to identify the type of damage on the one hand, and the damaged areas on the other, to decide and develop a suitable treatment.

One of the treatments commonly used in the oil industry is the injection of acid around a well. This injection makes it possible to reduce the damages, and thus, to improve the production of the well. The primary purpose of acid stimulation is to reduce the flow resistance of reservoir fluids due to damage. The injected acid dissolves the material in the reservoir matrix and creates channels that increase the permeability of the reservoir matrix. These channels are all the more frequent in carbonate rocks, ie rocks containing more than 50% of carbonaceous minerals (calcite, dolomite), such as limestones. The efficiency of this process depends on the type of acids used, the conditions of injection, the structure of the medium, the transfer between the fluid and the solid, the rate of reactions, etc. While the dissolution increases the permeability, the relative increase in permeability for injection of a given acid volume is observed to be strongly dependent on the injection conditions.

In sandstone reservoirs, the reaction fronts tend to be uniform and the flow channels are not observed. In carbonate reservoirs, depending on the injection conditions, multiple vermicular breakthroughs in rock, commonly referred to as "wormholes", may be created.

Presentation of the prior art

It is therefore very important for the oil industry to identify both the type of damage and the damaged areas, to optimize the parameters of acid stimulation, in order to produce vermicular breakthroughs. ("Whormholes") with optimum density and depth of penetration into the formation.

• L'échelle du pore qui est l'échelle à laquelle nous décrivons les mécanismes de la réaction chimiques :
• L'échelle de la carotte à laquelle apparaît l'instabilité des wormholes
• L'échelle du puits qui est l'échelle à laquelle nous pouvons apprécier la compétition entre les wormholes et l'impacte des hétérogénéités à cette échelle.
• L'échelle du réservoir à laquelle l'effet de la stimulation est mesuré par le skin.

Les figures 1A à 1D, où le milieu σ représente la roche et le milieu β l'eau et l'acide, illustre ces différentes échelles impliquées dans la stimulation acide :

• Figure 1A : échelle du pore (µm - mm)
• Figure 1B : échelle de la carotte (mm - cm)
• Figure 1C : échelle du puits (cm-m)
• Figure 1D : échelle du réservoir (m - km)

The study of the formation and behavior of vermicular breakthroughs to determine the parameters of the acid injection can be done according to four different scales:

• The scale of the pore which is the scale to which we describe the mechanisms of the chemical reaction:
• The scale of the carrot at which the instability of the wormholes appears
• The scale of the well which is the scale at which we can appreciate the competition between wormholes and the impact of heterogeneities on this scale.
• The scale of the reservoir at which the effect of stimulation is measured by the skin.

FIGS. 1A to 1D, where the medium σ represents the rock and the medium β the water and the acid, illustrates these different scales involved in the acid stimulation:

• Figure 1A: pore scale (μm - mm)
• Figure 1B: Carrot scale (mm - cm)
• Figure 1C: well scale (cm-m)
• Figure 1D: Tank scale (m - km)

Many models, such as the one presented by Wang, Y., Hill, AD, and Schechter, RS, "The Optimum Injection Rate for MAtrix Acidizing Carbonate Formations," SPE Paper 26578, SPE ATCE, Houston, 1993, have already been proposed to study the effect of fluid leakage, kinetic reaction, etc. on the rate of wormhole spread and the effect of neighboring wormholes on the dominant wormhole growth rate. The simple structure of these models offers the advantage of studying in detail the reaction, the mechanisms of diffusion and convection within vermicular breakthroughs. These models, however, can not be used to study the initialization of wormholes and the effects on the heterogeneities of the formation.

Models describing dissolution during acid injection were first set up to describe this phenomenon at the pore scale. Such a method is described for example in Hoefner, ML, Fogler, H., S., "Pore Ecolution and Channel Formation During Flow and Reaction in Porous Media", AIChE J, 34, 45-54 (1998) . However, the carrot-scale simulation from these models is difficult and requires a large computing capacity. But it is at this scale that instabilities due to wormholes appear.

The first model at the scale of the carrot capable of fully reproducing the mechanisms of dissolution has been proposed by Golfier F. et al., "A discussion of Darcy-Scale modeling of porous media dissolution in homogeneous systems", Computational Methods in Water Resources, 2, 1195-1202 (2002). . This simple mid-scale model is constructed from a mean-volume acquisition of the pore-scale equations. This modeling has been included in the international application WO 03/102362 , which extended the model to the case of a dissolution limited by the kinetics of reaction. These models are based on a description of the physics at the scale of the carrot, which requires mesh sizes of the order of a millimeter.

However, an acidic injection process is a well-scale process. It therefore seems necessary to model the formation and behavior of wormholes on this scale. Especially since, in the oil industry, the generalization of horizontal wells has led to an increase in the quantities of acids injected into a single well. The need for simulation means to increase the chances of successful treatment has increased. However, the modelizations described previously do not make it possible to simulate the acidification on a domain representing the section of a well and its surroundings (1 to 3m).

• Buisje,M.A. Understanding Wormholing Mechanisms Can Improve Acid Treatments in Carbonate Formations. (SPE 38166). 1997. SPE European Formation Damage Conférence.
• Buisje,M.A. & Glasbergen,G. (SPE 96892). 2005. SPE Annual Technical conference and Exhibition .
• Gdanski,R. A Fundamentally New Model of Acid Wormholing in Carbonates. (SPE 54719). 1999. SPE European Formation Damage Conference.

Ces méthodes reposent sur des considérations empiriques basées sur des observations faites en laboratoire, très éloignées des conditions et des dimensions réelles.Models designed to simulate acid treatment on a scale greater than that of carrots have already been proposed. For example:

• Buisje, MA Understanding Wormholing Mechanisms Can Improve Acid Treatments in Carbonate Training. (SPE 38166). 1997. SPE European Training Damage Conference.
• Buisje, MA & Glasbergen, G. (SPE 96892). 2005. SPE Annual Technical Conference and Exhibition .
• Gdanski, R. A Fundamentally New Model of Acid Wormholing in Carbonates. (SPE 54719). 1999. SPE European Training Damage Conference.

These methods are based on empirical considerations based on observations made in the laboratory, far removed from actual conditions and dimensions.

The method according to the invention is a method for modeling, on a metric scale, acidification in a porous medium following the injection of acid, making it possible to meet the needs of reservoir engineers, to define a scenario adapted from Acid stimulation of wells in the context of carbonated reservoirs.

The method according to the invention

1. a) on construit ledit modèle double milieu
   • en considérant un premier sous-milieu propice à des percées de dissolution, et un second sous-milieu non propice à des percées de dissolution ;
   • en réalisant, pour chacun desdits sous-milieux, une description à l'échelle métrique du transport de l'acide, de la conservation de la masse dudit sous-milieu et de la conservation de la masse dudit acide ;
   • en décrivant un transfert d'acide d'un sous-milieu à l'autre sous-milieu ;
2. b) on initialise ledit modèle double milieu à partir de calages expérimentaux ;
3. c) on modélise, à l'aide dudit modèle double milieu, ladite acidification, en déterminant des paramètres physiques représentatifs dudit milieu poreux et des paramètres physiques relatifs à l'acide injecté.

The invention relates to a method for modeling acidification in a porous medium following the injection of an acid, in which said medium is represented by a dual medium model, characterized in that the method comprises the following steps:

1. a) constructing said dual-medium model
   • considering a first sub-environment conducive to dissolution breakthroughs, and a second sub-medium not conducive to breakthroughs;
   • performing, for each of said sub-media, a metric scale description of the acid transport, the conservation of the mass of said sub-medium and the conservation of the mass of said acid;
   • describing acid transfer from one sub-medium to another sub-medium;
2. b) initializing said dual-medium model from experimental setups;
3. c) modeling, using said dual-medium model, said acidification, by determining representative physical parameters of said porous medium and physical parameters relative to the injected acid.

The representative physical parameters of the porous medium can be chosen, for each of the sub-media, from the following parameters: the average porosity, the metric scale permeability and the average total pressure. The physical parameters relating to the acid can be chosen, for each of the sub-media, from among the following parameters: the average concentration of the acid, the average Darcy speed.

According to the invention, the description can be performed using equations obtained by averaging the metric scale of equations describing the propagation of an acid in a single-scale model at a centimeter scale. These equations then preferably comprise a dissolution term. The latter can be defined as the product of an average concentration of the acid on the metric scale by a coefficient dependent on a local rate of the acid. It can also be defined as the product of a parameter by the divergence of a product between a concentration of the acid, a fractional flux function and a velocity vector. The parameter of this latter term of dissolution may depend on a standard of a local rate of acid and average porosity on the metric scale.

According to the invention, the calibration procedure can be either based on simulations at scales below the metric scale, or based on constant flow rate acid injection studies in a sample of the medium.

1. a) on construit un maillage dudit puits et de son voisinage ;
2. b) on définit des paramètres initiaux de l'injection d'acide ;
3. c) on détermine, en modélisant l'acidification due à l'injection de l'acide, au moins les paramètres physiques représentatifs dudit réservoir suivants : une porosité et une perméabilité dudit réservoir après injection de l'acide ;
4. d) on simule la production du puits en fonction de ladite porosité et de ladite perméabilité à l'aide d'un simulateur réservoir ;
5. e) on modifie lesdits paramètres initiaux et l'on réitère à l'étape c) jusqu'à obtenir un maximum de production.

In one embodiment, the porous medium may be a carbonated reservoir traversed by a well, the acid injection being carried out to stimulate hydrocarbon production by said well, and in which optimal parameters of the acid injection are determined, by performing the following steps:

1. a) constructing a mesh of said well and its vicinity;
2. b) initial parameters of the acid injection are defined;
3. c) determining, by modeling the acidification due to the injection of the acid, at least the following representative physical parameters of said tank: a porosity and a permeability of said reservoir after injection of the acid;
4. d) the production of the well is simulated according to said porosity and said permeability using a reservoir simulator;
5. e) modifying said initial parameters and repeating in step c) until a maximum production is obtained.

According to this embodiment, the initial parameters can be chosen from at least one of the following parameters: the injection rate of the acid, the initial injection speed, the volume of acid injected, the concentration of the acid used during stimulation, the areas to be treated.

Summary presentation of figures

• Figures 1A-1D show the different scales involved in acid stimulation.
• Figure 2 illustrates the different steps of the method according to the invention.
• Figure 3 illustrates the different stages of calibration of the parameter χ.
• FIG. 4 shows the distribution of volumes V _H and V _M in the double medium approach according to the invention.
• FIGS. 5A to 5C show the simplified representation of the distribution of volumes V _H and V _M used in the modeling of the exchange term.
• Figure 6 illustrates results using the dissolution terms g _{1 H} and g _{1 M.} It shows the pressure drop in the sample over time.
• Figure 7 illustrates results using the dissolution terms g _{1 H} and g _{1 M.} It shows the porosity of the sample as a function of the abscissa.
• Figure 8 illustrates results using the dissolution terms g _{2 H} and g _{2 M.} It shows the pressure drop in the sample over time.
• Figure 9 illustrates results using the dissolution terms g _{2 H} and g _{2 M.} It shows the porosity of the sample as a function of the abscissa.

Detailed description of the method

The method according to the invention makes it possible to model the acidification of a porous medium due to the injection of a chemical product, such as an acid. Acidification includes several phenomena, the main ones being: the dissolution of the medium by the acid and the transport (propagation) of the acid within the medium. To do this we build a dual-medium model to model these phenomena on a metric scale.

The description of the invention is carried out as part of the acid stimulation of production wells. This stimulation involves injecting acid around a well to increase hydrocarbon production. The method, when applied to a mesh domain representing the surroundings of a well to be stimulated, makes it possible to simulate the evolution of the porosity and the permeability of the rock, and thus to optimize the parameters of the acid stimulation such as the flow rate. injection and the treatment zone, to define the optimum acidification scenario for this well.

1. 1- Maillage du puits et de son voisinage (MAI)
2. 2- Modélisation de l'acidification due à la stimulation acide (MOD → ε, K)
3. 3- Simulation de la production du puits (SIM → Prod)
4. 4- Optimisation des paramètres d'injection d'acide (OPT)
5. 5- Stimulation acide optimisée du puits pour augmenter sa productivité (STIM)

Figure 2 illustrates the different steps of the method applied to acid injection around a well:

1. 1- Meshing of the well and its neighborhood (MAI)
2. 2- Modeling acidification due to acid stimulation (MOD → ε, K )
3. 3- Simulation of well production (SIM → Prod)
4. 4- Optimization of acid injection parameters (OPT)
5. 5- Optimized acid stimulation of the well to increase its productivity (STIM)

1- Meshing of the well and its neighborhood

To allow the effects of acid stimulation on a well and its direct environment to be modeled, this space (well + neighborhood) is discretized using a structured radial type mesh. This type of mesh, well known specialists, allows to take into account the radial directions of the flows around the wells, and thus improve the accuracy of calculations.

2- Modeling of acidification due to stimulation

This step requires firstly the definition of a dissolution and propagation (acidification) model allowing to model the formation and the behavior of all the dissolution figures at the well scale: compact front, conical wormhole, dominant wormhole, branched wormhole and uniform dissolution.

3.1- Acidification model at the well scale

To model the dissolution and propagation phenomena of the acid at the well scale, we build a model based on equations of fluid mechanics and laws of conservation of rock mass, fluid and acid. . According to the invention, this model is a double-medium model, constructed from a well-scale mean-volume uptake of equations describing the acid propagation in a single-scale model at the carrot scale ( cm-mm). These equations at the carrot scale have been developed by Golfier F. et al., "On The ability of a Darcy-scale model to capture wormhole training during the dissolution of a porous media. Journal of Fluid Mechanic. 547, 213-254. 2002 .

The model according to the invention thus makes it possible to use a radial mesh whose radial extension is of the order of one centimeter and one meter. Acidification can therefore be simulated on a scale large enough to reproduce the entire acid treatment and to evaluate the increase in permeability around the well. The simulation of the evolution of the permeability then makes it possible to simulate the production and then to optimize the acid injection parameters.

• le volume V_M contient la forte densité de wormholes dont la croissance s'achève rapidement et qui sont de petites tailles (mm-cm) (courbe noire sur la figure 4). Ce milieu est représentatif d'un régime compact dans lequel les wormholes ont une croissance courte ;
• le volume V_H contient les wormholes dominants, c'est-à-dire les wormholes pour lesquels la compétition s'étend sur des distances (dm-m) et des temps longs (figure 4). Ce milieu est propice au développement de wormholes et par conséquent est caractérisé par un front de dissolution rapide.

The double-medium model is defined by considering that the rock of the reservoir consists of two media, denoted H and M, of respective volumes V _H and V _M , characterized by two different dissolution regimes. Figure 4 illustrates these two environments where V _Section represents the volume of a section of a well, and the black curve represents vermicular breakthroughs (wormholes). According to the invention, the two regimes associated with the two media are defined as follows:

• the volume V _M contains the high density of wormholes whose growth ends rapidly and which are small (mm-cm) (black curve in Figure 4). This medium is representative of a compact diet in which the wormholes have a short growth;
• the volume V _H contains the dominant wormholes, that is to say the wormholes for which the competition extends over distances (dm-m) and long times (Figure 4). This medium is conducive to the development of wormholes and is therefore characterized by a rapid dissolution front.

Thus, the medium H is a medium conducive to the formation of breakthroughs: the injection of acid in such a medium causes the formation of breakthroughs (wormholes) of large sizes, generally greater than the decimeter. Medium M is a medium that is not conducive to the formation of breakthroughs: the injection of acid into such a medium does not cause the formation of large wormholes, and allows, at best, the formation of breakthroughs. small sizes, generally less than the decimetre.

$$\{\mathit{\phi }{\}}_H^H=\frac{1}{V_{\mathit{H}}}{\displaystyle \underset{V_H}{\int }}\mathit{\phi }dV\mathrm{et}\{\mathit{\phi }{\}}_M^M=\frac{1}{V_{\mathit{M}}}{\displaystyle \underset{V_M}{\int }}\mathit{\phi }dV$$

$${\varphi }_H=\frac{V_H}{V_{\mathit{Section}}}\mathrm{et}{\varphi }_H=\frac{V_M}{V_{\mathit{Section}}},{\mathrm{soit\; \varphi }}_H=1-{\varphi }_M$$

$$\mathit{\{}\mathit{\phi }{\mathit{\}}}_{\mathit{H}}\mathit{=}{\mathit{\phi }}_{\mathit{H}}\mathit{\{}\mathit{\phi }{\mathit{\}}}_{\mathit{H}}^{\mathit{H}}\mathrm{et}\mathit{\{}\mathit{\phi }{\mathit{\}}}_M\mathit{=}{\mathit{\phi }}_M\mathit{\{}\mathit{\phi }{\mathit{\}}}_M^M{\mathrm{avec\; \varphi }}_{\mathit{wh}}=\frac{V_{\mathit{wh}}}{V_{\mathit{Section}}}$$

$$\left\{\mathit{\phi }\right\}\mathit{=}{\mathit{\phi }}_H\mathit{\{}\mathit{\phi }{\mathit{\}}}_H^H+{\mathit{\phi }}_M\mathit{\{}\mathit{\phi }{\mathit{\}}}_M^M$$

To describe the well-scale acidification phenomena, a well-scale mean gain is taken from equations describing acid propagation in a single-scale model at the core scale. Thus, for any variable φ, which makes it possible to describe the acidification at the carrot scale, the application of the averaging theorem allows to calculate the following variables, allowing to describe the acidification on the scale of the well: $$\{\mathit{\phi }{\}}_H=\frac{1}{V_{\mathit{Section}}}{\displaystyle \underset{V_H}{\int }}\mathit{\phi }dV\mathrm{and}\{\mathit{\phi }{\}}_M=\frac{1}{V_{\mathit{Section}}}{\displaystyle \underset{V_M}{\int }}\mathit{\phi }dV$$

$$\{\mathit{\phi }{\}}_H^H=\frac{1}{V_{\mathit{H}}}{\displaystyle \underset{V_H}{\int }}\mathit{\phi }dV\mathrm{and}\{\mathit{\phi }{\}}_M^M=\frac{1}{V_{\mathit{M}}}{\displaystyle \underset{V_M}{\int }}\mathit{\phi }dV$$

$${\phi }_H=\frac{V_H}{V_{\mathit{Section}}}\mathrm{and}{\phi }_H=\frac{V_M}{V_{\mathit{Section}}},{\mathrm{be\; \phi }}_H=1-{\phi }_M$$

$$\mathit{\{}\mathit{\phi }{\mathit{\}}}_{\mathit{H}}\mathit{=}{\mathit{\phi }}_{\mathit{H}}\mathit{\{}\mathit{\phi }{\mathit{\}}}_{\mathit{H}}^{\mathit{H}}\mathrm{and}\mathit{\{}\mathit{\phi }{\mathit{\}}}_M\mathit{=}{\mathit{\phi }}_M\mathit{\{}\mathit{\phi }{\mathit{\}}}_M^M{\mathrm{with\; \phi }}_{\mathit{wh}}=\frac{V_{\mathit{wh}}}{V_{\mathit{Section}}}$$

$$\left\{\mathit{\phi }\right\}\mathit{=}{\mathit{\phi }}_H\mathit{\{}\mathit{\phi }{\mathit{\}}}_H^H+{\mathit{\phi }}_M\mathit{\{}\mathit{\phi }{\mathit{\}}}_M^M$$

• une équation de transport de l'espèce acide composée :
  • de termes convectifs contenant une fonction de type flux fractionnaire permettant de reproduire en partie la dispersion liée au wormholing
  • de termes réactifs
  • d'un terme d'accumulation
• une équation de Darcy
• une équation de conservation de la masse de roche
• une équation de conservation de la masse fluide.
• Une équation de fermeture du système reliant la perméabilité et la porosité.

The well-scale double-well acidification model according to the invention comprises for each of the two media M and H:

• a transport equation of the compound acid species:
  • of convective terms containing a fractional flow type function allowing to partially reproduce the dispersion related to wormholing
  • reactive terms
  • a term of accumulation
• an equation of Darcy
• a conservation equation of the mass of rock
• an equation of conservation of the fluid mass.
• A system closure equation linking permeability and porosity.

• Milieu H
  Équation du transport de l'espèce acide dans le milieu H $${\varphi }_H{\mathit{\epsilon }}^H\frac{\partial {\mathit{Cʹ}}^H}{\partial t}+\nabla \mathrm{.}\left({\varphi }_H{\mathit{Vʹ}}^H{\mathit{fʹ}}^H\right)-{\mathit{\psi ʹC}}_{H-M}\left({\mathit{Pʹ}}^M-{\mathit{Pʹ}}^H\right)=-{\varphi }_H{g}_H^{ʹ}$$
  
  
  Équation de Darcy appliquée au milieu H $${\mathrm{Vʹ}}^H=-{\mathrm{Kʹ}}^H\mathrm{.}\nabla {\mathit{Pʹ}}^H$$
  
  
  Équation de conservation de la masse de roche dans le milieu H $$\frac{\partial {\mathit{\epsilon }}^H}{\partial t}=N_{\mathit{ac}}{g}_H^{ʹ}$$
  
  
  Équation de conservation de la masse fluide dans le milieu H $$\nabla \mathrm{.}\left({\varphi }_H{\mathrm{Vʹ}}^H\right)-\mathit{\psi ʹ}\left({\mathit{Pʹ}}^{\mathit{M}}\mathit{-}{\mathit{Pʹ}}^{\mathit{H}}\right)=0$$
  
  
  Équation reliant la perméabilité et la porosité dans le milieu H $$K^H={\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">K</figure-callout>}}_0+\left(K_f-K_0\right)(\frac{{\mathit{\epsilon }}^H-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{1-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{)}^x$$
  
• Milieu M
  Équation du transport de l'espèce acide dans le milieu M $${\varphi }_M{\mathit{\epsilon }}^M\frac{\partial {\mathit{Cʹ}}^M}{\partial t}+\nabla \mathrm{.}\left({\varphi }_M{\mathit{Vʹ}}^M{\mathit{fʹ}}^M\right)+{\mathit{\psi ʹC}}_{H-M}\left({\mathit{Pʹ}}^M-{\mathit{Pʹ}}^H\right)=-{\varphi }_M{g}_M^{ʹ}$$
  
  
  Équation de Darcy appliquée au milieu M $${\mathrm{Vʹ}}^M=-{\mathrm{Kʹ}}^M\mathrm{.}\nabla {\mathit{Pʹ}}^M$$
  
  
  Équation de conservation de la masse de roche dans le milieu M $$\frac{\partial {\mathit{\epsilon }}^M}{\partial t}=N_{\mathit{ac}}{g}_M^{ʹ}$$
  
  
  Équation de conservation de la masse fluide dans le milieu M $$\nabla \mathrm{.}\left({\varphi }_M{\mathrm{Vʹ}}^M\right)+\mathit{\psi ʹ}\left({\mathit{Pʹ}}^{\mathit{M}}\mathit{-}{\mathit{Pʹ}}^{\mathit{H}}\right)=0$$
  
  
  Équation reliant la perméabilité et la porosité dans le milieu M $$K^M={\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">K</figure-callout>}}_0+\left(K_f-K_0\right)(\frac{{\mathit{\epsilon }}^M-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{1-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{)}^x$$
  

The double-medium model according to the invention is thus written as follows:

• Middle H
  Equation of the transport of the acidic species in the medium H $${\phi }_H{\mathit{\epsilon }}^H\frac{\partial {\mathit{VS}}^H}{\partial t}+\nabla \mathrm{.}\left({\phi }_H{\mathit{V\text{'}}}^H{\mathit{f\text{'}}}^H\right)-{\mathit{\psi \text{'}C}}_{H-M}\left({\mathit{P\text{'}}}^M-{\mathit{P\text{'}}}^H\right)=-{\phi }_H{\mathrm{boy\; Wut}}_H^{\text{'}}$$
  
  
  Darcy's equation applied to H medium $${\mathrm{V\text{'}}}^H=-{\mathrm{K\text{'}}}^H\mathrm{.}\nabla {\mathit{P\text{'}}}^H$$
  
  
  Conservation equation of the mass of rock in the medium H $$\frac{\partial {\mathit{\epsilon }}^H}{\partial t}={\mathrm{NOT}}_{\mathit{ac}}{\mathrm{boy\; Wut}}_H^{\text{'}}$$
  
  
  Conservation equation of the fluid mass in the medium H $$\nabla \mathrm{.}\left({\phi }_H{\mathrm{V\text{'}}}^H\right)-\mathit{\psi \text{'}}\left({\mathit{P\text{'}}}^{\mathit{M}}\mathit{-}{\mathit{P\text{'}}}^{\mathit{H}}\right)=0$$
  
  
  Equation linking permeability and porosity in H medium $$K^H={\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">K</figure-callout>}}_0+\left(K_f-K_0\right)(\frac{{\mathit{\epsilon }}^H-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{1-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{)}^x$$
  
• Middle M
  Equation of the transport of the acid species in the medium M $${\phi }_M{\mathit{\epsilon }}^M\frac{\partial {\mathit{VS}}^M}{\partial t}+\nabla \mathrm{.}\left({\phi }_M{\mathit{V\text{'}}}^M{\mathit{f\text{'}}}^M\right)+{\mathit{\psi \text{'}C}}_{H-M}\left({\mathit{P\text{'}}}^M-{\mathit{P\text{'}}}^H\right)=-{\phi }_M{\mathrm{boy\; Wut}}_M^{\text{'}}$$
  
  
  Darcy's equation applied to medium M $${\mathrm{V\text{'}}}^M=-{\mathrm{K\text{'}}}^M\mathrm{.}\nabla {\mathit{P\text{'}}}^M$$
  
  
  Conservation equation of the mass of rock in the medium M $$\frac{\partial {\mathit{\epsilon }}^M}{\partial t}={\mathrm{NOT}}_{\mathit{ac}}{\mathrm{boy\; Wut}}_M^{\text{'}}$$
  
  
  Conservation equation of the fluid mass in the medium M $$\nabla \mathrm{.}\left({\phi }_M{\mathrm{V\text{'}}}^M\right)+\mathit{\psi \text{'}}\left({\mathit{P\text{'}}}^{\mathit{M}}\mathit{-}{\mathit{P\text{'}}}^{\mathit{H}}\right)=0$$
  
  
  Equation linking permeability and porosity in the medium M $$K^M={\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">K</figure-callout>}}_0+\left(K_f-K_0\right)(\frac{{\mathit{\epsilon }}^M-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{1-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{)}^x$$
  

$${\mathit{Cʹ}}^H=\frac{\{C_{\mathit{A\beta }}{\}}_H^H}{C_0}{\mathit{Cʹ}}^M=\frac{\{C_{\mathit{A\beta }}{\}}_M^M}{C_0}$$

$${\mathrm{Vʹ}}^H=\frac{\{{\mathrm{V}}_{\mathit{\beta }}{\}}_H^H}{V_0}{\mathrm{Vʹ}}^M=\frac{\{{\mathrm{V}}_{\mathit{\beta }}{\}}_M^M}{V_0}$$

$$f^H\left(\{C_{\mathit{A\beta }}{\}}_H^H\right)=\frac{\{C_{\mathit{A\beta }}{\}}_H^H}{\frac{\{C_{\mathit{A\beta }}{\}}_H^H}{C_0}+\left(1-\frac{\{C_{\mathit{A\beta }}{\}}_H^H}{C_0}\right)/H^H}{f}^M\left(\{C_{\mathit{A\beta }}{\}}_M^M\right)=\frac{\{C_{\mathit{A\beta }}{\}}_M^M}{\frac{\{C_{\mathit{A\beta }}{\}}_M^M}{C_0}+\left(1-\frac{\{C_{\mathit{A\beta }}{\}}_M^M}{C_0}\right)/H^M}$$

$${\mathit{fʹ}}^H=\frac{f^H}{C_0}{\mathit{fʹ}}^M=\frac{f^M}{C_0}$$

$${\mathit{Pʹ}}^H=\frac{K_0}{\mathit{\mu }{V}_{0}L}\{P{\}}_H^H{\mathit{Pʹ}}^M=\frac{K_0}{\mathit{\mu }{V}_{0}L}\{P{\}}_M^M$$

$${\mathbf{Kʹ}}^H=\frac{{\mathbf{K}}^H}{{\mathbf{K}}_0}{\mathbf{Kʹ}}^M=\frac{{\mathbf{K}}^M}{{\mathbf{K}}_0}$$

$$g_H^{ʹ}=\frac{g_{H}L}{V_0{C}_0}{g}_M^{ʹ}=\frac{g_{M}L}{V_0{C}_0}$$

$$N_{\mathit{ac}}=\frac{{\mathit{\nu C}}_0}{{\mathit{\rho }}^{\mathit{\sigma }}}\begin{array}{}\end{array}\mathit{\psi ʹ}=\frac{{\mathit{\psi \mu L}}^2}{{\mathbf{K}}_0}$$

With the following variables used to adimensionalize the system: $$\mathit{t\; \text{'}}=\frac{V_0}{\mathrm{The}}\mathit{t}\mathit{x\; \text{'}}=\frac{x}{\mathrm{The}}$$

$${\mathit{VS}}^H=\frac{\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_H^H}{{\mathrm{VS}}_0}{\mathit{VS}}^M=\frac{\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_M^M}{{\mathrm{VS}}_0}$$

$${\mathrm{V\text{'}}}^H=\frac{\{{\mathrm{V}}_{\mathit{\beta }}{\}}_H^H}{V_0}{\mathrm{V\text{'}}}^M=\frac{\{{\mathrm{V}}_{\mathit{\beta }}{\}}_M^M}{V_0}$$

$$f^H\left(\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_H^H\right)=\frac{\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_H^H}{\frac{\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_H^H}{{\mathrm{VS}}_0}+\left(1-\frac{\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_H^H}{{\mathrm{VS}}_0}\right)/H^H}{f}^M\left(\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_M^M\right)=\frac{\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_M^M}{\frac{\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_M^M}{{\mathrm{VS}}_0}+\left(1-\frac{\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_M^M}{{\mathrm{VS}}_0}\right)/H^M}$$

$${\mathit{f\text{'}}}^H=\frac{f^H}{{\mathrm{VS}}_0}{\mathit{f\text{'}}}^M=\frac{f^M}{{\mathrm{VS}}_0}$$

$${\mathit{P\text{'}}}^H=\frac{K_0}{\mathit{\mu }{V}_0\mathrm{The}}\{P{\}}_H^H{\mathit{P\text{'}}}^M=\frac{K_0}{\mathit{\mu }{V}_0\mathrm{The}}\{P{\}}_M^M$$

$${\mathbf{K\text{'}}}^H=\frac{{\mathbf{K}}^H}{{\mathbf{K}}_0}{\mathbf{K\text{'}}}^M=\frac{{\mathbf{K}}^M}{{\mathbf{K}}_0}$$

$${\mathrm{boy\; Wut}}_H^{\text{'}}=\frac{{\mathrm{boy\; Wut}}_H\mathrm{The}}{V_0{\mathrm{VS}}_0}{\mathrm{boy\; Wut}}_M^{\text{'}}=\frac{{\mathrm{boy\; Wut}}_M\mathrm{The}}{V_0{\mathrm{VS}}_0}$$

$${\mathrm{NOT}}_{\mathit{ac}}=\frac{{\mathit{\nu C}}_0}{{\mathit{\rho }}^{\mathit{\sigma }}}\begin{array}{}\end{array}\mathit{\psi \text{'}}=\frac{{\mathit{\psi \mu L}}^2}{{\mathbf{K}}_0}$$

$$\begin{array}{l}\{{\mathrm{V}}_{\mathit{\beta }}{\}}_M^M=\mathrm{vitesse\; de\; Darcy\; moyenne\; dans\; le\; milieu}\begin{array}{}\end{array}M\left(\mathrm{m}\mathrm{/}\mathrm{s}\right)\\ \{{\mathrm{V}}_{\mathit{\beta }}{\}}_H^H=\mathrm{vitesse\; de\; Darcy\; moyenne\; dans\; le\; milieu}\begin{array}{}\end{array}H\left(\mathrm{m}\mathrm{/}\mathrm{s}\right)\end{array}$$

$$\begin{array}{l}\{P{\}}_M^M=\mathrm{pression\; totale\; moyenne\; dans\; le\; milieu}\begin{array}{}\end{array}M\left(\mathrm{Pa}\right)\\ \{P{\}}_H^H=\mathrm{pression\; totale\; moyenne\; dans\; le\; milieu}\begin{array}{}\end{array}H\left(\mathrm{Pa}\right)\end{array}$$

$$\begin{array}{l}\{{\mathit{\epsilon }}_{\mathit{\beta }}{\}}_M^M=\mathrm{porosité\; moyenne\; dans\; le\; milieu}\begin{array}{}\end{array}M\\ \{{\mathit{\epsilon }}_{\mathit{\beta }}{\}}_H^H=\mathrm{porosité\; moyenne\; dans\; le\; milieu}\begin{array}{}\end{array}H\end{array}$$

$$\begin{array}{l}{\mathrm{K}}^M=\mathrm{perméabilité\; dans\; le\; milieu}\begin{array}{}\end{array}\begin{array}{}\end{array}M\begin{array}{}\end{array}\mathrm{à\; lʹéchelle\; de\; la\; section}\left({\mathrm{m}}^2\right)\\ {\mathrm{K}}^H=\mathrm{perméabilité\; dans\; le\; milieu}M\mathrm{à\; lʹéchelle\; de\; la\; section}\left({\mathrm{m}}^2\right)\begin{array}{}\end{array}\end{array}$$

The outputs of the acidification model are: $$\begin{array}{l}\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_M^M=\mathrm{average\; concentration\; of\; acid\; in\; the\; medium}\begin{array}{}\end{array}M\left(\mathrm{kg}/{\mathrm{m}}^3\right)\\ \{{\mathrm{VS}}_{\mathit{Aß}}{\}}_H^H=\mathrm{average\; concentration\; of\; acid\; in\; the\; medium}\begin{array}{}\end{array}H\left(\mathrm{kg}/{\mathrm{m}}^3\right)\end{array}$$

$$\begin{array}{l}\{{\mathrm{V}}_{\mathit{\beta }}{\}}_M^M=\mathrm{average\; Darcy\; speed\; in\; the\; middle}\begin{array}{}\end{array}M\left(\mathrm{m}\mathrm{/}\mathrm{s}\right)\\ \{{\mathrm{V}}_{\mathit{\beta }}{\}}_H^H=\mathrm{average\; Darcy\; speed\; in\; the\; middle}\begin{array}{}\end{array}H\left(\mathrm{m}\mathrm{/}\mathrm{s}\right)\end{array}$$

$$\begin{array}{l}\{P{\}}_M^M=\mathrm{average\; total\; pressure\; in\; the\; medium}\begin{array}{}\end{array}M\left(\mathrm{Pa}\right)\\ \{P{\}}_H^H=\mathrm{average\; total\; pressure\; in\; the\; medium}\begin{array}{}\end{array}H\left(\mathrm{Pa}\right)\end{array}$$

$$\begin{array}{l}\{{\mathit{\epsilon }}_{\mathit{\beta }}{\}}_M^M=\mathrm{average\; porosity\; in\; the\; medium}\begin{array}{}\end{array}M\\ \{{\mathit{\epsilon }}_{\mathit{\beta }}{\}}_H^H=\mathrm{average\; porosity\; in\; the\; medium}\begin{array}{}\end{array}H\end{array}$$

$$\begin{array}{l}{\mathrm{K}}^M=\mathrm{permeability\; in\; the\; medium}\begin{array}{}\end{array}\begin{array}{}\end{array}M\begin{array}{}\end{array}\mathrm{at\; the\; section\; level}\left({\mathrm{m}}^2\right)\\ {\mathrm{K}}^H=\mathrm{permeability\; in\; the\; medium}M\mathrm{at\; the\; <figure-callout\; id="2"\; label="section\; level\; m"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">section\; level\; </figure-callout>}\left({\mathrm{m}}^{}\right)\left({\mathrm{}}^2\right)\begin{array}{}\end{array}\end{array}$$

V ₀ =

vitesse initiale d'injection de l'acide

C ₀ =

concentration de l'acide utilisé lors de la stimulation

ε₀ =

porosité initiale

K ₀ =

perméabilité initiale

L =

longueur caractéristique du problème (rayon de la zone acidifié)

µ =

viscosité cinématique (Pa.s)

ν =

coefficient stoechiométrique de la réaction de dissolution

ρ^σ =

masse volumique de la roche(Kg/m³)

K_f =

perméabilité dans le milieu dissout (m²)

The inputs of the acidification model are:

V ₀ =

initial rate of injection of the acid

C ₀ =

concentration of the acid used during the stimulation

ε ₀ =

initial porosity

K ₀ =

initial permeability

L =

characteristic length of the problem (radius of the acidified zone)

μ =

kinematic viscosity (Pa.s)

ν =

stoichiometric coefficient of the dissolution reaction

ρ ^σ =

density of the rock (Kg / m ³ )

K _f =

permeability in the dissolved medium (m ² )

These data are obtained from logs, core measurements or laboratory measurements. These parameters can also come from the geological knowledge of the specialist or from simulations. The initial porosity and permeability are then optimized during an optimization process based on acidification modeling following acid injection into the well.

K _f corresponds to the permeability in the wormhole and its value is therefore very large. It is calculated by analogy with a Poiseuille flow in a wormhole. Taking as characteristic wormholes b characteristic of 1 millimeter, we obtain: $$\begin{array}{l}{K}_f=\frac{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}{12}\begin{array}{}\end{array}\begin{array}{}\end{array}\begin{array}{}\end{array}=8,{33.10}^{-8}{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">m</figure-callout>}}^{\mathrm{2}}\mathrm{in}2D\left(\mathrm{flow\; between\; two\; parallel\; plates}\right)\\ K_f=\frac{{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}{8}=1,{25.10}^{-7}{\mathrm{<figure-callout\; id="2"\; label="m"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">m</figure-callout>}}^2\mathrm{in}3\mathrm{D}\left(\mathrm{flow\; in\; a\; tube}\right)\begin{array}{}\end{array}\end{array}$$

The accuracy of the value assigned to K _f is of little importance from the moment it is much greater than K ₀ .

Some parameters of the acidification model need to be determined prior to acidification modeling.

Concerning the dissolution coefficients g _M and g _H , two different formulations can be constructed through two different approaches: g _{1 M} , g _{1 H} on the one hand and g _{2 M} , g _{2 H} on the other hand. One is a function of concentration and porosity, and the other is dependent on velocity, porosity and local flux of acid.

Term of dissolution boy Wut _1H and g _1M .

Taking average volume gives dissolution terms g _H and g _M non-linear that it is therefore necessary to model. A first approach is to linearize these terms. The terms g _{1 M} and g _{1 H} are thus obtained which depend only on the parameter A. $$\begin{array}{l}{\mathrm{boy\; Wut}}_{}\end{array}$$$$\begin{array}{l}{}_{1H}={\mathit{<figure-callout\; id="1"\; label="\alpha "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_{1H}\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_H^{\mathrm{<figure-callout\; id="1"\; label="H\; boy\; Wut"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">H\; </figure-callout>}}& \\ {\mathrm{boy\; Wut}}_{}{}_{1M}={\mathit{<figure-callout\; id="1"\; label="\alpha "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_{1M}\{{\mathrm{VS}}_{\mathit{Aß}}{\}}_M^M\end{array}$$

With $$\begin{array}{l}{\mathit{<figure-callout\; id="1"\; label="\alpha "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_{1H}=\mathrm{AT}(1-{\mathit{\epsilon }}^H{)}^{2/3}\\ {\mathit{<figure-callout\; id="1"\; label="\alpha "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_{1M}=\mathrm{AT}(1-{\mathit{\epsilon }}^H{)}^{2/3}\end{array}$$

g _1H is the dissolution term for the medium H and g _1M that of the medium M. This expression of the coefficients α _1H and α _1M has the role of taking into account the evolution of the reaction surface by means of the variation of porosity.

After adimensionalisation we obtain: $$\begin{array}{l}{\mathrm{<figure-callout\; id="1"\; label="boy\; Wut"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">boy\; Wut</figure-callout>}}_{1H}^{\text{'}}=\frac{\mathrm{The}}{{\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">V</figure-callout>}}_0}{\mathit{VS}}^H\mathrm{AT}(1-{\mathit{\epsilon }}^H{)}^{2/3}\\ {\mathrm{<figure-callout\; id="1"\; label="boy\; Wut"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">boy\; Wut</figure-callout>}}_{1M}^{\text{'}}=\frac{\mathrm{The}}{{\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">V</figure-callout>}}_0}{\mathit{VS}}^M\mathrm{AT}(1-{\mathit{\epsilon }}^M{)}^{2/3}\end{array}$$

Term of dissolution boy Wut _2H and boy Wut _2M .

$$g_{2H}^{ʹ}=\frac{{\mathit{\alpha }}_{2H}ʹ}{{\varphi }_H}\nabla \mathrm{.}\left({\varphi }_H{\mathrm{Vʹ}}^H{\mathit{fʹ}}^H\right)$$

According to another embodiment, another modeling (term g _{2 M} and g _{2 H} ) based on the observation of the wormholing mechanism is presented. The coefficient g _{2 H} is the dissolution term for medium H and g _{2 M} that of medium M. Its principle is to define the term of dissolution according to the local balance of acid flows, ie of the convective term. This term is null in cases where there is no acid, no flow or when a wormhole passes right through the elementary volume at the well scale (the principle of elementary volume is linked to the scale to which the equation system refers). On the other hand, if a wormhole finishes growing in this volume, the acid flow balance becomes negative and there is dissolution. $${\mathrm{boy\; <figure-callout\; id="2"\; label="Wut"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">Wut</figure-callout>}}_{2M}^{\text{'}}=\frac{{\mathit{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_{2M}\text{'}}{{\phi }_M}\nabla \mathrm{.}\left({\phi }_M{\mathrm{V\text{'}}}^M{\mathit{f\text{'}}}^{\mathrm{<figure-callout\; id="2"\; label="M\; boy\; Wut"\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">M\; </figure-callout>}}\right)$$

$${\mathrm{boy\; Wut}}_{}^{}$$$${}_{2H}^{\text{'}}=\frac{{\mathit{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_{2H}\text{'}}{{\phi }_H}\nabla \mathrm{.}\left({\phi }_H{\mathrm{V\text{'}}}^H{\mathit{f\text{'}}}^H\right)$$

$${\mathit{\alpha }}_{2H}^{ʹ}=({\mathit{\epsilon }}_H{)}^{n1}(\frac{1}{\mathit{\gamma }\Vert {\mathrm{V}}_H^{ʹ}\Vert }{)}^{n2}$$

With $${\mathit{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_{2M}^{\text{'}}=({\mathit{\epsilon }}_M{)}^{\mathrm{not}1}(\frac{1}{\mathit{\gamma }\Vert {\mathrm{V}}_M^{\text{'}}\Vert }{)}^{\mathrm{not}2}$$

$${\mathit{<figure-callout\; id="2"\; label="\alpha "\; filenames="imgf0004.png,imgf0005.png"\; state="\{\{state\}\}">\alpha </figure-callout>}}_{2H}^{\text{'}}=({\mathit{\epsilon }}_H{)}^{\mathrm{not}1}(\frac{1}{\mathit{\gamma }\Vert {\mathrm{V}}_H^{\text{'}}\Vert }{)}^{\mathrm{not}2}$$

Thus, the parameters of the model that must be determined are:

• A, φ_H, H^M, H^H, Δy qui apparaissent dans le modèle double milieu lorsque l'on utilise les termes de dissolution g _1M et g _1H .
• n₁, n₂, γ, φ_H, H^M, H^H, Δy qui apparaissent lorsque l'on utilise les termes de dissolution g _2M et g _{2 H}.
• χ qui apparaît dans l'équation perméabilité / porosité
• ψ le coefficient d'échange entre les deux milieux
• C_H-M la concentration à l'interface entre les deux milieux (Kg/m³)

Thus, the parameters of the dual-medium acidification model that need to be determined are:

• A, φ _H , H ^M , H ^H , Δy which appear in the double-medium model when the terms of dissolution g _{1 M} and g _{1 H are used} .
• n ₁ , n ₂ , γ, φ _H , H ^M , H ^H , Δy appearing when the terms of dissolution g _{2 M} and g _{2 H are used} .
• χ that appears in the permeability / porosity equation
• ψ the exchange coefficient between the two environments
• C _HM concentration at the interface between the two media (Kg / m ³ )

3.2- Determinations of model parameters

All of these parameters are determined by calibration against core-scale simulation results or laboratory tests. These adjustments are presented below:

Calibration of the parameters of the dissolution coefficients

The parameters used in our model are determined by a calibration procedure against reference results covering a wide range of flow. These flows must be chosen from the range of flows for which wormholes are formed. These reference results are on the one hand the exact porosity at the scale of the core, averaged at the well scale, and secondly the pressure field, noted ΔP (t). These can come from either laboratory experiments, such as constant-flow injection studies in a rock sample, or simple core-scale simulations on small-scale domains (scale of carrot).

The determination of the parameters at the scale of the well, making it possible to reproduce the results obtained on the scale of the core, is carried out through an inversion method using a Levenberg-Maquart algorithm ( K. Madsen, HB Nielsen, O. Tingleff, Methods for Non-Linear Least Squares Problems. 2004. Informatics and Mathematical Modeling. Technical University of Denmark . This procedure makes it possible to determine the parameters A, φH, H ^M , H ^H , Δy that appear in the double-medium model using the dissolution terms g _{1 M} and g _{1 H.} In the same manner, this procedure makes it possible to determine the parameters n ₁ , n ₂ , γ , φ _H , H ^M , H ^H , Δy which appear in the double-medium model using the dissolution terms g _{2 M} and g _{2 H.}

These parameter determinations are performed against reference results covering a wide range of flow rates. For a different flow rate, the value assigned to each parameter for the well-scale model is determined by linear interpolation performed by comparing the speed at the scale of the section averaged with the volume V _section , with the speed of the injection simulations simple medium at the carrot scale.

The interpolation of these values as a function of flow allows us to perform large-scale simulations over a large area (well scale).

Setting the parameter χ

This parameter is also determined by a calibration procedure against reference results covering a wide range of selected flow rates in the range. flows for which wormholes are formed. These can come from either laboratory experiments, such as constant-flow injection studies in a rock sample, or simple core-scale simulations on small-scale domains (scale of carrot).

This calibration method is illustrated in Figure 3. To determine the coefficient χ, a calibration can be used with respect to results of simulations at the core scale at a constant rate (SimuC). The flow rate imposed on a sample of the size of a core must be chosen from the range of flows for which wormholes are formed. From these results obtained at the core scale, the porosity and the pressure field (ΔP (t)) are extracted, thus serving as reference results. For different instants of the dissolution, the average at the well scale, of the porosity obtained at the scale of the core, is calculated. This new _exact porosity ε is applied to the relationship between the permeability and the porosity described previously, K (ε, χ): $$K={\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">K</figure-callout>}}_0+\left(K_f-K_0\right)(\frac{{\mathit{\epsilon }}_{\mathit{exact}}-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{1-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{)}^x$$

The permeability thus calculated represents the average permeability (K) at the well scale. The pressure field (P) induced by this permeability is solved using the following relation: $$\nabla \mathrm{.}\left(K\nabla P\right)=0$$

To simplify the determination of the parameter χ, one can also use tests on a linear flow in a homogeneous medium so as to be able to solve the equation (41) in 1D analytically by using the following relation: $$P\left(\mathrm{The}\right)-\mathrm{<figure-callout\; id="0"\; label="P"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">P</figure-callout>}\left(0\right)=\mathit{U\mu }{\displaystyle {\int }_0^{\mathrm{The}}}\frac{1}{K\left(x\right)}dx$$

U is the injection speed.

Thus one obtains the pressure difference between the limits of the domain at different times, that is to say the pressure gradient across the sample (ΔP (t) _exp ). The difference between this gradient and the reference result is then calculated .DELTA.P (t). Depending on the error then measured, the parameter χ is modified (Δχ) accordingly.

Iteratively, an optimum value of χ is obtained which minimizes this error. This optimum value of χ is obtained for a given flow rate. The operation is repeated for different flow rates, chosen from the range of flows for which wormholes are formed. The K (ε) relationship thus parameterized is used for all flow rates during well-scale simulations. If the given flow rate was not used for the evaluation, a value interpolation is performed for the flow rates used to control the given flow rate.

Setting of the exchange parameter ψ between the two media H and M

Both H and M backgrounds interact through an exchange term depending on the pressure difference between these two media. This term makes it possible to model the diversion of the flow of acid towards the dominant wormholes to the detriment of those which is in the medium M.

To determine the exchange term ψ the model is applied to a particular case. The volume is represented by a medium in which acid is injected linearly. Cylindrical wormholes develop there, arranged periodically according to their size. The equivalence between this representation and reality is provided by a parameter Δy which must be determined by calibration. Δy here defines the distance between two dominant wormholes. FIGS. 5A, 5B and 5C show the simplified representation of the distribution of volumes V _H and V _M used in the modeling of the exchange term: FIG. 5A shows the actual dissolution figure, FIG. 5B illustrates the simplified representation, and Figure 5C illustrates the basic pattern. This periodic representation makes it possible to represent the terms of exchanges for the entire domain from its description on a basic pattern. In this description the volume V _section contains n times the basic pattern. $$\begin{array}{l}\frac{1}{V_{\mathit{section}}}{\displaystyle \underset{{\mathrm{AT}}_{H-M}}{\int }}{\mathbf{V}}_{\mathit{\beta }}\mathrm{.}{\mathbf{not}}_{M}ds=\frac{1}{V_{\mathit{section}}}{\displaystyle \underset{{\mathrm{AT}}_{H-M}}{\int }}\left(-\frac{\mathbf{K}}{\mathit{\mu }}\mathrm{.}\nabla P\right){\mathbf{not}}_{M}ds\\ \approx \frac{1}{\mathit{\Delta }\begin{array}{}\end{array}\mathit{x}\mathrm{.}\mathrm{not}\mathrm{.}\mathit{\Delta }\begin{array}{}\end{array}\mathit{there}}\left(-\frac{K_{{\mathit{eq}}_{-}\mathrm{there}}}{\mathit{\mu }}\frac{\partial P}{\partial \mathrm{there}}\right)2.\mathrm{not}\mathrm{.}\mathit{\Delta }\begin{array}{}\end{array}\mathit{x}\end{array}$$

The pressure gradient at the interface is evaluated by dividing the difference of the average pressures of the two media by the height Δ y / 2 of the base pattern (Figure 5C). The equivalent permeability K _{eq_y} is a variable calculated by making a harmonic mean of the transverse permeabilities ( K _{y_H} and K _{y_M} ) of the two media. $$\frac{1}{V_{\mathit{section}}}{\displaystyle \underset{{\mathrm{AT}}_{H-M}}{\int }}{\mathbf{V}}_{\mathit{\beta }}\mathrm{.}{\mathbf{not}}_{M}ds\approx \frac{4K_{{\mathit{eq}}_{\mathit{-}}\mathit{there}}\left(\{P{\}}_M^M-\{P{\}}_H^H\right)}{{\mathit{\mu }\begin{array}{}\end{array}\mathit{\Delta }\begin{array}{}\end{array}\mathit{there}}^2}$$

With $$K_{{\mathit{eq}}_{\mathit{-}}\mathit{there}}=\frac{K_{{\mathrm{there}}_{-}H}\mathrm{.}{K}_{{\mathrm{there}}_{-}M}}{{\phi }_M{K}_{{\mathrm{there}}_{-}H}+{\phi }_H{K}_{{\mathrm{there}}_{-}H}}$$

$$K_{{\mathit{y}}_{\mathit{-}}\mathit{M}}=\frac{K_0\mathrm{.}{K}_f}{\left(1-A_M\right)K_f+A_M{K}_0},A_M=\frac{{\mathit{\epsilon }}_M-{\mathit{\epsilon }}_0}{1-{\mathit{\epsilon }}_0}$$

We must now define the transverse permeabilities. For this we use an ideal representation of each medium, modeling them as blocks crossed by a certain amount of wormholes of constant section. By applying Darcy's law to this representation to determine K _{y_M} and K _{y_H} , we obtain: $$K_{{\mathit{there}}_{\mathit{-}}\mathit{H}}=\frac{{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">K</figure-callout>}}_0\mathrm{.}{K}_f}{\left(1-{\mathrm{AT}}_H\right)K_f+{\mathrm{AT}}_{\mathrm{<figure-callout\; id="0"\; label="H\; \; K"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">H\; </figure-callout>}}{K}_{}{}_0},{\mathrm{AT}}_H=\frac{{\mathit{\epsilon }}_H-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{1-{\mathit{\epsilon }}_0}$$

$$K_{{\mathit{there}}_{\mathit{-}}\mathit{M}}=\frac{{\mathrm{<figure-callout\; id="0"\; label="K"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">K</figure-callout>}}_0\mathrm{.}{K}_f}{\left(1-{\mathrm{AT}}_M\right)K_f+{\mathrm{AT}}_{\mathrm{<figure-callout\; id="0"\; label="M\; \; K"\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">M\; </figure-callout>}}{K}_{}{}_0},{\mathrm{AT}}_M=\frac{{\mathit{\epsilon }}_M-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}{1-{\mathit{<figure-callout\; id="0"\; label="\epsilon "\; filenames="imgf0005.png,imgf0006.png"\; state="\{\{state\}\}">\epsilon </figure-callout>}}_0}$$

One can finally write the term Ψ in the following form, according to the parameter Δy. $$\mathit{\psi }=\frac{4K_{{\mathit{eq}}_{\mathit{-}}\mathit{there}}}{{\mathit{\mu }\begin{array}{}\end{array}\mathit{\Delta }\begin{array}{}\end{array}\mathit{there}}^2}$$

Calibration of the concentration at the interface between the two media C _HM

$${\mathrm{Si}\begin{array}{}\end{array}\mathit{Pʹ}}^M<{\mathit{Pʹ}}^H{\mathrm{alors}\begin{array}{}\end{array}C}_{H-M}={\mathit{Cʹ}}^H$$

With regard to the C _HM concentration used with the exchange term in the double-medium model, either the concentration C ' ^M or the concentration C' ^{H are used} according to the values of P ' ^M and P' ^H : $${\mathrm{Yes}\begin{array}{}\end{array}\mathit{P\text{'}}}^M\ge {\mathit{P\text{'}}}^H{\mathrm{so}\begin{array}{}\end{array}\mathrm{VS}}_{H-M}={\mathit{VS}}^M$$

$${\mathrm{Yes}\begin{array}{}\end{array}\mathit{P\text{'}}}^M<{\mathit{P\text{'}}}^H{\mathrm{so}\begin{array}{}\end{array}\mathrm{VS}}_{H-M}={\mathit{VS}}^H$$

At each time step, we start by solving the pressure field. It is therefore possible to determine C _HM before calculating the transport of the acid species.

3.3- Modeling of acidification

avec :

- Q =

débit injecté dans la formation (m³.s^-1)

- k =

perméabilité dans le réservoir (m²)

- B =

facteur de volume

- r _w =

rayon du puits (m)

- r_e =

rayon du réservoir

- S =

skin

- ΔP =

différence de pression entre le puits et le réservoir

- µ =

viscosité cinématique (Pa.s)

Equations 6 to 15 define the acidification model according to the invention, the input data and the parameters of this model are determined experimentally. This model then makes it possible to determine the porosity and the permeability of the medium after the injection of acid into the well. From these new porosity and permeability, a factor called "skin" is determined. The skin measures the loss of load caused by the damage of a well of radius r _w . Consider these losses of loads limited to a radius r _s in which the permeability is worth k _s while the permeability of the reservoir is worth k. The skin S is calculated from the following formula: $$Q=\frac{2\mathit{\pi kh}}{\mathit{B\mu }}\frac{\mathit{\Delta }\begin{array}{}\end{array}\mathit{P}}{\ln \frac{r_e}{r_w}+S}$$

with:

- Q =

flow injected into the formation (m ³ .s ^-1 )

- k =

permeability in the tank (m ² )

- B =

volume factor

- r _w =

well radius (m)

- r _e =

tank radius

- S =

skin

- Δ P =

pressure difference between the well and the reservoir

- μ =

kinematic viscosity (Pa.s)

The parameters k, B, r _w , r _e of this equation being assumed to be known, and the simulator making it possible to know the injected flow rate Q and the pressure field ΔP, the skin S can be calculated from this formula. In general, the skin of a well is evaluated from a well test. When the skin is positive, the well is damaged. The treatment decreases the skin up to sometimes make it negative.

3- Simulation of the production of the well

From the skin thus obtained, a reservoir simulation is carried out, well known to specialists, using a reservoir simulator. This simulation provides, among other things, an estimate of well production.

4- Optimization of acid injection parameters

The reservoir simulation thus provides an estimate of the production from the skin, which has itself been obtained from the modeling of the acidification. To improve the production, it is sufficient to modify the input parameters of the acidification model at the well-scale, that is to say, the injection speed, the acid volume, the C ₀ concentration of the acid used during the stimulation, and the identification of the zones to be treated defined by their initial porosity ε ₀ and their initial permeability K ₀ .

5- Optimized acid stimulation of the well to increase its productivity

On the basis of the parameters thus optimized, that is to say, making it possible to obtain maximum production of the well, acid stimulation of the well is carried out by injecting acid under the optimum conditions in terms of injection speed, volume and concentration C ₀ of acid used during the stimulation, identification of the zones to be treated ...

Application example

According to an exemplary application of the method according to the invention, a well-rate injection of acid is simulated at a constant rate on a rock sample of a length of 2m, 40cm in width and 40cm in height.

Meshing and initialization:

• Vitesse initiale d'injection V₀ = 1.10^-4 m/s
• Concentration initiale C₀ = 210 Kg/m3
• Porosité initiale ε₀ = 0,36
• Perméabilité initiale K₀ = 2.318.10^-12 m²
• Viscosité cinématique µ = 1.10^-3 Pa/s
• Masse volumique de la roche ρ^σ = 2160 Kg/m3
• Perméabilité dans le milieu dissout K_f = 8,331.10^-8 m²
• Coefficient stoechiométrique massique ν = 1
• Longueur caractéristique du problème L = 0,1 m

After having meshed the sample using a Cartesian mesh (in this case of application the mesh is cartesian and non-radial as in the case of a well for example), one determines or defines the data of entry of the model:

• Injection initial speed V ₀ = 1.10 ^-4 m / s
• Initial concentration C ₀ = 210 Kg / m3
• Initial porosity ε ₀ = 0.36
• Initial permeability K ₀ = 2.318.10 ^-12 m ²
• Kinematic viscosity μ = 1.10 ^-3 Pa / s
• Density of the rock ρ ^σ = 2160 Kg / m3
• Permeability in the dissolved medium K _f = 8,331.10 ^-8 m ²
• Massive stoichiometric coefficient ν = 1
• Characteristic length of the problem L = 0.1 m

Experimental determination of the parameters:

Reference results are determined using core-scale simulations using the model developed by Golfier F. et al., "On The ability of a Darcyscale model to capture wormhole training during the dissolution of a porous media. Journal of Fluid Mechanic. 547, 213-254. 2002 . These simulations frame the flow rate used later for the well-scale simulation. A series of simulations is thus performed by using a single core-scale model on a domain that represents a small portion of the domain to be simulated over a range of flow sufficiently large to reproduce the different types of possible dissolution patterns. (compact front, conical wormhole, dominant wormhole, branched wormholes and uniform dissolution). The dimensions of the domain are 25 cm long, 40 cm high and 1 mm high.

To determine the coefficient χ related to the permeability / porosity relationship, the pressure and porosity results from the carrot-scale simulations to which the procedure detailed above is applied are used. This gives the optimum value χ = 3.08.

The well-scale model is then used on a field equivalent to that used on a small scale, applying the same injection conditions. For each core-scale simulation at a given rate, an optimization algorithm is used to determine the equivalent parameters used by our well-scale model for each of these flows. The results are shown in Tables 1 and 2.

Table 1 shows the values of the parameters of the model using the dissolution terms g _{1 M} and g _{1 H} for different injection speeds. V ₀ (m / s) AT φ _H Δy (m) H ^M H ^H 9,27E-08 0.0023 0.5 0.2 1 1 4,64E-06 0,001 0.05 0.0041 1.1 1.48 9,27E-06 0.0023 0.08459 0.005 1.1939 1,461 2,32E-05 0.04 0.08459 0.005884 1.16373 1.2 4,64E-05 0.04 0.0769 0.006 1.13 1,207 9,27E-05 0.04 0.07 0.006 1.13 1.3 1,85E-04 0.04773 0.0697 0.01 1 1,226 9,27E-04 .3936 0,136 0.1 1.0012 2.99 9,27E-03 7 0.2 .1366 1.5105 3

Table 2 shows the values of the parameters of the model using the dissolution terms g _{2 M} and g _{2 H} for different injection speeds. V ₀ (m / s) n ₁ φ _H Δy (m) H ^M H ^H γ n ₂ 9,27E-08 0 0.5 10 1 1 0.667 0 4,64E-06 0,265 0.499 .1783 3.8 4.3 0.755 0.83 9,27E-06 0,345 0.499 0.143 2.9722 3,962 0.71 0.8 2,32E-05 0,345 0.5 0.0938 2.5 3.5 0.68 0.55 4,64E-05 0.339 0.5 0.04859 2.47 3.35 0.667 0.5 9,27E-05 0.3 0.5 0.0344 2,479 3.3 0.705 0.45 1,85E-04 .2918 0.48 0.0678 2,522 3.15 0.8 0.35 9,27E-04 0.149 .4537 .1157 2.5742 2.8547 .8858 .2902 9,27E-03 0.0454 .4148 .1852 2,797 2,623 0.902 .2728

To determine the well-scale parameters to be used for the acid injection, interpolation of the values obtained in the previous step is performed. For an injection speed of 1.10-4 m / s, the following parameters are obtained:
- With the terms of dissolution g _{1 M} and g _{1 H} : A = 0.0406 φ _H = 0.07 H ^M = 1.192 H ^H = 1.447 Δy = 0.006314 - With the terms of dissolution g _{2 M} and g _{2 H} : n ₁ = 0.299 n ₂ = 0.4421 γ = 0.7124 φ _H = 0.498 H ^M = 2482 H ^H = 3.288 Δy = 0.037

Well-scale acidification modeling:

To model the acidification, the double-medium model according to the invention is used (equations 6 to 15). The sample is homogeneous in sectional porosity and permeability. For this reason the cross-sectional model is applied in one dimension, in the direction and direction of the injection.

Figures 6 and 8 show the pressure difference between the input and the output of the sample, ΔP expressed in pascal, as a function of time t expressed in seconds.

Figures 7 and 9 show the porosity ε of the sample as a function of the abscissa x of the sample in meters.

Figures 6 and 7 illustrate the results for the model using the dissolution terms g _{1 M} and g _{1 H.} The breakthrough time is 4 hours and 2 minutes. The volume of acid injected is 2.677.10 ^-1 m ³ .

Figures 8 and 9 illustrate the results for the model using the terms g _{2 M} and g _{2 H.} The breakthrough time is 3 hours and 53 minutes. The volume of acid injected is 2.244 × 10 ^-1 m ³ .

Both models show a sharp drop in pressure for a small increase in porosity, which is characteristic of wormholing. They also show that it takes approximately 4 hours of injection at the injection speed 1.10 ^-4 m / s to obtain a wormhole with a length of two meters, a characteristic length of a well stimulation by acid injection. .

Claims (12)

A method for modeling acidification in a porous medium following the injection of an acid, wherein said medium is represented by a dual medium model, characterized in that the method comprises the following steps: a) constructing said dual-medium model considering a first sub-environment conducive to dissolution breakthroughs, and a second sub-medium that is not conducive to breakthroughs; by carrying out, for each of said sub-media, a description on a metric scale of the acid transport, the conservation of the mass of said sub-mileu and the conservation of the mass of said acid; by describing an acid transfer from one sub-medium to the other sub-medium; b) initializing said dual-medium model from experimental setups; c) modeling, using said dual-medium model, said acidification, by determining representative physical parameters of said porous medium and physical parameters relative to the injected acid. The method of claim 1, wherein said representative physical parameters of said porous medium are selected, for each of said sub-media, from the following parameters: average porosity, metric scale permeability, and average total pressure. Method according to one of the preceding claims, wherein said physical parameters relating to the acid are selected, for each of said sub-media, from among the following parameters: the average concentration of the acid, the mean Darcy velocity. The method according to one of the preceding claims, wherein said description is performed using equations obtained by averaging the metric scale of equations describing the propagation of an acid in a single medium model to a centimeter scale. The method of claim 4, wherein said equations include a dissolution term. The method of claim 5, wherein said dissolution term is defined as the product of an average concentration of the acid at the metric scale by a coefficient dependent on a local rate of the acid. The method of claim 5, wherein said dissolution term is defined as the product of a parameter by the divergence of a product between a concentration of the acid, a fractional flow function and a velocity vector. The method of claim 5, wherein said parameter of said dissolution term is dependent on a standard of local acid velocity and metric scale average porosity. Method according to one of the preceding claims, wherein said calibration procedure is based on simulations at scales smaller than the metric scale. Method according to one of the preceding claims, wherein said calibration procedure is based on constant flow rate acid injection studies in a sample of said medium. Method according to one of the preceding claims, wherein said porous medium is a carbonated reservoir traversed by a well, the acid injection being carried out to stimulate hydrocarbon production by said well, and in which optimal parameters are determined. of the acid injection, by carrying out the following steps: a) constructing a mesh of said well and its vicinity; b) initial parameters of the acid injection are defined; c) determining, by modeling the acidification due to the injection of the acid, at least the following representative physical parameters of said tank: a porosity and a permeability of said reservoir after injection of the acid; d) the production of the well is simulated according to said porosity and said permeability using a reservoir simulator; e) modifying said initial parameters and repeating in step c) until a maximum production is obtained. The method according to claim 11, wherein said initial parameters are selected from at least one of the following parameters: the injection rate of the acid, the initial injection rate, the volume of acid injected, the concentration of the acid used during the stimulation, the areas to be treated.

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