0.6 Solution of the partial differential equations  (Page 12/13)

This equation expresses the velocity that a particular value of saturation propagates through the system, i.e., the saturation velocity , $v_S$ is equal to the slope of the fractional flow curve. It is also the slope of a trajectory of constant saturation (i.e., $dS=0$ ) in the $(x,t)$ space. Since we are assuming constant initial and boundary conditions, changes in saturation originate at $(x,t)=(0,0)$ . From there the changes in saturation, called waves, propagate in trajectories of constant saturation. We assume that $df/dS$ is a function of saturation and independent of time or distance. This assumption will result in the trajectories from the origin being straight lines if the initial and boundary conditions are constant. The trajectories can easily be calculated from the equation of a straight line.

Wave : A composition (or saturation) change that propagates through the system.

Spreading wave : A wave in which neighboring composition (or saturation) values become more distant upon propagation.

Indifferent waves : A wave in which neighboring composition (or saturation) values maintain the same relative position upon propagation.

Step Wave : An indifferent wave in which the compositions change discontinuously.

Self Sharpening Waves : A wave in which neighboring compositions (saturations) become closer together upon propagation.

Shock Wave : A wave of composition (saturation) discontinuity that results from a self sharpening wave.

Rule : Waves originating from the same point (e.g., constant initial and boundary conditions) must have nondecreasing velocities in the direction of flow. This is another way of saying that when several waves originate at the same time, the slower waves can not be ahead of the faster waves. If slower waves from compositions close to the initial conditions originate ahead of faster waves, a shock will form as the faster waves overtake the slower waves. This is equivalent to the statement that a sharpening wave can not originate from a point; it will immediately form a shock.

Mass balance across shock

We saw that sharpening wave must result in a shock but that does not tell us the velocity of a shock nor the composition (saturation) change across the shock. To determine these we must consider a mass balance across a shock. This is sometimes called an integral mass balance as opposed to the differential mass balance derived earlier for continuous composition (saturation) changes.

$\Delta f/\Delta S$ is the cord slope of the $f$ versus $S$ curve between $S_1$ and $S_2$ .

The conservation equation for the shock shows the velocity to be equal to the cord slope between $S_1$ and $S_2$ but does not in itself determine $S_1$ and $S_2$ . To determine $S_1$ and $S_2$ , we must apply the rule that the waves must have non-decreasing velocity in the direction of flow. The following figure is a solution that is not admissible . This solution is not admissible because the velocity of the saturation values (slope) between the $IC$ and $S1$ are less than that of the shock and the velocity of the shock (cord slope) is less than that of the saturation values immediately behind the shock.