Summary
Mass balances for two immiscible fluids and tracer and convective heat
balance form a system of three equations (nonisothermal Buckley-Leverett
Problem with tracers). Tracer component is considered to investigate the
propagation of a tracer to track the flood (or to track a miscible inert
contaminant introduced during drilling). We have solved the resulting
nonisothermal two-phase convective flow equation in porous media analytically,
including a tracer component (i.e., cold or hot waterflooding with and without
tracer). Method of characteristics (MOC) is used as a solution technique after
transforming the balance equations in a form that can be solved easily with two
Welge tangents. Our solution technique is valid for both radial- and
linear-flow models.
In practice, these solutions can be used
- To investigate the convective flow behavior around the wells (i.e., sudden
fluid losses, convective near-well tracer propagation, analyzing pressure
transients).
- To interpret formation-testing-tool responses by detecting the location of
the thermal front or to estimate the temperature-buildup time that is needed
for Horner-type analysis (for the identification of the formation
temperature).
- To calculate the location of the cold or hot water front (thermal water)
while injecting cold- or hot-water. This will yield the limit of the maximum
temperature disturbance around the well.
- To test/scale relative thermal effects of various systems against one
another.
- To test the accuracy of simulators and provide benchmark solutions.
- To interpret relevant laboratory experiments quickly.
Such solutions helped us to understand the depth of influence of the
temperature variations and their influence on the transport properties, in both
radial and linear systems. Solutions analytically proved that the thermal front
propagates much slower than the flood front. This explains why isothermal
black-oil simulators still work although the injected-water temperature is not
always equal to the reservoir temperature.
Furthermore, we have checked and verified our results against a commercial
thermal simulator and investigated the impact of numerical diffusion on the
thermal front as well as conductivity. This part of the work revealed that the
temperature front is more prone to numerical diffusion.
Introduction
Cooling resulting from sudden fluid losses (spurt loss) during drilling or
hot fluid/water (with some additives) treatments around the wellbore is a quite
common intervention for the near-wellbore region. In addition, waterflooding is
the most common secondary-recovery mechanism. Short-time behavior of such
systems is dominated by the convective terms as defined by the underlying
equations. As many engineers observe, most waterflood models can be history
matched with isothermal simulators, contrary to what one might expect (because
water has high heat capacity). Such black-oil simulations work because changes
in the temperature and heat capacity of water per unit volume of injection are
not enough to extract an excessive amount of heat (in the case of cold-water
injection) from the matrix and the in-situ fluids. The only way to extract
and/or inject heat is to take the advantage of phase transformation (i.e.,
latent heat as in the case of steam injection). As described in the section
that compares the numerical solutions with the analytical solutions proposed
here, the presence of the realistic heat losses as the front propagates away
from the injection well becomes dominant (as convective flow velocity decays
radially), further tapering off the thermal gradients induced by the injected
water. Therefore, the impact of cold water can be felt at a limited volumetric
region only around the well.
Another observation that can be made during the backflow phase of the
subsurface sampling is that contaminants and water propagate deeper into the
formation than the first-order (convective) thermal effects. The resulting
problem is analogous to the Buckley-Leverett (1942) problem, in which the
solution can be constructed by use of a series of tangents similar to Welge’s
(Welge et al. 1962; Johns and Dindoruk 1991; Dindoruk 1992; Johns 1992; Hovdan
1989; Bratvold 1989). The late-time temperature behavior, however, is dominated
by conduction (Platenkamp 1985; Roux et al. 1980; Hashem 1990). Such late-time
behavior is also demonstrated for the solutions presented here by use of a
commercial simulator (STARS manual 2004).
The classes of problems solved here are somewhat different from the early
solutions for nonisothermal displacement in porous media as described by Marx
and Langenheim (1959), Lauwerier (1955), and Rubinshtein (1959).
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
27 June 2006
- Meeting paper published:
24 September 2006
- Revised manuscript received:
9 December 2007
- Manuscript approved:
23 December 2007
- Version of record:
20 June 2008
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