**1. Introduction**

In recent years, with the increasing application of new materials such as aluminum alloy, high strength steel and composite materials in automobiles as well as the continuous development of multi-material hybrid design concept in the automobile industry, the traditional mechanical connection technology (such as welding and riveting) cannot meet the connection requirements between different materials [1,2]. As a new connection method, bonding technology has the advantages of uniform stress distribution, fatigue resistance, light weight, etc. [3–5]. In this case, the connection needs of dissimilar materials can be effectively realized. Therefore, compared with other connection technologies, increasing attention has been paid to bonding technology. As a tough adhesive, polyurethane adhesive is gradually being widely used in automobiles, since it not only has high tear strength, good impact resistance and excellent toughness, but also provides relatively uniform stress distribution due to its low elastic modulus.

However, as a kind of polymer material, adhesive relies on temperature to some extent. The change of temperature will directly affect the mechanical properties of the material, and its failure strength and failure form change with different temperatures [6]. In the process of service, the ambient temperature range of adhesive structure is large. During the process of vehicle operation, the adhesive structure needs to provide enough strength in the service temperature range. The performance of the bonding structure is closely related to the service temperature, and the bonding structure significantly affects the overall strength and fatigue characteristics of the car body. Therefore, the research on the influence of the whole service temperature field on the performance of the body

**Citation:** Liu, H.; Fan, Y.; Peng, H. Effect of Full Temperature Field Environment on Bonding Strength of Aluminum Alloy. *Crystals* **2021**, *11*, 657. https://doi.org/10.3390/ cryst11060657

Academic Editors: Yifeng Ling, Chuanqing Fu, Peng Zhang, Peter Taylor and Sergio Brutti

Received: 10 May 2021 Accepted: 5 June 2021 Published: 9 June 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

bonding structure is the technical guarantee to realize the lightweight design of the body structure. Scholars at home and abroad have carried out relevant studies on the static performance and strength-checking criteria of temperature bonding structures [7,8].

Temperature is the main factor affecting the performance of the adhesive, and the mechanical properties of the adhesive will change in different temperature ranges. The bonding strength, strain and fracture toughness show temperature sensitivity [9]. The joint strength of the bonding structure is determined by the performance change of the adhesive and the influence of thermal stress [10]. The effect of temperature on the properties of the bonding structure is obvious, especially when the temperature is close to the glass transition temperature (Tg) of the material [11,12]. In addition, when the temperature is higher than Tg, the adhesive is featured with high elasticity, and its failure strength and elastic modulus decrease rapidly, while the elongation increases; however, when the temperature is lower than Tg, its performance is opposite [13]. Adams et al. [14] tested the single lap joint at different temperatures and also compared and analyzed the influence of thermal stress caused by the difference of thermal expansion coefficients and shrinkage stress caused by curing on the joint performance, which lead to the change concerning the stress state of the lap joint; the stress/strain performance of polymer adhesive also changes with the change of temperature. Na et al. [15,16] studied the effect of temperature on the mechanical properties of basalt fiber-reinforced composite/aluminum alloy bonded joints and found that with the increase in temperature, the Young's modulus and tensile strength of the joint decreased, while the tensile strain increased. The closer the temperature to Tg, the more significant the change in mechanical properties. Silva et al. [17] conducted a test on the mechanical properties of the single lap joint at low temperature and high temperature and revealed that the adhesive was brittle at low temperature and ductile at high temperature. They also analyzed the effect of porosity on failure. Banea et al. [12] investigated the stress-strain properties of polyurethane and epoxy adhesives at −40 ◦C, room temperature and 80 ◦C. It was found that with an increase in temperature, the failure strength and the Young's modulus of epoxy adhesives decreased, while the failure strain increased, which resulted from the increase of adhesive toughness at high temperature. Zhang et al. [18] conducted a study on the tensile properties of double lap joints in the temperature range of −35~60 ◦C and found that the load-elongation response was mainly affected by the thermomechanical properties of the adhesive, while it was less affected by the adhesive base material. When the temperature was higher than Tg, the strength and stiffness of the joint decreased, while the elongation increased dramatically, and the failure mechanism changed with the increase in temperature. To be specific, crack growth rate is higher at low temperature. In addition, the critical strain energy release rate for crack initiation and propagation increases continuously with increasing temperature.

Adhesion technology provides technical support for mixed material body design, but it also brings some problems. The service temperature of the bonding structure used in vehicles varies to a great extent in practical application. As a macromolecule material, the performance of the bonding structure is greatly affected by temperature, which causes the mechanical properties of the bonding structure to change with temperature. In order to achieve the safety design of the vehicle, the bonding structure must ensure the reliability of the connection within the full temperature field of the vehicle service. Therefore, temperature is one of the important factors that must be considered in the design of a bonded structure. It is of great significance to study the changing rules of bonded joint performance at different temperatures and propose the failure prediction method of bonded joints under the full temperature field for guiding the design of bonded structures.

### **2. Material Selection and Specimen Design**

### *2.1. Adhesive and Substrate*

The experimental selective adhesive was a modified silane polyurethane adhesive widely used in the window bonding of cars, trucks and trains. ISR-7008 is produced by Bostik China Co., Ltd. The mechanical parameters of the adhesive and adhesive substrate are shown in Table 1. The working temperature range provided in the technical manual is −40 ◦C~−90 ◦C. A permanent elastomer is formed by reaction with moisture in the air. 6005A aluminum alloy was selected as the adhesive substrate; it is widely used in automotive body structures. Table 2 shows the main performance parameters of ISR-7008 adhesive (provided by suppliers).

**Table 1.** Mechanical property parameters of adhesive and adhesive substrate.

**Table 2.** Technical performance parameters of ISR-7008 adhesive.

## *2.2. Design and Processing of Specimens*

To investigate the durability of adhesive joints under different stress states, the single lap joint (SLJ), the scarf joint (SJ) and the butt joint (BJ) were selected. When adherends are isotropic metallic and when the bondline thickness is very thin, the stress of the adhesive is assumed to be uniform and equal to the average values [8,19]. Thus, the SLJ and the BJ represent the shear stress and normal stress, respectively, while the SJ refers to the combined shear and normal stress. Furthermore, the normal and shear stress components with an infinitesimal block of adhesive within the central region of the scarf joints are calculated by assuming the coordinate and stress system as shown in Figure 1. The ratio between the tensile force and shear force of the adhesive layer can be changed by changing the angle *α* between the adhesive interface and the axis of specimens. It can be seen from the force decomposition that *F* represents the tensile force on both ends of the specimen. *F* sin *α* is the tensile force component of *F* on the bonding interface, and *F* cos *α* indicates the shear force component of *F* on the bonding interface. The normal *σ* and shear stress *τ* components are given by Equation (1), where *F* is the uniaxial failure load, *A* denotes the bonding area and *α* refers to the scarf angle.

$$
\sigma = \frac{F \sin \alpha}{A}, \; \tau = \frac{F \cos \alpha}{A} \tag{1}
$$

The docking and lap specimens were designed to study the mechanical properties of the adhesive joints under tensile stress and shear stress, separately. The overall size of the butt joint is 201 × 25 × 25 mm3 , and the adhesive area is 25 × 25 mm2 , while the overall size of the lap joint is 175 × 25 × 11 mm3 , and the adhesive area is 25 × 25 mm2 . The butt joint and the shear joint are shown in Figure 2, where the thickness of the adhesive layer is 1 mm.

*Crystals* **2021**, *11*, x FOR PEER REVIEW 4 of 20

*Crystals* **2021**, *11*, x FOR PEER REVIEW 4 of 20

**Figure 2.** Schematic diagram of docking and lap joints (mm). **Figure 2.** Schematic diagram of docking and lap joints (mm). **Figure 2.** Schematic diagram of docking and lap joints (mm).

**Figure 2.** Schematic diagram of docking and lap joints (mm). In the actual service process, the adhesive structure is often affected by tensile stress In the actual service process, the adhesive structure is often affected by tensile stress and shear stress; thus, it is of great significance to study the failure behavior of adhesive joints under the coupling of tensile stress and shear stress through reasonable joint design, thus being conducive to establishing the failure prediction of adhesive joints under complex stress. In order to study the failure behavior of the adhesive layer under different stress conditions, scarf joints with adhesive angles of 15°, 30°, 45°, 60°and 75° were designed and processed (as shown in Figure 3). The adhesive thickness of all specimens is unified as 1 mm. In the actual service process, the adhesive structure is often affected by tensile stress and shear stress; thus, it is of great significance to study the failure behavior of adhesive joints under the coupling of tensile stress and shear stress through reasonable joint design, thus being conducive to establishing the failure prediction of adhesive joints under complex stress. In order to study the failure behavior of the adhesive layer under different stress conditions, scarf joints with adhesive angles of 15◦ , 30◦ , 45◦ , 60◦and 75◦ were designed and processed (as shown in Figure 3). The adhesive thickness of all specimens is unified as 1 mm. In the actual service process, the adhesive structure is often affected by tensile stress and shear stress; thus, it is of great significance to study the failure behavior of adhesive joints under the coupling of tensile stress and shear stress through reasonable joint design, thus being conducive to establishing the failure prediction of adhesive joints under complex stress. In order to study the failure behavior of the adhesive layer under different stress conditions, scarf joints with adhesive angles of 15°, 30°, 45°, 60°and 75° were designed and processed (as shown in Figure 3). The adhesive thickness of all specimens is unified as 1 mm.

and shear stress; thus, it is of great significance to study the failure behavior of adhesive

**Figure 3.** Schematic diagram of 15◦ , 30◦ , 45◦ , 60◦ and 75◦ scarf joints (mm).

### *2.3. Dumbbell Sample*

ISR-7008 is a kind of flexible polyurethane adhesive. As the completely cured adhesive sheet is extremely soft, it is not feasible to use machined tensile test samples. Therefore, molding technology is adopted to process the sample for further avoiding the scratch

problem in the cutting process and ensuring that the sample is obtained without defects. Apart from that, for achieving the above purpose, the related metal abrasive tool is designed as shown in Figure 4. The grinding tool consists of three parts: the lower part is the base to play a supporting role; the middle is the template to determine the sample size and shape; the upper part is the compression plate and bolts. problem in the cutting process and ensuring that the sample is obtained without defects. Apart from that, for achieving the above purpose, the related metal abrasive tool is designed as shown in Figure 4. The grinding tool consists of three parts: the lower part is the base to play a supporting role; the middle is the template to determine the sample size and shape; the upper part is the compression plate and bolts.

ISR-7008 is a kind of flexible polyurethane adhesive. As the completely cured adhesive sheet is extremely soft, it is not feasible to use machined tensile test samples. Therefore, molding technology is adopted to process the sample for further avoiding the scratch

*Crystals* **2021**, *11*, x FOR PEER REVIEW 5 of 20

*2.3. Dumbbell Sample*

**Figure 3.** Schematic diagram of 15°, 30°, 45°, 60° and 75° scarf joints (mm).

**Figure 4.** (**a**) Forming mold. (**b**) Dumbbell sample. **Figure 4.** (**a**) Forming mold. (**b**) Dumbbell sample.

During the process of making samples, in order to prevent adhesive from sticking to the mold, a layer of polytetrafluoroethylene (PTFE) material is spread on the upper surface of the base and the lower surface of the pressure plate, and a layer of release agent is applied on all the surfaces of the template. After the die groove is coated with adhesive, the pressure plate on the cover is pressed with bolts. ISR-7008 adhesive is moisture curing. In order to ensure complete curing, it is solidified in 25 °C/50%RH environment for 7 days to remove the pressure plate (the reference manufacturer provides curing conditions), as shown in Figure 4, and then curing continues for 21 days. The geometric dimensions of During the process of making samples, in order to prevent adhesive from sticking to the mold, a layer of polytetrafluoroethylene (PTFE) material is spread on the upper surface of the base and the lower surface of the pressure plate, and a layer of release agent is applied on all the surfaces of the template. After the die groove is coated with adhesive, the pressure plate on the cover is pressed with bolts. ISR-7008 adhesive is moisture curing. In order to ensure complete curing, it is solidified in 25 ◦C/50%RH environment for 7 days to remove the pressure plate (the reference manufacturer provides curing conditions), as shown in Figure 4, and then curing continues for 21 days. The geometric dimensions of the dumbbell stretch samples used refer to the NF ISO 527-2 standards.

the dumbbell stretch samples used refer to the NF ISO 527-2 standards.

### *2.4. Design and Manufacture of Fixture*

*2.4. Design and Manufacture of Fixture* There are two main difficulties to be overcome in the adhesive process of docking and scarf specimens: the accurate control of neutral and adhesive thickness of the upper and lower square test rods of the specimen. To address these, it is necessary to first design and make the corresponding adhesive fixture. While the fixture is being made, the upper and lower grooves of the fixture are milled with one knife, thus ensuring the neutrality of the upper and lower test rods in the adhesive process of the specimen. At the same time, there is a calibration line next to the groove on one side of the fixture, and it is employed to control the thickness of the adhesive layer. The metal strip in the upper part of the groove is used to fix the adhesive specimen, and the knob on one side is adopted to push There are two main difficulties to be overcome in the adhesive process of docking and scarf specimens: the accurate control of neutral and adhesive thickness of the upper and lower square test rods of the specimen. To address these, it is necessary to first design and make the corresponding adhesive fixture. While the fixture is being made, the upper and lower grooves of the fixture are milled with one knife, thus ensuring the neutrality of the upper and lower test rods in the adhesive process of the specimen. At the same time, there is a calibration line next to the groove on one side of the fixture, and it is employed to control the thickness of the adhesive layer. The metal strip in the upper part of the groove is used to fix the adhesive specimen, and the knob on one side is adopted to push the adhesive test rod to the bond, as shown in Figure 5. *Crystals* **2021**, *11*, x FOR PEER REVIEW 6 of 20

> There are also some difficulties in the adhesive process of lap specimens; the main ones are the parallelism of the two adhesive test rods and the warping of the upper and

All specimens are adhesive in a clean and stable environment (temperature: 25 ± 3

1. An 80 mesh sandpaper is used for the cross grind concerning the adhesive surface of aluminum alloy along the 45° direction to increase the surface roughness and facili-

2. Acetone is used to clean the specimen, thus removing the oil pollution and dust produced in processing. Wiping paper is dipped in acetone, and the adhesive surface is wiped in one direction until the surface of the tissue is clean. The surface is allowed

3. Surface pretreatment coating agent Primer M is used to clean the adhesive surface

4. As for adhesive ISR-7008, on its surface, the corresponding fixture described above is used to complete the specific adhesive work, and then it is solidified for a period

The cured fully adhesive joints are placed in the hot and humid environment box as shown in Figure 7 and are subject to the test temperature for two hours. To be specific, the temperature range is −40 °C ~ 150 °C, the humidity range is 20% RH ~ 99% RH, the temperature fluctuation is ±0.1 °C and the humidity fluctuation is ±1%. For each test joint, the

including the lateral fastening fixture and the gantry tightening fixture, are designed as shown in Figure 6. The parallelism of the adhesive specimen is guaranteed by the screw rotation clamping of the lateral fastening fixture, and the metal block between the two rods is used to control the lap width of the shear specimen, while the flatness of the upper and lower planes of the adhesive is ensured by the screw compression of the downward

**Figure 5.** Adhesion fixture for docking and scarf specimens. **Figure 5.** Adhesion fixture for docking and scarf specimens.

rotating gantry compression fixture.

*2.5. Bonding Process*

tate the adhesive.

to dry for 10 min.

**Figure 6.** Diagram of lateral fastening jig (**left**) and gantry clamping jig (**right**).

°C; relative humidity: 50 ± 5%). The preparation process is as follows:

again. The surface is again allowed to dry for 10 min.

*2.6. Testing of Strength under the Condition of the Full Temperature Field*

of 4 weeks in the experimental environment.

There are also some difficulties in the adhesive process of lap specimens; the main ones are the parallelism of the two adhesive test rods and the warping of the upper and lower surfaces. In order to solve these problems, in this paper, the corresponding fixtures, including the lateral fastening fixture and the gantry tightening fixture, are designed as shown in Figure 6. The parallelism of the adhesive specimen is guaranteed by the screw rotation clamping of the lateral fastening fixture, and the metal block between the two rods is used to control the lap width of the shear specimen, while the flatness of the upper and lower planes of the adhesive is ensured by the screw compression of the downward rotating gantry compression fixture. There are also some difficulties in the adhesive process of lap specimens; the main ones are the parallelism of the two adhesive test rods and the warping of the upper and lower surfaces. In order to solve these problems, in this paper, the corresponding fixtures, including the lateral fastening fixture and the gantry tightening fixture, are designed as shown in Figure 6. The parallelism of the adhesive specimen is guaranteed by the screw rotation clamping of the lateral fastening fixture, and the metal block between the two rods is used to control the lap width of the shear specimen, while the flatness of the upper and lower planes of the adhesive is ensured by the screw compression of the downward rotating gantry compression fixture.

*Crystals* **2021**, *11*, x FOR PEER REVIEW 6 of 20

**Figure 5.** Adhesion fixture for docking and scarf specimens.

**Figure 6.** Diagram of lateral fastening jig (**left**) and gantry clamping jig (**right**). **Figure 6.** Diagram of lateral fastening jig (**left**) and gantry clamping jig (**right**).

### *2.5. Bonding Process 2.5. Bonding Process*

All specimens are adhesive in a clean and stable environment (temperature: 25 ± 3 °C; relative humidity: 50 ± 5%). The preparation process is as follows: All specimens are adhesive in a clean and stable environment (temperature: 25 ± 3 ◦C; relative humidity: 50 ± 5%). The preparation process is as follows:


**1. Introduction**

Due to population growth, economic development and consumption upgrade, global water consumption has increased by six times, and it has been continuing to grow steadily at an annual rate of about 1% during the past 100 years [1]. All of these would lead to the water shortage problem that is already pessimistically even severer, and seriously hinders the sustainable development of social economy. Managing water resources is an effective way to deal with the above challenges. However, in the process of management, experts and governors have encountered a lot of problems [2–6], such as dynamic variability and uncertainty, which are thorny and inevitable. Besides, in areas with water shortage, when the available water cannot meet the needs, unreasonable water allocation will lead to conflicts among users [7]. Therefore, it is definitely necessary to put forward a comprehensive model to deal with dynamic variabilities and uncertainties in water resources system as well as the contradiction between different users, so as to improve the management efficiency and the users' satisfaction.

The water resources system is of great complexities involving many uncertain factors, such as water use efficiency, water demand, pollutant discharge, water supply capacity and so on, and these uncertain factors could affect the structure for the optimal allocation model of water resources and resulting solutions [8–11]. Previously, scholars in related fields have got fruitful achievements in dealing with uncertainties in water resources management. For the optimization under uncertainties, mathematical methods that are commonly used

Fuzzy-Interval Dynamic Optimization Model for Regional Water Resources Allocation under Uncertainty. *Sustainability* **2022**, *14*, 1096. https://doi.org/10.3390/ su14031096

**Citation:** Suo, M.; Xia, F.; Fan, Y. A

Academic Editors: Julian Scott Yeomans and Mariia Kozlova

Received: 12 November 2021 Accepted: 12 January 2022 Published: 18 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

include stochastic programming [12], fuzzy programming [13], interval programming [14], and various coupling programming methods [15]. Among them, fuzzy programming can deal with the conflicts among multi-objective functions well, making it a new flexible planning problem [16], while the interval programming can reflect the uncertain coefficients in the form of intervals, and convert the uncertain planning problems into deterministic planning problems [17]. Both of them are helpful to solve uncertain planning problems. Based on these models, a large number of integrated models have been developed to optimize the allocation of water resources, such as uncertain two-stage stochastic water resources optimal allocation model [18], improved interval linear optimal allocation model [19], chance constrained water resources optimal allocation model [20], multi-objective interval linear water resources optimal allocation model [21], and so on [22–24]. Li et al. [25] proposed a multi-objective water resources optimal allocation model under uncertainties by integrating constrained programming, semi-infinite programming, integer programming and interval linear programming. Suo et al. [26] presented an approach for interval multiobjective planning by coupling fuzzy programming and improved two-step method, and then proved the objectivity and stability of this method by comparing it with the weighted sum method. Li et al. [27] formulated a new two-stage random interval parameter fuzzy planning strategy model by considering various uncertainties in planning and management of water resources and water environment systems, which was then applied to reveal the relationship between local economic goals and environmental goals. The above-mentioned models can well deal with the uncertainty in data acquisition in the system. However, they are insufficient to handle the dynamic features in the allocation process of water resources.

In the process of water resources optimization, it is essential to give a full consideration to dynamic characteristics, and thus provide the best scheme for water distribution at different stages of the planning period [28]. Dynamic programming cannot only solve the optimization problem of multi-stage decision process in water resources allocation [29], but also obtain the optimal strategy of the whole process and the optimal sub-strategy of each stage [30–34]. Peng [35] established a multi-objective dynamic water resources allocation model to achieve a dynamic balance for the optimal water resources allocation by using a modified simplex method with the addition of a time variable. Feng [36] set up a multiobjective dynamic water resources optimization configuration model and introduced the satisfaction function to realize the dynamic balance of optimal allocation of water resources on the time scale. Ramírez et al. [37] used stochastic dynamic programming to provide release decisions for each stage, and combined genetic algorithm and reservoir operation simulation program to obtain the annual release curve. These models proposed above are able to solve the multi-stage decision-making problem and get satisfactory allocation results. However, they took less consideration for uncertainties in water resources system.

Therefore, in order to comprehensively consider uncertainties and dynamic variability in the water resources system, a fuzzy-interval dynamic (FIDP) optimal allocation model is proposed in this study by integrating fuzzy-interval linear programming (FILP) and dynamic programming (DP) into a general framework. In addition, in order to realize the fairness of water resources allocation, the satisfaction function is added as one objective function to reduce the contradictions among users. The main innovative points of this study can be summarized as: (i) By introducing FILP into the FIDP model, the uncertainty coefficients and constraints, such as water use efficiency and water demand in water resources system, can be reflected in the form of interval numbers, which would make the results more accurate and reasonable. (ii) By introducing DP into FIDP model, not only the annual optimization scheme, but also the detailed water distribution scheme of each stage in planning year can be obtained. (iii) For FIDP model, the principle of fairness for water users is added to the objective function, which can reduce the contradictions between government and water users, as well as among different water users. This model is then applied to Handan City, Hebei Province, China, where water volume is small and uneven, to pursue the maximization for social benefits, overall satisfaction of water users, and environmental benefits. Finally, the FIDP model is compared with the traditional FILP

model, to prove the dynamic superiority of the proposed model with stage changes. It is expected that this model would be helpful to optimize the allocation of regional water resources under uncertainties and dynamics to reduce water shortage and conflict, and promote sustainable development of local society and economy.

This paper is composed of the following parts. Section 2 expounds the generation process of the general FIDP method. Section 3 puts forward a specific FIDP model suitable for Handan city. Section 4 presents the result analysis, which briefly expounds the water consumption characteristics of users at different stages of the planning year, and then compares the proposed FIDP model with FILP model. Section 5 is the summary of this study.


**5. Digital rock physics**

Digital rock physics is based on the concept "image and compute," image rock at the pore scale (**Figure 19**) and then simulate in the computer various processes in such an image to arrive at a desired rock property. These simulations include viscous fluid flow to arrive at hydraulic permeability, electrical charge flow to arrive at electrical resistivity, as well as elastic deformation to arrive at the elastic moduli and velocities.

The advantage of such digital approach is that the same sample can be reused multiple times, unlike in physical experiments where a sample is altered after every test; the sample can be digitally altered by, e.g., introducing digenetic cementation, which is hardly possible in physical experiments, as well as subsampling of a digital volume to investigate how various rock properties vary within the volume and how relations between rock properties depend on the spatial scale of investigation.

**Figure 19.**

*Segmented digital images of loose sand (porosity about 30%), sandstone (porosity about 20%), and carbonate (porosity about 15%) showing the mineral matrix and pores.

Although the aforementioned concept is simple, its implementation is not. First, the imaging has to be conducted at the appropriate scale and resolution to reveal the salient features of natural rock relevant to the process under examination. Second, the image has to be segmented to separate minerals from pores and segregate various minerals within the solid matrix, as well as fluid phases inside the pores. Third, powerful computational engines have to be utilized and verified to simulate processes relevant to the physical experiment.

In spite of these complexities, during the last decade, DRP has emerged as a powerful technique complementing (if not replacing) physical testing, mostly due to the recent advances in imaging hardware and image processing and computational software, the latter combined with steadily improving computational power. Not only DRP has become a novel research tool in academia and national labs, but is has also been adopted by leading oil and service companies.

There is one more inherent feature of DRP that needs to be accounted for. Porescale rock images are only a few mm in size, and the higher the resolution needed to revel the salient features, the smaller the field of view. At the same time, these computational results have to be relevant at much larger spatial scales of feet for wireline measurement interpretations in the well or tens and hundreds of feet in seismic prospecting. Even such basic property as porosity may be different if measured on an inch-sized sample an on mm-sized fragment of the same sample.

One way out of this conundrum is instead of directly comparing data points generated by different methods of measurement, compare trends formed by such data points, such as permeability versus porosity trends. Dvorkin et al. [18] show that such trends are often hidden inside a very small digital sample and can be derived by subsampling it. Moreover, these computational trends often match

**Figure 20.** *Illustration of the subsampling approach.*

**Figure 21.**

*Permeability versus porosity in Fontainebleau sandstone. Left: Laboratory data matched with a Kozeny-Carman theoretical curve. Right: Multiple permeability versus porosity data points computed from a few digital Fontainebleau samples and subsamples thereof (adopted from Dvorkin et al. [18]).*

**Figure 22.** *Formation factor versus porosity computed on carbonate cuttings. The curves are from Archie's equation with the cementation exponent m 2.0, 2.5, and 3.0 (bottom to top) (adopted from Dvorkin et al. [18]).*

relevant physical trends and/or theoretical rock physics transforms, hence validating computational results and making them relevant at much coarser spatial scales.

The approach is to subsample a digital volume into 23 , 33 , or 43 subvolumes (**Figure 20**) and then compute the desired property pairs (e.g., porosity and permeability) on each of these subvolumes. Very often, the property pairs thus computed form a meaningful trend supported by physical measurements and/or theories (see examples in **Figures 21**–**23**). We can call this subsampling approach "to see the rock in a grain of sand."

These results open ways to a meaningful utilization of DRP in research and industry. Publications related to DRP are many and the number is growing. We refer the reader to Kameda and Dvorkin [19], Dvorkin et al. [20], Dvorkin and Derzhi [21], and Andra et al. [22, 23].

### **Figure 23.**

Vp *versus porosity for Fontainebleau sandstone as computed from a few digital samples and subsamples thereof (squares). Gray circles are from laboratory measurements of dry samples. The curve is from the stiff-sand model (adopted from Dvorkin et al. [18]).*