Chromatogram decomposition and corresponding calibration

Abstract

A method for calibrating a GC device includes measuring first and second chromatograms of corresponding first and second calibration samples, where the first calibration sample includes a first gas and the second calibration sample includes a first gas, a second gas, and a third gas. The first chromatogram is fitted with a basis function derived from a mass balance equation to obtain a first modeled chromatogram. The second chromatogram is fitted with the first affine transformation response, the second affine transformation response, and the third affine transformation response of the first modeled chromatogram to obtain a second modeled chromatogram. The second modeled chromatogram may be used to decompose a measured third chromatogram of the unknown gas sample and estimate its composition.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] Related Applications This application claims the benefit of EP Application No. 23306706.5, filed on October 4, 2023, entitled “CHROMATOGRAM DECOMPOSITION AND CORRESPONDING CALIBRATION,” the disclosure of which is incorporated herein by reference. Background Art

[0003] Gas chromatography (GC) is a widely used technique for separating and analyzing chemical compounds that can evaporate, such as organic compounds. Common applications involve quantitative and / or qualitative analysis of food ingredients (composition), natural products, food additives, flavor and aroma components such as essential oils, various conversion products and pollutants such as pesticides, fumigants, natural toxins, pharmaceuticals, packaging materials and environmental pollutants. GC technology is also used to evaluate the composition of gases released during downhole drilling operations, for example, including methane (C1), ethane (C2), propane (C3), butane (C4), pentane (C5), etc., as well as olefins and other compounds. Such measurements can provide valuable information to the logging instrument, and can provide information about the maturity and properties of hydrocarbons in the reservoir, the division of intervals in the reservoir being drilled and the quality of the oil, as well as information about the production area, lithology changes, reservoir accumulation history, sealing effectiveness and the environmental impact of drilling operations.

[0004] GC measurements are commonly used to separate and analyze individual gases in a gas sample (e.g., released gases obtained in a drilling operation). The measurements obtained from a GC are called a chromatogram. It can be expressed as a univariate function of the detector response relative to the retention time (or elution time), and can be approximated as a mixture of Gaussian functions (Gaussians) for large retention times. One way to characterize the individual gases in a gas sample is to decompose the chromatogram using a Gaussian mixture model (GMM). Although commonly used, the GMM approximation is generally insufficient to capture the asymmetric nature of the local maxima of the chromatogram and the asymptotic time decay behavior, and may therefore result in compositional errors. Improved chromatogram decomposition and calibration methods are needed, particularly for gas samples released during downhole drilling operations. BRIEF DESCRIPTION OF THE DRAWINGS

[0005] For a more complete understanding of the disclosed subject matter and advantages thereof, reference is now made to the following description taken in conjunction with the accompanying drawings, in which:

[0006] Figure 1 An example drilling rig is depicted that includes the disclosed gas chromatography apparatus for estimating the composition of formation gas.

[0007] Figure 2 Depicted Figure 1 Example GC apparatus shown.

[0008] Figure 3A A flow chart of an exemplary method for calibrating a GC measurement device is depicted.

[0009] Figure 3B A flow chart of one example method for estimating the composition of an unknown gas sample using GC is depicted.

[0010] Figure 4A An example chromatogram measured using a calibration gas sample comprising only C1 (solid line) and its modeled approximation (dashed line) are depicted.

[0011] Figure 4B Describes the use Figure 4A The corresponding components of the modeled approximate C1 response are depicted on

[0012] Figure 5 An example chromatogram measured using the calibration gas sample GB2 in Table 1 is depicted, with the individual Cx responses of the chromatogram decomposition annotated.

[0013] Fig. 6A , 6B , 6C and 6D (collectively referred to as Figure 6) depict the measured chromatograms (solid lines) and estimated chromatogram decomposition (dashed lines and labeled) for each gas of example gas samples GB3 (6A), GB4 (6B), GB5 (6C) and GB6 (6D).

[0014] FIG. 7A to FIG. 7G (collectively referred to as Figure 7) depicts the optimal m * Plots of estimated gas concentrations versus known concentrations for each of gas samples GB1 (7A), GB2 (7B), GB3 (7C), GB4 (7D), GB5 (7E), GB6 (7F), and GB7 (7G).

[0015] Figure 8 Depicted are the measured chromatogram (solid line) and the decomposed C1, C2 and C3 gas components (dashed and labeled) of gas sample GB7 (as a hypothetical unknown sample). DETAILED DESCRIPTION

[0016] Embodiments of the present disclosure include methods and systems for chromatogram decomposition and calibration. In an example embodiment, the disclosed method for estimating the composition of an unknown gas sample including at least a first, second, and third gas includes: calibrating a gas chromatography (GC) device using a first and a second calibration sample, wherein the first calibration sample includes a first gas and the second calibration sample includes the first, second, and third gases. Using the GC device, measure the first chromatogram and the second chromatogram of the corresponding first calibration sample and the second calibration sample. Fit the first chromatogram with a basis function derived from a mass balance equation to obtain a first modeled chromatogram. Fit the second chromatogram with a first affine transformation response, a second affine transformation response, and a third affine transformation response of the first modeled chromatogram to obtain a second modeled chromatogram. Use the GC device to measure a third chromatogram of the unknown gas sample. Decompose the third chromatogram into components corresponding to the first gas, the second gas, and the third gas using the second modeled chromatogram. Estimate the concentrations of the first, second, and third gases from the first, second, and third gas components of the decomposed third chromatogram.

[0017] Figure 1 An example drilling rig 20 is depicted, which includes a gas chromatography apparatus 100 for evaluating formation gas composition. The drilling rig 20 can be positioned above a subterranean formation (not shown). The drilling rig 20 can include, for example, a derrick and a hoisting apparatus (also not shown) for raising and lowering a drill string 30, which, as shown, extends into a wellbore 40 and includes, for example, a drill bit 32 and one or more downhole measurement tools 38 (e.g., logging while drilling tools or measurement while drilling tools) in a bottom hole assembly (BHA) above the drill bit 32. Suitable drilling systems (e.g., including drilling, steering, logging and other downhole tools) are well known in the art.

[0018] The drilling rig 20 also includes a surface system 50 for controlling the flow of drilling fluid used on the drilling rig (e.g., for drilling the wellbore 40). In the depicted example drilling rig, the drilling fluid 35 is pumped downhole (as depicted at 92) via a conventional mud pump 57, for example. The drilling fluid 35 may be pumped to the drill string 30, for example, in route through a riser 58 and a mud hose 59. The drilling fluid 35 typically exits the drill string 30 at or near the drill bit 32 and creates an upward flow 94 of mud that passes through the wellbore annulus 42 (the annular space between the drill string and the wellbore wall). The drilling fluid 35 then flows through a return conduit 52 to a mud pit system 56 where it may be recirculated. It should be understood that the terms drilling fluid and mud are used synonymously herein.

[0019] The circulating drilling fluid 35 is intended to perform a number of functions during the drilling operation, one of which is to transport drill cuttings 45 to the surface (in an upward flow 94). The drill cuttings 45 are typically removed from the return mud via a shale shaker 55 (or other similar solids control equipment) in a return conduit (e.g., immediately upstream of the mud pit 56). Formation gases released during drilling may also be carried to the surface in the circulating drilling fluid. These gases are typically removed from the fluid, for example, via a degasser or gas trap 54, which is located in or near a header box 53, which is immediately upstream of the shale shaker 55 in the example depiction. The drill cuttings 45 and gases are typically inspected at the surface to assess the formation through which the wellbore was drilled.

[0020] As is known to those of ordinary skill in the art, formation gas may be released into the wellbore 40 via the drilling process (e.g., by crushing the formation rock through the mechanical action of the drill bit), and may also migrate into the wellbore 40, for example, via fractures in the formation rock. Once in the wellbore, the formation gas may be transported to the surface via the drilling fluid (in the upwardly flowing fluid 94). The formation gas may be dissolved in the drilling fluid and / or in the form of bubbles, and may be sampled in the surface system, for example, via one or more drilling fluid degassers 54 and / or headspace gas probes. The disclosed embodiments are expressly not limited to how the gas is sampled.

[0021] Further references Figure 1 The drilling rig 20 may also include a testing facility 60 (e.g., a laboratory trailer including one or more instruments suitable for performing various measurements on drill cuttings and formation gas in the drilling fluid). In the illustrated embodiment, the testing facility 60 includes a gas chromatography apparatus 100 (described in more detail below) configured to receive formation gas and evaluate the composition of the formation gas. Of course, the testing facility 60 may include many other testing instruments known to those of ordinary skill in the art. Although the following embodiments are described with respect to drilling operations, it should be understood that the disclosed embodiments are not limited thereto and may be applied to the use of GC in other chemical applications.

[0022] While GC is a powerful chemical analysis technique that enables many (most) chemical species to be separated at the detector, there is still room for improvement. In many gas samples, such as those including light hydrocarbons, certain species can have asymmetric responses in time and partially or completely overlapping elution times. This results in overlapping peaks in the chromatogram and tends to complicate the quantitative determination of the gas composition. The disclosed embodiments can advantageously provide improved calibration and resolution of measured chromatograms, and thus can improve the accuracy of the estimated gas composition.

[0023] Of course it should be understood that Figure 1A land drilling rig 20 is depicted, but the disclosed embodiments are equally applicable to land drilling rigs or offshore drilling rigs. As known to those of ordinary skill in the art, an offshore drilling rig typically includes a platform deployed on top of a riser that extends from the seafloor to the surface. A drill string extends downward from the platform, through the riser, and through a blowout preventer (BOP) located on the seafloor into the wellbore. The disclosed embodiments are expressly not limited in these respects.

[0024] Figure 2 An example GC device 100 is depicted that includes a gas sample injection port 112 configured to feed a gas sample into a column assembly 120 including an optional pre-cut column 122 and a main GC column 126. The GC device 100 also includes a carrier gas supply 135, such as a supply of compressed nitrogen, argon, helium or air. The injected gas sample is mixed with the carrier gas and transported through the column assembly 120. The pre-cut column 122 can be configured to remove heavier hydrocarbon compounds with a carbon number above a threshold, such as C6, C8 or C10 and above. The main column 126 typically includes a stationary phase and is intended to separate various gas compounds in the gas sample so that they arrive at a detector 140 at different elution times, for example, so that C1 arrives before C2, C2 arrives before C3, and so on. The detector can include substantially any suitable GC detector, such as an FID detector, a TC detector or a mass spectrometer. It should be appreciated that while the example apparatus 100 includes a pre-cut post 122, the disclosed embodiments are not limited in this regard.

[0025] Continue to refer Figure 2 , the GC device 100 may also include an electronic controller 150 configured to control the detector 140 and a flow manifold (not shown). So arranged, the controller may be configured to cause the GC device to perform GC measurements and calibrate the device, for example, as described below with respect to Figure 3A and Figure 3B Described in more detail. Of course, it should be understood that the controller may include computer hardware and software configured to cause the GC device to perform these functions. The hardware may include one or more processors (e.g., microprocessors) that may be connected to one or more data storage devices (e.g., hard drives or solid-state memories) and a user interface. It will be further understood that the disclosed embodiments may include processor executable instructions stored in the data storage device. The disclosed embodiments are certainly not limited to the use or configuration of any particular computer hardware and / or software.

[0026] Figure 3AA flow chart of an exemplary method 200 for calibrating a GC measurement device is depicted. The method includes obtaining at least first and second calibration samples at 202, wherein the first calibration sample includes a first gas at a known concentration, and the second calibration sample includes a first gas at a known (same or different) concentration and at least first and second other gases (at least second and third gases) at known concentrations. In advantageous embodiments, the first calibration sample may nominally include only the first gas (e.g., no other active gas components), or may include at least the first and second other gases at negligible concentrations. In certain embodiments, the negligible concentrations are less than 10 ppm (e.g., less than 3 ppm or less than 1 ppm).

[0027] At 204, a GC device (e.g., Figure 1 and Figure 2 The apparatus 100 in the GC apparatus 100) evaluates a first gas sample and a second gas sample to measure corresponding first and second chromatograms. At 206, the first chromatogram is fitted using a basis function derived from a mass balance equation (MBE) to obtain a first modeled chromatogram. As described in more detail below, the first modeled chromatogram can be defined by the basis function and a corresponding first set of calibration parameters. It should be understood that the basis function can be advantageously asymmetric with the retention time of the gas sample in the GC apparatus.

[0028] At 208, a second chromatogram is fit using at least the first affine transformed response, the second affine transformed response, and the third affine transformed response (corresponding to the first gas and the first and second other gases) of the first modeled chromatogram (the basis functions and the first calibration parameters used to fit the first chromatogram) to obtain a second modeled chromatogram. As described in more detail below, the second modeled chromatogram may be defined by the first modeled chromatogram and a second set of calibration parameters characterizing the response of each gas in the second gas calibration sample.

[0029] Figure 3B A flow chart of an example method 220 for estimating the composition of an unknown gas sample using the GC device calibrated in 200 is depicted. At 222, the unknown gas sample is evaluated using the calibrated GC measurement device to measure a chromatogram (a third chromatogram). At 224, the measured chromatogram is decomposed into individual gas components using the second modeled chromatogram obtained at 208 of method 200 to obtain a third modeled chromatogram. As described in more detail below, the decomposition may include adjusting the amplitude of each of the individual gas components (peaks) in the second modeled chromatogram. The amplitudes of the individual decomposed gas components in the third chromatogram may then be used at 226 to estimate the concentration of each of the individual gases in the unknown gas sample.

[0030] It will be appreciated that the problem of material characterization in GC requires finding the decomposition of a given chromatogram f(t) for a positive time t As through θ m The superposition of parameterized predefined bases g(t,θ m ), for example, as follows:

[0031]

[0032] where a m represents the positive peak amplitude, m = 0, 1, ..., M and BL(t) represents the approximate baseline. For GC measurement, assume that f(t), BL(t) and g(t, θ m ) is a non-negative function. The baseline BL(t) is specific to the GC system and can be measured when no sample is present in the system. For simplicity, it can be assumed that the baseline can be approximated by a constant whose effect can be absorbed in a 0 and consider the decomposition of the chromatogram in the following form:

[0033]

[0034] In the disclosed embodiment, g(t,θ m ) is a basis function that can be advantageously asymmetric with respect to the elution time t and is derived from the solution of the mass balance equation (MBE). In an example embodiment, g(t,θ m ) can have the following forms:

[0035]

[0036] where H(t) is the Heaviside step function, which is equal to zero for t≤0 and equal to zero for t>0 and θ m =(t m ,s m ,p m ) is one, where t m Indicates the retention time of gaseous compounds in a GC column or GC equipment, s m represents the spatial diffusion (or dispersion) coefficient which may be inversely proportional to the GC column length, and 1≤p m ≤3 and represents the decay rate over time. The following discussion lists g MBE (t,θ m ) which may be useful for numerical implementation of certain embodiments of the disclosed embodiments.

[0037] g MBE (t,θ m ) Can meet:

[0038]

[0039] Equivalently:

[0040]

[0041] since

[0042]

[0043] Considering the positive roots of the quadratic equation we get:

[0044]

[0045] On the contrary, given the maximum and p m ,s m , we can calculate t m

[0046]

[0047] Note that t m Associated with the retention time and can be later than or equal to the maximum position Right now, if but

[0048] Basis function g MBE (t,θ m ) can also be dilated with respect to time t with a dilation factor a.

[0049]

[0050] where θ m As defined above, and γ m =(at m ,as m ,ap m ), which may be useful during implementation when time t needs to be rescaled.

[0051] g MBE (t,θ m ) is as follows:

[0052]

[0053] since

[0054]

[0055] Maximum This can be done by setting the above to zero and solving for t m To obtain.

[0056] g MBE (t,θ m ), for p m =1, can be obtained using the following identity:

[0057]

[0058] So that:

[0059]

[0060] g MBE (t,θ m ), for p m =3, a similar identity can be used to obtain:

[0061]

[0062] Make

[0063]

[0064] It will be appreciated that there is no unique solution to the decomposition problem expressed in equations (1) and (2), particularly when evaluating chromatograms containing combinations of gases whose responses partially or completely overlap. As described in more detail below, a satisfactory solution may require additional constraints that enable accounting for the underlying physical and operational phenomena associated with the gas samples of interest and the GC system. One way to achieve this solution is to measure the response of the GC system with respect to each gas of interest, for example, using calibration samples with known concentrations of each gas of interest. Once the individual responses are decomposed, the chromatogram can be represented as a combination of these responses. In this case, M represents the number of gases and g(t,θ m ) represents an approximation of the response of the GC system to the mth gas. However, a disadvantage of this approach is that it may require a large number of calibration samples corresponding to the number of expected gases.

[0065] When there is limited access to calibration samples that include only a single gas, alternative constraints may be considered. For example, in some embodiments, a single gas calibration sample may not be readily available for each expected gas in the gas sample. In this case, calibration can be achieved using calibration bottles that include a mixture of gases. Assuming that the GC signal behaves linearly with respect to the gas concentration range of interest and that the gas concentrations are linearly independent, the individual gas responses can be obtained by linear combinations of the corresponding chromatograms.

[0066] One aspect of the disclosed embodiments is the realization that in the case where the gas concentrations are linearly related to each other, it can be further assumed that the system response to a particular gas of interest is an affine transformation of each other. This assumption can advantageously enable system calibration, wherein the number of calibration bottles required is significantly reduced, and can further advantageously reduce the time required to calibrate a GC system. It should be understood that the term "affine transform" is used herein in the standard mathematical sense of a geometric transform or function that maps an affine space onto itself while maintaining the dimensionality of any affine subspace and the ratio of the lengths of parallel line segments. In other words, an affine transform preserves lines and parallelism, but does not necessarily preserve Euclidean distances and angles.

[0067] Table 1 lists the composition of example gas cylinders used in example embodiments of the disclosed calibration method. Gas cylinders GB1, GB2, GB3, GB4, GB5, GB6, and GB7 include a mixture of methane (C1), ethane (C2), propane (C3), isobutane (iC4), butane (nC4), isopentane (iC5), and pentane (nC5). Gas concentrations are listed in parts per million (ppm) (where 1 ppm equals 0.0001%).

[0068] Table 1

[0069] sample C1 C2 C3 iC4 nC4 iC5 nC5 GB1 300,000 0 0 0 0 0 0 GB2 100,000 25,000 25,000 10,000 10,000 2,500 2,500 GB3 50,000 12,500 12,500 8,500 8,500 2,000 2,000 GB4 10,000 2,500 2,500 2,500 2,500 1,000 1,000 GB5 1,000 250 250 250 250 250 250 GB6 100 25 25 25 25 25 25 GB7 7,500 25 25 0 0 0 0

[0070] In the example cylinders in Table 1, the gas concentrations are linearly related. Specifically, the concentrations of C2 and C3, iC4 and nC4, and iC5 and nC5 are pairwise related. In this example, there are seven cylinders and seven gases of interest, however, with the exception of C1, it is not possible to isolate the individual gas responses using linear inversion. In the example embodiment below, cylinders GB1 and GB2 are used to calibrate the GC system. Cylinders GB3, GB4, GB5, GB6, and GB7 are used for test calibration.

[0071] In the disclosed embodiments, it is assumed that the measured chromatogram can be represented as a linear combination of the individual gas responses, for example, as follows:

[0072]

[0073] where g Cx (t) represents the response of the GC system to gas Cx, Δt represents the unexplained time delay during the injection of the gas sample into the GC device, and G represents the gas set in calibration, for example, including C1, C2, C3, iC4, nC4, iC5 and nC5.

[0074] Reference again Figure 3A and Figure 3B, the C1 response is approximated at 206 using a first chromatogram measured using a first gas sample (e.g., a calibration sample including only C1, such as GB1 in Table 1). A second chromatogram is measured using a second gas sample (e.g., GB2) containing each of the other gases of interest. The responses of the other gases are modeled as affine transformed versions of the modeled C1 at 208. It should be understood that the gas samples GB3, GB4, GB5, or GB6 in Table 1 may be used in 208 instead. After modeling the individual gas responses and determining the system calibration (as described in more detail below), the chromatogram of the unknown gas sample may be decomposed at 224 into a linear combination of the individual gas responses modeled in 206 and 208 of method 200. In the following disclosure, an example calibration is described in which a GC device is calibrated using calibration samples GB1 and GB2. The calibration is then tested using gas samples GB3, GB4, GB5, GB6, and GB7 (as if they were unknown samples). Of course, the disclosed embodiments are not limited to calibrating a GC system using only the first calibration sample and the second calibration sample.

[0075] Further references Figure 3A Element 206, C1 response g of the GC device C1 (t) can be Approximate or model to obtain the first calibration parameters, for example, as follows:

[0076]

[0077] where a m represents the amplitude, and g MBE (t,θ m ) is defined as above and represents the model parameters θ m =(t m ,S m ,p m ). In this particular example, the first calibration parameter is a m ,t m ,s m and p m The number of first calibration parameters is made equal to 4M.

[0078] Continuing with equation (4), it is possible, for example, to calculate the cost function J[∈ 1 ]=(L p [∈ 1 ]) p The C1 response is modeled by fitting the measured histogram f(t) (measured chromatogram) for the calibration sample GB1 to minimize L for some p ≥ 1. Here L p By L p [∈ 1]=(∫|∈ 1 (t)| p dt) p-1 Limited error The p-norm of . Represents linear and nonlinear The combination of parameters corresponds to the approximation The optimal Θ is determined by the Θ that satisfies the following 1 gives:

[0079]

[0080] where Θ 1 Denotes the first model parameter. Note that the integral with respect to t should be considered in a general sense, where the integral can be performed over combinations of single, multiple, infinite, finite, connected, disjoint, continuous and discontinuous domains.

[0081] Figure 4A An example measured chromatogram of calibration sample GB1 (solid line) and its modeled approximation (dashed line) are depicted. Note the good fit provided by the model up to about 5.5 seconds. Figure 4B Describes Figure 4A The corresponding components of the modeled approximate C1 response are depicted on Where: m=1, 2, 3 and 4 are shown. In this example, the first calibration parameters include 16 model parameters (4·4). Note that for p=1, the approximate C1 response can be tailored towards the first 5.5 seconds of the chromatogram to avoid the observed sudden amplitude drop. The remaining calibration is based on this approximate C1 response.

[0082] Further references Figure 3A Element 208, assuming that the response g for Cx∈G Cx (t) is g C1 The affine transformed version of (t) yields the following:

[0083]

[0084] The affine transformation is given by the model parameters t Cx and Cx Characterization, the model parameters represent the retention time and spatial diffusion coefficient of each Cx.

[0085] Following equation (3), the measured second chromatogram f 2 (t) can be modeled as For example, as follows:

[0086]

[0087] where a CxIndicates the amplitude of the response of each Cx component gas and a 0 represents the detector response offset. Let the model parameter Θ = {a Cx ,t Cx ,s Cx} Cx∈G . corresponds to the modeled gas response The optimal Θ can be determined by satisfying the following Θ G gives:

[0088]

[0089] where Θ G As described in more detail below, system calibration may also include generating an amplitude a Cx and the concentration Cx of the gas (such as a linear correlation). It has been found that the amplitude advantageously varies linearly with the gas concentration.

[0090] Figure 5 An example measured chromatogram using calibration sample GB2 in Table 1 is depicted. The measured chromatogram is the solid line, and the modeled Cx gas response for p=1 is the dashed line. In this example, the second calibration parameters include 21 model parameters (three parameters for each of the seven gases). Note that the sum of the dashed Cx responses provides a good fit to the measured chromatogram, especially for the asymmetric C1, C2, and C3 peaks. The individual Cx responses are labeled in the figure.

[0091] Further references Figure 3B Element 224, chromatogram of unknown gas sample (eg, third chromatogram) f 3 (t) may be modeled by decomposing into individual gas responses (or components) obtained during calibration in 206 and 208 (eg, as described above) as For example:

[0092]

[0093] in represents the response of each component gas, Cx, Δt represents the time shift, and f 0 represents the detector response shift that converts the gas responses together (as a unit), and f Cx Represents the magnitude of the response of each gas in the modeled chromatogram Decomposition can include optimizing {f Cx} Cx∈G ,f 0 and Δt, for example, as follows:

[0094]

[0095] Fig. 6A , 6B , 6C and 6D (collectively, FIG. 6 ) depict measured chromatograms (solid lines) and modeled decomposition responses (dashed lines) for individual gases of gas samples GB3 (6A), GB4 (6B), GB5 (6C) and GB6 (6D). In these examples, GB3, GB4, GB5 and GB6 gas samples were considered unknowns. Note that the sum of the dashed line Cx responses provides a good fit for each chromatogram for all peaks and several orders of magnitude of Cx concentrations.

[0096] Further references Figure 3B The concentration of each gas in the gas sample can be estimated by a calibration curve, which can be constructed by, for example, comparing an injected gas sample with a known composition with a modeled chromatogram. Assuming that the GC signal is proportional to the amount of gas reaching the detector, the integral of the components of the decomposed chromatogram, or equivalently the decomposition coefficient, can be expected to be linearly proportional to the amount of the corresponding gas in the GC system.

[0097] It should be understood that the disclosed embodiments are not limited to embodiments where the calibration sample includes every gas in the unknown sample. For example, in applications where there are one or more gases (or gas components) that are not within the calibration, the additional gas components can be detected (and optionally quantified) as contamination that falls outside the estimated response function.

[0098] Let f GB (t) is the chromatogram obtained by injecting the gas sample GB into the GC, and x Cx [f GB ] is the concentration Cx of the gas in the gas sample GB. Given the approximate response make

[0099]

[0100] The decomposition of the chromatogram obtained from the GB can be calculated by Cx [f GB ] and coefficient (peak amplitude) The Pearson correlation coefficient between them is used to verify linearity. The Pearson correlation coefficient (PCC) can be calculated, for example, as follows:

[0101]

[0102] in:

[0103]

[0104] make:

[0105]

[0106]

[0107] Table 2 lists the concentrations of the calibration samples x Cx and the corresponding chromatogram decomposition (obtained from the chromatogram shown in Figure 6) The Pearson correlation coefficient between the peak values ​​is shown in Table 1. The correlation coefficients listed are all greater than 0.997, indicating a high degree of linearity between the decomposition amplitude and the individual gas concentrations. Therefore, it should be understood that the gas concentration can be obtained directly from the individual peak amplitudes (via direct linear correlation).

[0108] Table 2

[0109] C1 C2 C3 iC4 nC4 iC5 nC5 0.997189 0.999997 0.999983 0.999994 0.999996 0.999830 0.999866

[0110] The decomposition f(t) of a given chromatogram corresponds to the gas sample of interest:

[0111]

[0112] The linear calibration curve shows the estimated concentration And the coefficients of decomposition is relevant, for example, as follows:

[0113]

[0114] For some constant m, it can be chosen or calculated to best match the gas concentration of the calibration sample(s). When only a single calibration sample is used, the optimal m Cx is precisely given as When multiple calibration samples are used, calculate the optimal m Cx An example method is to minimize the normalized mean square error, for example, as follows:

[0115]

[0116] For x Cx [f CB ]>0. Note that equations (8) and (9) advantageously ensure that the estimated concentration is zero when the amplitude is zero. Using the optimal A plot of the estimated gas concentrations relative to the known concentrations in the calibration samples is shown in Figure 7. Note the high linearity as further indicated by the Pearson correlation coefficients given in Table 2.

[0117] The ratio test is used to test the sensitivity of the measurement and decomposition process by injecting the calibration sample GB7 into the GC measurement equipment. As shown in Table 1, GB7 has a high C1 to C2 and C1 to C3 concentration ratio (300:1). It is generally desired to detect low concentrations of C2 and C3 relative to C1 and provide an estimate of the corresponding concentrations within a reasonable error limit. Accurately detecting such low concentrations can be challenging when the peaks of the individual gases are close to each other in time (have similar elution times) as C1, C2, and C3. In addition, this can be particularly challenging when the GC equipment is adapted to perform fast GC measurements (e.g., including a shorter GC column or faster carrier gas flow, etc.).

[0118] Figure 8 Depicted are the measured chromatogram (solid line) of calibration sample GB7 (as a hypothetical unknown sample) and the decomposed C1, C2 and C3 components (dashed lines) calculated as described above (using the calibration parameters obtained with calibration samples GB1 and GB2 during the GC system calibration). Note the good fit provided by the decomposed C1, C2 and C3 components. Table 3 lists the actual x values ​​for C1, C2 and C3. Cx [f CB7 ] and estimate The concentrations of C1 and C3 and the C1 / C2 and C1 / C3 ratios are shown in Figure 2. Note that the C1 and C3 concentrations are estimated to be within 1%, and the C2 concentration is estimated to be within 5%. This result is particularly encouraging, especially for C2, whose peak is almost completely obscured by the asymmetric tail of the C1 peak.

[0119] Table 3

[0120]

[0121] It should be understood that the present disclosure includes many embodiments. These embodiments include but are not limited to the following embodiments.

[0122] In a first embodiment, a method for estimating the components of an unknown gas sample including at least a first gas, a second gas, and a third gas includes: calibrating a gas chromatography (GC) device using a first calibration sample and a second calibration sample, wherein the first calibration sample includes a first gas and the second calibration sample includes the first gas, the second gas, and the third gas, the calibration including measuring a first chromatogram and a second chromatogram of the corresponding first calibration sample and the second calibration sample using the GC device; fitting the first chromatogram with a basis function derived from a mass balance equation to obtain a first modeled chromatogram; and fitting the second chromatogram with a first affine transformation response, a second affine transformation response, and a third affine transformation response of the first modeled chromatogram to obtain a second modeled chromatogram, the first affine transformation response, the second affine transformation response, and the third affine transformation response corresponding to the first gas, the second gas, and the third gas; measuring a third chromatogram of the unknown gas sample using the GC device; decomposing the third chromatogram into components corresponding to the first gas, the second gas, and the third gas using the second modeled chromatogram; and estimating the concentrations of the first, second, and third gases from the first, second, and third gas components of the decomposed third chromatogram.

[0123] A second embodiment may include the first embodiment, wherein the basis functions are asymmetric with respect to retention time.

[0124] A third embodiment may include any one of the first to second embodiments, wherein the first calibration sample includes negligible concentrations of the second gas and the third gas.

[0125] A fourth embodiment may include any one of the first to third embodiments, wherein the unknown gas sample includes at least methane, ethane, propane, butane, and pentane.

[0126] A fifth embodiment may include any one of the first to fourth embodiments, wherein the first modeled chromatogram includes a first set of calibration parameters, the first set of calibration parameters including the retention time, spatial diffusion coefficient, and decay rate of a first gas in the GC device; and the second modeled chromatogram includes a second set of calibration parameters, the second set of calibration parameters including the retention time and spatial diffusion coefficient of each of the first gas, the second gas, and the third gas, and the amplitude of each of the first affine transformation response, the second affine transformation response, and the third affine transformation response of the first modeled chromatogram.

[0127] A sixth embodiment may include the fifth embodiment, wherein the first chromatogram is fitted using the following mathematical equation:

[0128]

[0129] in represents the first modeled chromatogram, a mrepresents the amplitude, and g MBE (t,θ m ) represents the basis function, where:

[0130]

[0131] where H(t) represents the Heaviside step function and θ m =(t m ,s m ,p m ), where t m represents the retention time of the first gas in the GC device, s m represents the spatial diffusion coefficient of the first gas in the GC apparatus, and p m Represents the decay rate of the first gas in the GC device.

[0132] A seventh embodiment may include the sixth embodiment, wherein the second chromatogram is fitted using the following mathematical equation:

[0133]

[0134] in represents the second modeled chromatogram, t Cx x and s Cx represents the retention time and spatial diffusion coefficient of the first, second and third gases Cx, and a Cx represents the first, second, and third affine transformed responses of the first modeled chromatogram The amplitude.

[0135] An eighth embodiment may include any one of the first to seventh embodiments, wherein concentrations of the first, second, and third gases are estimated from corresponding amplitudes of the first, second, and third gas components of the decomposed third chromatogram.

[0136] A ninth embodiment may include the eighth embodiment, wherein the third chromatogram is decomposed using the following mathematical equation:

[0137]

[0138] in The third chromatogram after decomposition is shown, represents the response of the first, second and third gases Cx in the second modeled chromatogram, Δt represents the time shift, and f 0 represents the detector response offset, f Cx Represents the amplitude of the first, second and third gas components.

[0139] The tenth embodiment may include any one of the first to ninth embodiments, wherein the calibration gas chromatography (GC) device also includes a first linear correlation, a second linear correlation, and a third linear correlation between corresponding amplitudes of a first affine transformation response, a second affine transformation response, and a third affine transformation response for generating a first modeled chromatogram and the concentrations of the first gas, the second gas, and the third gas.

[0140] In an eleventh embodiment, a system for estimating the composition of an unknown gas sample including at least a first, a second, and a third gas includes: a gas chromatography (GC) device including a sample injection port, a main column, and a GC detector; and a processor configured to fit a first chromatogram with a basis function derived from a mass balance equation to calculate a first modeled chromatogram, the first chromatogram being measured using a first calibration sample including the first gas; fit a second chromatogram with a first affine transformation response, a second affine transformation response, and a third affine transformation response of the first modeled chromatogram to calculate a second modeled chromatogram, the second chromatogram being measured using a second calibration sample including the first gas, the second gas, and the third gas, the first affine transformation response, the second affine transformation response, and the third affine transformation response corresponding to the first gas, the second gas, and the third gas; decompose a third chromatogram into a first gas component, a second gas component, and a third gas component using the second modeled chromatogram, the third chromatogram being measured using the unknown gas sample; and estimate the concentrations of the first, second, and third gases from the first, second, and third gas components of the decomposed third chromatogram.

[0141] A twelfth embodiment may include the eleventh embodiment, wherein the first chromatogram is fitted using the following mathematical equation:

[0142]

[0143] in represents the first modeled chromatogram, a m represents the amplitude, and g MBE (t,θ m ) represents the basis function, where:

[0144]

[0145] where H(t) represents the Heaviside step function and θ m =(t m ,s m ,p m ), where t m represents the retention time of the first gas in the GC device, s m represents the spatial diffusion coefficient of the first gas in the GC apparatus, and p m Represents the decay rate of the first gas in the GC device.

[0146] A thirteenth embodiment may include the twelfth embodiment, wherein the second chromatogram is fitted using the following mathematical equation:

[0147]

[0148] in represents the second modeled chromatogram, t Cx and Cx represents the retention time and spatial diffusion coefficient of the first, second and third gases Cx, and a Cx A first affine transformed response, a second affine transformed response, and a third affine transformed response representing a first modeled chromatogram The amplitude.

[0149] A fourteenth embodiment may include any one of the eleventh to thirteenth embodiments, wherein concentrations of the first, second, and third gases are estimated from corresponding amplitudes of the first, second, and third gas components of the decomposed third chromatogram.

[0150] A fifteenth embodiment may include the fourteenth embodiment, wherein the third chromatogram is decomposed using the following mathematical equation:

[0151]

[0152] in The third chromatogram after decomposition is shown, represents the response of the first, second and third gases Cx in the second modeled chromatogram, Δt represents the time shift, and f 0 represents the detector response offset, f Cx Represents the amplitude of the first, second and third gas components.

[0153] In a sixteenth embodiment, a method for calibrating a gas chromatography (GC) device to estimate the composition of an unknown gas sample including at least a first gas, a second gas, and a third gas includes: using the GC device to measure a first chromatogram and a second chromatogram of a corresponding first calibration sample and a second calibration sample, the first calibration sample including a first gas, and the second calibration sample including the first gas, the second gas, and the third gas; fitting the first chromatogram with a basis function derived from a mass balance equation to obtain a first modeled chromatogram; fitting the second chromatogram with a first affine transformation response, a second affine transformation response, and a third affine transformation response of the first modeled chromatogram to obtain a second modeled chromatogram, the first affine transformation response, the second affine transformation response, and the third affine transformation response corresponding to the first gas, the second gas, and the third gas; and generating a first linear correlation, a second linear correlation, and a third linear correlation between the corresponding amplitudes of the first affine transformation response, the second affine transformation response, and the third affine transformation response of the first modeled chromatogram and the concentrations of the first gas, the second gas, and the third gas.

[0154] A seventeenth embodiment may include the sixteenth embodiment, wherein the first calibration sample includes negligible concentrations of the second gas and the third gas.

[0155] An eighteenth embodiment may include any one of the sixteenth to seventeenth embodiments, wherein the first calibration sample includes methane; and the second calibration sample includes methane, ethane, propane, butane, and pentane.

[0156] A nineteenth embodiment may include any one of the sixteenth to eighteenth embodiments, wherein the first modeled chromatogram includes a first set of calibration parameters, the first set of calibration parameters including the retention time, spatial diffusion coefficient, and decay rate of a first gas in the GC device; and the second modeled chromatogram includes a second set of calibration parameters, the second set of calibration parameters including the retention time and spatial diffusion coefficient of each of the first gas, the second gas, and the third gas, and the amplitude of each of the first affine transformation response, the second affine transformation response, and the third affine transformation response of the first modeled chromatogram.

[0157] The twentieth embodiment may include a nineteenth embodiment, wherein the first chromatogram is fitted using the following mathematical equation:

[0158]

[0159] in represents the first modeled chromatogram, a m represents the amplitude, and g MBE (t,θ m ) represents the basis function, where:

[0160]

[0161] where G(t) represents the Heaviside step function and θ m =(t m ,s m ,p m ), where t m represents the retention time of the first gas in the GC device, s m represents the spatial diffusion coefficient of the first gas in the GC apparatus, and p m represents a decay rate of the first gas in the GC apparatus; and

[0162] The second chromatogram was fitted using the following mathematical equation:

[0163]

[0164] in represents the second modeled chromatogram, t Cx and Cx represents the retention time and spatial diffusion coefficient of the first, second and third gases Cx, and a Cx A first affine transformed response, a second affine transformed response, and a third affine transformed response representing a first modeled chromatogram The amplitude.

[0165] Although the chromatogram decomposition and corresponding calibration have been described in detail, it should be understood that various changes, substitutions and alterations can be made herein without departing from the spirit and scope of the disclosure as defined by the appended claims.

Claims

1. A method for estimating a composition of an unknown gas sample comprising a first gas, a second gas, and a third gas, the method comprising: Calibrating a gas chromatography (GC) device using a first calibration sample and a second calibration sample, wherein the first calibration sample includes a first gas and the second calibration sample includes the first gas, the second gas, and a third gas, the calibration comprising: Using a GC device to measure first and second chromatograms of the corresponding first and second calibration samples; fitting the first chromatogram using basis functions derived from the mass balance equation to obtain a first modeled chromatogram; and fitting the second chromatogram with first, second, and third affine transformed responses of the first modeled chromatogram to obtain a second modeled chromatogram, the first, second, and third affine transformed responses corresponding to the first, second, and third gases; Measuring a third chromatogram of the unknown gas sample using the GC device; decomposing the third chromatogram into components corresponding to the first gas, the second gas, and the third gas using the second modeled chromatogram; and Concentrations of the first gas, the second gas, and the third gas are estimated from the first gas component, the second gas component, and the third gas component of the decomposed third chromatogram.

2. The method of claim 1, wherein the basis function is asymmetric with respect to retention time.

3. The method according to claim 1, wherein: a first modeled chromatogram including a first set of calibration parameters including a retention time, a spatial diffusion coefficient, and a decay rate of a first gas in the GC instrument; and The second modeled chromatogram includes a second set of calibration parameters, which includes retention times and spatial diffusion coefficients for each of the first gas, the second gas, and the third gas, and amplitudes for each of the first affine transformed response, the second affine transformed response, and the third affine transformed response of the first modeled chromatogram.

4. The method of claim 1, wherein the concentrations of the first gas, the second gas, and the third gas are estimated from corresponding amplitudes of the first gas component, the second gas component, and the third gas component of the decomposed third chromatogram.

5. The method of claim 4, wherein the third chromatogram is decomposed using the following mathematical equation: in The third chromatogram after decomposition is shown, represents the response of the first, second and third gases Cx in the second modeled chromatogram, Δt represents the time shift, f0 represents the detector response offset, and f Cx Represents the amplitude of the first, second and third gas components. The response of the first modeled chromatogram and the concentrations of the first gas, the second gas, and the third gas.

6. A system for estimating a composition of an unknown gas sample comprising a first gas, a second gas, and a third gas, the system comprising: A gas chromatography (GC) device comprising a sample injection port, a main column and a GC detector; and A processor configured to: fitting a first chromatogram using basis functions derived from a mass balance equation to calculate a first modeled chromatogram, the first chromatogram being measured using a first calibration sample including a first gas; fitting a second chromatogram using a first affine transformation response, a second affine transformation response, and a third affine transformation response of the first modeled chromatogram to calculate a second modeled chromatogram, the second chromatogram being measured using a second calibration sample including a first gas, a second gas, and a third gas, the first affine transformation response, the second affine transformation response, and the third affine transformation response corresponding to the first gas, the second gas, and the third gas; decomposing a third chromatogram into a first gas component, a second gas component, and a third gas component using the second modeled chromatogram, the third chromatogram being measured using the unknown gas sample; and Concentrations of the first, second, and third gases are estimated from the first, second, and third gas components of the decomposed third chromatogram.

7. The system of claim 6, wherein the first chromatogram is fitted using the following mathematical equation: in represents the first modeled chromatogram, a m represents the amplitude, and g MBE (t,θ m ) represents the basis function, where: where H(t) represents the Heaviside step function and θ m =(t m ,s m ,p m ), where t m represents the retention time of the first gas in the GC device, s m represents the spatial diffusion coefficient of the first gas in the GC apparatus, and p m Represents the decay rate of the first gas in the GC device.

8. The system of claim 7, wherein the second chromatogram is fitted using the following mathematical equation: in represents the second modeled chromatogram, t Cx and Cx represents the retention time and spatial diffusion coefficient of the first, second and third gases Cx, and a Cx A first affine transformed response, a second affine transformed response, and a third affine transformed response representing a first modeled chromatogram The amplitude.

9. A method for calibrating a gas chromatography (GC) apparatus to estimate the composition of an unknown gas sample comprising a first gas, a second gas, and a third gas, the method comprising: measuring, using the GC device, first and second chromatograms of corresponding first and second calibration samples, the first calibration sample comprising a first gas and the second calibration sample comprising a first gas, a second gas, and a third gas; fitting the first chromatogram using basis functions derived from the mass balance equation to obtain a first modeled chromatogram; fitting the second chromatogram with first, second, and third affine transformed responses of the first modeled chromatogram to obtain a second modeled chromatogram, the first, second, and third affine transformed responses corresponding to the first, second, and third gases; and First, second, and third linear correlations are generated between corresponding amplitudes of the first, second, and third affine transformed responses of the first modeled chromatogram and concentrations of the first, second, and third gases.

10. The method according to claim 9, wherein: The first calibration sample includes methane; The first calibration sample includes negligible concentrations of the second gas and the third gas; and The second calibration sample included methane, ethane, propane, butane, and pentane.

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