3.1. D-Optimal Design for Response Surface Methodology
Manymethods have been proposed to conduct resolve uncertainty analysis [
41,
42,
43,
44,
45]. In this work, the RSM approach is applied to the optimization of the index SRA and SLA. RSM is utilized to approximate a response, in terms of maximum SRA and SLA, over a region of interest specified by the range of variability of input factors based on the least squares criterion. The RSM model can be in fully quadratic or linear. It can offer a cost-effective and efficient way to manage the uncertainties for shale gas reservoir development. A more detailed statistical and mathematical theories of RSM can be referred in the reports by Myers and Montgomery [
46]. The shale gas reservoir model is shown in
Figure 2. Seven uncertain parameters such as laminae thickness (T), laminae spacing (S), laminae dip angle (θ), laminae IFA, laminae cohesion (C), SD and IR are given a reasonable range with the actual maximum and minimum values or coded symbol of “+1” and “−1” respectively, as shown in
Table 2.
For the convenience of building up the numerical model, and discuss the influence of thickness on fracturing effectiveness, the minimum thickness is set as 2mm. According to the sevenvariables, 46 cases were required, based on the approach of D-Optimal design, which originated from the optimal design theory [
47,
48].
Table 3 shows the 46 combinations of the seven studied uncertain factors generated by the D-Optimal design. After the numerical simulation of each case, results of hydraulic fracturing are shown in
Figure 4 for partial selected cases with various designs in
Table 3. From
Table 3, in order to qualitatively evaluate the effect of hydraulic fracturing, two of the indices for the model responses were evaluated during the fluid injection. They are:
- (1)
SRA: defined as the interaction area of hydraulic fractures and silty laminae that has experienced a fluid pressure increase due to injection;
- (2)
SLA: defined as the area of silty laminae that has been damage during hydraulic fracturing.
SRA is the overall reflection of hydraulic fracturing effectiveness, and SLA reflects the hydraulic fracturing effectiveness of silty laminae. In
Table 3, the quantitative maximum stimulated area was listed in columns 9 and 10; they are the two responses in optimization. As there is no precise criteria for defining the SRA, a criteria based on fracture pressure change was employed in this work.
The interaction area of hydraulic fractures and silty laminae having a pore pressure increament below the maximum value of 5 MPa was considered as SRA, it refers to the “red” region in
Figure 5a. For SLA, during hydraulic fracturing, because of the lower mechanical properties and lower brittleness of the silty laminae, the silty laminae was easily damaged. From the simulation results, the picture labeled “black” (
Figure 5) corresponds to the silty laminae that experienced injection pressure. The total area of the “black” area was considered as SLA. Determination of SRA and SLA are based on digital image process (DIP) method. Taking case 14 for example, after gray segmentation, the area of SRA and SLA are shown in
Figure 5.
3.2. Response Surface MethodologyModel Analysis
According to the research results of [
10], the amount of free gas and solution gas in shale increases with the increase of silty laminae, and the amount of adsorbed gas decreases. Silty laminae plays a significant role in free-state shale gas formation, migration, and production. Therefore, in this work, not only SRA but also SLA is considered for the optimization of the fracturing design across multiple parameters to determine the optimal stimulation design. Once the maximum simulated area of 46 run cases was obtained, the RSM method is used to analyze the relationship between the response value and influential factors. To choose the siutable model, the statistical method was adopted to judge which polynomial fit the equation with a linear model, two factor model, interaction model (2FI), quadratic model, or cubic model, as shown in
Table 4 and
Table 5, which are response surface models for SRA and SLA, respectively. The criterion for selecting the appropriate model is choosing the highest polynomial model, where the additional terms are significant and the model is not aliased. Although the cubic model is the highest polynomial model, it is not selected because is it aliased. Aliasing is a result of reducing the number of experimental runs. When it occurs, several groups of effects are combined into one group and the most significant effect in the group is used to represent the effect of the group. Essentially, it is important that the model is not aliased. In addition, other criteria are needed to select the model that has the maximum “Adjusted R-Squared” and “Predicted R-Squared”. Thus, the fully quadratic model and linear model are selected to build the SRA and SLA response surface in the subsequent optimization process, respectively.
From the results of the anova for response surface quadratic model of SRA, the model F-value of 4.37 implies the model is significant. There is only a 0.86% chance that a “Model F-value” this large could occur due to noise. As shown in
Figure 6, values of “Prob > F” less than 0.05 indicates the model terms are significant. In this case, factor of A, D, E, and G, are significant model terms. The influential order of these seven factors is: G-IR > A-thickness > D-cohesion > E-friction angle > C-dip angle > B-spacing > F-SD.
The anova for the response surface linear model of SLA is shown in
Figure 7. The model F-value of 3.01 implies the model is significant. There is only a 1.29% chance that a “Model F-value” this large could occur due to noise. Values of “Prob > F” less than 0.05 indicates the model terms are significant. In this case A, B, D, and G, are significant model terms. The influential order of these seven factors is: G-IR > A-thickness > D-cohesion > B-spacing > F-SD > C-dip angle > E-friction angle.
From
Figure 6 and
Figure 7, it can be seen that the operational parameter of the IR is the most sensitive factor in the models. This indicates that IR is crucial to hydraulic fracturing effectives. This conclusion is consistent with Wang
et al. [
35,
36], King [
49], Gil
et al. [
50] and Nagel
et al. [
51]. With low IR, fluid is prone to leak into the natural fractures (bedding faces) despite the effect of fluid pressure and once the natural fractures accepts fluid, the pressure can rise far above the confining pressure with no induced fractures generation. With large IR, the hydraulic fracture is prone to cross the natural fractures resulting from the increase of the pressurization rate. In addition, the factor A (thickness) is also sensitive to both SRA and SLA. Silty laminae contains plenty of free gas, with the increase of thickness, the free gas recovery improves.
The equations fitted to the SRA and SLA response surface with coded symbols are presented below:
where
A is laminae thickness;
B is laminae length;
C is laminae dip angle;
D is laminae cohesion;
E is laminae IFA;
F is
in situ SD, and
G is IR.
The normal plot of residuals, reflecting the distribution of the residuals, for SRA and SLA are shown in
Figure 8. All the points in the “Normal Plot of Residuals” fall on a straight line, meaning the residuals are normally distributed.
Figure 9 shows the plot of “Predicted
versus Actual” for SRA and SLA, illustrating whether the generated equation of stimulated area response surface accurately predicts the actual SRA and SLA values. It can be seen that generated SRA and SLA response surface models provide reliable predicted SRA and SLA values, as compared with the actual values of SRA and SLA. This means that the generated SRA and SLA response surface models are reliable.
Figure 10 shows the 3D surface of the studied factors for SRA, the response surface represents a run case among the 46 cases. It shows the influential tread of the seven factors to SRA. As shown in
Figure 10a, SRA decreases with the increase of thickness, and increases with the increase of spacing.
Figure 10b shows the influence of spacing and dip angle on SRA, and SRA increases with increasing spacing.
Figure 10c shows the influence of cohesion and friction angle on SRA, and SRA increases with increasing cohesion and friction angle.
Figure 10d shows the relationship of SD, IR and SRA, SRA increases with increasing IR and decreasing SD. It can be seen that large IR, cohesion and friction angle, can lead to the increasing SRA. This can be better interpreted that with larger cohesion and friction angle, the strength of silty laminae is higher; during hydraulic fracturing, leak off is relative lower into laminae, and the injected fluid is mainly used to communicate silty laminae and shale matrix.
Figure 11 shows the 3D response surface of the studied factors for SLA.
Figure 11a plots the relationship of thickness, spacing, and SLA. SLA increases with increasing thickness and decreasing spacing.
Figure 11b presents the 3D surface of spacing and dip angle for SLA, and SLA increases with increasing dip angle.
Figure 11c presents the 3D surface of the cohesion and friction angle for SLA.While the cohesion and friction angle increase, SLA decreases gradually.
Figure 11d plots the relationship of SD, IR, and SLA, and SLA increases with increasing IR, and decreases with increasing SD. Similarly to SRA, IR is a positive factor resulting in the increase of SLA; however, cohesion and friction angle are negative factors leading to decreasing SLA. In silty laminae, shale gas is mainly in free state and solution state, the amount of free-state and solution gas in shale increases with increasing amount of silty laminae [
10].
The relatively large porosity that developed in silty laminae facilitates gas storage, and the mechanical properties are relatively weaker. This may be the reason why the hydraulic fracturing effectiveness is good for weak silty laminae, as studied in this paper.
3.3. Stimulated Area Optimization
The index of SRA indicates the total interaction area between hydraulic fractures and silty laminae, the larger the index, the better the fracturing network is. Also, the SLA index represents the production of free-state and solution gas. These two indices are very important to shale gas productivity in silty laminae formations, particularly in southeastern Ordos Basin, China. In this work, the numerical optimization was done to select the set of variables that leads to the maximum SRA and SLA value. A total of 71 optimal projects were generated after numerically optimization.
Table 6 shows the top ten optimal solutions maximizing the SRA and SLA simultaneously.
From
Table 6, it can be seen that the IR is a critical factor controlling the stimulated area of SRA and SLA. When SRA and SLA reach a maximum, the IR is 0.02 m
3/d/m. In addition, the smaller the SD, the maximum stimulated area can be reach for the SRA and SLA. Dip angle of silty laminae is about 60° when SRA and SLA reach the maximum. Mechanical properties control the damage and failure mechanism during hydraulic fracturing, according to Mohr-Coulomb shear failure criterion [
52], the fraction strength of the silty laminae affects the fracturing evolution to a large extent.
Figure 12a–d shows the 3D response surfaces for the optimal SRA and SLA with desirability of 0.780, respectively. The shape of the response surface displays a curved surface, which is different from the shape of SRA and SLA. This reflects the non-linear fracturing process when SRA and SLA are at a maximum simultaneously.
3.4. Discussion
From the RSM analysis of SRA and SLA, the influence factor of thickness presents a reverse trend. SRA has a negative correlation to laminae thickness, however, SLA has a positive correlation to laminae thickness. Also, laminae spacing is a negative factor for both SRA and SLA. Due to the weak mechanical properties of silty laminae compared to shale matrix, silty laminae has a larger porosity than shale matrix, the injected fluid is mainly used to reactive the silty laminae, and leak-off in silty laminae is most serious,thus, generation of hydraulic fractures is unfavorable. With increased spacing and SD, the degree of communicationbecomes weaker during hydraulic fracturing, which is unfavorable both for SRA and SLA. These two conclusions are consistent with the results of [
32]. Dip angle of silty laminae has a negative correlation with SRA, but a positive correlation with SLA. Generally speaking, the general opinion on the mechanism leading to the success of waterfrac in shale gas reservoirs is that a complex fracture network is created by the stimulation of pre-existing natural fractures and the silty laminae plays the role of natural fractures, which is actually a kind of cement-filled material. Fracture complexity is thought to be enhanced when pre-existing fractures are oriented at an angle to the maximum stress direction, or when both horizontal stresses and horizontal stress anisotropy are low, because the natural fractures in multiple orientations are prone to be stimulated with this combination of stress condition. The two mechanical parameters of cohesion and fraction angle, are both positively correlated with SRA and negative correlation to SLA. The reason for this result is that with lower mechanical strength, silty laminae are prone to damage and failure; once the laminae elements are damaged, it is different to generate hydraulic fractures, which reduces the interaction between hydraulic fractures and silty laminae. IR is a positive factor both for SRA and SLA, at a low IR, fluid tends to leak into the pre-existing discontinuities despite the influence of fluid pressure and once the fluid leak off into the natural fractures, the pressure can rise far above the confining pressure without inducing new fractures. At a large IR, the hydraulic fracture tends to cross the natural fracture due to the increase in the pressurization rate.