Experimental Measurement and Artificial Neural Network Prediction of Dew Point Pressure for Ultra-Deep Condensate Gas

Abstract

The development of oil and gas resources in global petroliferous basins has extended from shallow to deep reservoirs. Dew point pressure (Pd) is a vital parameter for fluid characterization and field development. Accurately and quickly obtaining Pd is crucial for the development of ultra-deep condensate gas reservoirs. The objective of this work is to predict the Pd of condensate gas by an artificial neural network (ANN) model. Ten ultra-deep condensate gas samples were analyzed using an experimental method and the Pd at reservoir temperature was obtained. A total of 113 datasets including 103 collected datasets and 10 measured datasets were adopted for ANN model training and testing. The results show that the average absolute percent relative error (AAPRE) of the developed ANN model between the measured and predicted values on the test set was 4.9589%. The predicted accuracy between the ANN model and widely used equations of state was compared. The results of statistical and graphical analysis show that the ANN model achieves the minimum prediction error. This ANN model can provide the necessary guidance for predicting the Pd for the development of different kinds of reservoirs.

1. Introduction

During the past several decades, technological breakthroughs have been witnessed across the globe in the development of deep and ultra-deep condensate gas reservoirs [1,2,3]. The proven recoverable reserves of oil and gas resources in deep (>4500 m) and ultra-deep (>6000 m) formations account for 40% and 49% of the total, respectively [4]. Condensate gas reservoirs have high economic value due to being oil–gas reservoirs and their lower refining cost [5,6,7]. The phase characteristics of condensate gas depend on pressure–volume–temperature (PVT) properties, and the dew point pressure (P_d) represents an important PVT property [8,9]. During the exploitation of condensate gas reservoirs, retrograde condensation occurs when the pressure declines below the P_d, and a gas–liquid miscible phase region forms [10,11,12]. This phenomenon can subsequently damage production facilities and transmission pipelines, leading a rapid decline in well productivity [13,14]. The accurate determination of P_d is essential for evaluating flow performance, simulating reservoir and well behavior, as well as designing production facilities and assessing reserves [15,16,17,18]. Therefore, accurately determined dew point pressure is a critical issue in development and management of condensate ultra-deep gas reservoirs.

In general, the main approaches to determining the dew point pressure of condensate gas are experimental methods and the thermodynamics equations of state (EOS) method. The determination of dew point pressure for high-pressure condensate gas through constant composition expansion (CCE) experiments has been widely reported in the literature [19,20,21,22,23]. Experimental measurement has the advantage of gaining precise and reliable results, but the costly sample transportation and time consumption often limit its application [24,25]. In the absence of experimental measurements, the EOS method is an option to estimate the P_d of condensate gas. During the EOS calculations, the temperature, components, C₇₊ molecular weight and other properties have impacts on the dew point pressure [26,27,28,29,30,31]. The prediction accuracy of the EOS method depends on determination of fluid characterization parameters, selected EOSs, and tuning procedures [32,33]. High-temperature and pressure conditions in ultra-deep reservoirs impose substantial challenges on conventional thermodynamic models, resulting in significant inaccuracies in dew point pressure predictions. With the continuous advancement of artificial intelligence, machine learning is widely applied to estimate the PVT properties under complex conditions [34,35,36,37,38,39,40]. As early as 1998, Elsharkawy and Foda [41] developed two general regression neural network (GRNN) models, which were applied to estimate dew point pressure and composition evolution in constant volume depletion (CVD) tests. Thereafter, González [42] improved the accuracy of dew point pressure prediction for retrograde gas reservoirs using a novel neural network model (NNM). Shokir [15] adopted genetic programming combined with an orthogonal least-squares algorithm (GP-OLS) to develop a dew point pressure prediction model for condensate gas. Based on 110 experimental data points, Nowroozi [43] developed a neural fuzzy system (NFS) to predict the dew point pressure. Over the past decade, a growing number of machine learning models were proposed to predict the dew point pressure of condensate gas, including: gene expression programming (GEP) [44,45], the radial basis function (RBF) network [46,47] genetic algorithms (GAs) [37], the adaptive neuro fuzzy inference system (ANFIS) [48], and the least square support vector machine (LSSVM) [49]. With the exploration and development of condensate gas resources advancing into ultra-deep formations, the temperature of most reservoirs has exceeded 180 °C, which poses a significant challenge to determining P_d by experimental method [50,51,52]. Therefore, it is highly significant to establish an efficient and accurate predictive model for dew point pressure in ultra-deep condensate gas reservoirs.

The main focus of this study is to develop a generally applicable artificial neural network (ANN) model to predict the P_d of ultra-deep condensate gas. A comprehensive and reliable ANN model was developed with multi-depth field datasets to predict the P_d. The correlation analysis was used to determine the input parameters of the ANN model. The measured P_d of ultra-deep condensate gas was employed to validate the applicability of the model. The model predictions are most reliable for ultra-deep condensate gas reservoirs with conditions comparable to those covered by the current data. In addition, the accuracy of the P_d predicted by the developed ANN model and three thermodynamic models was compared.

2. Experimental and Data Acquisition

2.1. Apparatus and Materials

The high temperature and pressure PVT setup, with a maximum operating temperature of 200 °C and pressure of 150 MPa, was used to measure the P_d of ultra-deep condensate gas. The accuracies in temperature and pressure are 0.01 K and 0.01 MPa, respectively. The phase behavior of ultra-deep condensate gas was observed through the sapphire window installed at the front of the PVT cell. In addition, a high-quality imaging system was used to capture and record the process of P_d formation. A densitometer (Anton Paar GmbH, Graz, Austria) and gas chromatograph (GC, Agilent Technologies, Santa Clara, CA, USA) were used to analyze the density and sample components. The schematic diagram of the P_d measurement is depicted in Figure 1.

Ten condensate gas samples used in this work were collected from distinct ultra-deep gas fields in northwestern China. As shown in

Table 1. Reservoir conditions of ultra-deep gas samples.

Sample	Depth (m)	Temperature (°C)	Pressure (MPa)	Oil Density (g/cm3)
S1	5235	133.70	48.83	0.7688
S2	5796	145.80	53.52	0.8197
S3	6346	124.50	90.00	0.7702
S4	6904	125.90	116.3	0.7863
S5	7375	158.63	146.07	0.8152
S6	7783	156.70	79.94	0.7805
S7	7956	164.99	92.52	0.7857
S8	7978	161.15	85.63	0.7838
S9	8254	163.36	89.41	0.7945
S10	8455	167.80	90.56	0.7827

, the basic sample properties represent a wide range of reservoir conditions. The reservoir depths range from 5235 to 8455 m, the reservoir temperatures from 124.50 to 167.80 °C, and reservoir pressures from 48.83 to 146.07 MPa.

2.2. Experimental Procedure

The CCE test was employed to measure the P_d of ultra-deep condensate gas. The P_d was determined by detecting the change in light transmittance of the system using an infrared laser. Above the dew point, the fluid is single-phase and affected by gas density and volume; the transmittance increases linearly with decreasing pressure. At the dew point, mist-like droplets appear and the transmittance-change rate changes abruptly, and the pressure at this inflection is taken as the dew point pressure. Detailed descriptions of the experimental procedure have been reported in our previous studies [53,54].

Data Acquisition

One of the core factors that dictates the reliability of the neural network model is the reliability and comprehensiveness of the training dataset [24]. In order to establish the correlation between depth and dew point pressure, reservoir depth was selected as an input parameter. According to the report, fluid compositions, reservoir conditions and fluid properties were key factors determining the P_d values [55,56]. Therefore, the input parameter datasets used in this research include reservoir depth, temperature, pressure, fluid composition and density of condensate oil. The fluid composition was classified into non-hydrocarbon fractions (N₂, CO₂), volatile hydrocarbon fractions (C₁–C₄), intermediate hydrocarbon fractions (C₅–C₆) and heavy fractions (C₇₊). The only desired output parameter is the P_d value. A total of 103 sets of P_d data were collected from numerous gas fields in China which cover several reservoir types. In addition, 10 sets of P_d data were obtained from laboratory PVT test results. All 113 datasets of data were employed for the development of a neural network model in this study.

Table 2. Ranges and corresponding statistical parameters of the dataset.

Parameters	Minimum	Maximum	Average	Standard Deviation
Depth (m)	2218.2	8630.0	5508.9	1705.9
Reservoir temperature (°C)	68.10	186.00	138.88	22.47
Reservoir pressure (MPa)	22.58	146.07	69.87	31.51
Non-hydrocarbon fractions (mol%)	0.08	42.54	4.85	5.10
Volatile hydrocarbon fractions (mol%)	49.98	98.25	88.51	7.08
Intermediate hydrocarbon fractions (mol%)	0.01	6.30	1.44	1.18
Heavy fraction (mol%)	0.46	15.91	5.20	3.61
Condensate oil density (g/cm3)	0.74	0.85	0.79	0.02
Dew point pressure (MPa)	21.96	65.36	43.40	8.66

shows the ranges and corresponding statistical parameters of the dataset.

3. Methodology

3.1. Artificial Neural Network

As a typical machine learning algorithm, ANN mimics human brain information processing to explore the relationship between inputs and outputs [57,58]. The network structure consists of parallel layers containing multiple neurons, which can be categorized into input layer, hidden layers, and output layer. The input layer takes input from the datasets, in which each input value corresponds to one neuron in the input layer. The hidden layers with a certain number of neurons are normally optimized by a trial-and-error process, and activation function is employed to transfer the data flow throughout the layers. The output layer is responsible for outputting a value for the exact problem, and the number of neurons is equal to the desired outputs. All neurons between different layers are connected by simple weighted links, and the weights can be either positive or negative. The neurons in the output layer receive signals from the hidden layers and produce final outputs according to relevant weight coefficients. The ANN model is trained using known input–output data to tune the weights between the different layers until the optimization criteria is met.

3.2. Model Architecture

The accuracy of the model is directly dependent on the architecture of the neural network [59]. The model constructed in this study adopts a four-layer structure, including an input layer, two hidden layers and an output layer. According to the sensitivity of P_d to the input parameter, the input layer is configured with 5 input parameters, namely condensate oil density, volatile hydrocarbon fractions, reservoir temperature, reservoir depth and reservoir pressure. Each of the 2 hidden layers contains 8 neurons. The optimal numbers of hidden layers and of neurons within each hidden layer were determined through repeated iterations until the minimum mean squared error (MSE) was reached. The network was trained using the Conjugate Gradient Method. Training was terminated when the mean squared error (MSE) reached the target value of 1 × 10⁻¹⁰. The connection weights were initialized using the Nguyen–Widrow method. The sigmoid function was used as a transfer function. The sigmoid function is expressed by the following:

$$f \left(\mathrm{S}\right) ={\displaystyle \frac{1}{1+e^{-S}}}$$

(1)

where S is expressed as follows:

$$S={\displaystyle \sum _{i=0}^n{x}_i{w}_i}$$

(2)

where x ($x_0$, $x_1$, $x_2$, … $x_n$) refers to the input vector and w ($w_0$, $w_1$, $w_2$, … $w_n$) refers to the corresponding weight vector. The n refers to the dimension of the input parameter.

To reduce the uncertainty associated with random weight initialization, the training procedure was repeated 10 times with different initial weights, and the network achieving the lowest validation MSE was selected as the final model. The output layer uses a linear function as the activation function for the estimation of the dew point pressure. The detailed parameters and general structure of the ANN model are shown in

Table 3. ANN model architecture and parameters.

Parameter	Value
Number of layers	4
Number of input layer neurons	5
Number of hidden layers	2
Number of neurons in each hidden layer	8
Training algorithm	Conjugate Gradient Method
The activation function of the hidden layer	Sigmoid function
The activation function of the output layer	Linear function

and Figure 2, respectively.

3.3. Data Splitting

The total 113 sets of data (measured and collected) were split into three classes. The first class, with 92 sets of data (80%), was allocated to the training phase for constructing and training the neural network model. The second class contained 11 sets of data for validating model reliability and accuracy. Meanwhile, 10 sets of experimental data were utilized to evaluate model performance. Before training, all input and output variables were normalized to the range [−1, 1] by Equation (3), using the statistics of the training set only; the same scaling factors were then applied to the validation and test sets to avoid data leakage.

$$X^{\prime }=2\left({\displaystyle \frac{X_{\mathrm{i}}-X_{\min }}{X_{\max }-X_{\min }}}\right)-1$$

(3)

4. Results and Discussion

4.1. Experimental Results

The experiment started at the reservoir pressure (higher than P_d), at which the system exhibited a single gas phase. The variation in fluid-phase behavior in the PVT cell is shown in Figure 3. When the pressure reduced to the P_d, the light transmittance of the fluid declined with the appearance of tiny liquid droplets, and the PVT cell became darkened. With a further decrease in pressure, these droplets gradually condensed and settled at the bottom, forming a clear gas-retrograde liquid interface.

The P_d experimental measurement results of the ultra-deep condensate gas sample are listed in

Table 4. The experimental data measured in this study.

Parameters	S1	S2	S3	S4	S5	S6	S7	S8	S9	S10
N2	4.55	6.64	1.70	0.37	2.60	1.51	1.85	2.65	1.29	3.97
CO2	5.06	0.35	0.27	0.30	0.04	9.73	6.96	7.01	5.45	1.62
CH4	76.58	91.58	86.45	89.36	91.31	79.46	80.02	82.13	77.69	75.27
C2H6	1.40	0.58	5.59	5.77	1.48	2.66	3.57	2.64	4.45	3.86
C3H8	0.66	0.08	1.46	1.32	0.23	0.87	1.07	0.80	1.77	1.43
i-C4H10	0.25	0.02	0.37	0.26	0.20	0.28	0.39	0.30	0.61	0.50
n-C4H10	0.48	0.04	0.48	0.27	0.23	0.43	0.50	0.42	0.83	0.86
i-C5H12	0.32	0.02	0.02	0.10	0.14	0.30	0.33	0.26	0.54	0.54
n-C5H12	0.40	0.02	0.17	0.06	0.16	0.31	0.32	0.25	0.50	0.68
C6H14	1.01	0.03	0.24	0.10	0.85	0.38	0.34	0.25	0.66	1.17
C7+	9.28	0.63	3.25	2.08	2.76	4.07	4.66	3.30	6.20	10.10
Pd (MPa)	38.20	53.52	48.19	53.07	53.76	50.08	43.80	41.90	40.50	40.65

. The P_d ranges from 38.20 to 53.76 MPa at reservoir temperature. Combined with the fluid components, it was found that P_d exhibits a nonlinear decreasing trend with an increase in the C₇₊ content. This is because increasing C₇₊ content raises the average molecular weight and critical temperature of the mixture, while strengthening the intermolecular attraction, making the gas more prone to condensation at a given temperature; thus, the dew point pressure decreases accordingly.

4.2. Variable Impact Analysis

Determining the sensitivity of the target parameter to each input parameter is crucial for improving model prediction accuracy. In this study, the influence of all dataset parameters on the dew point pressure was analyzed using grey relational analysis and the Pearson correlation coefficient. The grey relational degree ranges from 0 to 1, with values closer to 1 indicating a stronger correlation between the input variables and the target variable. The Pearson correlation coefficient ranges from −1 to +1, with values closer to 1 indicating a perfect positive correlation between input and target variables, and values closer to −1 indicating a negative correlation. The analysis results of grey relational analysis and the Pearson correlation coefficient are shown in Figure 4a,b. As shown in Figure 4b, condensate oil density, volatile hydrocarbon fractions, reservoir temperature, reservoir depth and reservoir pressure exhibit a positive correlation with P_d, while Figure 4a indicates that these parameters have a stronger correlation with P_d. Based on the combined results of the sensitivity analysis, these five parameters are the most significant factors influencing the dew point pressure, and are finally selected as input parameters for the ANN model.

4.3. Evaluated Parameters of the Models

The ANN model in this study was evaluated with different statistical quality measures. The statistical quality measures include average absolute percent relative error (AAPRE), average percent relative error (APRE), root mean square error (RMSE), standard deviation (SD) [8] and correlation coefficient (R²). The functions of these measures are shown in Equations (4)–(8).

$$\mathrm{AAPRE}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|\left(\frac{Exp.-Cal.}{Exp.}\right)\times 100\right|}$$

(4)

$$\mathrm{APRE}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left[\left(\frac{Exp.-Cal.}{Exp.}\right)\times 100\right]}$$

(5)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(Exp.-Cal.\right)}^2}}$$

(6)

$$\mathrm{SD}=\sqrt{{\displaystyle \frac{1}{n-1}}{{\displaystyle \sum _{i=1}^n\left(\frac{Exp.-Cal.}{Exp.}\right)}}^2}$$

(7)

$${\mathrm{R}}^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(Exp.-Cal.\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(Exp.-\overline{Exp}.\right)}^2}}}$$

(8)

4.4. Performance Evaluation of the Model

The statistical error evaluation for the ANN model in this study is provided in

Table 5. The statistical error analysis of the developed models.

Parameter	Train	Validation	Test	Total
APRE (%)	−1.2643	−3.0097	0.1872	−1.4245
AAPRE, %	8.9877	4.6519	4.9589	8.0904
RMSE (MPa)	4.8241	1.6182	1.8683	4.8372
SD	0.1123	0.0384	0.0401	0.1134
R2	0.7445	0.8112	0.7554	0.7620
Number of Datasets	92	11	10	113

Note: The “Total” values are computed from all 113 datasets combined, and are not an average of the three subsets.

. Table 5 shows the prediction deviations of the ANN model for the training set, validation set, test set and the entire dataset, respectively.

The APRE represents the direction and degree of the relative deviation between predicted and experimental data. As shown in Table 5, the APRE values for the training, validation and test sets were −1.2643%, −3.0097% and 0.1872%, respectively. The APRE represents the average direction and magnitude of the relative percentage deviation between predicted and target data. The AAPRE indicates the average absolute relative percentage error between predicted and experimental data, where a lower AAPRE value signifies more reliable prediction results. As shown in Table 5, the AAPRE values for the training, validation and test sets were 8.9877%, 4.6519% and 4.9589%, respectively. The results clearly demonstrate the capability of the utilized ANN model in predicting the P_d of condensate gas samples.

The cross plot of ANN model predictions versus the corresponding experimental values is shown in Figure 5. Almost all of the predicted data are located on the diagonal line, indicating that the ANN model had better accuracy and robustness. Moreover, the APRE of the ANN model is presented in Figure 6. As can be seen from Figure 6, the ANN model shows more data points distributed closely around the zero line, which confirms the accuracy of this model.

4.5. Comparison of ANN Model and Equations of State

In order to further examine the accuracy and stability of the ANN model in this study, the P_d of ten experimental data points (test sets) was calculated using EOS models. Three typical EOS models are adopted for comparative analysis, including the Peng–Robinson (PR) equation [60], Soave–Redich–Kwong (SRK) equation [61], and Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) equation [62]. The EOS calculations were performed using the fluid composition and reservoir temperatures. In the EOS calculations, fluid components were used to calculate the critical properties. Binary interaction parameters k_ij for $k_{{\mathrm{C}\mathrm{H}}_4-j}$, $k_{{\mathrm{C}\mathrm{O}}_2-j}$, and $k_{{\mathrm{N}}_2-j}$ were determined from critical volumes, while k_ij = 0 was assigned for hydrocarbon–hydrocarbon components. The C₇₊ fraction was split into single-carbon-number groups using Ahmed’s exponential decay function with Pedersen constraints, and then regrouped into 12 pseudo-components whose critical properties were obtained from the Kesler–Lee correlation. The dew point pressures were determined by solving the vapor–liquid fugacity equality at the reservoir temperature.

The detailed statistical results of different EOS models are reported in

Table 6. The statistics analysis for EOS models and the ANN model.

Parameter	PR	SRK	PC-SAFT	ANN Model
APRE (%)	23.0659	21.7289	21.6091	0.1872
AAPRE, %	31.7248	28.1352	29.9300	4.9589
RMSE (MPa)	10.9213	10.0889	10.0946	1.8683
SD	0.2294	0.2095	0.2191	0.0401

. Compared with the prediction accuracy of the ANN model, the EOS models exhibit lower accuracy in predicting the P_d of ultra-deep condensate gas. The AAPRE values for PR, SRK and PC-SAFT were 31.7248%, 28.1352% and 29.9300%, respectively, all of which are much higher than the 4.9589% of the ANN model. This indicates that the ANN model has superior performance in predicting P_d values.

The obtained cross plots for the ANN model and different EOSs are illustrated in Figure 7. According to Figure 7, the ANN model showed a more compact distribution of points around the X = Y line, confirming the higher precision of this method in comparison with EOS models. In addition, the APRE distribution (Figure 8) shows that points predicted by the ANN model are positioned in a small area (error less than 10%) around the zero line. Figure 7 and Figure 8 also confirm the higher predictive accuracy of the ANN model over other EOS models.

Under ultra-deep reservoir conditions (high temperature and pressure), the non-ideal nature of the gas phase is enhanced. The prediction of P_d by EOS relies on binary interaction parameters fitted to experimental data obtained at low temperatures and low pressures, which consequently leads to deviations in the predicted P_d. In contrast, the proposed ANN model directly captures the complex nonlinear relationships from experimental data, mapping five input parameters to the P_d through a multilayer architecture. Therefore, the ANN model achieves higher accuracy in predicting the P_d of ultra-deep condensate gas.

5. Conclusions

In this study, the measurement and prediction of dew point pressure for ultra-deep condensate gas were investigated. Experimental dew point data for ten ultra-deep condensate gas samples, at a depth range from 5235 to 8455 m, were measured using a high-temperature and pressure PVT setup. These experimental data along with 103 sets of data collected from ultra-deep gas fields were used to train and test the ANN model. Results of variable impact analysis showed that condensate oil density, volatile hydrocarbon fractions, reservoir temperature, reservoir depth and reservoir pressure were the most significant parameters governing the dew point pressure, these parameters were consequently adopted as input variables for the ANN model. The results of statistical and graphical analyses showed that the AAPRE of the developed ANN model between the measured and predicted values was 4.9589%. The AAPRE values for PR, SRK and PC-SAFT EOS models were 31.7248%, 28.1352% and 29.9300%, respectively. Compared with PR, SRK and PC-SAFT EOS models, the accuracies of the developed ANN model were verified. The ANN model developed in this article successfully predicts dew point pressure with high accuracy and is computationally convenient, which provides guidance for engineering calculations of dew point pressure in ultra-deep condensate gas reservoirs. It should be noted that the proposed ANN model was developed and validated within the range of the current dataset, and caution should be exercised when extrapolating predictions beyond this data range.