PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT PREDICTION OF SURFACE GAS CONCENTRATIONS IN DRILLING FLUID CROSS REFERENCE TO RELATED APPLICATIONS [0001] This application claims priority to European Patent Application No. 22306387.6, which was filed on September 21, 2022, and is incorporated herein by reference in its entirety. BACKGROUND [0002] When drilling a well for the production of hydrocarbons, drilling fluid is often circulated through the well for a number of purposes. For example, drilling fluid is commonly intended to provide hydrostatic pressure to the subterranean formation, cool and lubricate the drill bit, flush cuttings away from the drill bit and carry them to the surface, and provide hydraulic power to various downhole tools. Drilling fluids also commonly carry formation fluids and dissolved formation gasses to the surface. Such gasses may be liberated by the drill bit as it cuts the formation and may include various alkane gasses such as methane, ethane, propane, butane, pentane, and the like. [0003] It will be appreciated that the drilling fluid often begins to degas as it returns to the surface (and atmospheric pressure) and further degasses as it moves through various surface PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT equipment (including one or more degassers, pumps, tanks, etc.) before being re-circulated downhole. Gas concentration measurements are commonly made at one or more surface locations, for example, as the gas emerges from the wellbore and prior to being pumped back downhole. The measured gas concentrations are sometimes referred to in the industry as gas- out (in fluid emerging from the wellbore) and gas-in (just prior to the fluid being re-circulated downhole). [0004] Making gas concentration measurements in drilling fluid on a rig site is often equipment and labor intensive. For example, such measurements generally require deployment of a degasser in the fluid, an apparatus for collecting the gas, and a measurement instrument such as a gas chromatography or mass spectrometry instrument. Moreover, the measurements are commonly made repetitively (at some frequent time interval) and can require substantial manual labor. There is a need in the industry to reduce costs and reduce reliance on redundant equipment and excessive manual labor. 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 descriptions taken in conjunction with the accompanying drawings, in which: [0006] FIG.1 depicts an example drilling rig including a system for predicting drilling fluid gas concentrations. PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT [0007] FIG.2 depicts an example wellbore and surface system for predicting drilling fluid gas concentrations. [0008] FIGS. 3A and 3B (collectively FIG. 3) depict flow charts of example method embodiments for predicting drilling fluid gas concentrations. [0009] FIG. 4 depicts a flow chart (or block diagram) of another example method for predicting drilling fluid gas concentrations. [0010] FIG.5 depicts gas-out and gas-in peaks as a function of time for an example drilling operation. [0011] FIGS.6A-6G (collectively FIG.6) depict plots of concentration versus time for a first example in which the methane (6A), ethane (6B), propane (6C), butane (6D), iso-butane (6E), pentane (6F), and iso-pentane (6G) gas concentrations are plotted. [0012] FIGS. 7A-7C (collectively FIG. 7) depict plots of concentration versus time for a second example in which a gas concentrations from a first section of a wellbore are plotted (7A), gas concentrations from a second section of a wellbore are plotted using model parameters from the first section of the wellbore (7B), and gas concentrations from the second section of the wellbore are plotted using model parameters from the second section of the wellbore (7C). [0013] FIG.8 depicts plots of concentration versus time for a third example in which gas- out concentrations are back-predicted from gas-in measurements. PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT [0014] FIG. 9 depicts a plot of a gas-in to gas-out ratio versus gas-out concentration measurements for a fourth example in which various example models may be utilized to estimate gas-in concentrations. DETAILED DESCRIPTION [0015] Example ones of the disclosed embodiments may advantageously reduce drilling costs by reducing surface equipment and manual labor commonly needed to make gas measurements in drilling fluid. In one embodiment, a method for estimating surface concentrations of gas in a drilling fluid in use in a drilling rig includes measuring gas-out or gas-in concentrations while drilling a wellbore and processing the gas-out measurements or the gas-in measurements with a calibrated model to estimate corresponding gas-in concentrations or gas-out concentrations. The calibrated model may be obtained, for example, by measuring gas-out and gas-in while drilling another section of the wellbore or another wellbore and processing the gas-out and gas-in measurements with a model to obtain the calibrated model. [0016] FIG.1 depicts an example drilling rig 20 including a system 60 for predicting drilling fluid gas concentrations. The drilling rig 20 may be positioned over a subterranean formation (not shown). The rig may 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 wellbore 40 PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT and includes, for example, a drill bit 32 and one or more downhole measurement tools 50 (e.g., a logging while drilling tool and/or a measurement while drilling tool). [0017] Drilling rig 20 further includes a surface system 80 for controlling the flow of drilling fluid used on the rig (e.g., used in drilling the wellbore 40). In the example rig depicted, drilling fluid 35 is pumped downhole (as depicted at 92) via a conventional mud pump 82. The drilling fluid 35 may be pumped, for example, through a standpipe 83 and mud hose 84 in route to the drill string 30. The drilling fluid typically emerges from the drill string 30 at or near the drill bit 32 and creates an upward flow 94 of mud through the wellbore annulus (the annular space between the drill string and the wellbore wall). The drilling fluid then flows through a return conduit 88 and solids control equipment 95 (FIG. 2) to a mud tank system 81. It will be appreciated that the terms drilling fluid and mud are used synonymously herein. [0018] System 60 may be located on the rig site or at an offsite location. The system may include substantially any suitable computer hardware and software configured to process gas concentration measurements using a mathematical model. To perform these functions, the hardware may include one or more processors (e.g., microprocessors) which may be connected to one or more data storage devices (e.g., hard drives or solid state memory). The hardware may further include a network interface to enable communication with one or more gas measurement modules (e.g., modules 70 on FIG.2). Such computer hardware is well known and ubiquitous. It will, of course, be understood that the disclosed embodiments are not limited to the use of or the configuration of any particular computer hardware and/or software. PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT [0019] FIG. 2 depicts another embodiment of surface system 80 including solids control equipment 95 and mud tank system 81 (e.g., including a mud pit). Example surface equipment utilized in the solids control equipment 95 and/or mud tank system 81 may include a shale shaker, a desander, a desilter, a mud cleaner, one or more degassers, and/or a gas separator. The disclosed embodiments are, of course, not limited to the use of any particular surface equipment. As further depicted in FIG. 2, surface system 80 may include one or more gas measurement modules 70 (e.g., gas chromatographs and/or mass spectrometers) configured to measure concentrations of various alkane gases in the drilling fluid (such as methane, ethane, propane, butane, pentane, and the like). The measurement module(s) 70 may be located in a rig laboratory or may include portable instruments located adjacent to the surface equipment (e.g., adjacent to a degasser). The measurement module(s) 70 are generally configured to measure gas concentrations in the fluid as it exits the wellbore 40 (referred to herein as gas- out 72) and prior to re-entering the wellbore 40 (referred to herein as gas-in 74). [0020] FIGS. 3A and 3B (collectively FIG. 3) depict flow charts of example methods 100 and 110 for estimating a gas-in or a gas-out concentration from corresponding gas-out and gas- in measurements. In FIG. 3A, the method 100 includes measuring gas-out or gas-in concentrations while drilling a wellbore at 102. The measured concentrations may be processed with a calibrated model at 104 to predict or back-predict corresponding gas-in or gas-out concentrations. In FIG.3B, the method 110 includes making gas-out and gas-in measurements (e.g., gas concentration measurements) while drilling a first section of a wellbore at 112 and PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT processing the measurements with a model to obtain a calibrated model at 114. Gas-out or gas- in concentrations are measured while drilling a second section of the wellbore at 116 (or a second wellbore) and processed at 118 using the calibrated model to predict or back-predict corresponding gas-in or gas-out concentrations (by back-predict it meant that the corresponding gas-out concentrations are predicted back in time). It will be appreciated that the same calibrated model may be used to predict either gas-in or gas-out values. [0021] With continued reference to FIG.3, in certain example embodiments the calibrated model may be thought of as a transfer function that models (or predicts) degassing of drilling fluid at the surface (e.g., in the solids control equipment 95 and mud tank system 81 depicted on FIG.2). In particular, such a transfer function may be configured to output predicted gas- in values corresponding to gas-out measurements (or to back-predict gas-out values from corresponding gas-in measurements). As described in more detail below, calibration of the model may include determining model coefficients corresponding to a particular set of drilling conditions. The calibrated model may include substantially any suitable empirical model having substantially any suitable elementary mathematical functions (e.g., a linear function, a polynomial function, or an exponential decay function) or substantially any suitable physics based model that models degassing of the fluid (e.g., including one or more delay differential equations). The disclosed embodiments are not limited in this regard. [0022] FIG. 4 depicts a flow chart (or block diagram) of another example method 120 for estimating a gas-in concentration from a corresponding gas-out measurement. Method 120 PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT includes a calibration and learning routine 130 and a prediction routine 150. The calibration and learning routine 130 is intended to calibrate and train a model to predict gas-in values at a plurality of drilling conditions (e.g., over a range of normal drilling fluid properties, measured gas compositions, environmental conditions, and operational factors). The calibration and learning routine 130 may include identifying and measuring/determining drilling conditions ^^^^ at 132 that impact model calibration (e.g., impact the model parameters determined during the calibration). Gas-out 134 and gas-in 136 concentrations are measured for at least one gas composition (e.g., while drilling a first section of a subterranean wellbore or a first wellbore). The measured gas-out and gas-in concentrations may be processed with a model to determine calibration parameters at 138 (e.g., as described above with respect to FIG.3 and as discussed in more detail below for an example model). The model calibration parameters determined at 138 and the drilling conditions ^^^^ may be associated at 142 to train a parameter model. By train a parameter model, it is meant that a certain set of drilling conditions ^^^^ may be mapped to corresponding model parameters. It will be appreciated that learning block 142 may be repeated substantially any number of times to train the model at any number of distinct sets of drilling conditions. The learning block 142 generates a prediction operator ^^^^ _{^^^^ ^^^^} at 144 that is intended to correlate gas-out measurements with predicted gas-in values at a particular set of drilling conditions (and therefore model parameters). Again, it will be understood that the model calibration and learning routine 130 may be advantageously repeated a large number of PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT times in numerous wellbore sections drilled at various drilling conditions to obtain a robust prediction operator for predicting gas-in values. [0023] Method 120 further includes a prediction routine 150 configured to predict gas-in (or gas-out) values from gas-out (or gas-in) measurements made while drilling. The drilling conditions ^^^^ are measured and/or determined at 154 (e.g., while drilling) and processed using a parameter estimation model ^^^^ _{^^^^ ^^^^ ^^^^} to determine the model parameters at 156 (e.g., the parameter model generated at 142). Gas-out measurements are made at 152 while drilling and may be input into the degassing model ^^^^ _{^^^^ ^^^^} at 158 that may include the estimated model parameters obtained at 156 to in turn estimate the corresponding predicted gas-in values at 162. While FIG. 4 is described above with respect to predicting gas-in concentrations from corresponding gas-out measurements, it will be understood that the calibrated model obtained in 138 and/or 144 may also be utilized to back-predict gas-out concentrations from corresponding gas-in measurements as described above with respect to FIG. 3. Example degassing models are described in more detail below. [0024] One aspect of certain ones of the disclosed embodiments is the realization that the concentration of dissolved gas in circulating drilling fluid may be modeled as a mixture problem. With reference again to FIGS. 1 and 2, drilling fluid exiting the wellbore moves through solid control equipment 95 and a mud tank system 81 before being pumped back into the well. In the well, the fluid is pumped down through the drill string before being exposed to PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT the formation. The circulating drilling fluid interacts with the formation before exiting the well at the surface and completing a full cycle. [0025] The gas concentration measured at the head of the well, referred to herein as gas-out, generally decreases as it moves through the surface equipment, e.g., owing to its interaction with the solids control equipment and its exposure to the air (i.e., the fluid degasses). Pressure differences between the mud and air at the surface, as well as the mechanical interaction of the mud with the solids control equipment and other components of the circulation system facilitate degassing of the mud. By the time the mud is pumped out of the mud tank system and down into the wellbore, the gas concentration may be significantly reduced. As depicted in FIG.5, a gas peak 182 observed in the gas-out measurements may be delayed and widened as it traverses the surface equipment such that a corresponding gas-in peak 184 is delayed and widened and has a smaller amplitude than the gas-out peak 182. The observed delay between the peaks is referred to as the surface transit time (STT). [0026] Another aspect of certain ones of the disclosed embodiments was the realization that the concentration of gas being pumped into the well (the gas-in) may be modelled using an ordinary first order differential equation, for example, as follows: ^ _{^^^ =} ^{^^^^ − Δ ^^^^ ^^^^ ^^^^ ^^^^( ^^^^)} ^ _{^^^ ^^^^ −} [0027] where ^^̃^^ ₁ ( ^ _^^^ derivative of ^^̃^^ ₁ ( ^^^^) with respect to time, ^^^^ ₀ ( ^^^^) represents the measured gas-out, and ^^^^ ₁ ( ^^^^), PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT ^^^^ ₀₁ ( ^^^^), Δ ^^^^ _{^^^^ ^^^^ ^^^^} ( ^^^^), and ^^^^ _{^^^^ ^^^^ ^^^^} ( ^^^^) represent model parameters. In Equation 1, ^^^^ ₁ ( ^^^^) ≥ 0 and is related to a stationary degassing rate at the mud tank system. The parameter ^^^^ ₀₁ ( ^^^^) = ^^^^ ₀₁ · _^^^^1 ⁽ _^^^^ ⁾ _∈ ^[ _{0, ^^^^1} ⁽ _^^^^ ^)] _{and is related to a dilution rate of gas-out and thereby represents a fraction} gas-out contributing to gas-in. The parameter Δ ^^^^ _{^^^^ ^^^^ ^^^^} ⁽ ^^^^ ⁾ ≥ 0 represents the delay associated with a surface transit time of the drilling fluid. Larger ^^^^ ₁ ⁽ ^^^^ ⁾ values indicate higher stationary degassing rates while smaller ^^^^ ₀₁ ⁽ ^^^^ ⁾ values indicate degassing rates of gas- out until it mixes with gas-in. In addition to delay and dilution rates, the parameter ^^^^ _{^^^^ ^^^^ ^^^^} ≥ 1 is intended to accommodate dispersion that may result in peak widening of gas-in peaks with respect to corresponding gas-out peaks. [0028] Although it is not explicitly recited in Equation 1, the model parameters may implicitly depend on numerous drilling conditions (a set of drilling conditions referred to above as ^^^^ in FIG.4) including, for example, various mud properties (e.g., rheology, density, etc.), the specific gas being measured (e.g., methane, ethane, propane, butane, pentane, etc.), the environmental conditions (e.g., temperature, atmospheric pressure, etc.), and certain operational factors (e.g., flow rate, rig design, status of surface equipment, etc.). The model parameters may be associated with the set of drilling conditions, for example, as described above at 142 of FIG.4. This association may be represented mathematically, for example, as follows: ^^^^ ₁ = ^^^ ₁ ^( ^^^^); ^^^^ ₀₁ = ^^^ ₀ ^ ₁ ( ^^^^); Δ ^^^^ _{^^^^ ^^^^ ^^^^} = ^^^ _Δ ^ _t ( ^^^^); ^^^^ _{^^^^ ^^^^ ^^^^} = ^^^ _^ ^ _^^^ ( ^^^^) PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT [0029] where ^^^ ₁ ^( ^^^^), ^^^ ₀ ^ ₁ ( ^^^^), ^^^ _Δ ^ _t ( ^^^^), ^^^ _^ ^ _^^^ ( ^^^^) indicate that the model parameters ^^^^ ₁ , ^^^^ ₀₁ , Δ ^^^^ _{^^^^ ^^^^ ^^^^} , and ^^^^ _{^^^^ ^^^^ ^^^^} are functions of or are mathematically related to the drilling conditions ^^^^. For _{example, considering the four drilling conditions: flow rate ( ^^^^} ⁽ _^^^^ ⁾ _{∈ ℝ} ⁺ _{), density ( ^^^^} ⁽ _^^^^ ⁾ _{∈ ℝ} ⁺ _{), temperature ( ^^^^} ⁽ _^^^^ ⁾ _{∈ ℝ), and a binary degasser status ( ^^^^ ^^^^} ⁽ _^^^^ ⁾ _∈ ^{ _0,1 ^} _{), the model parameters} may be captured by the following relationship: ^^^^ ₁ ( ^^^^) = ^^^ ₁ ^� ^^^^( ^^^^), ^^^^( ^^^^), ^^^^( ^^^^), ^^^^ ^^^^( ^^^^)� � _� [0030] where ^^^ ₁ ^(∙), ^^^ ₀ ^ ₁ ^^^^ ₁ ( ^^^^), ^^^^ ₀₁ ( ^^^^), and Δ ^^^^ _{^^^^ ^^^^ ^^^^} ⁽ ^^^^ ⁾ are functions of or are related to the flow rate, density, and degasser status. In this example ^^^^ _{^^^^ ^^^^ ^^^^} ( ^^^^) is taken to be equal to unity such that there is no dispersion of the gas-out measurements. The disclosed embodiments are, of course, not limited in this regard. [0031] With continued reference to Equation 1, the model parameters may be estimated at 114 of method 110 (FIG.3B) or 138 of method 120 (FIG.4), for example, by minimizing the error between measured and modeled gas-in values (in certain embodiments minimizing the error may include reducing the error below a threshold error). For example, in certain embodiments it may be assumed that the model parameters ^^^^ ₀₁ ( ^^^^), ^^^^ ₁ ( ^^^^), Δ ^^^^ _{^^^^ ^^^^ ^^^^} ( ^^^^) and ^^^^ _{^^^^ ^^^^ ^^^^} ( ^^^^) are constant over a period of time ^^^^ ∈ [ ^^^^ _min , ^^^^ _max ], for example, a period of time over PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT which the above described drilling conditions remain substantially unchanged. In such an embodiment, the model parameters may be obtained by minimizing the least square error ^^^^( ^^^^ ₀₁ , ^^^^ ₁ ,Δ ^^^^ _{^^^^ ^^^^ ^^^^} , ^^^^ _{^^^^ ^^^^ ^^^^} ) between the measured gas-in values ^^^^ ₁ ( ^^^^) and modeled gas-in values ^{^^̃^^1( ^^^^) over the period of time, for example, as follows:} a _{rg min ^^^^} ⁽ _{^^^^01, ^^^^1,Δ ^^^^ ^^^^ ^^^^ ^^^^ , ^^^^ ^^^^ ^^^^ ^^^^ ^^^^} ⁾ 0 _{1, ^^^^1,Δ ^^^^ ^^^^ ^^^^ ^^^^, ^^^^ ^^^^ ^^^^ ^^^^} [0032] where ^ _^^^ ⁽ _{^^^^01, ^^^^1,Δ ^^^^ ^^^^ ^^^^ ^^^^ , ^^^^ ^^^^ ^^^^ ^^^^} ⁾ ₌ ^‖ _{^^^^1 − ^^̃^^1} ^[ _{^^^^01, ^^^^1,Δ ^^^^ ^^^^ ^^^^ ^^^^ , ^^^^ ^^^^ ^^^^ ^^^^} ^]‖2 2 [0033] It will an ordinary differential equation (ODE) which may be non-convex and include multiple local minima. As such, providing additional information relating to the physical bounds of the model parameters, for example, via additional measurements or controlled experiments may provide a more satisfactory solution. Further regularization of the minimization problem may be achieved by incorporation of causality and proximality into the solution. Moreover, an optimal dispersion rate may be achieved when there is no dispersion (i.e., when ^^^^ _{^^^^ ^^^^ ^^^^} = 1). While the disclosed embodiments are in no way limited in this regard, the remaining disclosure assumes that there is no dispersion and only considers numerical results with optimization of the model parameters ^^^^ ₁ , ^^^^ ₀₁ , and Δ ^^^^ _{^^^^ ^^^^ ^^^^} . PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT [0034] Over a fixed time period (e.g., ^^^^ ∈ [ ^^^^ _min , ^^^^ _max ] as described above), the delay differential equation may be discretized and solved using two point finite differences, for ^{example, as follows:} ^ _{^̃^^1( ^^^^ + Δ ^^^^) − ^^̃^^1( ^^^^)} = ^^^^ ₀ ( ^^^^ − Δ ^^^^ _{^^^^ ^^^^ ^^^^} ) ^^^^ ₀₁ − ^^̃^^ ₁ ( ^^^^) ^^^^ (3) Δ ^^^^ _{1 [0035] where ^^^^,} ⁽ consecutive or sequential time sample) within the fixed time period at which ^^^^ ₁ is measured. The delayed gas-out, ^^^^ ₀ ⁽ ^^^^ − Δ ^^^^ _{^^^^ ^^^^ ^^^^} ⁾ may be estimated, for example, via interpolating the sampled ^^^^ ₀ ( ^^^^) (e.g., using linear or other known interpolation techniques). The corresponding finite difference equation may be expressed, for example, as follows: ^ _^̃^^1 ⁽ _{^^^^ + Δ ^^^^} ⁾ _{= ^^^^0} ⁽ _{^^^^ − Δ ^^^^ ^^^^ ^^^^ ^^^^} ⁾ _{^^^^01 Δ ^^^^ + ^^̃^^1} ⁽ _^^^^ ⁾⁽ _{1 − ^^^^1 Δ ^^^^} ⁾ _{(4) [0036] where ^^^^,} ⁽ _{^^^^ + Δ ^^^^} ⁾ _∈ ^[ _{^^^^min , ^^^^max} ^] _{and with the initial condition that ^^̃^^1} ⁽ _^^^^min ⁾ _{= 0.} The disclosed embodiments are not limited to this initial condition, which implicitly makes the assumption that there is no gas present in the drilling fluid when the measurement starts. Invalidity of this assumption may be revealed in predictions around the beginning of the measurement period. Notwithstanding, this initial condition tends not to have a dominant effect in the optimization of the model parameters when the measurement period is long enough (as may be seen from the following examples). [0037] In a first example, the model parameters ^^^^ ₁ , ^^^^ ₀₁ , and Δ ^^^^ _{^^^^ ^^^^ ^^^^} were computed for seven distinct gases from a dataset including corresponding gas-in and gas-out measurements PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT collected over a 35-hour time interval. The gases included methane (C1), ethane (C2), propane (C3), butane (nC4), isobutane (iC4), pentane (nC5), and isopentane (iC5). Table 1 summarizes model notation for ^^^^ ₀ ( ^^^^), ^^^^ ₁ ( ^^^^) and ^^̃^^ ₁ ( ^^^^) corresponding to the above listed gasses (in which the subscript ‘out’ refers to the measured gas-out, the subscript ‘in’ refers to the measured gas- in, the subscript ‘in, pred’ refers to the predicted or modeled gas-in and the subscript ‘in, error’ refers to the error or difference between the measured and modeled gas-in). Table 1 Gas ^^^^ _^^^^ ( ^^^^) ^^^^ _^^^^ ( ^^^^) ^�^^^ _^^^^ ( ^^^^) ^^^^ _^^^^ ( ^^^^) − ^�^^^ _^^^^ ( ^^^^) C1 ^^^^1 _{^^^^ ^^^^ ^^^^} ^^^^1 _{^^^^ ^^^^} ^^^^1 _{^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^} ^^^^1 _{^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^} [0038] Th s listed in Table 1. The example model parameters are listed in Table 2 (for this particular data set). The model parameters may be optimal or suboptimal (depending on the nature of the data set). The disclosed embodiments are not limited in this regard. Table 2 Gas ^^^^ ₁ ^^^^ ₀₁ ^^^^ ^^^^ _{^^^^ ^^^^ ^^^^} (min) PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT nC4 0.0009 0.0009 9 iC4 0.0009 0.0009 9 [0039] FIGS.6A-6 tion versus time for each of the listed gases in the first example. The top row in each figure plots the measured gas-out values ( ^^^^ ₀ ( ^^^^)). The middle row plots the measured and predicted gas-in values ( ^^^^ ₁ ( ^^^^) and ^^̃^^ ₁ ( ^^^^)). And the bottom row plots the difference or error between the measured and predicted _{gas-in values ( ^^^^1} ⁽ _^^^^ ⁾ _{− ^^̃^^1} ⁽ _^^^^ ⁾ _{). In this example, the optimal parameters were obtained by minimizing error between ^^^^1} ⁽ _^^^^ ⁾ _{and ^^̃^^1} ⁽ _^^^^ ⁾ _{over the whole duration of the} measurement in which it was ensured by the rig personnel that mud and operational parameters were retained approximately constant. In each of the figures, note the generally good agreement between the measured and predicted gas-in values. [0040] In a second example, the dependency of the model parameters on the drilling conditions is demonstrated. This example includes first and second datasets in which the first data set includes methane gas measurements over a fifty-hour interval (hours zero to fifty) and the second dataset includes methane gas measurements over a subsequent forty-hour interval (hours sixty to one hundred). In this second example, the rig degasser was turned on between the fiftieth and sixtieth hour. The model parameters ^^^^ ₁ , ^^^^ ₀₁ , and Δ ^^^^ _{^^^^ ^^^^ ^^^^} were determined using the first dataset and the methodology described above. The computed model parameters are shown in Table 3. PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT Table 3 Gas ^^^^ ₁ ^^^^ ₀₁ ^^^^ ^^^^ _{^^^^ ^^^^ ^^^^} (min) C1 0.0002 0.00002 21 [0041] FIG.7A first dataset where the top, middle, and bottom rows are as described above with respect to FIG.6. [0042] FIG. 7B depicts plots of concentration versus time for the second dataset in which the predicted gas-in values were computed using the model parameters obtained from the first data set (shown in Table 3). Note that there is a significant difference (error) between the measured and predicted gas-in values plotted in the middle row. It is evident that turning the degasser on prior to the sixtieth hour significantly impacted the operation such that the model parameters determined for the first dataset were not applicable to modelling the second data set. [0043] In this second example, new model parameters ^^^^ ₁ , ^^^^ ₀₁ , and Δ ^^^^ _{^^^^ ^^^^ ^^^^} were determined using the second dataset and the methodology described above. These new model parameters are shown in Table 4. Note the change in model parameters after turning on the degasser between the fiftieth and sixtieth hour. In this example, ^^^^ ₁ increased by nearly a factor of four (from 0.00017 to 0.00060) while Δ ^^^^ _{^^^^ ^^^^ ^^^^} decreased by a factor of 3.5 (from 21 min to 6 min). Table 4 Gas ^^^^ ₁ ^^^^ ₀₁ ^^^^ ^^^^ _{^^^^ ^^^^ ^^^^} (min) PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT [0044] FIG. 7C depicts plots of concentration versus time for the second dataset in which the predicted gas-in values were computed using the model parameters obtained from the second data set (shown in Table 4). Note the generally good fit (low error) between the measured and predicted gas-in values plotted in the middle row obtained using the new model parameters. It is evident from this example that the most appropriate model parameters may depend on the drilling conditions. The use of machine learning, for example, as described above with respect to FIG.4 may advantageously enable appropriate model parameters to be selected for various drilling conditions. [0045] With reference again to FIG.3B, gas-out concentrations may be back-predicted from corresponding gas-in concentrations. For example, equation 1 may be rearranged (with ^^^^ _{^^^^ ^^^^ ^^^^} ( ^^^^) = 1), for example, as follows to compute (e.g., estimate or back-predict) gas-out: ^^̃^^ ₀ ( ^^^^) = ^^^^ ₀₁ ⁻¹ ∙ [ ^^^^ _^^^^ ^^^^ ₁ ( ^^^^ + Δ ^^^^ _{^^^^ ^^^^ ^^^^} ) + ^^^^ ₁ ( ^^^^ + Δ ^^^^ _{^^^^ ^^^^ ^^^^} ) ^^^^ ₁ ] (5) [0046] Note in measurements at time ^^^^ + Δ ^^^^ _{^^^^ ^^^^ ^^^^} . In other words, the computed gas-out values are indicative of (estimates of) the gas-out concentration at an earlier time than the gas-in measurements (and are therefore referred to herein as ‘back-predicted’). [0047] FIG.8 depicts plots of concentration versus time for methane gas (C1) in which the the top row plots back-predicted gas-out values from measured gas-in values ( ^^^^ ₁ ⁽ ^^^^ + Δ ^^^^ _{^^^^ ^^^^ ^^^^} ⁾ ). The middle row plots predicted gas-in values from measured gas-out values. The bottom row PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT plots error between measured and back-predicted gas-out values ( ^^^^ ₀ ( ^^^^) − ^^̃^^ ₀ ( ^^^^)). Note the good fit (low error) between the measured and back-predicted gas-out concentrations. [0048] In another example, surface degassing was modelled using empirical exponential decay functions. Gas-out and gas-in measurements were made while drilling a subterranean well. A surface transit time (STT) was estimated for a measured alkane gas in the drilling fluid. The estimated STT matched the average pump flow and was constant for all alkane gas compositions. The STT was then applied to the gas-in measurement to synchronize the gas-in and gas-out measurements. Filters were then applied on pump flows (for example to discard data when the pump flow was low and on gas data (for example, to discard low concentration points which may be irrelevant to the calibration). [0049] FIG. 9 depicts a plot of a gas-in to gas-out ratio versus gas-out for measurements made in the above-described example. A visual examination of the depicted scatterplot data may indicate an exponential decay (or multiple exponential decays) of the ratio with respect to the gas-out values. In certain embodiments, the depicted data (or portions thereof) may be fit using a mathematical model including an exponential function and/or other functions including various elementary functions such as linear or polynomial functions. An example exponential decay model is shown below: ^^^^ ^^^^ ^^^^ ^^^^ ^^^^/ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ = ^^^^ + ^^^^ ∙ ^^^^ ^^^^ ^^^^ ⁽ ^^^^ ∙ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ^^^^ ⁾ (6) [0050] where ^^^^, ^^^^, and ^^^^ represent the model parameters. PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT [0051] It will be appreciated that the scatterplot data shown on FIG. 9 may possibly be interpreted to indicate several exponential decay parameters (e.g., possibly indicative of a corresponding number of sets of drilling conditions utilized to drill the well). For example, the data indicated at 202, 204, 206, 208, 210, and 212 may be fit using distinct model parameters (e.g., distinct values of ^^^^, ^^^^, and ^^^^). In one example embodiment, the data indicated at 202 may be fit, for example, using model parameters ^^^^ ≈ 1, ^^^^ ≈ 10, and ^^^^ ≈ −5, the data indicated at 204 may be fit, for example, using model parameters ^^^^ ≈ 1, ^^^^ ≈ 40, and ^^^^ ≈ ^{−1.8, the data indicated at 206 may be fit, for example, using model parameters ^^^^ ≈ 1, ^^^^ ≈} _{70, and ^^^^ ≈ −1, the data indicated at 208 may be fit, for example, using model parameters} ^^^^ ≈ 1, ^^^^ ≈ 2600, and ^^^^ ≈ −0.001, and the data indicated at 210 may be fit, for example, using model parameters ^^^^ ≈ 15, ^^^^ ≈ 85, and ^^^^ ≈ −0.6. [0052] With continued reference to FIG.9, it will be appreciated that the depicted data may be fit (or partially fit) using models including mathematical functions other the exponential functions. For example, the data indicated at 214 (e.g., including higher gas-out values) may be fit using a linear equation. Moreover, the disclosed embodiments are not limited to models in which a ratio of gas-in to gas-out is related to gas-out. For example, suitable models may directly relate gas-in to gas-out (or gas-out to gas-in). In such models, gas-in (or gas-out) may be expressed as substantially any suitable function (or relation) of gas-out (or gas-in), for example, including a linear function, a polynomial function, or a function including a radical PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT gas-out term (e.g., a square root, cube root, etc.). Such models may plot gas-in versus gas-out, for example, or may plot gas-out versus substantially any suitable power of gas-out (such as gas-out squared, gas-out cubed, or the square root or cube root of gas-out). The disclosed embodiments are expressly not limited in this regard. [0053] It will be understood that the present disclosure includes numerous embodiments. These embodiments include, but are not limited to, the following embodiments. [0054] In a first embodiment, a method for estimating surface concentrations of gas in a drilling fluid in use in a drilling rig includes measuring first concentrations of the gas in the drilling fluid as it exits a wellbore (gas-out) or second concentrations of the gas in the drilling fluid as it is pumped downhole (gas-in) while drilling a wellbore; and estimating the gas-in concentrations or the gas-out concentrations by evaluating the gas-out measurements or the gas-in measurements with a calibrated model. [0055] A second embodiment may include the first embodiment, wherein the calibrated model is obtained by measuring gas-out concentrations and gas-in concentrations while drilling at least one other section of the wellbore or another wellbore; and evaluating the measured gas- out concentrations and gas-in concentrations with a model to obtain the calibrated model. [0056] A third embodiment may include the second embodiment, wherein the model is configured to account for degassing of the drilling fluid with time as the drilling fluid moves through surface equipment located on the drilling rig. PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT [0057] A fourth embodiment may include the third embodiment, wherein the surface equipment comprises at least one member of the group consisting of a mud pump, a degasser, a shale shaker, a desilter, a desander, and a mud pit in fluid communication with the drilling fluid. [0058] A fifth embodiment may include any one of the third through fourth embodiments, wherein the model comprises a delay first order ordinary differential equation. [0059] A sixth embodiment may include the fifth embodiment, wherein the model equates a first derivative of the gas-in concentrations with respect to time to a difference between a first product and a second product, the first product being a product of the gas-out concentrations at a time offset by a surface transit time and a first model parameter, and the second product being a product of the gas-in concentrations and a second model parameter. [0060] A seventh embodiment may include any one of the second through sixth embodiments, wherein the evaluating comprises determining a set of model parameters that provide a fit between the gas-out measurements and modeled gas-out concentrations. [0061] An eighth embodiment may include the seventh embodiment, further comprising repeating the measuring and the evaluating at a plurality of distinct sets of drilling conditions to obtain a corresponding plurality of sets of the model parameters; and using a learning algorithm to generate a parameter model that correlates the plurality of distinct sets of drilling conditions and the plurality of sets of the model parameters. PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT [0062] A ninth embodiment may include the eighth embodiment, wherein the evaluating comprises obtaining a set of drilling conditions for the other section; processing the set of drilling conditions with the parameter model to obtain a predicted set of model parameters and thereby obtain the calibrated model; and processing the gas-out measurements from the second section with the calibrated model to estimate the corresponding gas-in concentrations. [0063] A tenth embodiment may include any one of the first through ninth embodiments, wherein the gas-out measurements and the gas-in measurements comprise measurements of alkane gas concentrations. [0064] In an eleventh embodiment, a surface system configured for use on a drilling rig includes a gas measurement module configured to measure first concentrations of a gas in drilling fluid as it exits a wellbore (gas-out) and second concentrations of the gas in the drilling fluid as it is pumped downhole (gas-in) while drilling; and a processor configured to: receive gas-out and gas-in measurements made while drilling a first section of a subterranean wellbore; evaluate the gas-out and gas-in measurements with a model to generate a calibrated model; receive gas-out measurements or gas-in measurements made while drilling a second section of a subterranean wellbore; and estimate the gas-in concentrations or the gas-out concentrations by evaluating the gas-out measurements or the gas-in measurements with the calibrated model. [0065] A twelfth embodiment may include the eleventh embodiment, wherein the model is configured to account for degassing of the drilling fluid with time as the drilling fluid moves through surface equipment located on the drilling rig. PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT [0066] A thirteenth embodiment may include the twelfth embodiment, wherein the surface equipment comprises at least one of a mud pump, a degasser, a shale shaker, a desilter, a desander, and a mud pit in fluid communication with the drilling fluid. [0067] A fourteenth embodiment may include any one of the twelfth through thirteenth embodiments, wherein the model comprises a delay first order ordinary differential equation. [0068] A fifteenth embodiment may include the fourteenth embodiment, wherein the model equates a first derivative of the gas-in concentration with respect to time to a difference between a first product and a second product, the first product being a product of the gas-out concentration at a time offset by a surface transit time and a first model parameter, and the second product being a product of the gas-in concentration and a second model parameter. [0069] In sixteenth embodiment a method for estimating concentrations of gas in drilling fluid while drilling a wellbore includes drilling the wellbore at a set of drilling conditions; evaluating the set of drilling conditions with a parameter model to obtain a calibrated degassing model including a set of calibrated degassing model parameters; measuring first concentrations of a gas in the drilling fluid as it exits the wellbore (gas-out) or second concentrations of the gas in the drilling fluid as it is pumped downhole (gas-in) while drilling; and estimating the gas-in concentrations or the gas-out concentrations by evaluating the gas-out measurements or the gas-in measurements with the calibrated model. [0070] A seventeenth embodiment may include the sixteenth embodiment, wherein the parameter model is obtained via drilling a plurality of other sections of the wellbore or other PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT wellbores at a corresponding plurality of distinct sets of drilling conditions; making a plurality of sets of gas-out and gas-in measurements at a plurality of sets of drilling conditions; processing the plurality of sets of gas-out and gas-in measurements with a degassing model to obtain a corresponding plurality of sets of degassing model parameters; and correlating the plurality of sets of drilling conditions with the corresponding plurality of sets of degassing model parameters to obtain the parameter model. [0071] An eighteenth embodiment may include any one of the sixteenth through seventeenth embodiments, wherein the calibrated degassing model is configured to account for degassing of the drilling fluid with time as the drilling fluid moves through surface equipment on a drilling rig used to drill the wellbore. [0072] A nineteenth embodiment may include any one of the sixteenth through eighteenth embodiments, wherein the calibrated degassing model comprises a delay first order ordinary differential equation. [0073] A twentieth embodiment may include the nineteenth embodiment, wherein the calibrated degassing model equates a first derivative of the gas-in concentration with respect to time to a difference between a first product and a second product, the first product being a product of the gas-out concentration at a time offset by a surface transit time and a first model parameter, and the second product being a product of the gas-in concentration and a second model parameter. PATENT APPLICATION Attorney Docket No. IS20.2705-WO-PCT [0074] Although prediction of surface gas concentrations in drilling fluid has been described in detail, it should be understood that various changes, substitutions and alternations can be made herein without departing from the spirit and scope of the disclosure as defined by the appended claims.