EP2526400A1

EP2526400A1 - Measurement of parameters linked to the flow of fluids in a porous material

Info

Publication number: EP2526400A1
Application number: EP11704666A
Authority: EP
Inventors: Didier Lasseux; Yves Jannot
Original assignee: Centre National de la Recherche Scientifique CNRS; Total SE
Current assignee: Centre National de la Recherche Scientifique CNRS; TotalEnergies SE
Priority date: 2010-01-22
Filing date: 2011-01-21
Publication date: 2012-11-28
Also published as: RU2012136121A; AU2011208574C1; US20130054157A1; RU2549216C2; FR2955662B1; WO2011089367A1; CN102906556A; CA2787771A1; AU2011208574A1; FR2955662A1; AU2011208574B2; BR112012018093A2

Abstract

The invention relates to a method in which a sample (2) of the material to be studied is placed in a sealed cell (1) such that the upstream surface (3) thereof communicates with a first space (V₀) and the downstream surface (4) thereof communicates with a second space. The pressure in the first space is modulated and the variations over time of the respective pressures in the first space and in the second space are measured. By means of a differential equation taking as parameters the intrinsic permeability of the material, the porosity and the Klinkenberg coefficient thereof, the pressure variations measured are digitally analysed to estimate at least the intrinsic permeability and the Klinkenberg coefficient of the material, and advantageously the porosity of the material during the same experiment.

Description

MEASUREMENT OF PARAMETERS RELATED TO FLUID FLOW

IN POROUS MATERIAL

The present invention relates to the measurement of physical properties relating to the flow of a fluid phase in a porous material. It is particularly applicable to materials having very small diameters of pore-scale flow channels, that is to say materials having a high resistance to the flow of a fluid (inverse intrinsic permeability). Non-limiting examples of such materials include rocks of tight gas reservoirs, rocks of potential storage site covers, materials used in sealing devices, composite materials, etc.

The flow of a fluid through a porous medium depends, on the scale of a representative block of the material, three intrinsic physical characteristics in the middle that are:

- its liquid or intrinsic permeability k _| , expressed in m ² or more commonly in D (darcy: 1 D "0.987 ^χ ^10-12 m ² );

its Klinkenberg coefficient b, expressed in Pa, in the case of a low-permeability medium and a gas flow at a low pressure level, or its Forchheimer coefficient β, expressed in m ^-1 , also called the inertial resistance factor, when one is interested in high flow rates giving rise to inertial effects;

its porosity φ, equal to the ratio of the volume of the voids of the material to its total volume. [0004] No current method allows the simultaneous determination of these three parameters using a single experiment. In particular, the porosity is often measured separately from the other two parameters by a method of pycnometry (helium, mercury, ...) or weighing.

To measure the gas permeability of a material, there are steady-state methods and unsteady-state methods. See, for example, JA Rushing et al., Klinkenberg-corrected permeability measurements in tight gas sands: Steady-state versus unsteady-state techniques, SPE 89867 1 -1 1, 2004.

The steady-state method has the disadvantage of requiring a long time before obtaining the stationary flow regime for the acquisition of a measuring point. The time at which this steady state is reached varies with the inverse of k | and with the square of the thickness of the sample. It easily reaches several hours for very low permeabilities. Separate determination of intrinsic permeability k _| and the Klinkenberg coefficient b requires several measuring points and thus obtaining as many stationary states. This can become very long, so this method is poorly suited to the low permeability range. Furthermore, this technique requires the measurement of fluid flow, which can be tricky when the permeability is very low. To overcome these disadvantages, a measurement in transient condition is preferable. Typically, an unsteady state experiment consists in recording the evolution of the differential pressure ΔΡ (ί) between the ends of the sample. Each end of the sample is connected to a respective reservoir and one of them is initially subjected to a pressure pulse. This method is named "Puise decay". A variant where the downstream reservoir is of infinite volume (atmosphere) is named "Draw down".

The interpretation of ΔΡ (ί) leads to identify the permeability of the medium. Frequently, the technique does not take into account Klinkenberg effects.

In US Pat. No. 2,867,166, an approximate method has been proposed for experimentally determining porosity, apparent permeability (i.e. including Klinkenberg effects) and intrinsic permeability. In these works, k _| , b and φ are approximated by performing the same experiment three times with a constant ratio between the value of the initial pulse of pressure and the initial pressure of the sample. The first experiment is conducted by raising the decay time of the pressure pulse to a given fraction (for example 55%) of its initial value. The second experiment is identical to the first, but conducted by simply changing the pressure level of the pulse and the initial pressure of the sample so that the gap between them is the same as in the first experiment. There is still time for the pressure pulse to decrease to the same fraction (55%) of the initial value. The third experiment is identical to the other two by modifying the volume of the chamber used to generate the pressure pulse. The determination of k _| , b and φ from these three experiments is performed in an approximate way using a graphical chart and exploiting an empirical linear behavior. In fact, it is difficult to estimate the real impact of these approximations in the general case. In addition, note the experimental difficulty related to the device and the execution time required by conditioning the sample at different pressures.

In "A method for the simultaneous determination of permeability and porosity in low permeability cores", SPE 15379, 1-11, 1988, S.E. Haskett et al. have proposed a method for determining the permeability k |, the Klinkenberg effects being neglected, and the porosity φ. The method requires that the experiment be conducted to the final equilibrium of pressures in upstream and downstream reservoirs. It is based on the measurement over time of the pressure difference between the upstream and downstream reservoirs. This configuration is not very precise and is not optimal for determining the parameters.

[0011] In "A detailed analysis of permeability and Klinkenberg coefficient estimation from unsteady-state pulse-decay or draw-down experiments", Symp. Soc. Core Analysts, Calgary, 10-13 September, 5CA2007-08, 2007, Y. Jannot et al. have reconsidered the "Puise decay" method without particular simplifying assumptions. It has simply been considered that the sample constitutes a solid matrix that is indeformable under the effect of the flow of the measuring gas, and that the flow of the gas is weakly compressible, isothermal, and creeping. In this context, the physical problem describing the experience of "Puise decay" in the general case is expressed by: d

(P + b) - - for 0 <x <e and t> 0 (1)

; dx k | dt with the initial conditions:

P (0,0) = P _0i (2) Ρ (χ, Ο) = Pu for x> 0 (3) and the boundary conditions: k, S r. i dP, _ ôP

[P (0, t) + b] ¾- (0, t) = ¾- (0, t) (4) μνο dx ol [P (e, t) ₊ b] ¾e, t) = - ^ (e , t) (5) μν- | dx dt where: P is the pressure at time t and at position x along the sample, x = 0 corresponding to the upstream face of the sample, x = e at its downstream face, the pulse of pressure being applied at t = 0;

S is the section of the sample; e is the length of the sample;

V ₀ and V- _| are respectively the volumes of the upstream reservoir (high pressure) and the downstream reservoir (low pressure) communicating with the sample, where initially there are (at t = 0) respective pressures

Poi and P _{1 i;} μ is the dynamic viscosity of the gas, assumed to be constant.

In "Draw down" configuration, the second boundary condition is replaced by a classical Dirichlet condition: P (e, t) = P- _| = P- _] ,. It is assumed here that the sample is initially at ambient pressure with which it is normally at equilibrium.

A dead volume is necessarily present upstream of the sample, between the valve which isolates the sample from the upstream reservoir and the face upstream of the sample. It is desirable to have a very small volume V ₀ (ideally of the order of pore volume in the sample) to increase the sensitivity of the measurements to the porosity φ, but then its precise determination, to take into account in the condition (4), becomes very delicate because it supposes to know then the dead volume with precision. The existence of this dead volume therefore has a significant impact on the estimated values of k | and B. In addition, the opening of the valve at the moment when the "Puise decay" experiment begins produces a relaxation of the fluid in the dead volume which causes observable thermal and hydrodynamic perturbations but which is extremely difficult to accurately report in A model. The equations (1) to (5) above do not include these thermal and hydrodynamic effects.

An error in the value of the porosity φ has a considerable impact on the estimated values of the permeability k _| and the Klinkenberg coefficient b. A quality estimate of these two parameters therefore requires precise knowledge of φ if this quantity is provided as an input parameter. The pyknometric techniques used for this are time-consuming and lead to an estimation of the intrinsic porosity and not of the constrained porosity (storage coefficient) generally useful for the analysis of a real material.

[0015] There is a need for an experimental method to improve the estimation of the permeability k _| and the Klinkenberg coefficient b

(for low permeability ranges, replaced by the Forchheimer coefficient for strong permeability ranges). It is further desirable to allow the porosity φ to be estimated simultaneously using a single experiment.

There is provided a method for estimating physical parameters of a material, comprising:

placing a sample of the material in a sealed cell so that an upstream face of the sample communicates with a first volume and that a downstream face of the sample communicates with a second volume;

- cause a pressure modulation in the first volume;

measuring the variations over time of respective pressures in the first volume and in the second volume; and

- using a differential equation whose parameters are the intrinsic permeability of the material, the porosity of the material and at least one other coefficient specific to the material, and for boundary conditions the measured variation of the pressure in the first volume, to analyze numerically the measured variation of the pressure in the second volume to estimate at least the intrinsic permeability and said other coefficient.

To overcome the difficulties related to the dead volume upstream of the sample, the initial data is no longer considered to be only the value of a pressure pulse P ₀ j serving to simulate the evolution of P (0, t) to perform the inversion. On the contrary, a finite volume reservoir V- | the downstream side and consider two separate information both measured: the upstream pressure signal P (0, t) = Po (t) and the downstream pressure signal P (1, t) = P- | (T). The signal Po (t) can serve as an input signal to the analysis step which consists of a digital inversion of differential equation, performed on the downstream signal P- _| (T). Since Po (t) is no longer simulated, but measured, it can contain irregularities related to thermal events, the existence of a dead volume, etc., without this being a source of disturbance compared to the model. used in the inversion procedure. The other coefficient specific to the material estimated in conjunction with its intrinsic permeability k _| is typically the Klinkenberg coefficient b when it is known that the studied material is in a low range of permeability (less than about 10 ^{~ 6} m ² ). If the permeability is in a higher range, the other coefficient may be the Forchheimer β coefficient. There may be a range of permeability where both the Klinkenberg coefficient b and the coefficient of Forchheimer β are likely to be taken into account. account in the model.

In the case where the Klinkenberg coefficient b is estimated with intrinsic permeability k |, the analysis step consists of the digital inversion of (1) performed on the downstream signal P- _| (T). The boundary condition (4) is replaced by a Dirichlet condition under pressure P (0, t) = Po (t), where Po (t) is measured using a manometer in the first volume VQ. The physical problem no longer depends on V ₀ , nor a dead volume that does not need to be known any more.

The pressure modulation in the first volume is not applied simply instantaneously, but on a time scale greater than that of a pressure pulse. It is typically on a time scale depending on the permeability range of the material but generally greater than one minute. This pressure modulation in the first volume may in particular be caused by a succession of pressure pulses.

In one embodiment, the numerical analysis of the variations of the measured pressures comprises a follow-up of the evolution over time of the reduced sensitivity of the pressure P (t) measured in the second volume to the intrinsic permeability and of the evolution over time of the reduced sensitivity of Pi (t) to the Klinkenberg or Forchheimer coefficient. This makes it possible to verify that the pressure modulation has been applied in the first volume so as not to allow the ratio between these two sensitivities to be stabilized, which would not allow a proper estimation of the permeability and the coefficient in question. In a preferred embodiment, the numerical analysis of the variations of the measured pressures Po (t), P-i (t) is performed so as to estimate the porosity φ of the material in addition to its intrinsic permeability k | and its Klinkenberg coefficient b (or Forchheimer β).

In a typical "Puise decay" type experiment, the sensitivity of P- _| (t) at φ becomes rapidly constant, after a time too short for that this parameter can be correctly estimated. To increase this sensitivity, it is necessary to multiply the effects at short times so that the accumulation of fluid in the pores of the material occurs repeatedly over the duration of the experiment. Since the method comprises a measurement of Po (t) which becomes a data item for the inversion performed on P- _| (t), any variation of Po (t) imposed is possible. A series of pressure pulses is thus generated upstream of the sample so as to excite the capacitive behavior of the system to facilitate the estimation of the porosity.

The numerical analysis of the variations of the measured pressures can comprise a follow-up of the evolution over time of the reduced sensitivity of the pressure Pi (t) measured at the porosity. This makes it possible to verify that the pressure modulation has been applied in the first volume so as not to allow this sensitivity reduced to porosity to be stabilized, which would not make it possible to estimate the porosity φ properly.

To facilitate the convergence of the estimation of the parameters, it is possible in certain cases to carry out a pre-estimation of the intrinsic permeability k _| and the Klinkenberg coefficient b using the pressures measured in time intervals where the pressure in the second volume varies substantially linearly. An advantageous embodiment then comprises an examination of the evolution over time of the pressure in the second volume. When this examination shows that the pressure in the second volume varies substantially linearly with time, this pressure is allowed to vary substantially linearly to acquire values for the pre-estimation of the intrinsic permeability and the coefficient, and then apply a new pressure pulse in the first volume.

Other features and advantages of the present invention will become apparent in the following description of a nonlimiting exemplary embodiment, with reference to the accompanying drawings, in which: FIG. 1 is a diagram of an installation that can be used to implement a method for estimating physical parameters according to the invention;

FIG. 2 is a graph showing reduced sensitivities to permeability, Klinkenberg coefficient and porosity in one embodiment of the process;

FIG. 3 is a graph showing the evolution of the simulated pressure downstream of the sample in an exemplary implementation of the method;

FIG. 4 is a graph showing the evolution of the reduced sensitivities to the permeability, the Klinkenberg coefficient and the porosity in the example of FIG. 3;

FIG. 5 is a graph showing the evolution of the ratio between the reduced sensitivities to the permeability and to the Klinkenberg coefficient in the example of FIG. 3;

FIG. 6 is a graph showing the evolution of the ratio between the reduced sensitivities to the permeability and the porosity in the example of FIG. 3;

- Figures 7 to 10 are graphs similar to those of Figures 3 to 6 in another example of implementation of the method;

- Figures 1 1 to 14 are graphs similar to those of Figures 3 to 6 in yet another example of implementation of the method;

FIGS. 15 and 16 are graphs showing the evolution of the simulated pressures upstream and downstream of the sample in a test case of the method;

FIGS. 17 and 18 are graphs showing the evolution of the pressures measured upstream and downstream of the sample in a test on a pine sample;

FIG. 19 is a graph showing the pressure residue downstream of the sample in the test of FIGS. 17 and 18, the residue being the the difference between the pressure calculated by a model describing the physics of the test and the pressure measured during the test;

FIGS. 20 to 22 are graphs similar to those of FIGS. 17 to 19 in a first test on a rock sample;

FIGS. 23 to 25 are graphs similar to those of FIGS.

17 to 19 in a second test on the same rock sample;

- Figures 26 to 28 are graphs similar to those of Figures 17 to 19 in a third test on the same rock sample.

[0G28J The installation shown in Figure 1 comprises a Hassler cell in which is placed a sample 2 of material which is to determine physical parameters in the presence of a fluid flow. Without this being limiting, the fluid used is a gas such as nitrogen or helium.

In known manner, the Hassler cell is a sleeve in which the sample 2, of cylindrical shape of section S and length e, is sealed in order to force the flow of gas through the porous structure of the material. Sample 2 has an upstream face 3 and a downstream face 4 which communicate with two reservoirs 5, 6 whose volumes are respectively denoted VQ and V- _| .

Manometers 7, 8 can measure the pressures in the tanks 5, 6. The gas is passed through the sample from a bottle 10 connected to the upstream volume VQ through a valve 1 1 and a regulator 12. On the downstream side, the volume V- | is connected to a recovery bottle 15 via a valve 16 and a pressure reducer 17. Other valves 18, 19 are provided between the expander 12 and the upstream volume VQ and between the expander 17 and the volume downstream V- | for selectively communicating the regulators with the tanks 5, 6.

Another valve 20 is placed between the upstream reservoir 5 and the Hassler cell 1 in order to trigger pressure pulses at the upstream face 3 of the sample. To apply a first pressure pulse to the sample 2, the valve 19 is positioned to put the reservoir downstream 6 at a starting pressure Pn (eg atmospheric pressure) while the valve 20 is closed. Once the pressure equilibrium is reached, the valve 19 is closed. The valves 1 1, 18 are opened and the regulator 12 is adjusted to the desired pressure value for the pulse. The upstream volume VQ is thus filled with gas at the desired pressure. The valve 18 is then closed and the valve 20 is opened in order to apply the pressure pulse to the sample 2. Using the manometers 7 and 8, the pressure decrease in the upstream volume VQ is then observed. and the pressure increase in the downstream volume V < _| . The evolution of the pressures Po (t) and P- _| (t) thus measured is recorded in order to be analyzed numerically. To subsequently apply another pressure pulse to the sample 2, the regulator 12 is set to the new desired pressure value, the valve 18 is opened to fill the volume V ₀ to the desired pressure, and then closed again.

Before the application of the first pressure pulse, the valve 20 being closed, the sample 2 is in equilibrium with the downstream volume V- _| , so that the initial condition (3) applies. In the case where the Forchheimer effects can be neglected, the numerically solved physical problem for parameter estimation is then the following problem (1) - (3) - (4 ') - (5): a

(P + b) ^ ^' for 0 <x <e and t> 0 (1)

; 8x k _| with the initial condition:

P (x, 0) = Pu for x> 0 (3) and the boundary conditions:

P (0, t) = P ₀ (t) for t> 0 (4 ')

In the expression of this problem, the pressure Po (t) upstream of the sample 2 is a datum. The physical parameters of the material of sample 2 involved in the system are its porosity φ, its intrinsic permeability k _| , and its Klinkenberg coefficient b.

The feasibility of estimating parameters from a signal f (t) can be studied using a sensitivity study. In our case, we can for example be interested in the signal f (t) = P- _| (T). The sensitivity of f (t) to a parameter ψ that we want to estimate is defined by. For reasons practical, we use instead the reduced sensitivity Σ _ψ = ψ- ^, allowing δψ

to obtain these quantities in pressure units. The analysis of the evolution of these quantities as a function of time allows a diagnosis on the possibility of estimating the parameter ψ from the signal f (t). This estimate is possible if

- the variations of Σ _ψ = ψ-- are significant over an interval of ψ δψ

time large enough in front of the temporal sampling step of the signal. By significant is meant queΣ _ψ is greater than the accuracy of the measuring tool (pressure sensors 7, 8) used to raise f (t); if several parameters are searched (for example k _| , b, even φ), the sensitivities reduced to all these parameters must be decorrelated, which supposes that they are not proportional to each other. In the opposite case, this means that observed variations on f (t) can not be independently attributable to one or the other of the parameters, making their simultaneous estimation impossible from a single signal f (t).

FIG. 2 shows the evolution in time of the reduced sensitivities to the permeability k |, the Klinkenberg coefficient b and the porosity φ calculated in the case of a single pressure pulse ("Puise decay" type) under the conditions following: k _| = 10 ^{~ 9} m ² , b = 13.08 bar, φ = 0.02, e = 5 cm, sample diameter d = 5 cm, VQ = 10 ^{~ 3} m ³ , V- | = 2.5 × 10 ^-3 m ³ , with an initial pressure of 15 bar in the upstream volume VQ and 1 bar in the downstream volume V -1. These sensitivities were calculated from P- _| (t) simulated using the physical model (1) - (3) - (4 ') - (5). It can be seen that after a few tens of minutes, the reduced sensitivity Σψ to the porosity stabilizes so that after this time, the measurement of the pressures no longer informs the value of the porosity φ. Measurements made under the conditions of FIG. 2 may therefore be insufficient to determine the porosity φ. Nevertheless, they may be appropriate for determining the permeability k _| and the coefficient of

Klinkenberg b if we also know the value of porosity φ. These estimates of k | and b are obtained without having to determine with precision the volume VQ and the associated dead volume on the upstream side of the sample 2, and avoiding possible irregularities on P ₀ (t) which, being measured, are no longer need to be calculated.

To increase the sensitivity to the porosity φ and allow its estimation, it is necessary to multiply the effects at short times so that the accumulation of gas in the pores of the material occurs repeatedly over the duration of the experiment. . This is illustrated below using three examples.

Example 1 (Figures 3-6)

In this example, the sensitivity analysis was conducted in a simulation on a material of intrinsic permeability k _| = 10 ^{~ 7} m ² , Klinkenberg coefficient b = 2.49 bar, porosity φ = 0.02 over a duration of experimentation tf = 500 s. Three pressure pulses were applied successively, the first of 5 bar at t = 0, the second of 10 bar at t = t _f / 3 and the third of 15 bar at t = 2t _f / 3. The volume of the upstream reservoir 5 was VQ = 10 ^{~ 3} m ³ and that of the downstream reservoir 6 was V- | = 2.5 × 10 ^-3 m ³ .

FIG. 3 shows the evolution over time of the pressure P- | (t) downstream of the sample. Figure 4 shows the evolution over time of the reduced sensitivities Σ _{k |} , Σ _b and Σψ from the pressure P (t) to the permeability intrinsic k _| , the Klinkenberg coefficient b and the porosity φ. Figure 5 shows the evolution over time of the ratio between the reduced sensitivities Σ ^, Σb to the intrinsic permeability k | and the Klinkenberg coefficient b. Figure 6 shows the evolution over time of the ratio between the reduced sensitivities Σ ^, Σψ to the intrinsic permeability k | and porosity φ.

Example 2 (Figures 7-10)

In this example, the sensitivity analysis was conducted under the same conditions as for Example 1 on a material of intrinsic permeability k _| = 10 ^-7 m ² , Klinkenberg coefficient b = 2.49 bar, porosity φ = 0.1 over an experimentation time t _f = 200 s Figures 7 to 10 are graphs for Example 2 similar to those in Figures 3 to 6.

Example 3 (FIGS. 1 to 14)

In this example, the sensitivity analysis was conducted under the same conditions as for Examples 1 and 2 on a material of intrinsic permeability k _| = 10 ^{~ 9} m ² , Klinkenberg coefficient b = 13.08 bar, porosity φ = 0.02 over a duration of experimentation t _f = 13000 s. Figures 11 to 14 are graphs for Example 3 similar to those of Figures 3 to 6.

The three examples show, on three different materials, that the pressure lifts P- _| (t) downstream of the sample are measurable quantities even having chosen a volume V- | relatively large (2.5 liters). This volume was chosen excessively large to emphasize the fact that one can minimize the relative error on its measurement. A smaller volume choice leads to more significant increases and we can check that the sensitivities are not affected. The choice of a large volume for V- _| is not motivated, as in the "Draw down" method, by obtaining a large variation of the average pressure over time. Indeed, the Mean pressure variation, very significant, is here due to successive pulses of pressure.

In all cases, the sensitivities are very significant and well decorrelated two by two. This allows a simultaneous estimation of the three parameters k _| , b and φ. In particular, by comparing FIG.

12 in Figure 2, that the modulation of the upstream pressure with several successive pulses significantly improves the sensitivity to the porosity φ, thus facilitating its estimation.

To illustrate the capacity of the method to allow the simultaneous estimation of the three parameters k |, b and φ, a series of tests was carried out from signals generated numerically using the physical model, with P _0i = 1 bar. A random noise given by δΡ ₀ = 0.01 dP s Po _m ax ^{/ 3 and} δΡι = 0.01 x dP xsx Pi max / 3 was superimposed on the two simulated signals Po (t) and P- | (t) to better represent a real measure. These noises are such that s is a random number of unit deviations and dP is the error on P ₀ (t) and Pi (t) (in% of the measurement). The coefficient 3 was taken so that the intervals Po (t) ± ÔPQ and P- | (t) ± δΡ- | include 99.7% of the values if they were actually measured. In these simulations, dP = 0.1%, which is a typical value of a pressure sensor, was taken except for cases 14, 15 and 16 for which dP = 1% was taken.

Case kl b Φ v ₀ v ₁ P _0i dP tf

N M

No. (m ² ) (bar) (%) (m ³ ) (m ³ ) (bar) (%) (s)

1 10 ^"17 2.49 0.02 10- ⁵ 10-2 3 0.1 1000 1000 200

2 _{1 0} -17 2.49 0.02 10-2 10-2 3 0.1 1000 1000 200

3 10 ^"17 2.49 0.02 10- ⁵ 10-2 3 0.1 100 1000 200

4 _{1 0} -17 2.49 0.02 10- ⁵ 10-2 3 0.1 1000 300 200

5 10 ^"17 2.49 0.02 10- ⁵ 10-2 3 0.1 1000 1000 50

6 10 ^"17 2.49 0.1 10- ⁵ 10-2 3 0.1 1000 1000 200

7 _{1 0} -17 13.08 0.02 10- ⁵ 10-2 3 0.1 1000 100000 200

8 10 ^"17 2.49 0.02 10-2 10- ⁴ 3 0.1 1000 1000 200

9 10 ^"17 2.49 0.1 10-2 10- ⁴ 3 0.1 1000 1000 200

10 _{1 0} -19 13.08 0.02 10-2 10-4 3 0.1 1000 10000 200

11 _{1 0} -19 13.08 0.02 10-2 10-4 5 0.1 1000 10000 200

12 _{1 0} -17 2.49 0.02 10-2 10-4 3 1 1000 1000 200

13 10 ^"17 2.49 0.1 10-2 10-4 3 1 1000 1000 200

14 _{1 0} -19 13.08 0.02 10-2 10-4 5 1 1000 10000 200

Table I

The parameters used in the test series are those indicated in Table I, including the number N of measurement points of the pressures P ₀ (t) and P- _| (t), the duration tf of the experiment and the number M of space discretization steps on the thickness e of the sample used for the inversion of the problem (1) - (3) - (4 ') - (5). In each case, three pressure pulses causing the pressure in the upstream reservoir to j ₀ P, 2P and 3P ₀ _0i j were applied at times 0, t _f / 3 and 2t _f / 3. The pressure modulation adopted in this series of tests makes it possible to qualify the measurement method in this case of "Step Decay". Figures 15 and 16 show the evolution of the pressures Po (t) and P- | (t) in bar synthesized upstream and downstream of the sample in case # 1.

The results of the inversions are presented in Table II, with the differences dk _| , db, de)) in% between the starting values of k _| , b and φ and the values found by the inversion calculations. Case kl dk | b db Φ dφ

No. (m ² ) (%) (bar) (%) (%) (%)

1 1.007.10 ^"17 0.10 2,456 0.26 1.96 0.28

2 1.003.10 ^"17 0.11 2.469 0.34 1.97 0.50

3 1.067.10 ^"17 0.61 2.159 1.70 1.60 2.00

4 1.005.10 ^"17 0.09 2,460 0.31 1.98 0.21

5 1.006.10 ^"17 0.10 2,460 0.26 1.97 0.28

6 1.007.10 ^"17 0.09 2.444 0.28 9.93 0.18

7 1.030.10 ^"19 0.31 12.55 0.44 1.96 0.25

8 1.004.10 ^"17 0.11 2.460 0.35 1.95 0.53

9 1.002.10 ^"17 0.08 2.477 0.38 9.95 0.38

10 1.027.10 ^"19 0.21 12.58 0.47 1.96 0.30

11 1.007.10 ^"19 0.21 13.08 0.37 2.00 0.30

12 1.001.10 ^"17 1.07 2.49 3.47 2.07 5.11

13 1.003.10 ^"17 0.81 2.42 3.85 9.67 3.81

14 1,088.10 ^"19 1.81 11.24 3.37 1.78 2.99

Table II] These results lead to the following conclusions:

- the accuracy of the estimates is excellent (often better than 1%) for the three estimated parameters and remains quite acceptable with a measurement noise of 1% of the maximum value of the pressure (case n ° 12 to

14).

the accuracy depends little on the volume V ₀ . A volume of 0.1 to 10 liters, preferably 0.1 to 1 liter, is very suitable;

- the volume V- _| must be chosen so that the pressure rise is sufficient to be measured with good accuracy. A value of 0.1 liter leads to satisfactory results in the case of the studied materials and is sufficiently high to limit the error due to the dead volume downstream. A volume V- | 0.05 to 10 liters can generally be used; - working at higher pressure levels (5, 10 and 15 bar) leads to better accuracy in the case where b = 13.08 bar;

- a number of experimental points of 1000 seems adequate, the accuracy decreasing somewhat if we reduce this number to 100 (case 3). - the extension of the duration of the experiment beyond a certain limit does not significantly improve the accuracy (comparison of cases 7 and 10).

- acceptable measurement times for the estimation of the three parameters are 20 minutes for k _| around 10 ^{~ 7} m ² and 3 hours for k _| around 10 ^{~ 9} m ² . It is generally appropriate to apply the pressure modulation in the first volume on a time scale of a few tens of minutes to a few hours, and in any case greater than one minute. It will be observed that the pressure modulation by a succession of pulses is a special case and that other forms of modulation on a sufficient time scale may be suitable for the implementation of the invention since the pressure profile in FIG. upstream Po (t), which is measured, can have any shape.

A relatively large volume V ₀ has the advantage that the pressure

Po (t) varies slightly between two pulses. Moreover, if the volume V- _| is chosen large enough so that the rise of pressure is not significant in front of PQ, then the experiment takes place in a weakly unsteady state (quasi-stationary). Under these conditions, a good quality pre-estimation can be made by simple linear regression on the portions of P- _| (t) corresponding to each pressure pulse. This pre-estimation of good quality makes it possible to achieve an easier convergence of the estimate on the completeness of the signal with the complete model.

An automation of the progress of the experiment can be used. Indeed, the appearance of a linear or quasi-linear regime on P- | (t) for each pressure tap (FIG. 16) corresponds to a quasi-stationary regime with loss of sensitivity of P- | (t) porosity φ (by definition, in quasi-stationary regime, the accumulation effect in the pores of the sample disappears). The experiment can therefore be conducted in such a way that each pressure pulse has a duration allowing P- _| (t) to reach a quasi-linear behavior as a function of time. This quasi-linear regime is then allowed to last for a short period, making it possible to make a good estimate of k _| and B. The use of volumes VQ and V- _| large enough allows a direct control of the experiment to optimize the duration while obtaining well-converged results.

Laboratory tests were performed with V ₀ = 1, 02.10 ^"3 m ³ and V- | = 2.26 × 10 ^-3 m ³ and with the following experimental protocol:

- implementation of sample 2 in Hassler cell 1;

pressurizing the outer chamber of the Hassler cell;

closing the valve 20 and waiting for equilibrium;

- opening of the valve 18 and adjustment of the regulator 12 to obtain po ^{= P} 0i1>

- start recording pressure Po (t) and P- | (t);

closing the valve 18 and opening the valve 20;

adjusting the expander 12 to obtain P ₀ = P _{0i 2} ;

- after a time t < _| , opening valve 18 for a few seconds

adjusting regulator 12 to obtain P ₀ = P ₀ ₁₃ ;

after a time t ₂ , opening of the valve 18 for a few seconds;

- after a time t3, stop the recording, open the valve 19 and disassemble the sample 2.

For some of these tests, it was proceeded to a pre-estimation of the permeability k _| and the Klinkenberg coefficient b as follows: - estimate the slopes of the three parts of the curves P- _| (t), assimilated to straight lines, corresponding to the successive pressure pulses;

- deduce three values of the apparent permeability k _g . These values are used for PQ and P- | equal to the half sum of the values at the ends of each interval;

- plot the values of k _g as a function of 1 / P _moy = 2 / (P ₀ + P- _| ) and by linear regression, classically obtain a pre-estimate of k _| and b, knowing that k _g = k |. (1 + b / P _avg ).

These pre-estimated values of k | and b are then used as starting values for the final estimate of k _| , b and φ by inversion on the complete signal P- | (t), the signal Po (t) being an input data, using the physical model (1) - (3) - (4 ') - (5).

Example 4 (Figures 17-19)

Two tests were carried out according to the experimental protocol above on a sample of dry pine wood whose dimensional characteristics are d = 38.5 mm and e = 60 mm. The porosity (excluding stress) of the sample was measured by pycnometry as being φ = 0.27.

In the second test, the permeability k | and the coefficient of

Klinkenberg b were pre-estimated at 1.76 × ¹⁰ -16 m ² and at 0.099 bar The final results of the estimate are shown in Table III, with the relative standard deviations σ 1, σ and σ 2 on the three parameters estimated simultaneously.

Table III The evolutions of the measured pressures P ₀ (t) in bar and ΔΡ- _| (ί) = P- _| (t) -P- _| (0) in millibar are shown in FIGS. 17-18. Figure 19 shows the residue on P- _| (t), in millibar, after estimation. The estimates are of excellent quality as evidenced by the curves P- _| (t) measured and estimated and especially the residual curve, and as confirmed by the small relative standard deviations σ ^, σψ.

Example 5 (Figures 20-28)

Three tests were carried out according to the experimental protocol above on a rock core whose dimensional characteristics are d = 38 mm, e = 60.3 mm. The porosity (excluding stress) of the core was measured by pycnometry as being φ = 0.06.

[0058J Permeability k _| and coefficient of Klinkenberg b were préestimés to 3,34.10 ^"17 m ² and 1 to 47 bar in the second test and ^{3,86.10" 17} m ² and 0.97 bar in the third test. The final results of the estimate are shown in Table IV.

Table IV

The evolutions of the measured pressures Po (t) in bar and

ΔΡ- | (ί) = P- | (T) -P- | (0) in millibar are shown in Figures 20-21 for the first test, in Figures 23-24 for the second test and in Figures 26-27 for the third test. Residues on P- _| (t) after estimation, in millibar, are shown in figure 22 for the first test, in figure 25 for the second test and in figure 28 for the third test. Again, estimates are of excellent quality as evidenced by the curves P- _| (t) measured and estimated and the residual curves, and as confirmed by the small standard deviations of _{k |} , a _b , σψ.

Claims

1. A method of estimating physical parameters of a porous material with respect to a flow, comprising:

placing a sample (2) of the material in a sealed cell (1) so that an upstream face (3) of the sample communicates with a first volume (Vn) and a downstream face (4) of the sample communicates with a second volume (V- _| );

- cause a pressure modulation in the first volume;

measuring the variations over time of respective pressures (Po (t) _> Pi (t)) in the first volume and in the second volume; and

using a differential equation whose parameters are the intrinsic permeability (k |) of the material, the porosity (φ) of the material and at least one other coefficient (b, β) specific to the material and for the boundary condition measured variation of the pressure (Pn (t)) in the first volume, numerically analyzing the measured variation of the pressure (P- _| (t)) in the second volume to estimate at least the intrinsic permeability and said other coefficient.

2. The method of claim 1, wherein the pressure modulation in the first volume (V ₀ ) is applied on a time scale greater than that of a pressure pulse.

The method of claim 1, wherein the pressure modulation in the first volume (V ₀ ) is applied on a time scale greater than one minute.

4. A method according to any one of the preceding claims, wherein the modulation of the pressure in the first volume (V ₀ ) is caused by a succession of pressure pulses.

5. Method according to any one of the preceding claims, in which the numerical analysis of the variations of the measured pressures (Po (t), Pi (t)) comprises a follow-up of the evolution over time of the reduced sensitivity (Σ _k ) from the pressure (P- _| (t)) measured in the second volume (V- _| ) to the intrinsic permeability (k |) and from the evolution over time of the reduced sensitivity (Σ _b ) of the pressure (Pi (t)) measured in the second volume to said other coefficient (b, β).

6. Method according to any one of the preceding claims, wherein the numerical analysis of the variations of the measured pressures (Po (t), P-i (t)) is performed so as to further estimate the porosity (φ) of the material.

7. A method according to claim 6, wherein the numerical analysis of the variations of the measured pressures (Po (t), Pi (t)) comprises a follow-up of the evolution over time of the reduced sensitivity (Σψ) of the pressure. (P- | (t)) measured in the second volume (V- _| ) at the porosity (φ).

8. A method according to any preceding claim, wherein the numerical analysis of changes in the measured pressure (P ₀ (t), P _| (t)) comprises, in the times when the pressure intervals (P- _| (t)) in the second volume varies substantially linearly, a pre-estimation of the intrinsic permeability (k _| ) and said coefficient (b) to facilitate the convergence of the estimate.

9. The method of claim 8, wherein the evolution over time of the pressure (P- | (t)) in the second volume (V- |) is examined and when it is observed that the pressure in the second volume varies substantially linearly with time, this pressure is allowed to vary substantially linearly to acquire values for the pre-estimation of the intrinsic permeability (k _| ) and said coefficient (b), then applying a new pressure pulse in the first volume (VQ).

The method of any of the preceding claims, wherein the first volume (V ₀ ) is 0.1 to 10 liters.

A process according to any one of the preceding claims, wherein the second volume (V 1) is between 0.05 and 10 liters.

The method of any one of the preceding claims, wherein said further coefficient comprises a Klinkenberg coefficient (b).

The method of any one of the preceding claims, wherein said other coefficient comprises a Forchheimer coefficient (β).