Saturation discontinuities

The value of S0 decreases with increasing elevation. Zao, the interface between air and the LNAPL phase, may or may not coincide with Zu, the upper boundary of the aquifer. Typically, the saturation of the LNAPL phase extends over two distinct regions (see Figure 5.10). These are (1) water and LNAPL phase zone, and (2) water, LNAPL phase, and air zone. When a single homogeneous stratum is considered, O can be assumed constant. In a stratified medium, however, saturation discontinuities generally exist due to the variation in soil characteristics, and the determination of LNAPL volume based on Equation 6.22 may become much more involved. 

Before discussing fractional flow analysis, we first need to derive the moving velocity of a saturation discontinuity or shock. Figure 2.13 shows a saturation shock from 8 2 to S i. 8 2 moves from Xj to X2 during the time interval At = t2 - h. The total injection rate, qt, is constant, but the water cut changes from f i to f 2, which corresponds to 8 i and 8 2, respectively. Therefore, during the time interval. At, the total incremental water injected into the block from Xi to X2 is (q,)(At)(f 2-f i). Meanwhile, this incremental water injected results in the increase in saturation from 8 1 to 8 2. The material balance of water gives... 

Most of the integrated plant and control system design studies have used linear control systems such as multi-loop PI control, and do not accommodate actuator saturation discontinuities. However, promising recent approaches include strategies for incorporating actuator saturation into a simultaneous optimization framework [33], and controller parametrization that accommodates saturation behavior [34]. Application of these techniques to more complex problems within an integrated design and control framework, as well as the consideration of other more complex control systems, would be useful. 

The resulting boundary value formulations for Siw(x,t) and S2w(x,t) lead to the so-called Buckley-Leverett problem well known to reservoir engineers. Their solutions can contain shockwaves or steep saturation discontinuities, depending on the form of the fractional flow functions and the initial conditions. The basic issues are discussed in Collins (1961) and will not be repeated here. 

If the initial condition F(x) is such that F > 0, the denominator 1 -I-1F > 0 is positive and the gradient 5u/9x is well behaved. If F is negative, shockwave solutions with infinite values of 5u/5x form in a finite amount of time. These shocks are analogous to the water breakthrough (a.k.a., saturation discontinuity) phenomena familiar in waterflooding. 

Identify a reputable immiscible, two-phase flow simulator for use in this problem, and select a validated problem set (with available solutions) where consistent relative permeability and capillary pressure curves have been successfully tested against field data. Re-mn selected data sets. How do your solutions change as the absolute magnitude of capillary pressure change What happens when the capillary pressure vs. saturation curve is replaced by an approximate straight-line function What if the capillary pressure is set identically to zero In all three scenarios, note the position of the saturation discontinuity, its steepness, and the thickness of the front. Do your solutions oscillate in time If so, numerical instability is indicated. 

As discussed, we can expect shockwaves and steep saturation discontinuities to form in time, depending on the exact form and values of our fractional flow functions and initial conditions. We will assume that the particular functions do lead to piston-like shock formation very close to the borehole. The shock boundary value problem just stated can be solved in closed form, and, in fact, is the petroleum engineering analogue of the classic nonlinear signaling problem (Pt -I- c(p) Px = 0, p = po for X > 0, t = 0, and p = g(t) for t > 0, x = 0) discussed in the wave mechanics book of Whitham (1974). 

Diffusion in cake-dominated flows. Close to the well, immiscible flows containing propagating saturation discontinuities may exist. But very often, flows are obtained that do not contain shocks. These include immiscible flows with and without capillary pressure, and miscible flows governed by highly diffusive processes, where discontinuities never form. 

Large Qs model rapid influxes of injected water and should result in sharp saturation discontinuities for such problems, there is little smearing at the shock due to capillary pressure. This is not to say that capillary pressure is... 

Note that Multiple Factors That Influence Wireline Formation Tester Pressure Measurements and Fluid Contact Estimates, by M.A. Proett, W.C. Chin, M. Manohar, R. Sigal, and J. Wu, SPE Paper 71566, presented at the 2001 SPE Annual Technical Conference and Exhibition in New Orleans, Louisiana, September 30-October 3, 2001, extends the work in this chapter to higher order, ensuring that mass is accurately conserved at strong saturation discontinuities. For further information or a complimentary copy of the paper, the reader should write or contact the author directly at wilsonchin aol.com. 

Select several available immiscible two-phase flow simulators, and define conditions that would lead to water breakthrough in finite time. Assume different capillary pressure functions. How are breakthrough times and locations affected Is mass conserved across the saturation discontinuity Rerun your problem sets with capillary pressure identically zero and compare results. 

At every value of SjJ,., there is a corresponding value of at the saturation discontinuity. Thus, as the polymer-free water displaces the polymer slug, saturations 8%,. are overtaken by the drive water and new saturations, S r, evolve at the rear of the polymer bank. The saturations S r can be found graphically for each value... 

When the polymer front is overtaken by the drive water, the process becomes a waterflood. For example, when tpp=DJ2 (in Example 5.8), the polymer front is overtaken at =0.5/1.2377= 0.404. Fig. 5.61 shows the saturation distribution at this instant. Note that the oil bank created by the polymer front is present, with a saturation discontinuity from to 5 ,x at the rear of the oil bank. This saturation discontinuity is not stable. Because of miscibility, the velocity of this discontinuity is given by... 

In accordance with Eqs. 2 and 3, the connate water bank will break through at / = (S - 5,)/fy, and the polymer front and associated saturation discontinuity will break through at / = (S + b)/fg. Fig. 5 shows the oil recovery curve constructed from Eqs. 2 through 4 and, for comparison, the recovery curve for a normal waterflood as calculated by the Buckley-Leverett method. 

The sharp polymer front and the associated saturation discontinuity ah travel at the velocity... 

This saturation discontinuity will break through at X = L after injecting / = (S + b)/fg pore volumes of water. The leading edge of the oil bank, ic, travels at die rate... 

Peroxides. These are formed by aerial oxidation or by autoxidation of a wide range of organic compounds, including diethyl ether, allyl ethyl ether, allyl phenyl ether, dibenzyl ether, benzyl butyl ether, n-butyl ether, iso-butyl ether, r-butyl ether, dioxane, tetrahydrofuran, olefins, and aromatic and saturated aliphatic hydrocarbons. They accumulate during distillation and can detonate violently on evaporation or distillation when their concentration becomes high. If peroxides are likely to be present materials should be tested for peroxides before distillation (for tests see entry under "Ethers", in Chapter 2). Also, distillation should be discontinued when at least one quarter of the residue is left in the distilling flask. 

Weathering rates are most sensitive to the throughput of water. In soils, this is a decidedly discontinuous process. Typically, water flows through soil following rainfall or snowmelt. Once saturated, the flux of water is largely dependent on the physical properties of the soil and not on the rate of supply. Water that cannot be accommodated by flow through the soil. 

We can draw another inference from these models in regard to the flow of water through the membrane. When the concentration of the solute in the membrane increases abruptly with a small change in the lipophilicity, it is likely that the membrane would approach saturation, that is, the cavities among the membrane cells would be extensively occupied. Trauble has proposed that water and small solutes are carried across a membrane by occupying discontinuities or... 

Advise smokers to quit 2 weeks before therapy be judicious in use of IV fluids for hypotension discontinue therapy if requiring greater than 4 L 02 or 40% 02 mask for saturation greater than 95%. 

In solutions saturated (i.e., excess solid present) at some pH, the plot of log Co versus pH for an ionizable molecule is extraordinarily simple in form it is a combination of straight segments, joined at points of discontinuity indicating the boundary between the saturated state and the state of complete dissolution. The pH of these junction points is dependent on the dose used in the calculation, and the maximum value of log Co is always equal to log. Sb in a saturated solution. [26] Figure 2.2 illustrates this idea using ketoprofen as an example of an acid, verapamil as a base, and piroxicam as an ampholyte. In the three cases, the assumed concentrations in the calculation were set to the respective doses [26], For an acid, log Co (dashed curve in Fig. 2.2a) is a horizontal line (log Co = log So) in the saturated solution (at low pH), and decreases with a slope of —1 in the pH domain where the solute is dissolved completely. For a base (Fig. 2.2b) the plot of log Co versus pH is also a horizontal line at high pH in a saturated solution and is a line with a slope of +1 for pH values less than the pH of the onset of precipitation. 

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