2. Two-dimensional Flow

Graphical form of solutions to Laplace equation for two-dimensional seepage can be presented as flow nets. Two orthogonal sets of curves form a flow net:

• Equipotential lines connecting points of equal total head h
• Flow lines indicating the direction of seepage down a hydraulic gradient.

Flow nets are useful graphical tools used to visualize the flow of water through soils in two dimensions. They consist of two sets of curves: equipotential lines and flow lines. Equipotential lines connect points where water pressure is the same, indicating levels of total head. Flow lines, on the other hand, show the direction that water will flow, guiding us to understand how water moves through different materials under pressure. Importantly, flow lines and equipotential lines never intersect, which means that at any point in the flow net, the paths of flow and pressure are distinct and separate.

Imagine a road map where the equipotential lines are the roads at a constant altitude (like a flat bridge) and the flow lines are the paths vehicles take down a hilly road. You can see that roads (equipotential) at the same height do not cross each other, just like flow lines do not meet.

Two flow lines can never meet and similarly, two equipotential lines can never meet. The space between two adjacent flow lines is known as a flow channel, and the figure formed on the flownet between any two adjacent flow lines and two adjacent equipotential lines is referred to as a field.

The uniqueness of flow nets is that flow lines and equipotential lines maintain their separation. This separation creates flow channels, the spaces through which water flows. Additionally, the area enclosed by two adjacent flow lines and two adjacent equipotential lines is called a field. This field represents a specific volumetric area of soil where water is moving and can be analyzed for flow characteristics. Knowing this helps engineers determine how water will behave in specific areas, such as near foundations or embankments.

Think of a field as a kitchen countertop divided by strips of dough (flow lines) marked by rows of cookie cutters (equipotential lines). Each area between the strips represents a place where ingredients (like water) can flow but where the strips prevent mixing with other areas directly.

If standpipe piezometers were inserted into the ground with their tips on a single equipotential line, then the water would rise to the same level in each standpipe. The pore pressures would be different because of their different elevations. There can be no flow along an equipotential line as there is no hydraulic gradient.

Standpipe piezometers are tools used to measure water pressure at various depths within soil. When these devices are placed along an equipotential line, the water level in each standpipe will equalize due to no pressure difference (or hydraulic gradient) existing at that level. However, the actual pressure exerted by the water will change if the standpipes are at different heights, showcasing that even among equal heads, the water pressure can differ based on depth.

Imagine a group of drinking straws submerged in a glass of water at the same height. If you place your finger on the top of one straw, the water in that straw will stay at the same level as the water in the other straws (if all are at the same height). However, if you compare straws of different lengths, the water pressure in the longer straw will be greater even if the water levels are equal at the top.

Consider a field of length L within a flow channel. There is a fall of total head ∆h. The average hydraulic gradient is. As the flow lines are b apart and considering unit length perpendicular to field, the flow rate is.

To determine the flow rate through a channel, we must consider the field's length and the total change in head (or height) water experiences as it flows through. The hydraulic gradient is calculated as the difference in height divided by the distance over which this change occurs. The flow rate can then be estimated by considering how far apart the flow lines are (denoted as 'b') and how wide a unit length of the field is. Mathematically, this creates a clear understanding of how much water flows through a cross-section of this channel.

Think of water flowing down a hill where the height of the hill represents total head. The slope of the hill gives you an idea of how steep the descent is (the hydraulic gradient), while measuring how far apart the rivulets of water flow tells you how fast and how much water is moving down that slope.

There is an advantage in sketching flow nets in the form of curvilinear 'squares' so that a circle can be inscribed within each four-sided figure bounded by two equipotential lines and two flow lines.

Sketching flow nets as curvilinear squares allows for a geometric approach to solving flow problems. By inscribing circles within these shapes, it simplifies the math and helps visualize the flow's properties. This method maintains the relationship between flow and pressure while simplifying calculations involved in determining flow rates across these areas.

Picture a pizza with slices cut in a circular fashion, with each slice representing a segment of flow net. By analyzing just one slice (the curvilinear square), you can efficiently understand the flow characteristics without needing to comprehend the entire pizza—this helps narrow down complex problems into manageable parts.

For a complete problem, the flow net can be drawn with the overall head drop h divided into N so that ∆h = h / N.

When solving flow problems, it is essential to calculate the total flow through a network of flow nets. This is done by dividing the overall change in head (or total head drop) by the number of flow channels (N). This way, we average the head drop over several segments instead of calculating it for one large section, which makes for more manageable calculations and greater accuracy.

Imagine climbing stairs. Instead of calculating the height of a tall building in one go, you can think of it as a series of steps (flow channels) that you take one at a time. Each step represents a smaller change in height, making it easier to comprehend and manage your ascent.

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