Can you make sound out of light? In his presentation, Carl Meinhart answers this question by starting small, with photons and phonons. The idea is that when an infrared photon interacts with matter in some manner, it could create a Stokes’-shifted photon with a lower energy level. Simultaneously, the excess energy from the shift could generate an acoustic phonon. In this way, light can generate acoustics. But, as Meinhart notes in the keynote video, “it’s kind of a chicken-and-egg [scenario]; you need the acoustics and this scattered light to create each other, so they have to exist simultaneously.”

*From the video: Carl Meinhart discusses a theory behind converting light into acoustics.*

While the idea was originally predicted in the 1920s as *Brillouin scattering*, it wasn’t observed until the 1960s. Modern researchers can now turn to the COMSOL® software to analyze this theory and all of the relevant multiphysics phenomena. For a specific photonics example, Meinhart examines an innovative design from the Vahala Research Group at Caltech, a pioneer in this field. The Vahala Research Group designed an optical ring that uses whispering gallery modes for the ring instead of guided waveguides. Meinhart explains that when simulating this kind of device, “it’s very important to design the optics and the acoustics simultaneously,” a task that can be achieved with multiphysics simulation.

Through their research, the team found that their design has a very high Q factor. Research like this indicates that very sensitive high-Q resonators can be built by combining photons, phonons, and the concept of Brillouin scattering.

To try this sort of simulation yourself, download the example Meinhart mentions in his presentation, the Optical Ring Resonator Notch Filter tutorial.

Next, Meinhart turns to an industry example: maximizing the speed of a microfluidic valve. When looking to increase speed, a researcher’s first move is often to decrease inertia by making their design light and small. However, physical prototypes of small devices like microfluidic valves are expensive and time consuming to create and difficult to measure experimentally.

Instead, to analyze microfluidic devices, researchers can use the COMSOL Multiphysics® software, which Meinhart states is “an invaluable tool for this process” because “the only way you can really visualize what’s going on is through numerical simulation.”

*From the video: Carl Meinhart shares the example of a magnetically actuated microfluidic valve (left) and its approximate real-world size (right).*

For a concrete example, Meinhart considers a microfluidic valve being commercialized by Owl Biomedical, Inc. To increase their microvalve’s speed, the group tried using magnetic materials and thin silicon, which bends well and is a high-Q material. The resulting magnetically actuated device can be evaluated by importing the complicated geometry into COMSOL Multiphysics® using a product like LiveLink™ *for* SOLIDWORKS®. Then, researchers can analyze the design by combining nonlinear magnetics, fluid-structure interaction, and particle tracing simulation studies.

Initial results revealed that this microvalve design contained nonoptimal flow patterns. But, by using simulation to modify the shape over many iterations, researchers can balance the spring forces and optimize the flow and opening and closing speeds. The result? An incredibly fast microfluidic valve design that, when used to create a cell sorter, can sort 55,000 cells in 1 second or 200 million cells per hour. This optimized design has the potential to revolutionize cell sorting through Owl Biomedical’s cell sorter.

To learn more about how Carl Meinhart uses multiphysics simulation to study transport processes in photonics and microfluidics, watch the video at the top of this post.

*SOLIDWORKS is a registered trademark of Dassault Systèmes SolidWorks Corp.*

Echologics provides specialized services in water loss management, leak detection, and pipe condition assessment. They developed a permanent leak detection system for pipe networks, using acoustic technology. With this solution, Sebastien says, “the pipes can talk to you.”

The location of a leak is measured using the time delay between signals captured with two sensors placed on the pipe. The time delay is determined using the correlation function. This technique also requires knowledge of the mechanical behavior of the pipe and the propagation speed of acoustic waves to accurately locate the leak. To solve this problem, Sebastien created an app using the Application Builder, a built-in tool in the COMSOL Multiphysics® software, to find the exact location of pipe leaks.

He explains that the app is advantageous for Echologics because its user interface is designed for ease of use in the field. This includes app dimensions that could easily fit on a tablet device when accessed with the COMSOL Server™ product, for instance. This is particularly useful for Echologics, as their field engineers travel extensively.

With apps, engineers at Echologics can easily run and rerun analyses. For example, an engineer can predict a leak location in a pipe using the app and contact the client to tell them where the leak is located. If the client recently replaced that segment of the pipe with a different material, for example, the engineer can rerun the analysis through the app and provide the exact leak location based on the new information. This enables them to quickly respond to the customer with an updated location.

During his keynote talk, Sebastien discussed how Echologics designed their app so that users can easily navigate its interface. By separating the app into five tabs, field engineers only have to calculate the information they need. For example, if an engineer using the app has already measured the speed of sound in a certain pipe segment, they don’t need to use the *Speed Prediction* tab in the app. Instead, they can simply input the measured speed in the *Leak Location* tab that calculates the results.

*From the video: Sebastien Perrier demonstrates the custom app built by Echologics for predicting the location of a pipe leak.*

After all of the information is entered into the app, it reports the leak’s location in relation to the two closest sensors. Echologics’ app also includes a *Visualization* tab so that the app users can see their results. For Sebastien, the beauty of this app is that he can “visualize and confirm” when each sensor detects the leak.

Watch Sebastien Perrier give a demonstration of this app in the keynote video at the top of this post.

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Inkjet printers are widely used to provide high-resolution 2D printouts of digital images and text, where the printhead ejects small droplets of liquid from a nozzle onto a sheet of paper in a specific pattern. In addition to printing images onto paper, the inkjet technique is also common in 3D printing processes. The printhead moves over a certain type of powdered printing material and deposits a liquid through the nozzle onto the powder to effectively bind it into a predetermined 3D shape. (Tip: Check out the video on 3dprinting.com to see this process in action.) Inkjet printheads are also prevalent in life science applications for diagnosis, analysis, and drug discovery. The nozzles are used as part of a larger instrument to deposit microdroplets in a very precise fashion.

*An inkjet nozzle deposits an ink droplet, which travels through the air before reaching its target. The model was created using the COMSOL Multiphysics® software.*

No matter what device or machine relies on the inkjet printhead to deposit material, precision is crucial. Therefore, the quality of the end product hinges on the nozzle design.

The droplet size for an inkjet nozzle is a key design parameter. In order to produce the desired size, you need to optimize the design of the nozzle and the inkjet’s operating conditions. Rather than build nozzle prototypes and test them in a lab, you can use simulation software to understand the physics of the fluid ejection and determine the optimal design. COMSOL Multiphysics® is one such software package.

When you expand COMSOL Multiphysics with either the CFD or Microfluidics add-on module, you can create models that help you understand how the ink properties and nozzle pressure profile affect the droplet velocity and volume as well as the presence of satellite droplets.

*Model created using the level set method to track the interface between air and ink. The color plot around the droplet signifies the velocity magnitude in the air.*

What happens inside the inkjet nozzle when the liquid is emitted? First, the nozzle fills with fluid. Next, as more fluid enters the nozzle, the existing fluid is forced out of the nozzle. Finally, the injection is halted, which ultimately causes a droplet of liquid to “snap off”. Thanks to the force transmitted to the droplet by the fluid in the nozzle, it travels through the air until it reaches its target. In terms of physics, inside the nozzle, there is a single-phase fluid flow. When the liquid moves through the air, the flow becomes a two-phase flow.

We won’t go into the details of how to build this model here, because you can download the step-by-step instructions in the Application Gallery.

As the simulation specialist in your organization, you are a member of a small and rather exclusive group of people tasked with serving a larger pool of colleagues and customers who rely on your models to make important business and design decisions. Wouldn’t it be nice if these stakeholders could take on some of the work that goes into rerunning simulations for different parameter changes?

The COMSOL Multiphysics software comes with the built-in Application Builder, which enables you to wrap your sophisticated models in custom user interfaces. By building your own apps, you can give your colleagues or customers access to certain aspects of your models, while hiding other aspects that may be unnecessary to change and too complicated to expose. For example, suppose that your colleagues in design or manufacturing want to test the performance of an inkjet nozzle for different geometries and liquid properties. Instead of coming back to you each time they want a minor change to the underlying model, they can input different values in simple fields and click on a button to plot new simulation results in the app you provide them. Since they can run their own analyses, your time can be spent on new projects, models, and apps.

To show you what we mean — and to inspire you to make your own apps — we have made a demo app based on our inkjet tutorial model. In this example, the app user can analyze various nozzle designs to see which version produces the ideal droplet size. Contact angle, surface tension, viscosity, and liquid density are all taken into account in the app. As you can see in the screenshot below, an app user can adapt the nozzle shape and operation by changing different input parameters.

*An example of what an inkjet printhead design app might look like. In this demo app, users can modify liquid properties, the model geometry, and simulation time intervals.*

When you build apps, you can empower other stakeholders to make better decisions faster without actually giving them access to your full underlying model. The model simply powers the app and you, as the app designer, decide what inputs the users can modify. Your original model file stays safely untouched in your care, but a variety of results are accessible by those who rely on them most.

Get started by downloading the .mph file and accompanying documentation for the tutorial model and demo app from the Application Gallery.

All you need to download the documentation is a COMSOL Access account. To get the .mph file, you will also need a valid COMSOL Multiphysics® software license or trial. Note that you can access these files directly within the product as well, via the Application Libraries.

- Watch a keynote video on industrial inkjet printheads from the COMSOL Conference 2014
- Get an introduction to modeling separated three-phase flow with COMSOL Multiphysics
- Learn about modeling piezoelectric actuators

Eli Lilly and Company leverages first principles — the fundamental rules governing the behavior of a system or process — in the design and development of their pharmaceutical products. First principles thinking enables the company’s design engineers to predict and explain why, for a specific set of circumstances, they get one type of behavior and not another.

The approach, as Bernard McGarvey explains, involves identifying a specific design decision and applying a first principle to a generic model of the design. After applying computational methods, such as a simulation in the COMSOL Multiphysics® software, the generic solution that is found is translated into a more specific solution to the original design challenge.

*Eli Lilly’s first principles approach to designing pharmaceutical products. Reproduction based on Bernard McGarvey’s keynote talk from the COMSOL Conference 2016 Boston.*

Using first principles when designing a product makes the entire process more efficient and effective. This is because there is no need to justify the first principles involved — such as the ideal gas law or the Navier-Stokes equations — as they are already proven. McGarvey summarizes the benefits of this process, saying: “You’re taking a specific situation and, by taking advantage of its generic principles, you make it efficient.”

In his keynote talk, Bernard McGarvey discusses a simple example of an insulin pen, one focus of Eli Lilly’s pharmaceutical development work. For this specific single-use autoinjector pen, two key design considerations are:

- Keeping the needle gauge small enough to deliver a large dose with minimal backpressure
- Keeping the needle gauge large enough to maintain patient comfort and reduce injection pain

The need for both patient comfort and device efficiency creates a natural opposition, which leads to a challenging problem for the engineers working on the design of these systems.

*From the video: Bernard McGarvey of Eli Lilly and Company discusses the needle-based drug delivery design.*

First principles for this problem is achieved by describing the physical parameters of this application through the Hagen-Poiseuille equation — what McGarvey refers to as “a design engineer’s worst nightmare.” He says this because one variable in the equation is particularly hard to control. This accounts for the delivery of a certain volume of a substance in a certain amount of time and, for the case of the needle design, any change in the needle’s inner diameter (ID) greatly affects the backpressure.

To address this challenge, a needle vendor presented them with an elegant solution: a tapered needle that might cut the required delivery force significantly, while not increasing the pain from the injection for the patient. Eli Lilly worked with the vendor to evaluate a tapered needle design instead of a straight cannula needle. This design reduces backpressure while maintaining the comfort level of the patient during injection. The company used COMSOL Multiphysics to investigate how much of a reduction in backpressure can be expected with the tapered design. They found a 40–50% decrease in backpressure is possible as compared to a straight cannula needle. This is a significant reduction in backpressure and provides Eli Lilly with a design option for future systems.

Want the full story of how Eli Lilly uses modeling and first principles thinking for product design and development? Watch the video of the keynote presentation at the top of this post.

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In fluid mechanics, pressure represents the force per unit area applied to a surface by a fluid. Using COMSOL Multiphysics, we can solve the governing equations for fluid flow, the Navier-Stokes equations, to determine the velocity and pressure fields that describe the flow.

There are two main ways that we can talk about pressure for CFD problems: absolute pressure and relative pressure.

Absolute pressure is the direct measurement of a fluid’s pressure against vacuum. For instance, if we measure the pressure outside on a typical day with a barometer, we see an absolute pressure reading of about 1 atm or 101.325 kPa, which is the atmospheric pressure at sea level. An absolute pressure of zero corresponds to vacuum.

*This barometer measures the outdoor pressure from 950 to 1050 mbar (1 mbar = 100 Pa). Image by Langspeed. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

Relative pressure refers to a fluid’s pressure with respect to a reference pressure level. Gauge pressure is the pressure measured relative to ambient pressure; i.e., the relative pressure using ambient pressure as a reference. Typically, relative pressure is used to characterize the pressure levels in closed systems. It can be measured using a manometer, which relates the internal pressure to the surrounding pressure.

*Manometers measuring the relative pressure in a pressure control station. Notice how the dials start at zero, which represents the system pressure equaling the reference pressure level. Image by Holmium — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

Absolute pressure and relative pressure are related according to:

*p _{A}* =

If we use vacuum as the reference pressure, then the absolute pressure and the relative pressure are equal. In most cases, the reference pressure is set to the atmospheric pressure, which usually is the ambient pressure.

Let’s connect these definitions of pressure to what we see in COMSOL Multiphysics. When we compute the solution to a fluid flow problem, the COMSOL® software solves for the components of velocity (u,v,w) and the relative pressure (p). As we explain later in this blog post, by using the relative pressure instead of the absolute pressure as the dependent variable, we can improve the accuracy of the description of pressure in our simulation. We can then use the values of relative pressure in the initial values and boundary conditions in the model, which we will see in the following example.

Let’s take a look at an example that illustrates how to use the variables for relative pressure and absolute pressure appropriately in a COMSOL Multiphysics model. To demonstrate these concepts, we use a simple model of air flowing into a channel with an inlet velocity of 1 m/s and exiting to an absolute outlet pressure of 1 atmosphere. The top and bottom boundaries represent the no-slip channel walls, except for two short inlet sections where we assume symmetry. The inlet sections are there to avoid inconsistent boundary conditions, which would be the case if we were to define a straight inlet velocity profile adjacent to a no-slip boundary.

*A schematic of a channel with air flowing through it.*

In this model, the variable name for relative pressure is p and the variable name for absolute pressure is spf.pA. In the settings for the *Laminar Flow* interface, we see that the dependent variables to be solved are the components of velocity (u,v,w) and the relative pressure (p).

*Settings window for the dependent variables.*

We can see in the figure below that the reference pressure level is set to 1[atm] by default. This reference pressure level is used to calculate the absolute pressure: spf.pA = p + spf.pref.

We also set the compressibility to *Weakly compressible flow*, which means that the density of air depends on temperature and reference pressure. To learn more about different compressibility settings, take a look at this previous blog post.

*The compressibility and reference pressure settings.*

Now we can specify our boundary conditions. At the inlet, we assign the normal velocity to be 1 m/s. For the initial conditions and outlet boundary condition, we need to enter the relative pressure, since we are using the default setting; i.e., to use a reference pressure. When we add the outlet condition, we see that the default value for the relative pressure is p = 0, which is equal to 1 atm in absolute pressure for the default reference pressure.

*Settings window for the boundary conditions, showing the relative pressure (left) and absolute pressure (right).*

You might be wondering, then, why the COMSOL® software calculates the variable for absolute pressure, spf.pA. The absolute pressure is used when calculating the density of a compressible fluid. For instance, if we navigate to the material properties for the air in our channel, we will see that the density is defined using the ideal gas law, where pA is the absolute pressure and T is the temperature. Since the ideal gas law is calculated based on the absolute pressure, we have to add the reference pressure to the relative pressure (p) to calculate the density. However, in this case, the relative pressure is such a small fraction of the total pressure (0.00025%; see below) that we may as well use the reference pressure to calculate density, which is what we get when using the *Weakly compressible flow* option. In systems with larger pressure variations, we can select the *Compressible flow* option.

*Defining density using the ideal gas law in the settings window.*

Now that we have defined the boundary conditions for our problem, we can compute the solution and visualize the velocity profile with streamlines.

*Velocity profile with streamlines and a vector plot of flow through a channel.*

We can also look at the pressure profile along the inlet (along the *y*-axis at the left vertical boundary). We can see in the plot below that the pressure variation along the inlet is around one tenth of a Pascal compared to the reference pressure, which is of the order of magnitude of 1·10^{5} Pa. This means that the reference pressure is about one million times larger than the variations in the inlet pressure!

*Relative pressure along the vertical inlet boundary.*

The default way to solve fluid flow problems in COMSOL Multiphysics is by using the relative pressure as a dependent variable and adding the reference pressure when an absolute pressure is required; for example, to compute the density of the fluid. This improves the accuracy of the description of the fluctuation of the pressure field around the reference pressure and the description of the gradients of the pressure field.

Let’s return to our channel example and calculate the pressure drop. If we use the *Line Average* feature to evaluate the relative pressure at the inlet, we will determine the pressure to be about p_{inlet} = 0.26 Pa.

Now, imagine that we solved our problem using absolute pressure instead. The absolute inlet pressure would be 101,325.26 Pa and the absolute outlet pressure would be 101,325.00 Pa. The relative change of the pressure field between the inlet and the outlet is 0.000253814%. As shown in the inlet pressure plot, the variations at the inlet are even smaller: one millionth of the absolute pressure. This is a very small relative change to look for when we solve the equations.

Since we are solving this problem numerically, we are approximating the real pressure field. This is defined at every point, with a numerical approximation defined at a relatively few number of points. We introduce a numerical error due to truncation and interpolation errors. In addition, the numerical equations can only be solved to a given tolerance. This boils down to a relative error in the computed numerical approximation of the pressure field that would disturb the relatively small fluctuations that we are looking for. By using a reference pressure, we can better resolve the gradients of pressure and the fluctuations around atmospheric pressure at viable values for the relative numerical error in the pressure field, compared to the case with the absolute pressure.

Now that we understand why the COMSOL Multiphysics software uses the relative pressure to solve fluid flow problems, we can also appreciate the importance of specifying an accurate reference pressure level. Obviously, a reference pressure level of 1 atm is appropriate for systems working around atmospheric pressure. For very high or low pressure systems, we should use a reference pressure level that matches the pressures expected in the flow.

For instance, in a traditional incandescent light bulb, lower-pressure argon is housed in a glass bulb to prevent oxidation of the filament. In the Application Gallery tutorial model, we see that the reference pressure level changes to match the pressure of this gas (50 kPa). In the *Initial Values* section, the relative pressure is set to p = 0, which corresponds to an absolute pressure of 50 kPa due to the updated reference pressure level.

*A simulation of the free convection of argon within a light bulb.*

For very low pressure systems, it’s important to check that fluid can still be considered a continuum. You can calculate the Knudsen number, which is the ratio of the mean free path to the length scale of the device, to determine if the flow is best solved using molecular flow physics.

In today’s blog post, we explained how the absolute pressure of a system is a direct measurement of pressure, while the relative pressure describes the pressure with respect to a reference pressure level.

COMSOL Multiphysics solves CFD problems using relative pressure to improve the numerical accuracy of the pressure field. This means that the initial conditions and boundary condition should be defined using relative pressure values. However, when calculating the density of a gas, the absolute pressure is used and the reference pressure is added to the relative pressure automatically. For high or low pressure systems, the reference pressure level should be changed to match the average pressure in the system.

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Natural convection is a type of transport that is induced by buoyancy in a fluid. This buoyancy is in turn caused by the fluid’s variations in density with temperature or composition.

You may be familiar with the concept of natural convection in indoor climate systems. In this scenario, hot air rises to the ceiling close to heat sources and cool air sinks to the floor close to cold surfaces, such as the windows during winter.

Electronics cooling is another type of process that often depends on natural convection in order to work. For example, we do not want to use noisy fans to cool the amplifiers and TVs in home cinema systems. Electronic devices that need to operate in quiet environments often rely on natural convection to circulate air over their built-in heat sinks.

*Free convection around a splayed pin fin heat sink that is heated from below. The animation shows the value of the velocity in the air around the heat sink.*

Less obvious natural convection problems are found in industries such as chemical and food processing. Environmental sciences and meteorology also involve natural convection problems, as scientists and engineers try to predict and understand transport in air and water.

In all of the cases mentioned above, it is important for engineers and scientists to understand and design systems to control natural convection. In this context, mathematical modeling is the perfect tool. In the latest version of COMSOL Multiphysics, it is easier to define and solve problems involving natural convection. We have introduced a number of new capabilities for this purpose.

The *Weakly compressible flow* option for the fluid flow interfaces neglects the influence of pressure waves, which are seldom important in natural convection. It allows for larger time steps and shorter solution times for natural convection problems.

The *Incompressible flow* option with the Boussinesq approximation for buoyancy-driven flow linearizes density using a coefficient of thermal expansion. This option includes the density variation only as a volume force in the momentum equations. This implies an even larger simplification compared to the *Weakly compressible flow* option, but it still gives an excellent and efficient description for systems with small density variations. This simplification is almost always valid for free convection in water subjected to small temperature differences.

The *Gravity* feature makes it easy to define a reference point for hydrostatic pressure and also automatically accounts for hydrostatic pressure variations at vertical boundaries.

Let’s learn more about these new features and how you can apply them in your natural convection modeling problems.

The *Nonisothermal Flow* interface includes the *Weakly compressible flow* option, which simplifies flow problems by neglecting density variations with respect to pressure. This option also eliminates the description of pressure waves, which requires a dense mesh and small time steps to resolve, thus also a relatively long computation time. In natural convection, there is usually very little influence of pressure waves, which means that we lose very little fidelity in the model’s description of reality by making this simplification.

The continuity equation for a compressible fluid looks as follows:

(1)

\[\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot \left( {\rho {\mathbf{u}}} \right) = 0\]

where *ρ* denotes density and **u** is the velocity vector.

For a gas, density is proportional to pressure and temperature. For example, for an ideal gas, this gives:

(2)

\[\frac{{\partial \rho }}{{\partial t}} = \frac{{\partial p}}{{\partial t}}\frac{M}{{RT}} - \frac{{\partial T}}{{\partial t}}\frac{{pM}}{{R{T^2}}}\]

If we neglect the dynamic effects of the density changes, we get:

(3)

\[\frac{{\partial \rho }}{{\partial t}} = - \frac{{\partial T}}{{\partial t}}\frac{{pM}}{{R{T^2}}}\]

If we use the expression for the density of an ideal gas and neglect the influence of pressure on density, we obtain the following continuity equation:

(4)

\[\frac{{\partial T}}{{\partial t}} + \nabla T \cdot {\mathbf{u}} - T\nabla \cdot {\mathbf{u}} = 0\]

This means that variations of density are taken into account only in terms of temperature variations. The variations in density may cause an expansion of the fluid, but the direct dynamic effects of those expansions on the pressure field are neglected when using the *Weakly compressible flow* settings.

In addition to the density expression in the continuity equation, selecting the gravity check box in the settings for the fluid flow interface adds a volume force in the momentum equation in the direction of gravity. By default, this is the negative *z*-direction. This force looks as follows:

(5)

\[{F_z} = -g\rho \left( T \right)\]

where density, ρ, is a function of temperature.

For an ideal gas, density is inversely proportional to temperature.

We can find the settings for the *Weakly compressible flow* option by selecting the *Nonisothermal Flow* interface or the *Conjugate Heat Transfer* interface. Selecting the *Fluid Flow* interface node in the Model Builder shows the settings window below. Selecting the *Weakly compressible flow* option removes the dependency between pressure and density, while selecting gravity automatically adds the volume force of buoyancy in the momentum equation.

*Settings window for the fluid flow interface showing the* Weakly compressible flow *option and gravity feature.*

The figure below shows the flow between two vertically positioned circuit boards. Only the unit cell of one circuit board is shown in the figure. The second circuit board is placed just in front, with its back facing the board that is visible. The flow is completely driven by buoyancy; i.e., there is no fan.

The flow rate at the inlet is around 0.2 m/s and around 0.3 m/s at the outlet. There is no inlet of air from the sides, which means that the difference in flow rate is due to the expansion caused by the increase in temperature along the height of the channel between the circuit boards.

*Buoyancy-driven flow between vertical circuit boards. The expansion is seen in the color legend for the arrows, where the flow velocity is around 0.2 m/s at the inlet and 0.3 m/s at the outlet.*

When the changes in density are negligible in terms of the influence of expansion on the velocity field, we can use the *Incompressible flow* option with the Boussinesq approximation for natural convection. This implies that the continuity equation is simplified even more than with the *Weakly compressible flow* option by treating the fluid as incompressible. In this case, the continuity equation becomes as follows:

(6)

\[\nabla \cdot {\mathbf{u}} = 0\]

Instead, a small change in density is accounted for in a volume force, which is introduced in the momentum equation in the opposite direction of gravity; by default, the *z*-direction. The small change in density is obtained by linearizing the fluid’s density at a reference temperature. The *z*-component of the volume force becomes as follows:

(7)

\[{F_z} = g{\rho _{{\text{ref}}}}\alpha \Delta T\]

Where *g* is the gravity constant, is the density at a given reference temperature, α is the coefficient of thermal expansion of the fluid, and Δ*T* is the temperature difference measured against the reference temperature.

The advantage of using the Boussinesq approximation for buoyancy-driven flow is that the nonlinearities in the fluid flow equations are reduced and the problem becomes easier to solve numerically, requiring less iterations and allowing for larger time steps for time-dependent problems.

A typical example where the Boussinesq approximation can give a realistic description of the flow is for the modeling of liquid water subjected to relatively small temperature differences. The figure below shows natural convection in a glass of water heated from below. Here, we obtain a very complex flow pattern with an upward flow close to the middle and bottom of the glass and with downward flows between the vertical walls and the middle.

*Natural convection in a glass of water. The plot shows the velocity field in the glass and the temperature distribution in the walls of the glass.*

We can obtain the *Incompressible flow* option with the Boussinesq approximation for buoyancy-driven flow by selecting the settings shown in the figure below for the fluid flow interfaces in COMSOL Multiphysics.

*Selecting the* Incompressible flow *option, Gravity feature, and reduced pressure gives the Boussinesq approximation for a natural convection problem.*

When modeling fully compressible flow, the pressure’s time dependency is included in the continuity equation, since density is a function of pressure for compressible fluids. This also means that it is usually sufficient to include an initial condition for the pressure in order to get a well-posed problem, even when we do not prescribe pressure at a boundary.

For weakly compressible and incompressible flows, the time-dependent pressure term in the continuity equation is neglected according to the discussions above. If there are no boundary conditions that set the pressure, the pressure field becomes undetermined, unless we set it in some point in the domain.

In COMSOL Multiphysics, we can use a so-called pressure point constraint in order to avoid an undetermined pressure field. The absence of a reference pressure point is often the source of problems with convergence when solving natural convection problems.

*The settings for the pressure point constraint in the water glass example.*

The equations that describe natural convection usually involve the momentum equation, the continuity equation, and the energy transport or mass transport equation. If buoyancy is driven by temperature differences, then the energy equation is fully coupled with the fluid flow equations (the Navier-Stokes equations). For natural convection, this coupling is fairly tight. This means that the most robust way to solve the equations is to use the fully coupled solver in COMSOL Multiphysics.

*The solver branch in the model tree with the fully coupled solver option.*

For very large problems, a segregated approach may be a preferable option. For example, if there are many chemical species and if buoyancy is caused by variations in density due to chemical composition, then a segregated approach may be the only viable option for getting decent memory consumption in the solution process.

I would like to end this blog post with one more natural convection problem. I often think about natural convection when I smoke a cigar. Although I do not want to promote smoking, my favorite natural convection problem is the smoke from a cigar on a cold winter day. The figure below shows a lighted cigar resting on an ashtray with the flow distribution caused by the heat from combustion.

*Natural convection (with a small forced component) around a lighted cigar resting on an ashtray.*

Some of the flow caused by the lighted cigar is actually forced convection, since a large part of the tobacco goes to smoke, changing the density from around 500 to 1000 kg/m^{3} down to 1 kg/m^{3}. This can be described as an inlet for the flow at the boundary between the ash and the air surrounding the cigar.

- Learn more about natural convection and fluid flow modeling on the COMSOL Blog:

After a TDD patch is applied to the skin, it continuously delivers a low-level drug dose. This is beneficial when dealing with drugs with a rapid onset and short duration, like the pain medication fentanyl, because the patch releases the drug gradually over time. TDD patches are also more effective and convenient than traditional drug delivery methods.

*Transdermal drug delivery patch. Image by RegBarc — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

Designing TDD patches is a challenge for many reasons. One design challenge is that the skin is a little *too* good at protecting the body from foreign substances. To get drugs past the body’s natural defenses, the patch needs to contain a chemical permeation enhancer. The stronger the enhancer, the better it is at transporting the drug. But a stronger permeation enhancer is also more likely to irritate the skin. Therefore, when designing an optimal TDD patch, we must consider both its effectiveness and patient comfort.

To find a balance between these factors, we can analyze the drug diffusion process in a TDD patch with simulation. Alireza Kermani and Nagi Elabbasi from Veryst Engineering, a COMSOL Certified Consultant, demonstrated this by modeling a TDD patch in COMSOL Multiphysics and comparing the results to those from an experiment.

To model the TDD patch, the researchers at Veryst Engineering first set up a 2D axisymmetric model. For their model, the skin and patch both have a thickness of 50.8 μm and the radius of the patch is 0.9 cm. The team also assumed that the drug and enhancer dissolve uniformly.

*A cross section of Veryst Engineering’s TDD patch model, displaying the normalized initial drug concentration (not to scale). Image by A. Kermani and N. Elabbasi and taken with permission from their COMSOL Conference 2016 Boston paper.*

The team then set up the appropriate physics and boundary conditions to accurately model the drug’s movement from the patch into the skin. They used a *pointwise constraint* in COMSOL Multiphysics to enforce the flux continuity and partitioning of the drug and enhancer at the interface. They also accounted for the nonlinear diffusion caused by the coupling of the drug and enhancer and specified the drug’s diffusion, which varies linearly with the enhancer’s concentration. Since drugs do not exit the top or sides of TDD patches, the team added boundary conditions to their model to stop the drug flux in those areas.

The lower boundary of the skin acts as a sink for both the drug and enhancer. Therefore, the concentration was set to zero at that boundary. This represents the drug and enhancer leaving the skin. Using a sink boundary condition means that the concentration is zero.

Next to the lower boundary of the skin is the dermis layer, which is not modeled in this research. However, the researchers still considered its effect. The dermis layer undergoes blood microcirculation, so when a drug reaches the lower boundary of the skin, it is removed via microcirculation and transferred to the rest of the body. The team assumed that the concentration of the drug or enhancer is zero at the skin’s lower boundary and added a sink boundary condition.

The group at Veryst Engineering tested their model to see how it performed in three different cases:

- When there is no permeation enhancer in the patch
- When the permeation enhancer’s initial concentration is 0.08 g/cm
^{3} - When the permeation enhancer’s initial concentration is 0.12 g/cm
^{3}

In all cases, the drug had an initial concentration of 0.06 g/cm^{3}.

*The drug diffusion process for the 2D model of the TDD patch. Animation courtesy of Veryst.*

The simulation results show that the drug flux increases when there’s a permeation enhancer present, especially when it has a higher initial concentration. The plot below shows the normalized drug flux over time for the three enhancer concentrations.

*The normalized drug flux in the skin for the three levels of permeation enhancer concentration. Image courtesy of Veryst.*

The Veryst Engineering team validated their model by comparing the results to a previous experiment. The experiment used the drug fentanyl at a concentration of 0.06 g/cm^{3} and the permeation enhancer lauryl pyroglutamate at an initial concentration of 0.12 g/cm^{3}.

Veryst’s model accounts for the maximum flux value in the TDD patch as well as how this value increases with a higher concentration of enhancer. However, the model doesn’t account for the flux’s broad peak and quick decay, which are investigated in the experiment. The simulation also does not predict the drug flux accurately over long periods of time.

*Comparison of the simulation results and experimental results for different permeation enhancer concentrations. Image courtesy of Veryst.*

The engineers at Veryst suspect several factors may contribute to the difference in results. For instance, the Fickian diffusion model does not represent drug diffusion over a long period of time. Also, the assumption that drug diffusion increases linearly with a higher enhancer concentration is too simple to describe the drug diffusion process, which is time dependent. This means that the linear increase is not accurate for longer time periods.

Other components of the model, such as the boundary condition at the skin’s bottom layer, also need further investigation. A sink boundary condition for the enhancer may not be the right approach, since the solubility of the enhancer is not significant in the skin. On the other end of the spectrum, the team could have assumed that the enhancer has zero flux at the bottom boundary of the skin. The Zero Flux boundary condition increases the concentration of the enhancer in the skin, therefore increasing the drug flux. The true approach to describing this boundary is neither of these boundary conditions, but instead, something in between.

Another aspect to consider moving forward is the hydration in the skin and patch. The skin sample in the experiment is fully hydrated and when the patch is applied, the patch hydration level increases. The patch begins to swell, changing the concentrations of the drug and enhancer. This effect is not accounted for in the model.

The model designed by the team at Veryst Engineering demonstrates that, with additional information, it’s possible to simulate TDD patches in COMSOL Multiphysics. According to the team, the COMSOL software made it easy to include the continuity of flux, partitioning of the drug and enhancer, and the effect of coupling the drug’s diffusion coefficient with the concentration of the enhancer.

To get an accurate representation of the diffusion process, more research needs to be done on selecting the appropriate boundary conditions as well as choosing the correct factors to investigate, including hydration.

After building an optimized TDD patch model for future research, it is possible to couple it with other types of physics. For example, we can account for heat transfer in the patch model to determine how heat affects the drug diffusion process.

- Learn more about Veryst Engineering
- Read the full paper from the COMSOL Conference 2016 Boston: Transdermal Drug Delivery with Permeation Enhancer
- Browse the COMSOL Blog to see more about how Veryst uses simulation:

Wedged between layers of porous or fractured rock formations are two of the natural resources that we rely on most: oil and gas. Recovering resources from these areas of accumulation, known as reservoirs, is a key point of focus for the oil and gas industry. For an efficient recovery process, engineers working in this field can design wells that effectively recover the oil and gas that fills the reservoir formations.

*Wells are drilled at specific locations to recover valuable resources. Image by Caitlin Heryford. Licensed under CC BY 2.0, via Flickr Creative Commons.*

One way to establish flow communication between the reservoir and the well is through the use of perforations. After drilling a well and adding a casing to provide further protection and stability, holes can be shot through the casing and cement at specific locations via a perforating gun. This produces a path that allows fluid to flow from the reservoir to the well. Perforated wells are particularly beneficial in the sense that the perforations can be sealed individually, providing greater control over the well itself.

*An image showing perforations in steel. Image by Mr Thinktank. Licensed under CC BY 2.0, via Flickr Creative Commons.*

The safety and productivity of a perforated well is greatly impacted by its design, from the number of perforations that are included to their size and orientation. Therefore, to prevent damage to equipment and injuries to personnel, as well as to maximize the production of hydrocarbons, it’s important to first analyze the fluid flow in the well and obtain a better understanding of the fluid intake at each perforation. Numerical modeling apps, as we’ll highlight in the next section, offer an optimized solution for doing just that.

Our Perforated Well demo app is based on a model that simulates Darcy flow within a perforated wellbore (the hole that forms the well). Modeling this process requires performing a full 3D analysis in the COMSOL Multiphysics® software. The plot below shows the pressure at the wellbore and pumping rates for a given set of parameters.

*Simulation plot depicting the pressure at the wellbore as well as the pumping rates. Note that annotations of values can be directly included in the plot.*

Say a user now wants to compare these initial results with the results from a new set of design parameters. Rather than turning to simulation experts to run further rounds of tests, these users now have the power to run their own analyses with an easy-to-use app. By hiding the complexity of the underlying model behind a simplified interface, apps enable those with little knowledge of simulation processes to achieve accurate simulation results quickly.

To get a better idea of the functionality that can be included in such an app, let’s go back to our perforated well example. This demo app provides users with the option to modify a number of parameters. With regards to the well, they can specify its radius, the number of perforations, the perforation radius and length, and the pressure at the perforations. Users can also modify the viscosity and density of the fluid as well as the radius, porosity, thickness, permeability, and pressure of the reservoir. Keep in mind that when creating an app of your own, you have control over which parameters users have the ability to change — as well as the structure and organization of the app.

*The user interface (UI) of the Perforated Well app.*

As highlighted in the screenshot above, the results generated by the app are shown in the Graphics window. In addition to a plot of the pressure at the perforations and the pumping rates (as referenced earlier), the app also plots the flow field and pressure drop in the reservoir. With all of the underlying model’s physics included in the app and controlled parameters, users can be confident in the reliability and usefulness of their simulation results.

Perforated wells offer a viable approach for accessing oil and gas trapped in reservoirs. Ensuring the safety and productivity of these wells is a more streamlined process thanks to simulation apps, which empower a larger group of people to run their own simulation tests. Extending the scope of simulation capabilities paves the way for further advancements in the field of well design, which could potentially help recover oil and gas from even more complex rock formations with smaller permeability.

- Get a quick introduction on how to turn your COMSOL Multiphysics model into an app by watching this video
- Gain inspiration and guidance for building your own apps by browsing blog posts in our Applications category

Humans have used drying as a method for preserving food since ancient times. Since then, the drying process has expanded from open-air drying or sun drying to other drying techniques, such as solar drying, freeze drying, and vacuum drying. Drying is also a key process in many other application areas, from the pharmaceutical industry to plastics.

Today, we’ll focus on the chemical process of vacuum drying, which is particularly useful when drying heat-sensitive materials such as food and pharmaceutical drugs. Vacuum dryers, commonly called vacuum ovens in the pharmaceutical industry, also offer other benefits. Because they require lower temperatures to operate, vacuum dryers use less energy and therefore, reduce costs. They also recover solvents and avoid oxidation.

*A rotary vacuum dryer. Image by Matylda Sęk — Own Work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
*

Vacuum dryers remove water and organic solvents from a wet powder. The dryer operates by reducing the pressure around a liquid in a vacuum, thereby decreasing the liquid’s boiling point and increasing the evaporation rate. As a result, the liquid dries at a quicker rate — another major benefit of this process.

For vacuum drying to be effective, we need to decrease drying times without harming the products, which means that we need to maintain a strict control of the operating conditions. To balance these goals and to understand how operating conditions influence the product, you can use the multiphysics modeling capabilities of COMSOL Multiphysics.

Today, we’ll analyze the vacuum drying process of a Nutsche filter-dryer model. The dryer works by heating a wet cake from the bottom and the side walls of a container and by decreasing the pressure in the gas phase on the top of the cake. This example is based on a paper published by Murru et al. (Ref. 1 in the model documentation).

Let’s start by taking a closer look at our model. The vacuum dryer is comprised of a cylindrical drum filled with wet cake, which consists of three different phases: solid powder particulates, a liquid solvent, and a gas. As such, the cake’s material properties need to include the properties of all three individual phases, which vary depending on the proportion of each phase in the cake. The portion of each phase is determined by the volume fraction, which is one of our modeled variables.

The cake is modeled as a rectangular geometry with a radius of 40 cm and height of 10 cm in a 2D axisymmetric component. At the top, our model is exposed to a low-pressure head space. Meanwhile, heat flux boundary conditions at the filter dryer’s side and bottom boundaries account for a 60°C heating fluid.

*The vacuum drying process in an axisymmetric Nutsche filter dryer.*

Moving on, our tutorial combines evaporation and heat transfer modeling in order to study the cake’s liquid phase profiles and temperature. We calculate the cake’s solvent volume fraction with the *Coefficient Form PDE* interface and simulate heat transfer with the *Heat Transfer in Solids* interface. To solve the moisture transport in porous media, we use a predefined multiphysics interface in the Heat Transfer Module. We also include solvent evaporation by using both a heat-sink and mass-sink term and approximate the solvent transport as a diffusion process.

Our model makes the following assumptions:

- Evaporation stops when the value of the liquid phase reaches zero, indicating that the liquid is fully evaporated.
- Evaporation stops when the local vapor pressure is less than the head-space water vapor pressure, indicating that evaporation has no driving force.
- Diffusion in the solvent stops when the liquid phase’s volume fraction dips below the critical value.

In these situations, we can use a step function to smoothly ramp both the evaporation rate and diffusion coefficient down to zero.

We see that our simulation results are as predicted. Let’s start by examining our analysis of the cake after 30 hours have passed. As seen below, the cake’s temperature is close to that of the heating fluid (60°C) at both the side and bottom boundaries, and the liquid phase’s volume fraction is lowest near these heated boundaries and highest at the cake’s center. Additionally, the apparent moisture diffusivity is highest at the cake’s center and almost zero in places where the liquid phase has evaporated. Considering our model’s assumptions, these results are all expected.

*The cake’s temperature (left), volume fraction of the liquid phase (middle), and apparent moisture diffusivity (right) after 30 hours.*

Switching gears, let’s expand our timescale to look at the evaporation rate after 10, 20, and 30 hours. This study also yields expected results, since it shows evaporation beginning at the heated walls and decreasing when the amount of solvent at these boundaries lessens. During this process, the evaporation front shifts toward the cake’s center.

*The evaporation rate after 10 (left), 20 (middle), and 30 (right) hours.*

The quantitative results generated by our simulation study are in good agreement with previous research, confirming their validity. As such, we can use this model to accurately predict how dry a product is as a function of time. Using this information, we can minimize the amount of time that a product is exposed to elevated temperatures. Additionally, we can change the dryer’s size if we want to reduce the drying time when working with heat-sensitive products. Through multiphysics simulation, we can design more efficient and effective vacuum dryers for use in a variety of industries.

- Try it yourself: Download the Vacuum Drying tutorial featured in this blog post
- Check out these related blog posts:

Industrial mixers are a key element in many fields, from the pharmaceutical and food industries to consumer products and plastics. Further, the purpose of mixers can vary greatly. Mixers are not only used to combine elements and create homogeneous mixtures, but to also reduce the size of particles and generate chemical reactions.

*An industrial mixer. Image by Erikoinentunnus — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

Mixers are required for efficient and timely production as well as for producing a uniform product quality within a batch and between batches. In some cases, mixers are required for the safe operation of systems, for example, in exothermic reactions that may create hot spots and runaway reactions (explosions) under poor mixing. With modeling, we can run inexpensive and streamlined experiments with different mixer designs in order to optimize the mixing process, avoid poor product quality, and meet safety requirements.

To resolve these issues, you can turn to COMSOL Multiphysics, which provides you with the tools for testing a wide assortment of mixers. In the next section, we’ll discuss three different mixer design examples that speak to the versatility of COMSOL Multiphysics.

A typical batch mixer generally consists of two main components: a vessel and an impeller, both of which can vary in type and shape. Baffles can also be added to the device to improve the mixing by suppressing the bulk’s main vortex formation.

The importance of the baffles depends on the type of impeller. Radial impellers, for instance, require baffles to work. Otherwise, the solution will rotate like a merry-go-round and mixing will not be achieved. Here, the impellers will only create vertical mixing as the solution hits the walls of the vessel. Axial impellers, on the other hand, create a vertical mixing flow at the impeller, which means they do not require baffles to achieve mixing. However, axial impellers also have a radial component, so baffles can be used to increase radial mixing in axial impellers, if desired.

Let’s take a look at a mixer’s vessel, shown below, which is often modeled as either a vertical cylinder with a dish-shaped or flat bottom.

*Side views of a flat-bottom mixer (above) and a dished-bottom mixer (below).*

Within the vessel, the fluid is mixed by a rotating impeller. The rotation and design of the impeller determines the axial and radial direction in which the liquid is discharged. As such, impellers come in many different designs, enabling them to be used for a variety of different industrial purposes. Here, we will investigate a six-blade Rushton disc turbine, which is a radial impeller used for high-shear mixing, and a more general-purpose pitched-blade impeller, which is an axial impeller.

*A Rushton disc turbine with six blades (left) and a pitched-blade impeller with four blades (right).*

By combining these two common types of vessels with two types of impellers, we create two separate geometries (shown below) and three separate studies. All three studies use the *Frozen Rotor* study type and the *Rotating Machinery, Fluid Flow* interface.

The first study involves the laminar mixing of silicon oil in a baffled flat-bottom mixer that contains a Rushton turbine with six blades rotating at 40 rps. While we focus on the highest of three rotation rates in this example, you can easily adjust the rotation to simulate the slower rotation rates. This first example is based on a PhD thesis by M.J. Rice entitled *High Resolution Simulation of Laminar and Transitional Flows in a Mixing Vessel* (see Ref. 1 in the model documentation) and includes comparisons from the PhD thesis *Study of Viscous and Visco-elastic Flows with Reference to Laminar Stirred Vessels* by J. Hall (see Ref. 2 in the model documentation).

*Two mixer geometries, one combining a baffled flat-bottom mixer and a Rushton turbine (left) and one with a baffled dished-bottom mixer and a four-blade pitched impeller (right).*

Moving on, our next two examples deal with the turbulent mixing of water within a baffled dished-bottom mixer. This mixer contains a pitched four-blade impeller that rotates at 20 rpm. It’s possible to reduce the computational time required to solve these models by using periodicity and only simulating a quarter of the domain.

Our turbulent mixing examples enable you to explore how different models affect your results. Here, we compare a *k-epsilon (k-ε)* model, which has a quick convergence rate, to a *k-omega (k-ω)* model, which works better for flows with recirculation regions.

Let’s begin by looking at the velocity magnitude and in-plane velocity vectors for our three models. These results provide a general view of the circulation patterns in the mixing vessels for all three of our examples.

For our first mixer model, the laminar mixing example, we can see that the fluid is discharged radially outward by the Rushton turbine, creating two zonal vortices. The resulting compartmentalization phenomenon, which is common for radial impellers, is also displayed in our simulation results. This leads to mixing in the top and bottom vortices, albeit less intensely than inside each individual vortex.

*The velocity magnitude (*xz*-plane) and in-plane velocity vectors (*yz*-plane) for the laminar mixing example.*

On the other hand, the velocity magnitude and vector projection for the turbulent flow *k-ε* model indicate that the fluid is expelled axially and radially by the pitched-blade impeller. As a result, a large zonal vortex is generated from the top to the bottom of the vessel. Additionally, a small zonal vortex appears below the impeller, which can aggregate the heavy dispersed particles in this area.

*The velocity magnitude (*xz*-plane) and in-plane velocity vectors (*yz*-plane) for the *k-ε* turbulence model example.*

The third study reveals that the turbulent flow *k-ω* model has a large zonal vortex, similar to the *k-ε* example. However, this time, the core is more vertically stretched. For its part, the smaller zonal vortex located beneath the impeller is stretched in the radial direction. Another difference lies with the torque and power draw values, which are both higher than the *k-ε* model. While the *k-ω* model is a good model to use for these types of flows, we still need to determine if its results are actually more valid than the *k-ε* model. Comparing simulation results to experiments is, therefore, a necessary next step.

*The velocity magnitude and in-plane velocity vectors for the *k-ω* turbulence model example.*

Finally, our simulations reveal that all three examples generate good approximations for at least a few averaged flow quantities. Our results from the frozen rotor simulation for the laminar mixing study can be easily used as initial conditions for a new time-dependent study.

It’s easy to modify the mixer geometries presented here to fit a wide assortment of mixer designs and conditions. Simply change the parameters in the supplied model to alter the types of components and properties of the geometry. For further customization, you can also add your own subsequences into the mix. With this, you can create a customized model to fit your specific application.

For more information on how to improve your mixer simulations, check out the resources in the next section.

- Try out the tutorial featured in this blog post: Modular Mixer-Turbulent Mixing (k-omega)
- Read a few blog posts on simulating mixers in COMSOL Multiphysics:
- Watch this in-depth archived webinar: Simulating Mixers and Non-Ideal Reactors
- Download a related tutorial: Laminar Flow in a Baffled Stirred Mixer

When modeling thin fractures within a 3D porous matrix, you can efficiently describe their pressure field by modeling them as 2D objects via the *Fracture Flow* interface. Significant fracture flux calculation issues, however, may arise for systems of practical interest, such as hydraulic fractures contained within unconventional reservoirs. See how a hybrid approach overcomes such difficulties.

To model an actual fracture as a 2D object using the Subsurface Flow Module, you first have to solve for the pressure field (through a tangential form of Darcy’s law) within the internal surface representing the fracture’s lateral extent. You can then calculate, in principle, the corresponding fluid flux through the actual fracture cross section by multiplying the component normal to an edge of the velocity vector (delimiting the 2D fracture object) by the fracture’s thickness. This approach is much more computationally efficient, as a very thin but otherwise ample 3D object can now be described as a 2D object, one that only needs to be meshed as a surface.

Say you have a 2D fracture object with the following characteristics:

- It has an edge contained within the porous matrix.
- It is significantly more permeable than its surrounding formation.
- It features a very large aspect ratio between its lateral and cross-sectional dimensions.

In a system of interest such as this, significant fracture flux computational errors can occur. Let’s take a look at one such example.

Note: With the latest version of the COMSOL Multiphysics® software — version 5.2a — you can also model heat transfer in thin fractures. This is made possible via the

Heat Transfer in Fracturesinterface.

The system shown below features a 3D penny-shaped hydraulic fracture embedded in a reservoir block and connected to a horizontal well. The inlet for this simplified system consists of the two reservoir boundaries shown in green at the top and the back of the block. The only outlet is through the narrow boundary where the fracture disc connects to the wellbore. Both the inlets and the outlet are set as pressure boundary conditions, with the values of ΔP and 0, respectively. The geometry only considers one quarter of the actual system, as it takes advantage of existing symmetry.

*A 3D penny-shaped fracture (shown in blue) embedded in a reservoir block and hydraulically connected to a horizontal well (shown in red). The two reservoir inlet boundaries are highlighted in green.*

Note that the dimensions of the above system are not representative of cases of practical interest. The dimensions are scaled down to allow for adequate 3D meshing of the discoidal fracture, which has a radius of 7.62 m (25 ft) and a thickness of 1.27 cm. (Properly meshing 3D fractures with radii of hundreds of feet, as encountered in field applications, would be quite computationally expensive.) The wellbore radius is 12.7 cm (5 in), while the reservoir block’s dimensions are approximately 8 m x 15 m x 15 m (25 ft x 50 ft x 50 ft). The entire mesh consists of 2,246,298 tetrahedral elements, 657,720 of which are used for the discoidal fracture domain alone. The minimum and average element quality values of the latter are 0.148 and 0.700, respectively, while the average quality for the entire mesh is 0.673.

*Outlet boundaries for the 3D (shown in green) and 2D (shown in red) realizations of the actual hydraulic fracture of thickness d _{HF}.*

Darcy’s law is used to solve for the pressure field *p* in incompressible, single-phase, and stationary flow parametric studies for various values of the drawdown ΔP. The fluid is a light liquid hydrocarbon with a dynamic viscosity value of 0.26 cP. The permeability of the reservoir matrix is taken as 1 mD, while that of the (propped) hydraulic fracture is 45.6 Darcy.

The fracture flux calculation issue referenced above is depicted in the following figure. This figure shows the inlet and outlet flow rates as functions of the drawdown ΔP when the hydraulic fracture (HF) is described as either a 3D or 2D object. While the first three curves (for the inlet flow rates and the 3D outlet flow rate) overlap as expected, the outlet flow rate for the 2D case represents only a quarter of the inlet flow rate. The fluxes for the first three curves were calculated as integrals of the normal component of the fluid velocity vector over the respective inlet and outlet surfaces (). The outlet flow rate for the 2D fracture, meanwhile, was calculated as an integral of along the outlet edge, multiplied by the fracture thickness *d _{HF}*: .

The flux calculation issue remains regardless of the applied meshing and no matter how is probed, with its integrand expressed as (dl.nx*dl.u + dl.ny*dl.v + dl.nz*dl.w), (sys1.e_n1*dl.u + sys1.e_n2*dl.v + sys1.e_n3*dl.w), dl.bndflux/dl.rho, or as (root.nx*dl.u + root.ny*dl.v + root.nz*dl.w). The ‘dl.’ identifier stands for the applied interface (*Darcy’s Law* interface); {nx,ny,nz} are the Cartesian components of the unit vector normal to the edge ; and {u,v,w} are the Cartesian components of the fluid velocity vector .

*Inlet and outlet flow rates as functions of the drawdown ΔP when the actual HF is modeled as either a 3D or 2D object for the respective system.*

Notice that when the hydraulic fracture is described as a 2D object, the discoidal fracture (3D) domain is omitted from the model and is considered instead only through its inner lateral boundary. Otherwise, the geometry and mesh are identical between the 2D and 3D descriptions. This simplification greatly reduces the size of the system and thus represents one of the most attractive elements of the *Fracture Flow* interface: it enables the modeling of much larger fracture surfaces with proper meshing. As such, it would be quite useful if there was a way to work around the 2D fracture outlet flux issue.

A hybrid approach, which combines a 2D description of the fracture away from the wellbore with a 3D one in its immediate vicinity, makes this possible. The figure below shows the meshed geometry of the hybrid implementation. The 3D component of the fracture is represented by the blue domain, while the 2D component is represented by a red surface, which depicts the boundary toward the porous matrix of the actual fracture. Note that the 3D part of the actual fracture that corresponds to the 2D component is excluded from the model.

*Meshed geometry of the hybrid fracture implementation. The blue domain represents the 3D component of the fracture, while the red surface represents the 2D component. The latter is chosen as the inner lateral boundary of the actual fracture (toward the matrix).*

In the hybrid approach, the pressure field continues to be properly accounted for at any point within the actual fracture, while the flux through the outlet boundary is computed without the shortcoming of the 2D description. The following table compares relevant quantities for the 3D, 2D, and hybrid realizations of the hydraulic fracture. These computations were performed using a direct solver on a machine with an Intel® Core™ i74770 processor and 32 GB RAM.

Hydraulic fracture | Degrees of freedom | Memory (GB) | Time per iteration (s) | ||
---|---|---|---|---|---|

3D | 3,231,747 | 23.74 | 247.5 | 1 | 1.00026 |

2D | 2,354,490 | 15.98 | 153.5 | 0.99948 | 0.24992 |

Hybrid | 2,397,891 | 16.50 | 158.0 | 0.99941 | 0.99967 |

*Comparison of relevant quantities for the 3D, 2D, and hybrid realizations.*

The plot below shows, in logarithmic scale, the pressure profiles along a diagonal line within the *YZ*-plane containing the 2D fracture surface for a drawdown of 100 psi. This probe line is delimited by the outlet (wellbore) at the lower-right part of the surface and by the inlet of the reservoir block at the other end. The white line at the inset of the plot highlights the probe line. The surface color of the inset corresponds to the pressure value within the probed *YZ*-plane, and the guiding arrows help map important graphing points on it. The graph’s curves overlap for all three cases, indicating that the pressure field solution is practically identical among the three fracture descriptions: 3D, 2D, and hybrid.

*Pressure profiles along a face diagonal line within the *YZ*-plane.*

Flux calculation issues can occur with a solely 2D description of a fracture. As we’ve demonstrated here today, the proposed hybrid approach for describing an actual fracture provides a viable solution. As such, this technique can be applied to various systems of practical interest that feature a greater number of arbitrarily thin fractures.

Ionut Prodan is the principal of Boffin Solutions, LLC, a COMSOL Certified Consultant. Prior to his founding of Boffin Solutions, Ionut worked within upstream technology at Shell and Marathon Oil. He earned his doctorate in physics from Rice University, where he conducted research on the photoassociation of ultra-cold atoms and computational solid-state chemistry.

*Intel and Intel Core are trademarks of Intel Corporation or its subsidiaries in the U.S. and/or other countries.*