Applying Reservoir Simulation and Artificial Intelligence Algorithms to Optimize Fracture Characterization and CO₂ Enhanced Oil Recovery in Unconventional Reservoirs: A Case Study in the Wolfcamp Formation

Abstract

Reservoir simulation for unconventional reservoirs requires proper history matching (HM) to quantify the uncertainties of fracture properties and proper modeling methods to address complex fracture geometry. An integrated method, namely embedded discrete fracture model–artificial intelligence–automatic HM (EDFM–AI–AHM), was used to automatically generate HM solutions for a multistage hydraulic fracturing well in the Wolfcamp Formation. Thirteen scenarios with different combinations of matrix and fracture parameters as variables or fixed inputs were designed to generate 1300 reservoir simulations via EDFM–AI–AHM, from which 358 HM solutions were retained to reproduce production history and quantify the uncertainties of matrix and hydraulic fracture properties. The best HM solution was used for production forecasting and carbon dioxide (CO₂)-enhanced oil recovery (EOR) strategy optimization. The results of the production forecast for primary recovery indicated that the drainage area for oil production was difficult to extend further into the low-permeability reservoir matrix. However, CO₂ EOR simulations showed that increasing the gas injection rate during the injection cycle promoted incremental oil production from the reservoir matrix, regardless of minimum miscibility pressure. A gas injection rate of 25 million standard cubic feet per day (MMscfd) resulted in a 14% incremental oil production improvement compared to the baseline scenario with no EOR. This paper demonstrates the utility of coupling reservoir simulation with artificial intelligence algorithms to generate ensembles of simulation cases that provide insights into the relationships between fracture network properties and production.

1. Introduction

Reservoir simulation plays an important role in the evaluation, optimization, and management of unconventional reservoirs because of the complex fluid flow behavior in the low-permeability matrix (nano- to micro-Darcy range) and the complex fracture network comprising natural and hydraulic (stimulated) fractures. The presence of natural and hydraulic fractures creates challenges for an effective simulation study when a detailed sensitivity analysis is required to investigate the effect of fracture network properties on production. For conventional reservoir simulation, reservoir properties such as reservoir permeability and porosity can be obtained from well log and core data. However, for unconventional reservoir simulation, the fracture network geometry is highly uncertain and difficult to observe or measure. A diagnostic fracture injection test (DFIT) can provide fracture geometry information, but the interpretation is limited by data obtained over a short period [1]. Microseismic is another approach to characterize fracture geometry; however, the fracture geometry described by microseismic is stimulated reservoir volume (SRV), which is larger than the effective fracture geometry that contributes to production [2]. In addition, the fracture conductivity, or the ability of the fracture network to transmit fluids, cannot be obtained from microseismic interpretation. Reservoir simulation provides a quantitative method for evaluating the effect of the fracture network properties on production when the geometry and conductivity of the fracture network are uncertain. There are two critical aspects for properly characterizing the fracture network that contributes to production. First, proper history matching (HM) is needed to identify the fracture network properties that yield the closest match between simulated and observed production history. Second, the fractures need to be efficiently embedded in the simulation model to ensure that the simulation model can run smoothly over multiple cases.

1.1. Automatic History Matching

HM is an inverse problem that uses observed production behavior to evaluate the reservoir matrix and fracture properties [3]. One of the challenges with HM is that multiple combinations of reservoir matrix and fracture properties may provide sufficient HM results. Therefore, a single HM solution may not correctly describe reservoir conditions and fracture properties. Instead, an ensemble of solutions should be evaluated for better assessment of the reservoir matrix and fracture network uncertainties and better predictions of future production. Manual HM requires sound engineering judgment, and the process is labor- and time-intensive. To efficiently generate multiple HM solutions for better characterization of the reservoir and fractures, automatic history matching (AHM) is needed. In this context, AHM refers to a process where computer codes and algorithms sample a user-defined parameter space for the matrix and fracture network properties, execute reservoir simulations with the parameter values, and repeat the process until the iterations converge towards the best estimates for the matrix and fracture network properties.

The key to AHM is the algorithm for uncertainty characterization. The Markov chain Monte Carlo (MCMC) method is one of the most effective sampling methods for uncertainty characterization and HM; however, performing MCMC for AHM is time-consuming because it requires the realization of multiple reservoir models that are computationally expensive and require many hours to run one simulation case depending on the computing power [3,4,5]. The randomized maximum likelihood (RML) method has been demonstrated to generate acceptable uncertainty quantification results for some HM problems with less computational effort than MCMC [5,6]. However, the posterior samples provided by the RML method could be biased for some nonlinear relationships between reservoir model variables and production data, leading to unsuccessful HM performance [7,8]. The ensemble Kalman filter (EnKF) has been proven to be efficient and flexible for uncertainty quantification; however, EnKF fails to provide reasonable quantification of uncertainties for strongly nonlinear problems [5,7]. To improve the computational efficiency while maintaining the accuracy of uncertainty quantification, a workflow was proposed to replace the reservoir simulator with proxy models obtained from artificial neural networks (ANNs), one of several types of artificial intelligence (AI) or machine learning (ML) algorithms widely used for complex input–output relationships between features and target variables [9]. The proposed procedure significantly reduces the computational effort and makes the application of MCMC sampling in HM feasible. Tripoppoom et al. improved the original ANN–MCMC workflow by replacing a single proxy model with multiple proxy models for MCMC sampling to minimize the nonlinearity of the HM [10]. The multiple proxy-based MCMC approach with neural networks (NNs) as the proxy models was chosen in this study to perform AHM (hereafter NN–MCMC).

1.2. Embedded Discrete Fracture Model

To successfully perform an automatic history matching (AHM) procedure for unconventional reservoirs, it is crucial to model multistage hydraulic fractures and complex natural fracture geometry. A conventional method to include fractures in a reservoir model is the dual-porosity, dual-permeability (DP/DK) approach. The DP/DK approach is efficient in providing solutions for models of small scale or with uniformly distributed natural fractures, but the accuracy is low for modeling fractures with irregular geometry and high heterogeneity [11,12,13]. Local grid refinement (LGR) and unstructured grid (UG) methods provide higher accuracy for simulating complex fracture geometry; however, they require longer computational time because a heavy and complicated gridding structure is needed for modeling fractures [14,15,16]. The embedded discrete fracture model (EDFM) method combines the advantages of DP/DK, LGR, and UG methods. The accuracy of the EDFM method is comparable to the LGR and UG methods, but the EDFM method requires less computing time. The EDFM method was adopted in this study to construct fractures that were then integrated into a compositional reservoir simulation model for flow simulation.

1.3. Production Forecast

After better understanding matrix and fracture properties and obtaining the optimum reservoir model via the AHM workflow, the simulation model can be used for production forecasts (predictions of future oil, gas, and water production) and the optimization of well operations or sets of well operations.

1.4. Study Goal and Objectives

The goal of this study was to illustrate a workflow for applying the NN–MCMC approach and the EDFM method to characterize the matrix and fracture network properties in an unconventional reservoir and generate production forecasts. The study design included the following objectives: (i) select a test site for an unconventional reservoir, (ii) develop a reservoir simulation model for a representative well from the test site, (iii) apply the EDFM method to describe the fracture network around the selected well, (iv) implement the NN–MCMC approach to generate an ensemble of simulation cases for the test well and to match historical (observed) production, and (v) generate production forecasts to estimate primary recovery oil production and to explore different strategies for enhanced oil recovery (EOR) with carbon dioxide (CO₂) injection.

2. Test Site Description

2.1. Reservoir Overview

The test site studied in this work is a hydraulic fracturing test site, HFTS-1-Midland, located in the Wolfcamp Formation of the Midland Basin in Reagan County, Texas. The Midland Basin is part of the Permian Basin, which is one of the largest oil and gas production basins in the United States. In 2021, the Permian Basin contributed to 42% of crude oil production and more than 20% of natural gas production in the United States [17]. As of 2016, the technically recoverable oil, gas, and natural gas liquid (NGL) of the Wolfcamp Formation in the Midland Basin were 20 billion barrels, 16 trillion cubic feet, and 1.6 billion barrels, respectively, making the Midland Basin one of the largest hydrocarbon plays in the United States [18].

The Wolfcamp Formation in the Midland Basin consists of carbonate, evaporite, siliciclastic materials, lime mudstone, and dark shale [19,20]. Large amounts of organic materials that were accumulated in the Midland Basin during the early sedimentation period were later converted to oil and gas. Core sample analysis indicates that the Wolfcamp Formation in the Midland Basin has total organic carbon (TOC) of up to 8.7%, indicating high potential for oil and gas generation [21,22].

2.2. Production Overview

The HFTS-1-Midland site consists of 11 wells: 3SU, 4SM, 4SU, 5SM, 5SU, 6SM, 6SM, 7SM, 7SU, 8SM, and 8SU, as shown in Figure 1. The suffixes “SM” and “SU” refer to wells completed in the Middle Wolfcamp and the Upper Wolfcamp, respectively. The well spacing is approximately 660 feet within the same layer, and the range of lateral offset between two adjacent “SM” and “SU” wells is from 290 to 370 feet [23]. The wells are located at a depth of approximately 7700 feet. A review of the production history of the 11 wells revealed similar trends in oil, water, and gas production within the same production period. Well 7SU was selected as the candidate for reservoir simulation to investigate the production dynamics of the site and to develop the reservoir simulation model and EDFM fracture network.

2.3. HFTS Data Analysis

Through research conducted at HFTS-1-Midland (separate from the current study), extensive data were acquired from the Middle and Upper Wolfcamp Formations at the test site to characterize fracture geometry, identify interwell interference, analyze pressure response, and evaluate production performance [24]. Downhole microseismic mapping (MSM) data were obtained from approximately 434 fracturing stages. The MSM data suggest that the fracture geometry is relatively consistent for all the wells, and most of the hydraulic fractures appear to have relatively symmetric fracture growth in fracture length [25]. The core and image logs show that natural fractures exist in the test site, but more than 70% of the natural fractures are sealed, and the remaining natural fractures are possibly induced during hydraulic fracturing, drilling, or core handling [26]. These observations imply that while a few non-sealed natural fractures provide an insignificant amount of hydrocarbon production, the induced hydraulic fractures contribute to most of the hydrocarbon production. Based on these observations, the modeling and characterization of hydraulic fractures constitute the primary focus of this study.

High-frequency bottomhole pressure (BHP) data were collected via two pressure interference tests (PITs) performed at 7 and 18 months of production time to identify interwell communication between the active and observation wells [23]. Before well 4SU’s shut-in at the time of $t=25.9 \mathrm{h}$, the pressure buildup of Well 3SU follows a power law trend baseline. After Well 4SU’s shut-in, the pressure buildup of Well 3SU deviates from the power law trend baseline [23]. The pressure deviation $\Delta P$ between the buildup pressure and power law trend line is used to calculate Chow Pressure Group ($CPG$), which is defined as:

$$CPG=\frac{\Delta P}{2\left(\Delta P^{\prime }\right)}$$

(1)

and $\Delta P^{\prime }$ is the natural logarithmic derivative of $\Delta P$ calculated by:

$$\Delta P^{\prime }=\frac{d\left(\Delta P\right)}{d\left(\ln \left(\Delta t\right)\right)}$$

(2)

where $\Delta t$ is the time difference between the actual time and the shut-in time of Well 4SU [27,28]. The curves of $\Delta P$, $\Delta P\prime$, and $CPG$ against $\Delta t$ are presented on a log–log plot (Figure 2) to quantify the magnitude of pressure interference (MPI). The MPI value of 0.625 is obtained by reading the stabilized $CPG$ value (the blue solid diamonds show straight-line character) for the interference of Well 4SU shut-in on Well 3SU in Figure 2. This same approach is applied to estimate MPI values between other wells, as shown in Figure 3. The MPIs have ranges from 0 to 1. Higher MPI values indicate strong pressure interference caused by interwell fracture connection.

Comparing Figure 3a,b, the MPIs among the Upper Wolfcamp wells decreased from the first to second pressure interference test (PIT), while the change in MPIs among the Middle Wolfcamp wells was not significant. The results imply that the fractures connecting the Upper Wolfcamp wells were closing from the first to second PIT, while the fracture closure effect was not significant for the Middle Wolfcamp wells from the first to 2 s PIT. The PIT results also show that the pressure interference from other wells to Well 7SU is the smallest because no arrow points to Well 7SU. Despite the pressure data for Wells 8SU and 8SM being lost for the second PIT, the information provided in Figure 3a and the fracture closure effect together imply that the pressure interference from other wells to Well 7SU during the second PIT was not significant.

3. Reservoir Simulation Model Development

3.1. Equation of State (EOS) Modeling

Crude oil collected from Well 6SU was used to measure pressure, volume, and temperature (PVT) data. The oil was relatively light, as 82.9 mol % of the components were hydrocarbons below decane (C10) and 53.2 mol % of the components were gas components, including C1 (methane), C2 (ethane), and C3 (propane), as shown in Figure 4.

A series of pressure–volume–temperature (PVT) experiments including separator test, constant composition expansion (CCE) test, and differential liberation (DL) test were performed to measure the phase behavior of the crude oil in a wide range of pressure and temperature conditions. Based on the experimental data, an equation of state (EOS) was regressed using the Peng–Robinson method.

Table 1. Components and mole fractions of the EOS model.

Index	Component Name	Mole Fraction, %
1	N2	1.19
2	CO2	0.24
3	CH4	38.34
4	C2H6	7.26
5	C3H8	7.62
6	IC4 to NC4	5.19
7	IC5 to C7	13.99
8	C8 to C12	11.03
9	C13 to C19	7.05
10	C20 to C31+	8.09

“IC” refers to iso-alkanes, with C equal to the carbon number. “NC” refers to normal alkanes. “C” with no “I” or “N” prefix refers to either iso- or normal alkanes within the specific carbon number range.

shows the ten components used in the regressed EOS, which satisfactorily matched the experimental data, as illustrated in Figure 5. The EOS was integrated into the geologic model for dynamic reservoir simulation.

3.2. EDFM Method

The recently developed nonintrusive EDFM method allows complex fracture geometry to be simulated via structured grids; thus, the obtained complex fracture geometry can be integrated into a regular compositional reservoir model to perform simulation for unconventional reservoirs [29,30]. The EDFM method assists reservoir simulation via inserting fracture cells into the reservoir model using the non-neighboring connection (NNC) approach [29,30,31].

Three types of non-neighboring connections (NNCs) are: (i) NNC Type I—matrix and fracture connection, (ii) NNC Type II—fracture segments connection in a single fracture, and (iii) NNC Type III—a connection between two intersecting fracture segments. If a new fracture segment is added to the physical domain, then a non-neighboring new fracture cell will be added to the computational domain accordingly to calculate fluid flow inside fractures or between fractures and the rock matrix. The NNC approach allows the simulation of three-dimensional (3D) nonplanar and slanted fractures using additional two-dimensional (2D) fracture cells. Therefore, the EDFM method can maintain simulation accuracy while significantly improving computational efficiency [29,30,31,32,33,34].

3.3. Description of Reservoir Simulation Model

The goal of this study was to illustrate a workflow for applying the NN–MCMC approach and the EDFM method to characterize the matrix and fracture network properties and not to match production for all 11 HFTS-1-Midland wells. Thus, a single-well reservoir simulation model for Well 7SU was built using the reservoir parameters, the EOS model, and fractures created using EDFM. Figure 6 shows an oblique view of the reservoir simulation model with horizontal wellbore and hydraulic fractures, and

Table 2. Basic reservoir and fracturing data in the reservoir simulation model.

Parameter	Unit	Value
Model length	ft	11,234
Model width	ft	1395
Model height	ft	802
Matrix permeability	mD	0.00046
Wellbore lateral length	ft	10,091
Number of stages	–	43
Number of hydraulic fractures	–	150
Average fracture spacing	ft	67
Fracture half-length	ft	400
Fracture height	ft	100
Fracture conductivity	mD-ft	200
Fracture aperture	ft	0.1

shows the basic reservoir, wellbore, and fracturing data for the simulated Well 7SU. Natural fractures were not simulated in this study because of a lack of available data and observations by other HFTS-1-Midland researchers that natural fractures minimally contribute to the fracture network fluid flow as compared to hydraulic fractures [25]. The grid size of the model was 41 ft in both X- and Y-directions. The grid size varied from 26.4 to 118.7 ft/cell in the Z-direction to represent the different thicknesses of layers in the vertical direction. The purpose was to refine the region containing Well 7SU and its hydraulic fractures so the fracture–matrix and matrix–matrix flow calculations could be more accurate. The number of grids in the X-, Y-, and Z-directions was 274, 34, and 16, respectively. The length (in the Y-direction), width (in the X-direction), and height (in the Z-direction) of the model were 11,234 ft, 1394 ft, and 802 ft, respectively. An additional four grids were added to the X-direction to represent non-neighboring hydraulic fractures. The added grids were only used for flow calculation without impacting the material balance in the model.

3.4. Manual History Matching (HM) Results

Prior to running the automatic history matching (AHM), a manual HM was performed to evaluate the reservoir simulation response against measured BHP, oil rate, and water rate for Well 7SU. For all the HM results presented in Figure 7, the field and simulation data were normalized upon operator request [35]. The oil and water rate data were normalized by dividing the data value by the field maximum oil rate. The well BHP data were normalized by dividing the data value by the initial reservoir pressure.

The purpose of manual HM was to generate a base case rather than to obtain a high-quality history match. The results of manual HM show that the BHP, oil rate, and water rate could be partially captured, but the simulated curves still had a deviation from the historical data (Figure 7). The deviation may stem from uncertainties in matrix and hydraulic fracture properties.

4. Automatic HM with EDFM–AI–AHM

HM is one of the most time-consuming tasks in the simulation process for unconventional reservoirs, especially when considerable uncertainties exist for fractures. Recently, an innovative method, EDFM–AI–AHM, was developed to assist the HM process by tuning fracture and matrix parameters automatically based on a set of parameters with ranges provided [36]. The method utilizes neural network (NN)-based proxy models by sampling stochastic reservoir model realizations and updating the proxy models using MCMC [37,38,39,40,41,42,43]. The approach has been applied to several field cases [10,43,44,45]. The performance of the EDFM–AI–AHM method is affected by the initial selection and ranges of the uncertainty parameters. Inappropriate selection or ranges of uncertainty parameters could lead to insufficient HM solutions [43]. In this study, multiple EDFM–AI–AHM scenarios with different selections and ranges of uncertainty parameters were designed to achieve an ensemble of HM solutions.

Figure 8 illustrates the general workflow employed in each EDFM–AI–AHM scenario [42]. The first step is to screen and identify uncertainty parameters that need to be included in the automatic history matching (AHM). One matrix parameter (matrix compaction coefficient) and seven hydraulic fracture parameters (fracture efficiency, fracture half-length, fracture height, fracture aperture or width, fracture conductivity, fracture water saturation, and fracture compaction coefficient) were identified as either fixed or uncertainty parameters and used to tune the EDFM–AI–AHM process.

These parameters were chosen because similar projects demonstrated that these parameters are often important for the HM process [39,40,41,42,43]. The ranges (minimum and maximum values) of the eight matrix and hydraulic fracturing parameters were established based on available HFTS-1-Midland field data, and these ranges are summarized in

Table 3. Ranges for the eight matrix and hydraulic fracturing parameters for EDFM–AI–AHM to generate HM solutions.

ML Variable Name	Parameter Name	Unit	Minimum	Maximum
X1	Fracture efficiency	Dimensionless	0.3	1
X2	Fracture half-length	ft	50	500
X3	Fracture height	ft	50	200
X4	Fracture aperture (or width)	ft	0.1	0.5
X5	Fracture conductivity	mD-ft	10	1300
X6	Fracture water saturation	frac	0.1	0.9
X7	Fracture compaction coefficient	Dimensionless	0.01	0.05
X8	Matrix compaction coefficient	Dimensionless	0.1	0.2

. The eight matrix and hydraulic fracturing parameters were assumed to follow a uniform prior distribution between the minimum and maximum values. The prior distribution assumed no central tendency, and then the MCMC process was used to converge toward a posterior distribution.

Initial cases that covered the input ranges of the uncertainty parameters were generated by Latin hypercube sampling (LHS) [46]. Fracture models were generated by EDFM and then integrated into simulation models for these initial cases, which were executed to generate simulation results. In the first iteration, the inputs and results of the initial simulation cases were extracted to train an initial set of neural networks (NNs). Next, the NN–MCMC algorithm [42] was run to perform HM automatically and select acceptable HM solutions. The obtained simulation inputs and results were then used to retrain the NN in the second iteration. The workflow would stop if no significant improvement was detected in the latest iteration (stopping criteria); otherwise, the workflow would continue until the maximum number of iterations was reached.

The quality of the HM was determined by the deviation between simulation results and historical (observed) data. The response variables (parameters) were historical oil rate, water rate, and well BHP, which were used to define the global objective function (GOF) as:

$$\mathrm{GOF}=\frac{{{\displaystyle \sum }}_{j=1}^m{{\displaystyle \sum }}_{i=1}^n{\left[{\displaystyle \frac{X_{ij,model}-X_{ij,history}}{X_{ij,history}}}\times 100\right]}^2\times w_{ij}}{{{\displaystyle \sum }}_{j=1}^m{{\displaystyle \sum }}_{i=1}^n{w}_{ij}}$$

(3)

where i and j are the indices for data points and response parameters, respectively; m and n are the indices for the number of response parameters and data points, respectively; and $X_{ij,model}$, $X_{ij,history}$, and $w_{ij}$ are the modeling data, historical data, and weight of data at index i of response parameter j, respectively.

The definition of the GOF in Equation (3) allows one global value to represent multiple local errors. During each iteration, the EDFM–AI–AHM workflow proposed a random number of samples, for example, 10,000 cases, to train and test NNs. The realizations with the lowest GOF values were retained and validated with reservoir simulations [42].

EDFM–AI–AHM Scenarios

Thirteen scenarios with different combinations of variable and fixed input parameters were designed, as shown in

Table 4. Scenarios with different uncertainty parameter combinations to generate cases in EDFM–AI–AHM *.

Scenario	Uncertainty Parameters	Scenario	Uncertainty Parameters
1	X1, X2, X3, X7, X8	8	X1, X4, X5, X7, X8
2	X1, X2, X4, X7, X8	9	X1, X5, X6, X7, X8
3	X1, X2, X5, X7, X8	10	X2, X3, X4, X7, X8
4	X1, X2, X6, X7, X8	11	X2, X3, X5, X7, X8
5	X1, X3, X4, X7, X8	12	X2, X3, X6, X7, X8
6	X1, X3, X5, X7, X8	13	X2, X4, X6, X7, X8
7	X1, X3, X6, X7, X8

* X1 (fracture efficiency), X2 (fracture half-length), X3 (fracture height), X4 (fracture aperture or width), X5 (fracture conductivity), X6 (fracture water saturation), X7 (fracture compaction coefficient), and X8 (matrix compaction coefficient).

. X7 (fracture compaction coefficient) and X8 (matrix compaction coefficient) were included as variables in all the scenarios because they could characterize pressure-dependent matrix and fracture permeability changes. Providing a range of X7 and X8 promotes better bottomhole pressure HM performance. The other six fracture parameters were divided into three variables and three fixed parameters in each scenario with different arrangements so that the HM solutions would not be biased towards a single scenario of matrix and fracture property setups.

In each scenario, 40 initial cases were generated via Latin hypercube sampling (LHS) and simulated via reservoir simulation with EDFM to train the initial neural network (NN). Twenty cases with the lowest GOFs were generated for each iteration step to be validated with reservoir simulations and retrain the NN. To reduce computational time, the maximum number of iterations was set to three; hence, a total of 100 cases were generated for each scenario (40 initial + [3 iterations × 20 cases]). Each scenario was run through the full set of cases, so the total number of cases generated from the 13 scenarios was 1300 simulations.

Although the EDFM–AI–AHM workflow could potentially improve HM quality by retaining realizations with the lowest GOFs during each iteration, some retained cases could still possibly have high GOF values, and it would be less flexible to apply a single GOF threshold to treat simulations in all the scenarios. Therefore, the individual objective function (IOF) that measured the error for each response parameter was calculated after the iterations were completed in each scenario to further screen the optimum HM solutions, as shown in Equation (4).

$${\mathrm{IOF}}_j=\frac{{{\displaystyle \sum }}_{i=1}^n{\left[{\displaystyle \frac{X_{ij,model}-X_{ij,history}}{X_{ij,history}}}\times 100\right]}^2\times w_{ij}}{{{\displaystyle \sum }}_{i=1}^n{w}_{ij}}$$

(4)

where ${\mathrm{IOF}}_j$ is the individual objective function for the response parameter $j$ [47].

The IOF threshold for each response parameter was decided by comparing the EDFM–AI–AHM-generated 100 simulation results with the historical data for each scenario. A higher IOF value allows more cases to be further retained. In this study, the IOF threshold for well BHP was set to be from 60% to 80%, while that for oil and water rates was set to be from 30% to 50%. The higher threshold value was assigned to well BHP because the match of well BHP was not easy to achieve due to the assumption that pressure interference was not considered in the model.

Among the 1300 simulation results, 358 simulation cases that met the IOF criteria for the response parameters of the 13 scenarios were retained and used to generate posterior distributions of the matrix and fracture properties and to obtain a “best fit” reservoir simulation model used for the production forecasts. After the iterations were completed in each scenario, the data for model setup, including the range of uncertainty inputs, proxy locations, and number of cases in each iteration, were collected and saved. The prior and posterior distribution data were generated along with descriptive statistics including the 10th percentile (P10), the 50th percentile (P50) (or median), and the 90th percentile (P90) values that were determined from the posterior distributions. The solution curves set containing the historical data and the simulation solutions were also collected and saved.

5. HM Improvement Using EDFM–AI–AHM

5.1. Initial Simulation Cases

Parameter values generated via Latin hypercube sampling (LHS) were used to create 40 initial simulation cases for Well 7SU BHP, oil rate, and water rate (Figure 9). The field and simulation data in Figure 9 were normalized using the same method as that applied for Figure 7. In each panel, the curves show the simulation results and the hollow circles show the historical production data (history). As shown in the figures, the simulation results covered the full range of production history.

Table 5. EDFM–AI–AHM parameters and ranges (minimum/maximum or fixed value) for Scenario 1.

ML Variable Name	Parameter Name	Type	Unit	Minimum	Maximum	Value
X1	Fracture efficiency	Uncertain	Dimensionless	0.3	1	–
X2	Fracture half-length	Uncertain	ft	50	500	–
X3	Fracture height	Uncertain	ft	50	200	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X4	Fracture aperture	Fixed	ft	–	–	0.1
X5	Fracture conductivity	Fixed	mD-ft	–	–	150
X6	Fracture water saturation	Fixed	frac	–	–	0.5

shows the parameters and their ranges to set up Scenario 1 in EDFM–AI–AHM. For Scenario 1, the uncertainty parameters used to tune the automatic history matching (AHM) process were fracture efficiency (X1), fracture half-length (X2), fracture height (X3), fracture compaction coefficient (X7), and matrix compaction coefficient (X8). The other three parameters—fracture aperture (X4), fracture conductivity (X5), and fracture water saturation (X6)—were set as fixed parameters with values of 0.1 ft, 150 mD-ft, and 0.5, respectively. The input parameter ranges for Scenarios 2–13 can be found in

Table A1. Ranges of input variables for Scenario 2 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X1	Fracture efficiency	Uncertain	Dimensionless	0.3	1	–
X2	Fracture half-length	Uncertain	ft	50	500	–
X4	Fracture aperture	Uncertain	ft	0.1	0.5	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X3	Fracture height	Fixed	ft	–	–	106
X5	Fracture conductivity	Fixed	mD-ft	–	–	200
X6	Fracture water saturation	Fixed	frac	–	–	0.4

,

Table A2. Ranges of input variables for Scenario 3 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X1	Fracture efficiency	Uncertain	Dimensionless	0.3	1	–
X2	Fracture half-length	Uncertain	ft	50	500	–
X5	Fracture conductivity	Uncertain	mD-ft	10	300	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X3	Fracture height	Fixed	ft	–	–	106
X4	Fracture aperture	Fixed	ft	–	–	0.1
X6	Fracture water saturation	Fixed	frac	–	–	0.5

,

Table A3. Ranges of input variables for Scenario 4 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X1	Fracture efficiency	Uncertain	Dimensionless	0.3	1	–
X2	Fracture half-length	Uncertain	ft	50	500	–
X6	Fracture water saturation	Uncertain	frac	0.1	0.9	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X3	Fracture height	Fixed	ft	–	–	106
X4	Fracture aperture	Fixed	ft	–	–	0.1
X5	Fracture conductivity	Fixed	mD-ft	–	–	200

,

Table A4. Ranges of input variables for Scenario 5 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X1	Fracture efficiency	Uncertain	Dimensionless	0.3	1	–
X3	Fracture height	Fixed	ft	50	200	–
X4	Fracture aperture	Fixed	ft	0.1	0.5	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X2	Fracture half-length	Uncertain	ft	–	–	220
X5	Fracture conductivity	Uncertain	mD-ft	–	–	110
X6	Fracture water saturation	Fixed	frac	–	–	0.3

,

Table A5. Ranges of input variables for Scenario 6 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X1	Fracture efficiency	Uncertain	Dimensionless	0.3	1	–
X3	Fracture height	Uncertain	ft	50	200	–
X5	Fracture conductivity	Uncertain	mD-ft	150	700	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X2	Fracture half-length	Fixed	ft	–	–	250
X4	Fracture aperture	Fixed	ft	–	–	0.1
X6	Fracture water saturation	Fixed	frac	–	–	0.6

,

Table A6. Ranges of input variables for Scenario 7 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X1	Fracture efficiency	Uncertain	Dimensionless	0.3	1	–
X3	Fracture height	Uncertain	ft	50	200	–
X6	Fracture water saturation	Uncertain	frac	0.1	0.9	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X2	Fracture half-length	Fixed	ft	–	–	150
X4	Fracture aperture	Fixed	ft	–	–	0.1
X5	Fracture conductivity	Fixed	mD-ft	–	–	175

,

Table A7. Ranges of input variables for Scenario 8 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X1	Fracture efficiency	Uncertain	Dimensionless	0.3	1	–
X4	Fracture aperture	Uncertain	ft	0.1	0.5	–
X5	Fracture conductivity	Uncertain	mD-ft	100	500	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X2	Fracture half-length	Fixed	ft	–	–	85
X3	Fracture height	Fixed	ft	–	–	150
X6	Fracture water saturation	Fixed	frac	–	–	0.3

,

Table A8. Ranges of input variables for Scenario 9 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X1	Fracture efficiency	Fixed	Dimensionless	0.3	1	–
X5	Fracture conductivity	Fixed	mD-ft	10	150	–
X6	Fracture water saturation	Uncertain	frac	0.5	0.9	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X2	Fracture half-length	Uncertain	ft	–	–	189
X3	Fracture height	Fixed	ft	–	–	106
X4	Fracture aperture	Uncertain	ft	–	–	0.1

,

Table A9. Ranges of input variables for Scenario 10 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X2	Fracture half-length	Uncertain	ft	50	200	–
X3	Fracture height	Uncertain	ft	50	200	–
X4	Fracture aperture	Uncertain	ft	0.1	0.5	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X1	Fracture efficiency	Fixed	Dimensionless	–	–	1
X5	Fracture conductivity	Fixed	mD-ft	–	–	126.5
X6	Fracture water saturation	Fixed	frac	–	–	0.324

,

Table A10. Ranges of input variables for Scenario 11 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X2	Fracture half-length	Uncertain	ft	50	500	–
X3	Fracture height	Uncertain	ft	50	200	–
X5	Fracture conductivity	Uncertain	mD-ft	200	1000	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X1	Fracture efficiency	Fixed	Dimensionless	–	–	0.5
X4	Fracture aperture	Fixed	ft	–	–	0.3
X6	Fracture water saturation	Fixed	frac	–	–	0.7

,

Table A11. Ranges of input variables for Scenario 12 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X2	Fracture half-length	Uncertain	ft	50	500	–
X3	Fracture height	Uncertain	ft	50	200	–
X6	Fracture water saturation	Uncertain	frac	0.5	0.9	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X1	Fracture efficiency	Fixed	Dimensionless	–	–	0.6
X4	Fracture aperture	Fixed	ft	–	–	0.2
X5	Fracture conductivity	Fixed	mD-ft	–	–	150

and

Table A12. Ranges of input variables for Scenario 13 to run AHM workflow.

ML Variable	Parameter Name	Type	Unit	Minimum	Maximum	Value
X2	Fracture half-length	Uncertain	ft	50	500	–
X4	Fracture aperture	Uncertain	ft	0.1	0.5	–
X6	Fracture water saturation	Uncertain	frac	0.5	0.9	–
X7	Fracture compaction coefficient	Uncertain	Dimensionless	0.01	0.05	–
X8	Matrix compaction coefficient	Uncertain	Dimensionless	0.1	0.2	–
X1	Fracture efficiency	Fixed	Dimensionless	–	–	0.5
X3	Fracture height	Fixed	ft	–	–	130
X5	Fracture conductivity	Fixed	mD-ft	–	–	170

in Appendix A.

Parameter values generated via LHS were used to create 40 initial simulation cases for Well 7SU BHP, oil rate, and water rate (Figure 9). The field and simulation data in Figure 9 were normalized using the same method as that applied for Figure 7. In each panel, the curves show the simulation results and the hollow circles show the historical production data (history). As shown in the figures, the simulation results covered the full range of production history. While 31 cases had similar oil and water production trends compared to the historical data, only three cases presented a slight deviation from the BHP; thus, up to three cases can be retained in the initial iteration.

The gas rate was not used as a matching parameter since only a single well was considered in the model, while gas may flow across multiple wells in the actual field because of the higher mobility of gas compared to that of oil and water. The inputs and results of these 40 initial simulation cases for all the match points were used to train the initial neural network (NN). Then, the NN–MCMC algorithm was used to perform the automatic history matching (AHM).

5.2. NN–MCMC and AHM

For Scenario 1, 22 solutions that met the IOF criteria for response parameters were screened by the NN–MCMC algorithms during the iterations (Figure 10). The field and simulation data of BHP, oil rate, and water rate in Figure 10 were normalized with the same approach as that applied to Figure 7 and Figure 9. The IOF thresholds for oil rate, water rate, and well BHP are 30%, 30%, and 60%, respectively. Compared with the manual HM results shown in Figure 7 and the initial simulation results shown in Figure 9, the HM quality significantly improved after implementing the EDFM–AI–AHM workflow. The simulation results of BHP, oil rate, and water rate matched well with the historical data, as illustrated in Figure 10a–c, respectively. Compared to manual HM, the results were significantly improved by utilizing the EDFM–AI–AHM method.

5.3. Fracture Parameter Property Distributions

Figure 11 shows the prior and posterior property distribution of fracture half-length, fracture height, hydraulic fracturing (HF) efficiency, matrix compaction coefficient, and fracture compaction coefficient for the 22 solutions of Scenario 1 that had an acceptable GOF criterion. The results show that the posterior distribution (orange bars in the figure) of all five uncertainty parameters was different from the prior distribution (gray bars in the figure, which denote a uniform distribution between the minimum and maximum values).

Higher histogram probability was observed for the posterior distribution in a smaller range of uncertainty parameters compared to the prior distribution. The results indicate that the initial range of uncertainty parameters could be optimized to improve HM quality. For example, Figure 11a shows the 40 initial simulation cases had a uniform distribution of fracture height in the range of 50–200 ft, whereas the posterior distribution of fracture height had posterior P10 to P90 values of 90 and 129 ft, respectively, and a median of 110 ft. The results indicate that modifying the range of fracture height from 50–200 ft to 90–129 ft could potentially improve HM quality. Similarly, the posterior P10 and P90 values in Figure 11b imply that modifying the range of fracture half-length from 50–500 ft to 127–327 ft could potentially improve HM quality. Figure 11b also shows that the posterior distribution is not compact within the range between P10 and P90, and the posterior probability is 0 for fracture half-length in the range of approximately 208–298 ft. Running more iterations to generate more solutions may be an approach to further eliminate the 0-probability range between P10 and P90.

The best case for Scenario 1 was observed from the combination of fracture half-length, fracture height, HF efficiency, matrix compaction coefficient, and fracture compaction coefficient of 189.36 ft, 109.67 ft, 0.481, 0.11, and 0.038 respectively, as illustrated by the red vertical line in Figure 11. The prior and posterior distributions of the input parameters for Scenarios 2–13 can be found in Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7, Figure A8, Figure A9, Figure A10, Figure A11 and Figure A12 in Appendix A.

Table 6. Number of HM solutions for models with different variable combinations.

Scenario	Variable Combination	Number of Retained HM Solutions
1	X1, X2, X3, X7, X8	22
2	X1, X2, X4, X7, X8	20
3	X1, X2, X5, X7, X8	30
4	X1, X2, X6, X7, X8	6
5	X1, X3, X4, X7, X8	43
6	X1, X3, X5, X7, X8	37
7	X1, X3, X6, X7, X8	42
8	X1, X4, X5, X7, X8	59
9	X1, X5, X6, X7, X8	15
10	X2, X3, X4, X7, X8	4
11	X2, X3, X5, X7, X8	31
12	X2, X3, X6, X7, X8	8
13	X2, X4, X6, X7, X8	41
	Total	358

summarizes the number of retained HM solutions for all 13 scenarios, which shows that a total of 358 HM solutions were retained from the studied scenarios and 1300 reservoir simulation cases. These 358 reservoir simulations from EDFM–AI–AHM were a set of input/output (I/O) matrices from physics-based (i.e., reservoir simulations) solutions to well production performance, given values of the matrix and fracture network properties. These physics-based I/O matrices provide insights into the uncertainty in the matrix and fracture network properties, which can inform the subsequent production forecast and EOR strategy optimization. In addition, the I/O matrices can be further used as training and testing data for other ML workflows outside of EDFM–AI–AHM.

6. Production Forecast and EOR Strategy Optimization

The results of cumulative oil production for the 358 retained solutions were compared with the historical data to evaluate the HM performance after the EDFM–AI–AHM workflow, as shown in Figure 12a. While the EDFM–AI–AHM workflow significantly improved HM results compared to the manual HM and initial simulation cases for the production forecast and EOR optimization, the solutions were further screened to improve the accuracy of simulated oil production using the following two criteria:

(1)

The error $E_h$ of cumulative oil production between the simulated solutions and the historical data at the end of production history is less than 10%.

(2)

The root mean square error (RMSE) between the simulated solutions and the historical data for cumulative oil production is less than 10 Mstb (thousand stock tank barrels).

The $E_h$ and $RMSE$ values are calculated using the following equations.

$$E_h=\frac{|Q_{se}-Q_{he}|}{Q_{he}}\times 100\%$$

(5)

$$RMSE=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n{\left[Q_s\left(i\right)-Q_h\left(i\right)\right]}^2}{n}}$$

(6)

where $Q_{se}$ and $Q_{he}$ are cumulative oil production at the end of production history for simulation and historical data, respectively; $Q_s\left(i\right)$ and $Q_h\left(i\right)$ are cumulative oil production at data point $i$ for simulation and historical results, respectively; and $n$ is the total number of cumulative production data points.

The calculated $E_h$ and RMSE values for manual HM are 34% and 38%, respectively. The results show that the manual HM does not meet the $E_h$ and RMSE criteria. As shown in Figure 12a, most retained HM solutions show better HM performance than that of the manual HM.

Of the 358 retained EDFM–AI–AHM solutions, 57 were selected using the additional criteria of $E_h$ and RMSE. The results of the 57 selected solutions after further screening are shown in Figure 12b. These 57 selected solutions can be used in future works to further investigate the effects of different fracture and matrix properties on production prediction and enhanced oil recovery (EOR) performance. The best match representing the solution that has the minimum RMSE (red line in Figure 12b) was used for production forecast and EOR optimization in this study.

6.1. Production Prediction via Primary Recovery

Based on the best-match HM case, a predictive case for pressure depletion (with minimum BHP fixed at 300 psi for 10 years) was simulated to observe the oil production and drainage behavior of the well. The pressure distribution around Well 7SU at different time steps is shown in Figure 13, which indicates that most of the pressure depletion happened around the region with hydraulic fractures in the first several years of production. The difference between 5 and 10 years of prediction is small (Figure 13b,c) compared to the difference between the end of production history and the first 5 years of prediction (Figure 13a,b), as the drainage area (yellow and green area close to the blue lines with A pressure decline greater than 500 psi) is difficult to extend into the matrix because of the low permeability of the Upper Wolfcamp geological zone.

6.2. EOR via CO₂ Huff ‘n Puff (HnP)

Since the drainage area is difficult to extend into the reservoir matrix for primary production because of the depletion of pressure in the near-wellbore region, CO₂ HnP was explored as an EOR method to improve oil production. Five predictive cases were designed based on the best HM case to investigate the effects of gas injection rate on EOR performance. The injection rate varied from 2 to 25 million standard cubic feet per day (MMscfd), with injection, soaking, and production times of 30, 0, and 30 days, respectively, and a total cycle time of 60 days (

Table 7. Input parameters for different HnP cases.

Parameter	Value
EOR Operational Time	2 years
Injection Rate	2, 4, 8, 18, 25 MMscfd
Injection Time	30 days/cycle
Soaking Time	0 days/cycle
Production Time	30 days/cycle
Injection Gas	CO2
Maximum Gas Injection Pressure Constraint	7500 psi
Minimum Production Pressure Constraint	300 psi

). EOR performance was compared against the baseline case of primary production with no EOR. Two years of CO₂ HnP were simulated for each of the five cases. Figure 14 shows that an increased gas injection rate contributes to more incremental oil production. This is because a high gas injection rate is beneficial to providing pressure support in the hydraulic fractures and the drainage area near fractures so that the injected gas can penetrate the low-permeability reservoir matrix to interact with oil for EOR. While the 2, 4, and 8 MMscfd cases were not significantly different from the baseline case with no EOR (gray, blue, and green lines in Figure 14), the 18 and 25 MMscfd cases showed significantly greater cumulative oil production over the 2-year period (orange and red lines in Figure 14). For example, the cumulative oil production over the 2-year period for the baseline case with no EOR (pressure depletion) was 144 Mstb, while the cumulative oil production over the 2-year period for the 18 and 25 MMscfd cases was 153 and 164 Mstb, respectively, for increases of 6.4% and 14% over the baseline case.

In conventional high-permeability reservoirs, CO₂ injection is commonly performed at the pressure of slightly above minimum miscibility pressure (MMP), since increasing pressure above MMP does not improve oil recovery [48,49,50]. In contrast, for CO₂ injection in unconventional reservoirs, the oil recovery is a strong function of injection pressure, and increasing injection pressure, regardless of MMP, contributes to higher oil recovery [50,51,52,53,54,55,56]. This is because the pressure above MMP allows gas to enter the nanoscale pores to “unlock” and extract the residual oil. Figure 15 shows that a higher CO₂ injection rate contributes to higher injection pressure in each cycle.

Table 8. Simulation results for different CO₂ HnP cases.

Case No.	Injection Rate, MMscfd	Maximum Injection Pressure, psi	Total Oil Production, Mstb	Incremental Oil Recovery, % *
0	–	–	144	–
1	2	909	140	−2.6
2	4	1227	143	−0.4
3	8	1620	149	3.5
4	18	3052	153	6.4
5	25	4989	164	14

* Incremental oil recovery is calculated by comparing the total oil production of each case with that of Case 0.

summarizes the simulation results of maximum injection pressure during the HnP process, total oil production after the end of simulation, and incremental oil recovery for the five HnP cases. Case 0 corresponds to the pressure depletion scenario as the base case. The incremental oil recovery is negative for Cases 1 and 2 because the oil increase during the production period cannot compensate for the loss of production during the injection period.

The MMP for CO₂ and Wolfcamp oil tested by different researchers ranges from 1620 to 3706 psi [57,58,59]. Figure 16 shows that the simulated maximum injection pressure covers the range of MMP from the cited references [57,58,59]. The simulation results shown in Table 8 and Figure 16 further confirm that it does not matter if the maximum injection is above or below MMP; a higher maximum injection pressure contributes to more oil production. Another conclusion that can be drawn from the simulation results is that a gas injection rate of greater than 4 MMscfd is required to achieve meaningful incremental oil production, as suggested in Table 8.

7. Conclusions

A reservoir simulation model was developed based on the geological properties of the Wolfcamp Formation from the geologic model and fracture properties generated by an EDFM method. The reservoir simulation model was processed by the EDFM–AI–AHM approach to generate a set of reservoir simulations that explores the parameter space of the fracture network properties and generates physics-based solutions to well production given the fracture network properties.

A total of 358 HM solutions were obtained from 1300 reservoir simulations generated by 13 EDFM–AI–AHM scenarios. These solutions ensure better characterization of the matrix and fracture uncertainties. The matrix and property information and the corresponding simulated production results can be used as the input and output parameters for the development of other machine learning (ML) algorithms. The retained 358 HM solutions were further screened using the cumulative oil production error ($E_h$) and root mean square error (RMSE) criteria to obtain 57 solutions. The selected 57 solutions are suitable candidates for production prediction and EOR optimization.

The solution with the best HM performance was used for further simulations of production forecast and CO₂ EOR strategy optimization. The results of the production forecast via primary recovery indicate that the drainage area is difficult to extend into the low-permeability reservoir matrix to generate sustainable oil production. The simulation results of CO₂ EOR showed that a higher gas injection rate contributes to more oil production regardless of MMP. A gas injection rate of greater than 4 MMscfd is required for the well to improve oil production. A gas injection rate of 25 MMscfd could improve oil production by 14% compared to the pressure depletion (no EOR) scenario.