WO2023084265A1

WO2023084265A1 - Process for determining up-scaling parameters of an acidification injection in a geological formation, related use and system

Info

Publication number: WO2023084265A1
Application number: PCT/IB2021/000765
Authority: WO
Inventors: Priyank MAHESHWARI; Oussama GHARBI; Vincent Peyrony
Original assignee: Totalenergies Onetech
Priority date: 2021-11-12
Filing date: 2021-11-12
Publication date: 2023-05-19

Abstract

The process comprises repeating for the at least one acid radial injection flow rate, at successive time steps, the sub-steps of a three-dimensional radial injection simulation algorithm, until an acid breakthrough condition occurs in the block. Once the acid breakthrough condition is met in the block, the process comprises obtaining a volume of acid injected at breakthrough, a porosity profile at breakthrough, a permeability profile at breakthrough and/or a velocity profile at breakthrough, and obtaining at least an up-scaling parameter of the acidification reaction from the volume of acid injected at breakthrough, from the porosity profile at breakthrough, from the permeability profile at breakthrough and/or from the velocity profile at breakthrough.

Description

Process for determining up-scaling parameters of an acidification injection in a geological formation, related use and system

The present invention concerns a process for determining up-scaling parameters of an acidification injection carried out in a geological formation from a well, the process being carried out by a determination system.

The geological formation preferentially contains carbonate rocks.

Acid stimulation is widely used in fluid exploitation of hydrocarbons to enhance the transport of fluid from a reservoir to a production well and hence increase fluid production.

In particular, the acid stimulation removes wellbore damage, and enhances the permeability near the wellbore region.

Acidification of carbonate rocks is a common practice to remove near wellbore damage. In this process, an acidic solution (typically hydrochloric acid, emulsified acid, and/or polymeric acid) is injected through a wellbore in the formation where it reacts with the carbonate rock.

The injection of acidic solution into carbonate porous media is a reactive transport phenomenon. Chemical dissolution kinetics are coupled with transport dynamics (advection and diffusion) in heterogeneous (carbonate) porous media.

Depending on the injection flowrate, hence the interstitial velocities, a non-uniform dissolution characterized by highly conductive channels could be formed. These channels are commonly referred to as “wormholes”. As a result, the wellbore becomes well connected to the reservoir and hydrocarbons can easily flow back to the wellbore through these channels.

In order to design acid stimulation treatments operations, it is known to simulate the effect of acid stimulation at well scale before the actual acid injection is being carried out.

However, given the complex phenomena which are involved in the well, in the formation block around the well, and within the formation, the simulation at well scale does not always provide reliable results.

In prior art, the physical parameters for the simulation are calibrated in particular using a pore volume to breakthrough (PVBT) and a transient pressure determined from a linear flow experiment using a small core of geological formation.

The linear flow experiment generally accurately determines the diffusion-convection and dissolution phenomena occurring at a laboratory scale in a linear injection mode.

However, the experiment is poorly representative of the physical chemical phenomena actually occurring at greater scales, in particular in a well in which acid injection occurs radially from the well. In particular, within the formation, a pattern of radial channels generally forms around the injection point in the well. The linear behavior of acid transport in the core is not adapted to model the radial injection of acid which occurs in a real well.

Acid stimulation in carbonates (in particular reactive transport) is a scale-dependent phenomenon: it notably depends on heterogeneities, effective reaction rate, surface area available for reaction, etc.

Therefore, the experimental up-scaling parameters fed to the well scale model are often not accurate, and give results which, in some instances, may not match the actual experiments that are being carried out.

In order to improve the modelling, WO2020/219629 suggests scaling the measured fluid fraction and effective reaction rate constants from linear flow to radial flow to produce scale radial flow data and using the scale radial flow data in the axial acid simulation models. The latter method takes into account the radial flow. However, it is still not totally satisfactory.

Indeed, the model which is used in WO2020/219629 is a mechanistic model which is based on a mass balance in wormholes generated in the formation. It only calculates acid concentration and fluid velocity at the wormhole tip. The latter model is moreover applicable only to wormhole regime, whereas many other regimes may occur when acid diffuses within the formation. Moreover, wormholes are only described as tubular reactors with a constant diameter which is not an accurate representation. Moreover, the model is not applicable to non-Newtonian fluids and assumes a constant velocity into wormholes with negligible transverse pressure gradients.

Other methods are disclosed in US20160025895, US7561998 or EP1832711 .

The accuracy of the up-scaling to well scale hence remains unsatisfactory and is an area for potential improvement. This is, in particular, applicable to reactive transport which is a scale-dependent phenomenon.

One aim of the invention is to obtain a process producing acid stimulation up-scaling parameters which more realistically reflect the physical chemical behavior in a formation around a well, so that the design of acid stimulation in a real well can be more realistic and efficient.

To this aim, the subject matter of the invention is a process of the above-mentioned type, comprising the following steps:

- defining at least one acid radial injection flow rate into a block of geological formation comprising a center injection hole and a surrounding formation region surrounding the center injection hole; - repeating for the at least one acid radial injection flow rate, at successive time steps, the following sub-steps of a three-dimensional radial injection simulation algorithm, until an acid breakthrough condition occurs in the block:

* obtaining an updated pressure profile in the block at a further time step, from a permeability profile in the surrounding region obtained from a porosity profile in the surrounding region at a former time step, from an acid concentration profile in the surrounding region at a former time step and from coefficients representative of the geological formation and representative of acid rheology and reactivity;

* determining from the updated pressure profile in the block and from the permeability profile, an updated acid velocity profile at the further time step;

* determining an updated acid concentration profile in the block at the further time step, and an updated porosity profile at the further time step, from diffusion-convection of the acid and on dissolution reactions occurring in the surrounding region;

* determining an updated permeability profile in the block at the further time step, using a correlation with a correlation equation correlating the updated permeability profile to the updated porosity profile, to a predetermined pore broadening parameter and to a poreconnectivity parameter;

- once the acid breakthrough condition is met in the block, obtaining a volume of acid injected at breakthrough, a porosity profile at breakthrough, a permeability profile at breakthrough and/or a velocity profile at breakthrough, and

- obtaining at least an up-scaling parameter of the acidification reaction from the volume of acid injected at breakthrough, from the porosity profile at breakthrough, from the permeability profile at breakthrough and/or from the velocity profile at breakthrough.

The process according to the invention may comprise one or more of the following feature(s), taken solely, or according to any technical feasible combination:

- the pore broadening parameter and the pore connectivity parameters are obtained through calibration using at least two linear flow experiments in which an injection of acid is carried out linearly into a sample of geological formation;

- the correlation comprises a calculation of the ratio of a updated local permeability at any point in the surrounding region to a former local permeability at any point in the surrounding region, from at least a first characteristic ratio calculated from the updated local porosity at any point in the surrounding region and from a former local permeability at any point in the surrounding region powered to the pore broadening parameter and a second characteristic ratio calculated from the initial former porosity at any point in the surrounding region and from the updated porosity at any point in the surrounding region powered to the second pore connectivity parameter; - the coefficients representative of the geological formation include rock density, solidfluid interfacial surface area, and coefficients representative of the acid rheology and reactivity include viscosity, dissolving power of the acid, local mass transfer coefficient, and reaction rate constant;

- the local mass transfer coefficient is calculated at each successive time step from a mean pore radius updated at each time step using the pore broadening parameter, the solid fluid interfacial surface area being updated at each successive time step using the pore broadening parameter;

- the velocity profile at the further time step is determined from the updated pressure profile at the further time step and from the permeability profile through Darcy’s law;

- the determination of the updated concentration profile and of the updated porosity profile at the further time step comprises an operator splitting including first using a diffusion convection operator, taking into account acid diffusion-convection without taking into account acid reaction, and second using a reaction operator, taking into account acid reaction without taking into account acid diffusion-convection;

- the diffusion-convection operator is given by the following equations:

where D_e is the effective dispersion tensor of the acid, Cf^n+1/2 is the acid concentration profile at an intermediate time step between the former time step and the further time step, u_n+i is the velocity profile at the further time step, £^n+1/2 is the porosity profile at the intermediate time step, p_s is the geological formation (14) density and a_c is the dissolving power of the acid;

- the reaction operator is given by the following equations:

d£ⁿ⁺¹ _ k_sk_ca_va_{c n+1} dt (k_s + k_c)p_s ^r where Cfⁿ⁺¹ is the acid concentration profile at the further time step, £ⁿ⁺¹ is the porosity profile at the further time step, p_s is the geological formation density, a_c is the dissolving power of the acid, k_s is an acid reaction rate constant, k_c is a local mass-transfer coefficient, and a_v is a solid-fluid interfacial area;

- the sub-steps of the radial injection simulation algorithm are carried out for a plurality of acid injection flow rates, the obtaining of an up-scaling parameter comprising, obtaining a pore volume to breakthrough associated with each radial injection flow rate and determining from each of the pore volume to breakthrough value an optimum pore to breakthrough value, an optimum interstitial velocity to breakthrough, and/or at least a rock dimension, the at least one up-scaling parameter being chosen among the optimum pore to breakthrough value, the optimum interstitial velocity to breakthrough, and/or the at least a rock dimension;

- the process comprises determining at least one radial dissolution pattern from the porosity profile at breakthrough, and calculating a wormhole density, a wormhole diameter, and/or a radial wormhole penetration length from the radial dissolution pattern, the at least one up-scaling parameter being chosen among the wormhole density, the wormhole diameter, and the radial wormhole penetration length;

- the sub-steps of the radial injection simulation algorithm are carried out for a plurality of acid injection flow rates, the up-scaling parameter obtaining comprising calculating a fitting parameter to correlate wormhole density, wormhole diameter, and/or wormhole radial penetration length to interstitial injection flow rate, from the wormhole density, wormhole diameter, and/or wormhole radial penetration length calculated from dissolutions patterns obtained at the plurality of acid injection flow rates;

- the acid injected in the three-dimensional radial injection simulation algorithm is a non-Newtonian acid, the updated pressure profile in the block at a further time step including a calculation of an acid effective viscosity of the non-Newtonian acid.

The invention also relates to the use of up-scaling parameters obtained from the process for determining up-scaling parameters of an acidification reaction as defined above, to determine a simulated acidification pattern in a geological formation from a well and advantageously to inject acid in the geological formation from the well, from the simulated acidification pattern in the well.

The invention also relates to a system for determining up-scaling parameters of an acidification injection carried in a geological formation from a well comprising :

- a initialization module configured to define at least one acid radial injection flow rate into a block of geological formation comprising a center injection hole and a surrounding formation region surrounding the center injection hole; - a breakthrough calculation module configured to repeat, for the or each acid radial injection flow rate, the following sub-steps of a three-dimensional radial injection simulation algorithm at successive time steps, until an acid breakthrough condition occurs in the block:

* determining an updated permeability profile in the block at the further time step, using a correlation equation, the correlation equation correlating the updated permeability profile to the updated porosity profile, to a predetermined pore broadening parameter and to a pore connectivity parameter;

- an up-scaling parameter obtaining module configured, once the acid breakthrough condition is met in the block obtaining a volume of acid injected at breakthrough, a porosity profile at breakthrough, a permeability profile at breakthrough and/or a velocity profile at breakthrough, and configured to obtain at least an up-scaling parameter of the acidification reaction from the volume of acid injected at breakthrough, from the porosity profile at breakthrough, from the permeability profile at breakthrough and/or from the velocity profile at breakthrough.

The system according to the invention may comprise one or more of the following feature(s), taken solely, or according to any technical feasible combination:

- the up-scaling parameter obtaining module is configured to activate the breakthrough determining module at several acid radial injection flow rates to provide a pore volume to breakthrough value at each acid radial injection flow rate and is configured to determine an optimum pore volume to breakthrough value and an optimum interstitial velocity from the different pore volume to breakthrough values determined by the breakthrough determining module;

- the up-scaling parameter obtaining module is configured to obtain a radial dissolution pattern from the porosity profile determined at breakthrough by the breakthrough determining module and to calculate at least one up-scaling parameter chosen among a wormhole density, a wormhole diameter, a wormhole radial penetration depth, from the radial dissolution pattern.

The invention will be better understood, upon reading of the following description, given solely as an example, and made in reference to the appended drawings, in which:

- Figure 1 is a schematic view of a system according to the invention to carry out the process according to the invention;

- Figure 2 is a view of a formation sample in which a radial acid injection is simulated;

- Figure 3 is a flow chart explaining a calculation step of the process according to the invention;

- Figure 4 is a pore volume to breakthrough versus interstitial acid injection velocity obtained by carrying out the calculation step of figure 3 for various interstitial acid injection velocities;

- Figure 5 is an illustration of a dissolution pattern obtained through the process of the invention.

A first system 10 for determining up-scaling parameters of an acidification injection carried out by acid injection in a geological formation 14 from a well 12 is illustrated in figure 1.

The well 12 is for example a well of a fluid exploitation installation, in particular of an oil and gas production installation. It is here represented vertical. However, it can comprise inclined and/or horizontal regions.

The well 12 extends from a surface of the ground into the geological formation 14, the surface of the ground being onshore or offshore, below a body of water.

The geological formation 14 advantageously comprise a fluid reservoir from which fluid production in the well can be enhanced by acid stimulation.

At the location of the geological formation 14, the well 12 is for example an open bore in the geological formation 14. In a variant, the well 12 has an inner casing (not shown), which is perforated at least a location to allow acid to flow from the well 12 towards the geological formation 14.

Acid stimulation can be carried out in the well 12 by transporting a flow of acid in the well 12 to the geological formation 14 where it is injected at a radial injection flow rate IFR. The term “radial” is used with regards to the local axis of the well at the injection point. In a simplified version, if the well is vertical, the flow is purely radial. In a horizontal well, the injection is radial but the flow can then be pseudo-radial.

During acid stimulation, it is assumed here that acid will radially flow from the well 12 into the geological formation 14. At first, the acid flow is hindered by the geological formation 14, which creates a counter pressure. The geological formation 14 is for example a reservoir containing calcites, dolomites and anhydrites. It contains carbonates susceptible of being acidified and dissolved to increase the porosity.

When a “breakthrough” occurs, the permeability of the geological formation 14 substantially increases, and in some occurrences, a network of channels, referred to as “wormholes”, is created in the geological formation 14.

The amount of acid which must be injected in the geological formation 14 to reach breakthrough is quantified by a known parameter referred to as pore volume to breakthrough or “PVBT”. The pore volume to breakthrough is equal to the ratio of total volume of acid which is injected at breakthrough, over the initial volume of pores.

The determination system 10 is intended to provide up-scaling physical chemical parameters to an acid stimulation system 16.

As shown schematically in figure 1 , the acid stimulation system 16 comprises a simulator 16A able to numerically simulate the acid injection process and the effect in the geological formation 14, at the well scale, from the up-scaling parameters obtained from the determination system 10, at a scale of a geological formation block.

The acid stimulation system 16 further comprises an acid injector 16B for injecting acid in the well, from the results of the acid stimulation simulation carried out by the simulator 16A. The simulation is from an acidification model.

The acid stimulation simulator 16A for example comprises a calculator having a processor and a memory comprising a simulation software able to be executed by the processor. It is able to provide parameters such as transverse pressure response to an acid injection, wormhole fronts, overall skin, injectivity index versus time, pressure, injection rate versus time, total skin versus time, fluid flux versus distance from heel, wormhole penetration versus distance from heel, acid volume versus distance from heel, skin versus distance from heel.

An acid stimulation simulator 16A is well-known in the art. Example of simulators are for example available from various companies for example at the following websites: https://carboceramics.com/products/software-platforms-data-management/stimpro https://www.slb.com/completions/stimulation/stimulation-optimization/kinetix-matrix- stimulation-design-software https://www.halliburton.com/en/products/stim-2001 https://www.simworx.com.br/matrix-treatment-simulator-matrix/ The determination system 10 comprises a radial injection calculator 18 and an up- scaling parameters calculator 19 able to calculate up-scaling parameters from three- dimensional radial injection simulations carried out by the radial injection calculator.

The radial injection calculator 18 comprises at least a processor 20 and at least a memory 22 comprising software modules able to be executed by the processor 20.

The radial injection calculator 18 further comprises a man-machine interface 23. The man-machine interface 23 includes for example at least a screen, a keyboard, and a controller such as a mouse or touchscreen to interact with a user of the radial injection calculator 18, for the initialization of the system, and to provide the results of the calculation carried out by the radial injection calculator 18.

As described below, the radial injection calculator 18 is configured to carry out three dimensional simulations of a radial acid injection from a center hole 26 provided in a block 24 of geological formation 14, until breakthrough occurs in the block 14. The simulations can be carried out taking into account the acid is a Newtonian fluid or preferably taking into account the acid is a non-Newtonian fluid.

The dimensions of the block 24 are at the meter scale. This means that the block 24 advantageously has a maximum radius from the center hole 26 smaller than 1 m, for example between 1 cm and 30 cm, in particular between 1 cm and 20 cm.

The block 24 also has a maximum height smaller than 1 m, in particular smaller than 60 cm.

As shown in figure 2, the block 24 of geological formation 14 in which the simulation of radial acid injection is simulated has a center hole 26 having the same radial dimensions as the well 12 in the geological formation 14.

The center hole 26 is surrounded by a surrounding region 28 of geological formation 14. The radial injection of acid in the surrounding region 28 is advantageously creating wormholes at breakthrough in the surrounding region 28. In variant, it does not create wormholes at breakthrough. The determination system 10 is able to simulate radial acid injection in the surrounding region 28 at different regimes of acid injections, for example face, conical, dominant and uniform.

As shown in figure 1 , the radial injection calculator 18 comprises an initialization module 30, able to acquire initialization parameters comprising geometric parameters of the block 26, geological formation 14 parameters, acid rheology and reaction parameters, and calibrated values of pore connectivity parameter y and pore broadening parameter p, the latter parameters being obtained from two linear flow calibration experiments.

The radial injection calculator 18 further comprises a breakthrough calculation module 32 which is able to carry out three-dimensional radial flow simulations at least a radial injection flow rate IFR, from the initialization parameters acquired by the initialization module 30 until a breakthrough is achieved.

The breakthrough calculation module 32 is able to produce a set of physical values in the simulated block 24 at breakthrough, including in particular local porosity £BT(P, 0, z) at any position p, 0, z in the block 24, and local permeability K_BT(P, 0, z) at any position in the block 24, as well as total acid injected volume TABT.

The set of local porosities £BT(P, 0, z) is referred to as a porosity profile. The set of local permeabilities KBT(P, 0, z) is referred to as a permeability profile.

The up-scaling parameters calculator 19 also comprises at least a processor 33A and a memory 33B comprising software modules able to be executed by the processor 33A.

The up-scaling parameters calculator 19 further comprises a man-machine interface 33C. The man-machine interface 33C includes for example at least a screen, a keyboard, and a controller such as a mouse or touchscreen to interact with an user of the up-scaling parameters calculator 19, for the initialization of the system, and to provide the results of the calculation carried out by the up-scaling parameters calculator 19.

The up-scaling parameters calculator 19 comprises an up-scaling parameter obtaining module 34 able to use the results of the breakthrough calculation module 32 to determine up-scaling parameters, including an optimum pore volume to breakthrough value PVBT_opt, an optimum interstitial acid injection velocity value IV_opt,(or interval of values if a plateau is present), optimized rock dimension parameters, and if applicable, a wormhole density, wormhole diameter and wormhole penetration length radially inside the block 24.

The up-scaling parameter obtaining module 34 is able to transmit at least one of these up-scaling parameters to the simulator 16A of the acid stimulation system 16, to be used in order to simulate an acid stimulation in the well 12, and further, to carry out an actual acid stimulation in the well 12 via the injector 16B, from the results of the simulation by the simulator 16A.

In particular, the results of simulations are able to provide an optimum injection flow rate to produce the most efficient breakthrough in the geological formation 14 in order to increase fluid production, with the minimal required acid volume. The acid stimulation system 16 can then be used in the axial well 12 to carry out the acid stimulation at the optimum injection flow rate.

The initialization module 30 is able to acquire initialization parameters, including block 26 geometric parameters, geological formation 14 parameters, acid injection parameters, acid rheology and reactivity parameters, and calibrated values of pore connectivity parameter y and pore broadening parameter p. These parameters for example are acquired by input of a user using the man-machine interface 23, and/or from downloading parameters from an external source.

The geometric parameters include the dimensions of the simulated block 24, and of the center hole 26 made in the block 24, as defined above.

The geological formation parameters advantageously include a rock type chosen for example between quarry and reservoir, a mineralogy type chosen for example between limestone and dolomite, an anhydrite content, a mercury injection capillary pressure type, and/or a fracture type.

The geological formation parameters further include an average pore radius, a pore size distribution, an average porosity with maximal and minimal deviation from the average porosity. These parameters allow the definition of an initial permeability profile KINIT(P, 9, z), and an initial porosity profile £INIT(P, 6, z) in the block 24. By “profile,” it is meant a set of values of porosities and/or permeabilities, at each coordinate p, 0, z, taken in a radial system of coordinates centered in the block 24.

The geological formation parameters also include the density p_s of the geological formation 14 in the block 24, which is constant depending on the mineralogy of the rock, generally between 2.7 g/cm³ and 3.0 g/cm³ (function of the rock mineralogy, e.g 2.71 g/cm³ for a calcite, 2.84 g/cm³ for a dolomite) and the interfacial area a_v available for reaction per unit volume of the porous medium in the block 24.

The acid injection parameters include an acid type, an inlet acid concentration (percentage), a molecular weight (g/mol) a molecular Diffusivity (cm²/s), dissolving power a_c of the acid, a Sherwood number, a Newtonian viscosity or non-Newtonian viscosity, a brine viscosiy (cP) and a kinematic viscosity. Thus, the effective dispersion tensor D_e of the acid, the local mass transfer coefficient k_c , and the reaction rate constant (or dissolution rate) constant k_s , can be obtained.

The acid injection parameters also include flow parameters such as the radial injection flow rate IFR of the acid,

The initialization module 30 is also able to acquire a pore broadening parameter p and a pore connectivity parameter y. The pore-broadening parameter translates the increase in the pore radius due to the acid/rock interaction. The pore-connectivity parameter y is defined to consider the overcoming of the pore-throat restriction.

The pore broadening parameter p and the pore connectivity parameter y are obtained from two linear flow experiments on two real cores of geological formation. Each core is generally cylindrical and extends along a longitudinal axis. It has a volume which is smaller than the volume of the block 24. The volume of the core is generally lower than 500 cm³, and is comprised generally between 100 cm³ and 200 cm³.

The calibration experiments are carried out in a calibration system 40 comprising at least a cell in which the core is contained, an axial flow injector, to inject an acid front in the cell which propagates along the longitudinal axis.

The experiments are carried out at several injection flow rates (at least two), to match the pore volume to breakthrough along with the transverse pressure response with the experimental data.

Examples of experiments are given in the following publications :

• Hoefner, M. L., & Fogler, H. S. (1988). Pore evolution and channel formation during flow and reaction in porous media. AIChE Journal, 34(1 ), 45-54.

• Fredd, C. N., & Fogler, H. S. (1998). Influence of transport and reaction on wormhole formation in porous media. AIChE journal, 44(9), 1933-1949.

• Bazin, B. (2001 ). From matrix acidizing to acid fracturing: a laboratory evaluation of acid/rock interactions. SPE Production & Facilities, 16(01 ), 22-29.

The calibrated value of the effect of the pore broadening parameter and pore connectivity parameter y are used by the breakthrough calculation module 32 to determine a modified Carman-Kozeny correlation between an initial local permeability K_o, an updated local permeability K, an initial local porosity £o, and an updated local porosity E, according to the following equations:

As will be described below, the modified Carman-Kozeny correlation is used in the radial injection simulation algorithm carried out by the breakthrough calculation module 32.

The breakthrough calculation module 32 is able to carry out a radial injection simulation algorithm, from the center hole 26 in the surrounding region 28 of the block 24 to determine breakthrough porosity £BT(P, 6, z) at any position p, 0, z in the block 24, and breakthrough permeability KBT(P, 6, z) at any position in the block 24, as well as total acid injected volume TABT, from at least a radial injection flow rate IFR.

To this end, the breakthrough calculation module 32 is able to initiate radial injection simulation algorithm by acquiring from the initialization module 30, an initial acid concentration profile CINIT(P, 6, z) in radial coordinates, an initial porosity profile £INIT(P, 6, z) in radial coordinates, and inlet velocities uiiNiT. from the predetermined injection flow rate at the inlet face of the surrounding region 28 in the center hole 26, and on the initialization parameters defined by the initialization module 30.

The acid injection flow rate at the inlet face and the pressure at the exit face of the surrounding region are for example fixed as follows:

P(r, 6, z) = P(r, 6 + 2n, z), and Cf(r, 6, z) = Cf(r, 6 + 2n, z) (3) where p is the effective viscosity of the acid, K is the permeability tensor, u_o is the axial component of injection velocity of acid, Co is the inlet concentration of acid in the acid solution, P_exit is the pressure at the exit boundary of the outer face of the surrounding region 28, r_w is the inner radius of the surrounding region 28 and r_e is the outer radius of the surrounding region.

For species balance equation for acid, Danckwerts’ inlet condition with zero flux exit condition in the axial direction is used as stated in equations 4 and 5: dC_f uoCo = urCr — eD — — at r = rw (4)

^J dr

where uo is the axial component of velocity at the injection face; C_o is the inlet concentration of the acid. At transverse boundaries, zero-flux boundary conditions are used for both pressure and concentration as n ■ VP = n- Cf = 0 at transverse boundaries, (6) where n is the normal vector perpendicular to the transverse boundary.

The surrounding region is considered to be initially saturated with brine solution, i.e., no acid is present in the core or C_f (t = 0) = 0.

The heterogeneity of rock is accounted by uniformly distributing the initial porosity profile with an average porosity e₀ and heterogeneity magnitude As₀ , i.e., s = s_o + s ' at t = 0, (7) where E' is random fluctuation in the initial porosity profile. This random number (s') is assumed to be uniformly distributed in the interval [ -As₀ to As₀] and is added to the mean value of porosity s₀ .

Then, from the above describe initialization step 50, the breakthrough calculation module 32 is able to repeat a series of steps 52 to 58 of the radial injection simulation algorithm described below, at a plurality of sequential time steps t_n, t_n+i until a breakthrough condition is reached.

A first step is a determination of a pressure profile pⁿ⁺¹ = p(p, 0, z, t_n+i) at a further time step t_n+i, in the block 24, from a permeability profile K(p, 0, z, t_n) at the former time step t_n, and from the acid concentration profile Cfⁿ = Cf(p, 0, z, t_n) at the former time step t_n

The pressure profile pⁿ⁺¹ at time step t_n+i is obtained by the following equation:

where p is the acid effective viscosity, k_s is the reaction (or dissolution) rate constant, k_c is is the local mass-transfer coefficient, a_v is the solid-fluid interfacial surface area, a_c is the dissolving power of the acid (defined as grams of solid dissolved per mole of acid consumed), and p_s is the geological formation density.

For a Newtonian acid, the acid effective viscosity p is a constant. For a non-Newtonian acid, the acid effective viscosity p can be determined with the following equation:

where K is the magnitude of local permeability at each position; II UH being the norm of the Darcy velocity vector; and E is the porosity at each position, H_o is the consistency factor at a reference temperature (T_o), n is the power-law index.

Ho and n are obtained from the literature or through rheological measurements (viscosity vs shear rate).

Typical values of H_o are between 5 and 600, and typical values for n are between 0.2 and 0.6.

The mass transfer coefficient k_c is determined with the following equation :

Sh = ^2kcrp = Sh_m + 0.7 Re „ ^1/2 Sr- ^1/3 , (8b)

D_m ^{00 P} where r_p is the mean pore radium, Sh is the Sherwood number or dimensionless mass-transfer coefficient; Sh _ is the asymptotic Sherwood Number; Re_p is the pore scale

Reynolds number defined as Re , null being the norm of the Darcy velocity vector; v is the kinematic viscosity; D_m is the effective molecular diffusivity of acid; Sc is the

Schmidt number defined as Sc = — .

^Dm

The values of Sh°° depend mainly on the shape of the pore to be considered it ranges between 2.98 (square pore) and 3.66 (triangle), with an average of around 3. For a packed bed of spheres, Sh°° is 2.

For liquids the values of Schmidt number (ratio between the kinetic viscosity and the molecular diffusion coefficient) is around 1000.

A velocity profile uⁿ⁺¹ = u(p, 0, z, t_n+i) at a further time step t_n+i , is also calculated from the permeability profile K(p, 0, z, t_n) at the former time step t_n and from the pressure profile pⁿ⁺¹ at time step tⁿ⁺¹, using Darcy’s equation :

The above-mentioned equations (8) and (9) are solved by finite volumes discretization, using simple algebraic multigrid libraries (SAMG).

A further step 54 is a determination of an updated acid concentration profile Cfⁿ⁺¹ = Ct(p, 0, z, t_n+i) at a further time step tn+1 and an updated porosity profile e(p, 0, z, t_n+i) at a further time step t_n+i , taking into account the diffusion-convection of the acid in the block 24, and the reactions of the acid with the geological formation in the block 24.

Preferably, an operator splitting scheme is used to solve a species balance equation by using first a diffusion-convection operator at an intermediate time step tn+1/2 between the former time step t_n and the further time step t_n+i , without taking into account the reaction of the acid with the geological formation 14, and then, by using an acid reaction operator, without taking into account the diffusion-convection of the acid.

As per the first step 52, the second step 54 also uses a finite volume discretization using simple algebraic multigrid libraries (SAMG).

The first diffusion-convection operator is for example written as below:

where D_e is the effective dispersion tensor of the acid.

The first diffusion-convection operator allows the calculation of an intermediate acid concentration Cf^n+1/2 at time step tn+1/2 from the velocity profile u_n+i at time step t_n+i and of an intermediate porosity profile £^n+1/2 = e(p, 0, z, tn+1/2) at time step tn+1/2. Longitudinal and transverse dispersion coefficients D_eX, D_eT of the effective dispersion tensor D_e can be calculated from the following equations :

Where HUH is the norm of the Darcy velocity vector, D_m is the effective molecular diffusivity of the acid, r_p is the mean pore radius, £ is the local porosity, a_os, A_x, AT are constants which depend on the fluid/rock system. For example, A_x, AT are respectively 0.5 and 0.1 for a packed bed of spheres.

The different parameters are more generally described in the following article :

Panga, M. K., Ziauddin, M., & Balakotaiah, V. (2005). Two-scale continuum model for simulation of wormholes in carbonate acidization. AIChE journal, 51 (12), 3231 -3248.

Then, the reaction operator is solved, to determine the acid concentration profile Cfⁿ⁺¹ = ct(p, 0, z, t_n+i) at all radial coordinates, at the further time step t_n+i, and the porosity profile £ⁿ⁺¹ = £(p, 0, z, tn+i) at every radial coordinate, at the further time step t_n+i. The reaction operator is for example according to the following equations:

deⁿ⁺¹ _ k_sk_ca_va_{c n+1} dt (k_s+k_c)p_s f (13)

Both diffusion-convection operator and reaction operator implicitly solve the variables in time.

A further step 56 of the algorithm is a breakthrough condition test. The breakthrough condition test is for example a test on the pressure, for example a test determining that the difference of pressure AP(t_n+i) at the further time step t_n+i between the inlet pressure Pi in the center hole 26 at the interface with the surrounding region 28 and the outlet pressure P_o at the outside radial surface of the surrounding region is smaller than a predetermined value, or that a ratio of the difference of pressure AP(t_n+i ) at the further time step t_n+i to the initial difference of pressure AP(to) is smaller than a predetermined value.

The predetermined value is for example between 0.15 and 0.05.

If the breakthrough condition is not met, the radial injection simulation algorithm comprises a step 58 of determining an updated permeability profile Kⁿ⁺¹ = K(p, 0, z, t_n+i) at the further time step t_n+i from the permeability profile Kⁿ = K(p, 0, z, t_n) at the prior time step t_n , from the porosity profile £ⁿ⁺¹ = £ (p, 0, z, t_n+i ) at the further time step t_n+i , from the porosity profile £ⁿ = £ (p, 0, z, t_n) at the prior time step t_n+i , and from the pore broadening parameter P and the pore connectivity parameter y, based on equation (1 ) rewritten as

„n+l /_Cn+1\ Y /_cn+l_/-1

( fc ) <^{l 4}> The relative changes in the mean pore radius r_p and in the solid fluid interfacial surface area a_v are also updated using the following characteristic ratios :

The updated permeability profile Kⁿ⁺¹ = K(p, 0, z, t_n+i) is then used to reiterate steps 52, 54 and 56, until the breakthrough condition is met.

When the breakthrough condition is met, the breakthrough calculation module 32 is able to produce a set of simulated data including a total volume of acid injected to breakthrough TVIBT , a porosity profile £BT(P, 0, z) at breakthrough at each radial coordinate and a permeability profile £BT(P, 0, z) at breakthrough at each radial coordinate.

The porosity profile £BT(P, 0, z) and the permeability profile £BT(P, 0, z) can also be obtained at any time before breakthrough at each radial coordinate.

The up-scaling parameter obtaining module 34 is able to retrieve the results of the breakthrough calculation module 32 at each injection flow rate IFR to obtain a set of up- scaling parameters to be used by the simulator 16A.

The up-scaling parameters for example comprise an optimum PVBT value and an optimum interstitial velocity at breakthrough IVBT.

The latter up-scaling parameters are obtained by determining a curve of simulated PVBT versus acid injection velocity from values of PVBT and injection velocities obtained by activating the breakthrough calculation module 32 at different acid injection flow rates.

An example of curve which can be determined by the up-scaling parameter obtaining module is shown in figure 4.

The optimized PVBT and the optimized interstitial velocities IVBT are obtained at the absolute minimum of the curve.

The up-scaling parameters obtaining module is also able to recover the rock dimensions which were used to carry out the simulation.

Advantageously, the up-scaling parameter obtaining module 34 is able to build a dissolution pattern representation at at-least one acid injection flow rate IFR, from the porosity profile at breakthrough £BT(P, 0, z) obtained from the breakthrough calculation module 32 at the injection flow rate IFR.

To this aim, a dissolution porosity threshold is predefined. When the local porosity at a particular radial coordinate is greater than the predefined porosity threshold, the local aspect of the geological formation at the radial coordinate is considered to be a void. On the contrary, when the local porosity at a particular radial coordinate is smaller than the predefined porosity threshold, the local aspect of the geological formation at the radial coordinate is considered to be solid.

From this, the up-scaling parameter obtaining module 34 is able to build a tridimensional map of voids into the surrounding region 28, as shown in figure 5. Figure 5 depicts a particular dissolution pattern representation corresponding to a radial injection of acid at an injection flow rate IFR at breakthrough. In the example of figure 5, the map shows wormholes 70.

From the dissolution pattern representation, the up-scaling parameter obtaining module 34 is able to calculate a wormhole diameter, a wormhole density, and a wormhole radial penetration length.

The latter parameters also constitute up-scaling parameters to be used in the acid stimulation simulator 16A.

Preferably, several wormhole densities and wormhole penetration lengths (for example at least four wormhole densities and at least four wormhole penetration lengths) are determined from the dissolution pattern representations obtained at different injection flow rates.

The up-scaling parameter obtaining module 34 is then able to carry out a regression to calculate the best fitting parameters to correlate wormhole density as a function of acid injection flow rate and wormhole penetration length as a function of acid injection flow rate.

A determining method according to the invention will be described below.

Initially, at step 50, the initialization module 30 acquires the initialization parameters, including block 26 geometric parameters, geological formation 14 parameters, acid injection parameters including acid rheology and reactivity parameters, and calibrated values of pore connectivity parameter y and pore broadening parameter p ad described above.

The breakthrough calculation module 32 then carries out a radial injection simulation algorithm comprising sub steps 52 to 56, to simulate an acid radial injection from the center hole 26 in the surrounding region 28 of the block 24 in order to determine breakthrough porosity £BT(P, 0, z) at any position p, 0, z in the block 24, and breakthrough permeability KBT(P, 0, z) at any position in the block 24, as well as total acid injected volume TABT, from at least an radial injection flow rate IFR.

To this end, the breakthrough calculation module 32 acquires from the initialization module 30, an initial acid concentration profile CINIT(P, 0, z) in radial coordinates, an initial porosity profile £INIT(P, 0, z) in radial coordinates, and inlet velocities UIINIT, from the predetermined injection flow rate at the inlet face of the surrounding region 28 in the center hole 26, and on the initialization parameters defined by the initialization module 30. The acid injection flow rate at the inlet face and the pressure at the exit face of the surrounding region are for example fixed as described above.

Then, the breakthrough calculation module 32 repeats the above mentioned series of steps 52 to 58 described below, at a plurality of sequential time steps t_n, t_n+i until a breakthrough condition is reached.

If the breakthrough condition is not met, the algorithm also comprises a step 58 of determining an updated permeability profile Kⁿ⁺¹ = K(p, 0, z, t_n+i) at the further time step t_n+i from the permeability profile Kⁿ = K(p, 0, z, t_n) at the prior time step t_n , from the porosity profile eⁿ⁺¹ = e (p, 0, z, t_n+i) at the further time step t_n+i , from the porosity profile £ⁿ = £ (p, 0, z, t_n) at the prior time step t_n+i , and from the pore broadening parameter p and the pore connectivity parameter y, based on equation (14).

When the breakthrough condition is met, the breakthrough calculation module 32 produces a set of simulated data including the total volume of acid injected to breakthrough TVIBT, the porosity profile £BT(P, 0, z) at breakthrough at each radial coordinate and the permeability profile £BT(P, 0, z) at breakthrough at each radial coordinate.

The up-scaling parameter obtaining module 34 retrieves the results of the breakthrough calculation module 32 at each injection flow rate IFR to obtain, a set of up- scaling parameters.

The up-scaling parameters for example comprise an optimized PVBT value and an optimized interstitial velocity at breakthrough IVBT.

The latter up-scaling parameters are obtained as described above by determining a curve of simulated PVBT versus acid injection velocity from values of PVBT and injection velocities obtained by activating the breakthrough calculation module 32 at different acid injection flow rates IFR.

The optimized PVBT and the optimized interstitial velocities IVBT are obtained at the absolute minimum of the curve of figure 4, or as a range if the minimum is a plateau.

The up-scaling parameters obtaining module also determines rock dimensions.

Advantageously, the up-scaling parameter obtaining module 34 builds a dissolution pattern representation at at least one acid injection flow rate IFR, from the final porosity profile at breakthrough £BT(P, 0, z) obtained from the breakthrough calculation module 32 at the injection flow rate.

A dissolution porosity threshold is predefined. When the local porosity at a particular radial coordinate is greater than the predefined porosity threshold, the local aspect of the geological formation at the radial coordinate is considered to be a void. As described above, the up-scaling parameter obtaining module 34 builds a tridimensional map of voids into the surrounding region 28, as shown in figure 5, which constitutes a particular dissolution pattern representation. In the example of figure 5, the map shows wormholes 70.

From the dissolution pattern representation, the up-scaling parameter obtaining module 34 calculates a wormhole diameter, a wormhole density, and a wormhole radial penetration length.

The latter parameters constitute up-scaling parameters which are used in the acid stimulation simulator 16A.

Preferably, several wormhole densities and wormhole penetration lengths, for example at least four wormhole densities and at least four wormhole penetration lengths are obtained from dissolution pattern representations obtained at different injection flow rates.

A regression is then made to obtain the best fitting parameters to correlate wormhole density as a function of acid injection flow rate and wormhole penetration length as a function of acid injection flow rate.

The up-scaling parameters obtained by the determination system 10 from the implementation of the process according to the invention are then used by the simulator 16A, to determine a simulated acidification pattern in a geological formation 14 from a well 12. Advantageously, the acid injector 16B injects acid in the geological formation 14 from the well, from the simulated acidification pattern in the well 12.

Thanks to the method according to the invention, it is possible to obtain very accurate up-scaling parameters which are much more representative of the real behavior of acid in a geological formation as compared to the traditional linear flow experiments carried on cores of geological formation.

By carrying out radial simulations in a block 24 of geological formation and determining from the radial simulations, a full porosity profile and a full permeability profile in the block 24, it is possible to access to a more accurate simulated PVBT versus interstitial acid injection velocity curve which can help determine an optimum PVBT and an optimum interstitial acid injection velocity.

Moreover, obtaining of a full porosity profile in the whole block 24 allows a determination of a dissolution pattern representation, which can be used to more accurately determine wormhole density, wormhole diameter, wormhole penetration length, representative of the real behavior in a well.

Hence, the up-scaling parameters obtained by the system and method according to the invention can be used to more accurately simulate an acid stimulation process, and from the results of the simulation, carry out a more efficient acid stimulation in a real well 12.

These results are obtained with a set of simulations, which does not require using large blocks of a real rock to carry out experiments. On the contrary, only two linear flow experiments are enough to calibrate the method and to allow the determination system 10 to simulate final porosity profiles and acid quantities.

A database of behavior of particular acids in several types of geological formations 14 can therefore easily be built and used in various acid stimulation simulations and operations in real wells 12. A better understanding and control of acid stimulation is thus obtained. Moreover, cost reductions can be obtained by injecting the right quantity of acid at the most appropriate flow rate.

Claims

22 CLAIMS

1.- A process for determining up-scaling parameters of an acidification injection carried in a geological formation (14) from a well (12), the process being carried out by a determination system (10) and comprising the following steps:

- defining (50) at least one acid radial injection flow rate into a block (24) of geological formation (14) comprising a center injection hole (26) and a surrounding formation region (28) surrounding the center injection hole (26);

- repeating for the at least one acid radial injection flow rate, at successive time steps, the following sub-steps of a three-dimensional radial injection simulation algorithm, until an acid breakthrough condition occurs in the block (24):

* obtaining an updated pressure profile in the block (24) at a further time step, from a permeability profile in the surrounding region (28) obtained from a porosity profile in the surrounding region (28) at a former time step, from an acid concentration profile in the surrounding region (28) at a former time step and from coefficients representative of the geological formation and representative of acid rheology and reactivity;

* determining from the updated pressure profile in the block (24) and from the permeability profile, an updated acid velocity profile at the further time step;

* determining an updated acid concentration profile in the block (24) at the further time step, and an updated porosity profile at the further time step, from diffusion-convection of the acid and on dissolution reactions occurring in the surrounding region (28);

* determining an updated permeability profile in the block (24) at the further time step, using a correlation with a correlation equation correlating the updated permeability profile to the updated porosity profile, to a predetermined pore broadening parameter and to a pore-connectivity parameter;

- once the acid breakthrough condition is met in the block (24), obtaining a volume of acid injected at breakthrough, a porosity profile at breakthrough, a permeability profile at breakthrough and/or a velocity profile at breakthrough, and

2.- The process according to claim 1 , wherein the pore broadening parameter and the pore connectivity parameters are obtained through calibration using at least two linear flow experiments in which an injection of acid is carried out linearly into a sample of geological formation (14).

3.- The process according to claim 1 or 2, wherein the correlation comprises a calculation of the ratio of a updated local permeability at any point in the surrounding region (28) to a former local permeability at any point in the surrounding region (28), from at least a first characteristic ratio calculated from the updated local porosity at any point in the surrounding region (28) and from a former local permeability at any point in the surrounding region (28) powered to the pore broadening parameter and a second characteristic ratio calculated from the initial former porosity at any point in the surrounding region (28) and from the updated porosity at any point in the surrounding region (28) powered to the second pore connectivity parameter.

4.- The process according to any one of the preceding claims, wherein the coefficients representative of the geological formation include rock density, solid-fluid interfacial surface area, and wherein coefficients representative of the acid rheology and reactivity include viscosity, dissolving power of the acid, local mass transfer coefficient, and reaction rate constant.

5.- The process according to claim 4, wherein the local mass transfer coefficient is calculated at each successive time step from a mean pore radius updated at each time step using the pore broadening parameter, the solid fluid interfacial surface area being updated at each successive time step using the pore broadening parameter.

6.- The process according to any one of the preceding claims, wherein the velocity profile at the further time step is determined from the updated pressure profile at the further time step and from the permeability profile through Darcy’s law.

7.- The process according to any one of the preceding claims, wherein the determination of the updated concentration profile and of the updated porosity profile at the further time step comprises an operator splitting including first using a diffusion convection operator, taking into account acid diffusion-convection without taking into account acid reaction, and second using a reaction operator, taking into account acid reaction without taking into account acid diffusion-convection.

8.- The process according to claim 7, wherein the diffusion-convection operator is given by the following equations:

g_£n+l/2

- = 0 dt where D_e is the effective dispersion tensor of the acid, Cf^n+1/2 is the acid concentration profile at an intermediate time step between the former time step and the further time step, u_n+i is the velocity profile at the further time step, £^n+1/2 is the porosity profile at the intermediate time step, p_s is the geological formation (14) density and a_c is the dissolving power of the acid.

9.- The process according to claim 7 or 8, wherein the reaction operator is given by the following equations:

dEⁿ⁺¹ _ k_sk_ca_va_{c n+1} dt (k_s + k_c)p_s ^r where Cfⁿ⁺¹ is the acid concentration profile at the further time step, eⁿ⁺¹ is the porosity profile at the further time step, p_s is the geological formation (14) density, a_c is the dissolving power of the acid, k_s is an acid reaction rate constant, k_c is a local mass-transfer coefficient, and a_v is a solid-fluid interfacial area.

10.- The process according to any one of the preceding claims, in which the sub-steps of the radial injection simulation algorithm are carried out for a plurality of acid injection flow rates, the obtaining of an up-scaling parameter comprising, obtaining a pore volume to breakthrough associated with each radial injection flow rate and determining from each of the pore volume to breakthrough value an optimum pore to breakthrough value, an optimum interstitial velocity to breakthrough, and/or at least a rock dimension, the at least one up- scaling parameter being chosen among the optimum pore to breakthrough value, the optimum interstitial velocity to breakthrough, and/or the at least a rock dimension.

11.- The process according to any one of the preceding claims, comprising determining at least one radial dissolution pattern from the porosity profile at breakthrough, and calculating a wormhole density, a wormhole diameter, and/or a radial wormhole penetration length from the radial dissolution pattern, the at least one up-scaling parameter being chosen among the wormhole density, the wormhole diameter, and the radial wormhole penetration length.

12.- The process according to claim 1 1 , wherein the sub-steps of the radial injection simulation algorithm are carried out for a plurality of acid injection flow rates, the up-scaling parameter obtaining comprising calculating a fitting parameter to correlate wormhole density, wormhole diameter, and/or wormhole radial penetration length to interstitial 25 injection flow rate, from the wormhole density, wormhole diameter, and/or wormhole radial penetration length calculated from dissolutions patterns obtained at the plurality of acid injection flow rates.

13.- The process according to any one of the preceding claims, wherein the acid injected in the three-dimensional radial injection simulation algorithm is a non-Newtonian acid, the updated pressure profile in the block (24) at a further time step including a calculation of an acid effective viscosity of the non-Newtonian acid.

14.- An use of up-scaling parameters obtained from the process for determining up- scaling parameters of an acidification reaction according to any one of the preceding claims, to determine a simulated acidification pattern in a geological formation (14) from a well (12) and advantageously to inject acid in the geological formation (14) from the well (12), from the simulated acidification pattern in the well (12).

15.- A system (10) for determining up-scaling parameters of an acidification injection carried in a geological formation (14) from a well (12) comprising :

- a initialization module (30) configured to define at least one acid radial injection flow rate into a block (24) of geological formation (14) comprising a center injection hole (26) and a surrounding formation region (28) surrounding the center injection hole (26);

- a breakthrough calculation module (32) configured to repeat, for the or each acid radial injection flow rate, the following sub-steps of a three-dimensional radial injection simulation algorithm at successive time steps, until an acid breakthrough condition occurs in the block (24) :

* determining an updated permeability profile in the block (24) at the further time step, using a correlation equation, the correlation equation correlating the updated permeability profile to the updated porosity profile, to a predetermined pore broadening parameter and to a pore connectivity parameter; 26

- an up-scaling parameter obtaining module (34) configured, once the acid breakthrough condition is met in the block (24), obtaining a volume of acid injected at breakthrough, a porosity profile at breakthrough, a permeability profile at breakthrough and/or a velocity profile at breakthrough, and configured to obtain at least an up-scaling parameter of the acidification reaction from the volume of acid injected at breakthrough, from the porosity profile at breakthrough, from the permeability profile at breakthrough and/or from the velocity profile at breakthrough.

16.- The system according to claim 15, wherein the up-scaling parameter obtaining module (34) is configured to activate the breakthrough determining module (32) at several acid radial injection flow rates to provide a pore volume to breakthrough value at each acid radial injection flow rate and is configured to determine an optimum pore volume to breakthrough value and an optimum interstitial velocity from the different pore volume to breakthrough values determined by the breakthrough determining module (32).

17.- The system according to any one of claims 15 or 16, wherein the up-scaling parameter obtaining module (34) is configured to obtain a radial dissolution pattern from the porosity profile determined at breakthrough by the breakthrough determining module (32) and to calculate at least one up-scaling parameter chosen among a wormhole density, a wormhole diameter, a wormhole radial penetration depth, from the radial dissolution pattern.