METHOD FOR OPERATING A HYDROCARBON DEPOSIT BY INJECTION OF A GAS IN FOAM FORM

Abstract

Method for operating a hydrocarbon deposit by injection of gas in foam form, comprising a step of determination of a model of displacement of the foam, this model being a function of an optimal mobility reduction factor of the gas and of at least one interpolation function dependent on a parameter and constants to be calibrated.

The mobility reduction factor of the gas is determined and the constants of at least one interpolation function are calibrated from experimental measurements comprising injections of gas in non-foaming form and in foam form into a sample of the deposit for different values of the parameter relative to the function considered, and measurements of headloss corresponding to each value of the parameter of the interpolation function considered. The calibration of the constants is performed interpolation function by interpolation function.

Applicable notably to oil exploration and operation.

Claims

1. Method for operating an underground formation comprising hydrocarbons, by means of an injection of an aqueous solution comprising a gas in foam form and a flow simulator relying on a displacement model of said gas in foam form, said displacement model being a function of an optimal mobility reduction factor of said gas and of at least one interpolation function of said optimal mobility reduction factor, said interpolation function being a function of at least one parameter relating to at least one characteristic of the foam and of at least one constant, characterized in that, from at least one sample of said formation, measurements of conventional relative permeabilities to said gas in non-foaming form and measurements of conventional relative permeabilities to said aqueous phase: A. said displacement model of said simulator is determined according to at least the following steps: i. a plurality of values of said parameter is defined relative to at least one of said interpolation functions, an injection is performed into said sample of said gas in non-foaming form and of said gas in foam form according to said values of said parameter relative to said function, and a headloss with foam and a headloss without foam are measured respectively for each of said values of said parameter relative to said function; ii. from said measurements of headloss relative to said interpolation function, an optimal value of said parameter relative to said function is determined, said optimal value making it possible to maximize a ratio between said headlosses without foam and said headlosses with foam measured for said function; iii. from said measurements of headloss performed with said optimal value determined for at least said interpolation function, from said measurements of conventional relative permeabilities to said gas in non-foaming form and from said measurements of conventional relative permeabilities to said aqueous phase, said optimal mobility reduction factor is determined; iv. for at least said interpolation function, from said optimal mobility reduction factor, from said measurements of headloss relative to said interpolation function, from said measurements of conventional relative permeabilities to said gas in non-foaming form and from said measurements of conventional relative permeabilities to said aqueous phase, said constants of said interpolation function are calibrated; Bfrom said displacement model and from said flow simulator, an optimal operation scheme for said deposit is determined and said hydrocarbons are exploited.

2. Method according to claim 1, in which said displacement model of the foam is expressed in the form:
k.sub.rg.sup.FO(S.sub.g)=FM k.sub.rg(S.sub.g) in which k.sub.rg.sup.FO(S.sub.g) is the relative permeability to said gas in foam form for a given gas saturation value S.sub.g, k.sub.rg(S.sub.g) is the relative permeability to said non-foaming gas for said gas saturation value S.sub.g, and FM is a functional expressed in the form: $F.Math..Math.M=\frac{1}{1.Math.\left(M^{\mathrm{opt}}-1\right)*\underset{k}{.Math.}.Math..Math.F_k}$ in which M.sup.opt is said optimal mobility reduction factor of said gas and F.sub.k is one of said interpolation functions, with k1.

3. Method according to claim 1, in which there are four of said interpolation functions and said parameters of said functions are a foaming agent concentration, a water saturation, an oil saturation and a gas flow rate.

4. Method according to claim 1, in which said interpolation function F.sub.k of a parameter V.sub.k is written in the form: $F_k\left(V_k\right)=\frac{M_k^{\mathrm{opt}}-1}{M^{\mathrm{opt}}-1}$ in which M.sup.opt is said optimal mobility reduction factor and M.sub.k.sup.opt is an optimal mobility reduction factor for said parameter V.sub.k.

5. Method according to claim 1, in which, prior to the step iii), optimal conditions are defined as corresponding to said optimal values determined for each of said interpolation functions, said gas in non-foaming form and said gas in foam form are injected into said sample according to said optimal conditions, and a headloss with foam and a headloss without foam are respectively measured.

6. Method according to claim 1, in which said constants of at least one of said interpolation functions are calibrated by a least squares method, such as an inverse method based on the iterative minimization of a functional.

Description

BRIEF DESCRIPTION OF THE FIGURES

[0031] FIG. 1 presents an example of the trend of a mobility reduction factor R as a function of the gas flow rate Q.

DETAILED DESCRIPTION OF THE METHOD

[0032] In general, one of the objects of the invention relates to a method for operating an underground formation comprising hydrocarbons, by means of an injection of an aqueous solution comprising a gas in foam form, and in particular the determination of an optimal operating scheme for the underground formation studied. In particular, the method according to the invention targets the determination of the parameters of a model of displacement of the gas in foam form. Hereinbelow, foam describes a phase dispersed in another phase by the addition of a foaming agent in one of the two phases. One of the phases can be water and the other phase is a gas, such as natural gas, nitrogen or CO.sub.2.

[0033] The method according to the invention requires the availability of: [0034] a sample of the underground formation studied, taken by in situ coring for example; [0035] a flow simulator relying on a model of displacement of the gas in foam form (see below); [0036] measurements of conventional relative permeabilities to the gas in non-foaming form and measurements of conventional relative permeabilities to the aqueous phase: these can be measurements performed expressly for the requirements of the method according to the invention (the expert has perfect knowledge of how to conduct such laboratory experiments), but they can also be analytical functions calibrated from correlations well known to the expert.

[0037] Thus, the method according to the invention requires the availability of a flow simulator comprising a model of displacement of the foam. According to the invention, the model of displacement of the foam relies on the hypothesis that the gas present in foam form has its mobility reduced by a given factor in set conditions of formation and of flow of the foam. The formulation of such a model, used by many flow simulators, consists in modifying the relative permeabilities to the gas when the gas is present in foam form which, for a given gas saturation S.sub.g, is expressed according to a formula of the type:

k.sub.rg.sup.FO(S.sub.g)=FM k.sub.rg(S.sub.g)(1)

in which k.sub.rg.sup.FO(S.sub.g) is the relative permeability to the gas in foam form, that is expressed as the product of a function FM by the relative permeability to the non-foaming gas k.sub.rg(S.sub.g) for the same gas saturation value S.sub.g (later denoted S.sub.rg.sup.FO). One assumption underpinning the current foam models is that the relative permeability to water (or to liquid by extension) is assumed unchanged, whether the gas is present in continuous phase form or in foam form. Given this assumption, the gas mobility reduction functional, hereinafter denoted FM, is expressed according to a formula of the type:

[00003]$\begin{array}{cc}F.Math..Math.M=\frac{1}{1+\left(M_{\mathrm{mod}}^{\mathrm{opt}}-1\right)*\underset{k}{.Math.}.Math..Math.F_k\left(V_k\right)}& \left(2\right)\end{array}$

in which: [0038] M.sub.mod.sup.opt is the optimal mobility reduction factor, that is to say the ratio of the relative permeabilities to the gas (k.sub.rg) and to the foam (k.sub.rg.sup.FO) in optimal conditions for reducing the mobility of the gas, that is to say the conditions in which the terms F.sub.k(V.sub.k) defined hereinbelow have the value 1, i.e.:

[00004]$\begin{array}{cc}{M}_{\mathrm{mod}}^{\mathrm{opt}}=\frac{k_{\mathrm{rg}}\left(S_{g,\mathrm{opt}}^{\mathrm{FO}}\right)}{k_{\mathrm{rg}}^{\mathrm{FO}}\left(S_{g,\mathrm{opt}}^{\mathrm{FO}}\right)}=\frac{1}{F.Math..Math.M_{\mathrm{opt}}}& \left(3\right)\end{array}$ [0039] the terms F.sub.k(V.sub.k) (with k equal to or greater than 1) are the values of the interpolation functions F.sub.k of the mobility reduction factor between the value M.sub.mod.sup.opt and 1, which each depend on a parameter V.sub.k relative to at least one characteristic of the foam, and which involve a certain number of calibration constants to be calibrated as explained hereinbelow.

[0040] In order to provide a model of displacement of the foam to the simulator that is representative of the reality, the method according to the invention aims to determine, reliably from representative displacement measurements, the following modelling data: [0041] the optimal mobility reduction factor of the gas M.sub.mod.sup.opt as defined according to the equation (3); [0042] the calibration constants of each of the functions F.sub.k considered in the definition of the model of displacement of the foam according to the equations (1) and (2).

[0043] According to one implementation of the invention, the parameter V.sub.k can notably be the foaming agent concentration C.sub.s.sup.w, the water saturation S.sub.w, the oil saturation S.sub.o, or even the gas flow rate u.sub.g.

[0044] According to one implementation of the invention, the gas mobility reduction functional, denoted FM, comprises four interpolation functions F.sub.k(V.sub.k) and each of these functions comprises two constants to be calibrated from experimental data. According to an implementation of the invention in which the gas mobility reduction functional comprises four interpolation functions F.sub.k(V.sub.k), the following are defined: [0045] the interpolation function F.sub.1 relative to the parameter V.sub.1=C.sub.s.sup.w (foaming agent concentration C.sub.s.sup.w) by a formula of the type:

[00005]$\begin{array}{cc}{F}_1={\left(\frac{\mathrm{Min}\left(C_s^w,C_s^{w.Math.\text{-}.Math.\mathrm{ref}}\right)}{C_s^{w.Math.\text{-}.Math.\mathrm{ref}}}\right)}^{e_s}& \left(4\right)\end{array}$ [0046] and for which the constants to be calibrated are the exponent e.sub.s, and the constant C.sub.s.sup.w-ref which corresponds to the foaming agent concentration in reference optimal conditions; [0047] the interpolation function F.sub.2 relative to the parameter V.sub.2=s.sub.w (water saturation), by a formula of the type:

[00006]$\begin{array}{cc}{F}_2=\left[0.5+\frac{\arctan \left[f_w\left(S_w-S_w^{*}\right)\right]}{}\right]& \left(5\right)\end{array}$ [0048] and for which the constants to be determined are the constant f.sub.w which governs the transition (according to the water saturation) between the foaming and non-foaming states and the constant S.sub.w* which represents the transition water saturation between stable and unstable foaming states; [0049] the interpolation function F.sub.3 relative to the parameter V.sub.3=S.sub.o (oil saturation) by a formula of the type:

[00007]$\begin{array}{cc}{F}_3={\left(\frac{\mathrm{Max}\left[0;S_o^{*}-S_o\right]}{S_o^{*}}\right)}^{e_0}& \left(6\right)\end{array}$ [0050] in which S.sub.o* is the oil saturation beyond which the foam loses all its faculties to reduce the mobility of the gas, and the exponent e.sub.o is a constant to be determined; [0051] the interpolation function F.sub.4 relative to the parameter V.sub.4=u.sub.g (gas flow rate) by a formula of the type:

[00008]$\begin{array}{cc}{F}_4={\left(\frac{N_c^{*}}{\mathrm{Max}\left(N_c,N_c^{*}\right)}\right)}^{e_c}.Math..Math.\mathrm{with}.Math..Math.N_c=\frac{{}_g.Math.u_g}{{}_{\mathrm{vg}}\left(C_s^w\right)}& \left(7\right)\end{array}$ [0052] in which N.sub.c* is the reference value of the capillary number N.sub.c calculated for the reference optimal flow rate. The variables involved in the calculation of N.sub.c are the velocity of the gas u.sub.g, the porosity of the formation considered, the water-gas interfacial tension .sub.gw (which is dependent on the foaming agent concentration C.sub.s.sup.w), and the viscosity of the gas .sub.g. The exponent e.sub.c is also a constant to be calibrated.

[0053] Generally, it can be shown that any interpolation function F.sub.k of the parameter V.sub.k can be written in the form:

[00009]$\begin{array}{cc}{F}_k\left(V_k\right)=\frac{\frac{1}{F.Math..Math.M}-1}{\frac{1}{F.Math..Math.M_{\mathrm{opt}}}-1}=\frac{M_{\mathrm{mod}}\left(V_k\right)-1}{M_{\mathrm{mod}}^{\mathrm{opt}}-1}& \left(8\right)\end{array}$

in which M.sub.mod(V.sub.k) is the reduction of mobility for a value V.sub.k of the parameter k affecting the foam (and for optimal values of the other parameters V.sub.j, j being different from k) and in which M.sub.mod.sup.opt=M.sub.mod(V.sub.k.sup.opt) is the reduction of mobility obtained for the optimal value V.sub.k.sup.opt of the parameter V.sub.k. The method according to the invention thus consists, for each parameter V.sub.k affecting the foam, in determining the factors M.sub.mod(V.sub.k) for various values of this parameter, and M.sub.mod.sup.opt, then in determining, from these factors, the constants of the interpolation function F.sub.k considered.

[0054] According to an implementation of the invention in which the functional FM defined in the equation (2) involves the interpolation functions F.sub.1, F.sub.2, F.sub.3 and F.sub.4 defined in the equations (4) to (7), the determination of the model of displacement of the foam entails calibrating the 8 constants: C.sub.s.sup.w-ref, e.sub.s, f.sub.w, S.sub.w*, S.sub.o*, e.sub.o, N.sub.c.sup.ref, e.sub.c.

[0055] According to the invention, the determination of the constants of the interpolation functions F.sub.k involved in the equation (2) is performed via a calibration, interpolation function by interpolation function (and not globally, for all the functions), from experimental measurements relative to each of the interpolation functions, performed in the optimal conditions established for the other interpolation functions.

[0056] The method according to the invention comprises at least the following steps, the step 1 being able to be repeated for each of the interpolation functions of the model of displacement of the foam, and the step 2 being able to be optional:

[0057] 1. Laboratory measurements relative to an interpolation function [0058] 1.1. Definition of values of the parameter relative to the interpolation function [0059] 1.2. Injections with/without foam and measurements of headlosses [0060] 1.3. Determination of an optimal parameter value

[0061] 2. Laboratory measurements according to optimal conditions

[0062] 3. Determination of the foam displacement model [0063] 3.1. Determination of the optimal mobility reduction factor [0064] 3.2. Calibration of the constants of the interpolation functions

[0065] 4. Operation of the hydrocarbons of the formation

[0066] The various steps of the method according to the invention are detailed hereinbelow.

1. Laboratory Measurements Relative to an Interpolation Function

[0067] During this step, laboratory experiments are performed relative to a given interpolation function F.sub.k of the model of displacement of the foam defined according to the equations (1) and (2). According to one implementation of the invention, this step is repeated for each of the interpolation functions involved in the model of displacement of the foam defined according to the equations (1) and (2). It should be noted that the model of displacement of the foam can however comprise only a single interpolation function (case for which k=1). Optimal values are adopted for the other parameters affecting the foam in such a way that the other interpolation functions F.sub.j, j different from k, have the value 1 or are invariant in these experiments relative to the interpolation function F.sub.k.

[0068] During this step, applied to each interpolation function independently of one another, a plurality of values of the parameter is defined relative to the interpolation function considered, then an injection into said sample of the gas in non-foaming form and of the gas in foam form is performed according to the values of the parameter relative to the interpolation function considered, and a headloss with foam and a headloss without foam are measured respectively for each of the values of the parameter relative to this function. This step is detailed hereinbelow for a given interpolation function F.sub.k.

[0069] 1.1. Definition of Parameter Values Relative to the Interpolation Function

[0070] During this substep, the aim is to define a plurality of values V.sub.k,i (with i lying between 1 and I, and I>1) of the characteristic parameter V.sub.k of the interpolation function F.sub.k considered.

[0071] According to one implementation of the invention, the aim is to define a range of values of this parameter and a sampling step for this range.

[0072] According to one implementation of the invention, the plurality of values of the parameter V.sub.k relative to the interpolation function F.sub.k considered are defined from the possible or realistic values of the parameter considered (for example, a mass concentration of foaming agent less than 1% in all cases) and so as to sample in an ad hoc manner the curve representative of the interpolation function considered (an interpolation function that has a linear behavior does not need a high number of measurements, unlike other types of function). The expert in foam injection-assisted recovery has a perfect knowledge of how to define a plurality of ad hoc values of the parameters of each of the interpolation functions F.sub.k. According to an implementation of the invention in which the interpolation function considered relates to the fluid flow rate (parameter V.sub.4 of the function F.sub.4 of the equation (7)), an injection flow rate on coring of between 10 and 40 cm.sup.3/h is for example chosen, with a step of 10 cm.sup.3/h.

[0073] 1.2. Injections with/without Foam and Headloss Measurements

[0074] During this substep, at least two series of experiments are carried out on at least one sample of the underground formation for the interpolation function F.sub.k considered: [0075] injection of gas in non-foaming form (more specifically, a co-injection of water and of gas in non-foaming form) into the sample considered for each of the values V.sub.k,i of the parameter V.sub.k relative to the function F.sub.k considered. The flow rates of gas and of water adopted for each of these co-injections are the same as the flow rates of gas and of water injected in foam form in the tests which follow these co-injections. For example, in the case of the interpolation function F.sub.4 of the equation (7), the flow rate only is made to vary in the sample considered, the parameters of the other interpolation functions F.sub.1, F.sub.2, F.sub.3 (for example, the foaming agent concentration, the quality of the foam and the oil saturation) being fixed. During each of the experiments of this first series, a headloss (that is to say a pressure difference) is measured, that is denoted P.sub.k,i.sup.NOFO, for each value V.sub.k,i; [0076] injection of foam: the same experiment is repeated, for the same values of the parameter considered (for example the flow rate for the interpolation function F.sub.4 according to the equation (7)), but by this time injecting the water and the gas in foam form. During each of the experiments of this second series, a headloss (that is to say a pressure difference) is measured, that is denoted P.sub.k,i.sup.FO, pr for each value V.sub.k,i;

[0077] According to one implementation of the invention, the injections of gas in non-foaming form and in foam form are performed on samples of the formation initially saturated with a liquid phase (such as water and/or oil), the latter being able to be mobile or residual depending on the history of the coring and the measurement objectives (checking mobility of the gas in secondary or tertiary injection, after injection of water). The displacements studied are then draining processes in which the saturation of the gas phase increases in all cases.

[0078] According to a variant implementation of the invention, it is possible to measure, in addition to the headlosses, the productions of liquid phase (water and/or oil) and of gas, and, possibly, the gas saturation profiles during the transitional period of the displacement and in the steady state. These optional measurements make it possible to validate the model once the interpolation functions F.sub.k are calibrated.

[0079] 1.3. Determination of an Optimal Parameter Value

[0080] During this substep, the aim is to determine the value V.sub.k.sup.opt, that will hereinafter be called optimal value, maximizing the ratio between the headlosses without foam P.sub.k,i.sup.NOFO and the headlosses with foam P.sub.k,i.sup.FO relative to the interpolation function F.sub.k considered and measured during the preceding substep. Thus, if M.sub.lab.sup.k,i is used to denote the ratio of the headlosses measured in the presence and in the absence of foam for the value V.sub.k,i of the parameter V.sub.k, i.e.

[00010]$M_{\mathrm{lab}}^{k,i}=\frac{.Math..Math.P_{k,t}^{\mathrm{FO}}}{.Math..Math.P_{k,i}^{\mathrm{NOFO}}}=\frac{k_{\mathrm{rg}}\left(S_{g\left(k,i\right)}^{\mathrm{NOFO}}\right)}{k_{\mathrm{rg}}^{\mathrm{FO}}\left(S_{g\left(k,i\right)}^{\mathrm{FO}}\right)},$

it is then possible to define the optimal value V.sub.k,iopt as the value V.sub.k,i which maximizes M.sub.lab.sup.k,i whose value is then denoted as follows:

[00011]$\begin{array}{cc}{M}_{\mathrm{lab}}^{k,\mathrm{iopt}}=M_{\mathrm{lab}}^{\mathrm{kopt}}=\underset{i}{\mathrm{Max}}.Math.M_{\mathrm{lab}}^{k,i}& \left(9\right)\end{array}$

[0081] According to a preferred implementation of the invention, the step 1 as described hereinabove is repeated for each of the parameters V.sub.k relative to each of the interpolation functions F.sub.1, taken into consideration for the implementation of the method according to the invention. Thus, at the end of such a repetition, an optimal value V.sub.k.sup.opt is obtained for each parameter V.sub.k.

[0082] Subsequently, optimal conditions is the term used to denote the set of the values V.sub.k.sup.opt determined on completion of the step 1, the latter if necessary being repeated for each of the interpolation functions taken into consideration for the implementation of the method according to the invention.

2. Laboratory Measurements According to Optimal Conditions

[0083] During this step, the aim is to perform two types of experiments on at least one sample of the underground formation, by injecting gas in non-foaming form, and gas in foam form, similarly to the substep 1.2, but this time in the optimal conditions determined on completion of the substep 1.3, this substep being repeated if necessary for each of the interpolation functions taken into consideration for the definition of the model of displacement of the foam according to the equations (1) and (2)). In other words, the following measures are carried out: [0084] injection of gas in non-foaming form (more specifically, a co-injection of water and of gas in non-foaming form) into the sample considered, this injection being performed in the optimal conditions (defined by the set of the optimal values V.sub.k.sup.opt determined for each parameter V.sub.k) determined on completion of the step 1. During this first experiment, a headloss (that is to say a pressure difference) is measured, that is then denoted P.sub.opt.sup.NOFO; [0085] injection of foam (that is to say an injection of gas and of water, with an addition of a foaming agent into one of the water or gas phases) into the sample considered, this injection being performed in the optimal conditions (defined by the set of the optimal values V.sub.k.sup.opt determined for each parameter V.sub.k) determined on completion of the step 1. During this second experiment, a headloss (that is to say a pressure difference) is measured, that is then denoted P.sub.opt.sup.FO.

[0086] Subsequently, M.sub.lab.sup.opt will be used to denote the optimal mobility reduction factor relative to the laboratory measurements, defined by a formula of the type:

[00012]$\begin{array}{cc}{M}_{\mathrm{lab}}^{\mathrm{opt}}=\frac{.Math..Math.P_{\mathrm{opt}}^{\mathrm{FO}}}{.Math..Math.P_{\mathrm{opt}}^{\mathrm{NOFO}}}=\frac{k_{\mathrm{rg}}\left(S_{g,\mathrm{opt}}^{\mathrm{NOFO}}\right)}{k_{\mathrm{rg}}^{\mathrm{FO}}\left(S_{g,\mathrm{opt}}^{\mathrm{FO}}\right)}& \left(10\right)\end{array}$

[0087] This step is not necessary in practice if the precaution to perform the experiments of the step 1 relative to each of the parameters V.sub.k by adopting optimal values V.sub.j,jk.sup.opt of the other parameters V.sub.j affecting the foam has indeed been taken. This step nevertheless makes it possible to refine the value of M.sub.lab.sup.opt, if the assumed optimal conditions of the parameters V.sub.j,jk were not perfectly satisfied.

3. Determination of the Foam Displacement Model

[0088] 3.1. Determination of the Optimal Mobility Reduction Factor

[0089] During this substep, the aim is, from the headloss measurements performed in the optimal conditions, from measurements of conventional relative permeabilities to the gas in non-foaming form and from measurements of conventional relative permeabilities to the aqueous phase, to determine an optimal mobility reduction factor, that is to say the factor of reduction of the relative permeabilities to the gas when, present at a given saturation within the porous medium, it circulates in foam form or in continuous phase form (in the presence of water).

[0090] According to one implementation of the invention, the optimal mobility reduction factor is determined according to at least the following steps: [0091] from the conventional relative permeabilities to the gas k.sub.rg and to the aqueous phase k.sub.rw, the gas saturation is calculated in steady state conditions of flow of non-foaming gas and water S.sub.g.sup.NOFO according to a formula of the type:

[00013]$\begin{array}{cc}{S}_g^{\mathrm{NOFO}}={\left(\frac{k_{\mathrm{rg}}}{k_{\mathrm{rw}}}\right)}^{-1}.Math.\left(\frac{f_g}{1-f_g}.Math.\frac{{}_g}{{}_w}\right)& \left(11\right)\end{array}$ [0092] in which f.sub.g is the fractional gas flow rate (ratio of the gas flow rate to the total flow rate), .sub.g and .sub.w are, respectively, the viscosity of the gas and of the water; [0093] from the ratio of headlosses measured in the optimal conditions as defined on completion of the step 1 (the step 1 being able to be repeated if necessary for each of the interpolation functions F.sub.k considered), from the gas saturation in steady state conditions of flow of non-foaming gas and water S.sub.g.sup.NOFO, the gas saturation in the presence of foam S.sub.g.sup.FO is calculated according to a formula of the type:

[00014]$\begin{array}{cc}{S}_{g,\mathrm{opt}}^{\mathrm{FO}}=1-{\left(k_{\mathrm{rw}}\right)}^{-1}.Math.\left\{\frac{k_{\mathrm{rw}}\left(S_w^{\mathrm{NOFO}}=1-S_g^{\mathrm{NOFO}}\right)}{M_{\mathrm{lab}}^{\mathrm{opt}}}\right\}& \left(12\right)\end{array}$ [0094] This relationship devolves from the known assumption of invariance of the functions of relative permeability to water flowing in the form of foam films or in conventional continuous form. [0095] from the gas saturation in steady state conditions of flow of non-foaming gas and water S.sub.g.sup.NOFO from the gas saturation in the presence of foam S.sub.g.sup.FO, in the optimal conditions, from the factor M.sub.lab.sup.opt determined in the optimal conditions (see step 2), the mobility reduction factor M.sub.mod.sup.opt is determined according to a formula of the type:

[00015]$\begin{array}{cc}{M}_{\mathrm{mod}}^{\mathrm{opt}}=M_{\mathrm{lab}}^{\mathrm{opt}}.Math.\frac{k_{\mathrm{rg}}\left(S_{g,\mathrm{opt}}^{\mathrm{FO}}\right)}{k_{\mathrm{rg}}\left(S_{g,\mathrm{opt}}^{\mathrm{NOFO}}\right)}& \left(13\right)\end{array}$

[0096] 3.2. Calibration of the Constants of the Interpolation Functions

[0097] During this substep, the constants of each of the interpolation functions F.sub.k considered are calibrated, from the optimal mobility reduction factor M.sub.mod.sup.opt, from the headloss measurements relative to the interpolation function considered, from the measurements of conventional relative permeabilities to the gas in non-foaming form and from the measurements of conventional relative permeabilities to the aqueous phase.

[0098] According to one implementation of the invention, the procedure described in the substep 3.1 is applied beforehand to the ratios of the headlosses M.sub.lab.sup.k,i measured in the presence and in the absence of foam for the different values V.sub.k,i of the parameter V.sub.k. Thus, the mobility reduction factors M.sub.mod.sup.k,i relative to the values V.sub.k,i of the parameter V.sub.k are determined according to a formula of the type:

[00016]$\begin{array}{cc}{M}_{\mathrm{mod}}^{k,i}=M_{\mathrm{lab}}^{k,i}.Math.\frac{k_{\mathrm{rg}}\left(S_{g,\left(k,i\right)}^{\mathrm{FO}}\right)}{k_{\mathrm{rg}}\left(S_{g\left(k,i\right)}^{\mathrm{NOFO}}\right)}& \left(14\right)\end{array}$

in which the gas saturation in the presence of foam S.sub.g(k,i).sup.FO for the values V.sub.k,i of the parameter V.sub.k is obtained according to a formula of the type:

[00017]$\begin{array}{cc}{S}_{g\left(k,i\right)}^{\mathrm{FO}}=1-{\left(k_{\mathrm{rw}}\right)}^{-1}.Math.\left\{\frac{k_{\mathrm{rw}}\left(S_{w\left(k,i\right)}^{\mathrm{NOFO}}=1-S_{g,\left(k,i\right)}^{\mathrm{NOFO}}\right)}{M_{\mathrm{lab}}^{k,i}}\right\}& \left(15\right)\end{array}$

Advantageously, this operation is repeated for each of the interpolation functions F.sub.k. Then, the constants of each of the interpolation functions F.sub.k considered are calibrated, from the optimal mobility reduction factor M.sub.mod.sup.opt and from the values of the mobility reduction factors M.sub.mod.sup.k,i relative to each interpolation function determined as described above. In the case of the function F.sub.4 for example, a value of the exponent e.sub.c is determined which most closely adjusts the values of M.sub.mod.sup.4,i corresponding to the values V.sub.4,i of the parameter studied (flow rate in this example), which is formulated as follows:

[00018]$F_4\left(V_{4,i}\right)={\left(\frac{N_c^{*}}{\mathrm{Max}\left(N_{c,i},N_c^{*}\right)}\right)}^{e_c}=\frac{M_{\mathrm{mod}}^{4,i}-1}{M_{\mathrm{mod}}^{\mathrm{opt}}-1}$

[0099] According to one implementation of the invention, this calibration, interpolation function by interpolation function, can be performed by a least squares method, such as for example an inverse method based on the iterative minimization of a functional. The expert has a perfect knowledge of such methods. Advantageously, the implementation of a least squares method and in particular the iterative minimization of a functional, is performed by means of a computer.

[0100] According to another implementation of the invention, such a calibration is carried out, interpolation function by interpolation function, graphically. The expert has a perfect knowledge of such methods for calibrating constants of a function from a series of values of said function.

[0101] Thus, on completion of this step, there is a model of displacement of the foam that is calibrated and suitable for use by an ad hoc flow simulator.

4. Operation of the Hydrocarbons

[0102] During this step, the aim is to define at least an optimal scheme for operating the fluid contained in the formation, that is to say an operating scheme that allows for an optimal operation of a fluid considered according to technical and economic criteria predefined by the expert. It can be a scenario offering a high rate of recovery of the fluid, over a long period of operation, and requiring a limited number of wells. Then, according to the invention, the fluid of the formation studied is operated according to this optimal operation scheme.

[0103] According to the invention, the determination of said operational scheme is performed using a flow simulation that makes use of the foam displacement model established during the preceding steps. One example of flow simulator that makes it possible to take account of a foam displacement model is the PumaFlow software (IFP Energies nouvelles, France).

[0104] According to the invention, at any instant t of the simulation, the flow simulator solves all the flow equations specific to each mesh and delivers solution values of the unknowns (saturations, pressures, concentrations, temperature, etc.) predicted at that instant t. The knowledge of the quantities of oil produced and of the state of the deposit (distribution of the pressures, saturations, etc.) at the instant considered devolves from this resolution. According to one implementation of the invention, different schemes for operating the fluid of the formation studied are defined and the flow simulator incorporating the foam displacement model determined on completion of the step 3 is used to estimate the quantity of hydrocarbons produced according to each of the different operating schemes.

[0105] An operating scheme relative to a foam injection-assisted recovery can notably be defined by a type of gas injected into the formation studied and/or by the type of foaming agent added to this gas, by the quantity of foaming agent, etc. An operating scheme is also defined by a number, a geometry and a layout (position and spacing) of the injecting and producing wells in order to best take account of the impact of the fractures on the progression of the fluids in the reservoir. In order to define an optimal operating scheme, various tests of different production scenarios can be performed using a flow simulator. The operating scheme that offers the best fluid recovery rate for the lowest cost will for example be preferred. By selecting various scenarios, characterized for example by various respective layouts of the injecting and producing wells, and by simulating the fluid production for each of them, it is possible to select the scenario that makes it possible to optimize the production of the formation considered according to the technical and economic criteria predefined by the expert. The operating scheme offering the best fluid recovery rate for the lowest cost will for example be considered as the optimal operating scheme.

[0106] The experts then operate the fluid of the formation considered according to the scenario that makes it possible to optimize the production from the deposit, notably by drilling injecting and producing wells defined by said optimal operating scheme, and to produce the fluid according to the recovery method defined by said optimal operating scheme.

Production Example

[0107] The features and advantages of the method according to the invention will become more clearly apparent on reading about the following exemplary application.

[0108] More specifically, the present invention has been applied to an underground formation in which the reservoir rock consists of sandstone, of the Berea sandstone type. An assisted recovery of the hydrocarbons contained in the reservoir based on an injection of foaming CO2 is trialed.

[0109] For this example, a functional FM of the foam displacement model is used according to the equation (2) defined by the four interpolation functions according to the equations (4) to (7). As prescribed in the method according to the invention, the calibration of the constants of the interpolation functions is carried out interpolation function by interpolation function. Only the calibration of the interpolation function F.sub.4 (see equation (7)) is detailed hereinbelow, but the same principle can be applied to the other interpolation functions.

[0110] According to the step 1.2 described above, a series of co-injections of gas and of water and of injections of foam were carried out in the laboratory, on a sample of the reservoir rock originating from the formation studied. The characteristics of this sample and the measurement conditions are presented in Table 1. A non-dense gaseous mixture consisting of 62% CO2 and 38% methane at a temperature of 100 C. and a pressure of 100 bar was injected. These displacements were performed with fixed fractional gas flow rate (equal to 0.8) and for different successive total flow rates (10, 20, 30 and 40 cm.sup.3/h). The oil is absent for this series of tests and the headlosses in steady state conditions of flow of water and of gas on the one hand, of foam on the other hand, were measured in the same conditions.

[0111] The conventional relative permeabilities required to resolve the equations (11) and (12) are analytical functions defined as power functions (called Corey functions) with exponents equal to approximately 2.5 for the gas and 3.9 for the water with an irreducible drainage water saturation equal to 0.15, and limit points equal to 0.2 for the gas and 1 for the water, i.e.:

[00019]$k_{\mathrm{rg}}\left(S_g^{\mathrm{NOFO}}\right)=0.2.Math.{\left(\frac{S_g^{\mathrm{NOFO}}}{0.85}\right)}^{2.5}$$k_{\mathrm{rw}}\left(S_w^{\mathrm{NOFO}}\right)={\left(\frac{S_w^{\mathrm{NOFO}}-0.15}{0.85}\right)}^{3.9}$

These curves of relative permeabilities were estimated beforehand from literature data and checked afterwards by comparison of the headloss values calculated and measured during the co-injections of gas and of water. In this way the model of relative permeability to the foam does indeed return the gas mobility reductions, i.e. the ratios of relative permeability in the absence and in the presence of foam, but without necessarily well reproducing the real diphasic behavior (transient states in particular).

[0112] Table 1 presents the headlosses (pressure gradient) with and without foam for four values of the parameter V.sub.4=u.sub.g of the equation (7). From these values, the value of

[00020]$M_{\mathrm{lab}}^{\mathrm{kopt}}=\underset{i}{\mathrm{Max}}.Math..Math.M_{\mathrm{lab}}^{k,i}$

is deduced therefrom. This value (M.sub.lab.sup.4opt in this example), equal to 83, was obtained for a flow rate V.sub.4.sup.opt equal to 20 cm.sup.3/h (see substep 1.3). These laboratory experiments were repeated for the other parameters of the other interpolation functions F.sub.1, F.sub.2, and F.sub.3. The optimal conditions are then determined for all of the interpolation functions.

[0113] According to the step 2, measurements are performed with and without foam in the duly determined optimal conditions. The optimal mobility reduction factor M.sub.mod.sup.opt is then determined, in accordance with the step 3.1, as are the values of the mobility reduction factors M.sub.mod.sup.k,i relative to the sampled values V.sub.k,i of the parameter V.sub.k, in accordance with the step 3.2. The calibration of each of the interpolation functions is then carried out. In particular, the constant e.sub.c of the function F.sub.4 was calibrated and a value close to 0.6 was determined.

[0114] FIG. 1 shows by a solid line the trend of the mobility reduction factor R as a function of the parameter V.sub.4 of the function F.sub.4 (flow rate Q) deduced from the method according to the invention. The comparison with the mobility reduction factor deriving from the flow simulation (dotted line curve) shows a good consistency with the foam displacement model of according to the invention.

[0115] Thus, the method according to the invention allows for a reliable determination of the foam displacement model from experimental data produced and processed according to a sequential and systematic approach, parameter by parameter, and not by the overall adjustment of a set of measurements varying one or more parameters simultaneously. Moreover, given the parametric complexity of the behavior of the foams, the experiments according to the method according to the invention are carried out in conditions as close as possible to the reservoir conditions.

TABLE-US-00001 TABLE 1 Berea sandstone Total flow rate [cm3/h] 10 20 30 40 P = 100 bar P.sub.k,i.sup.FO 5.2 10.3 11.9 13.6 T = 100 C. P.sub.k,i.sup.NOFO 0.07 0.124 0.18 0.235 L = 15 cm A = 12.56 cm2 = 0.19 Kw = 120 mD w = 0.28 cp g = 0.02 cp g = 0.125 g/cm3 [00021]$M_{\mathrm{lab}}^{k,i}=\frac{.Math..Math.P_{k,i}^{\mathrm{NOFO}}}{.Math..Math.P_{k,i}^{\mathrm{FO}}}$ 74 83 66 58