[6.1.1.1] As emphasized above (see also footnote V)
the validity of the generalized Darcy law (23)
requires path connected fluids, i.e. fluid configurations
that percolate from inlet to outlet.
[6.1.1.2] Application of from (28)
to water flooding desaturation experiments therefore requires
that also the oil configuration
is percolating
at the initial time
, if the generalized Darcy law
is assumed to describe the reduction of oil saturation.
[6.1.1.3] An appropriate desaturation protocol consists of
steps with
![]() |
(36a) | ||||
![]() |
(36b) | ||||
![]() |
![]() |
(36c) | |||
![]() |
![]() |
(36d) |
with and
is chosen
such that
![]() |
(37) |
holds for every fixed .
[6.1.1.4] Here
are constant and
denotes the volumetric production rate
(outflow) of oil. Its support
is the
set of time instants
for which
holds.
[6.2.0.1] Condition (37) means that the
oil production has stopped.
[6.2.0.2] During the experiment the oil phase is kept
at a sufficiently high ambient pressure so that,
depending on the pressure drop across the sample,
also oil can enter the sample during the water flood.
[6.2.0.3] The desaturation protocol (36) is a continuous
mode displacement where water is injected into continuous oil.
[6.2.0.4] It will be referred to as CO/WI for short.
[6.2.1.1] The CO/WI-protocol (36) requires to
clean the sample after each step and refill it with oil.
[6.2.1.2] This is costly and time consuming.
[6.2.1.3] Many capillary desaturation experiments
are therefore performed in discontinuous mode.
[6.2.1.4] In discontinuous mode the water injection rate is increased
in steps, and the initial configuration of
step
is the final configuration of step
.
[6.2.1.5] The initial oil configuration
may or may not be percolating.
[6.2.1.6] The desaturation protocol
![]() |
![]() |
(38a) | |||
![]() |
![]() |
(38b) | |||
![]() |
![]() |
(38c) | |||
![]() |
![]() |
(38d) | |||
![]() |
![]() |
(38e) |
will be referred to as DO/WI (discontinuous oil/water injection).
[6.2.1.7] Here is again chosen such that
condition (37) holds
i.e. one waits sufficiently long until the oil
production
after step
has ceased.
[6.2.1.8] For nonpercolating fluid configurations
the applicability of eq. (23)
and (28) is in doubt
as emphasized in [40]
and known from experiment [10].
[6.2.2.1] To exclude gravity effects the water flow direction is
usually oriented perpendicular to gravity.
[6.2.2.2] In addition the sample’s thickness parallel to gravity
is chosen much smaller than the width of the capillary
fringe where
is
the mass density of water and
the acceleration
of gravity to minimize saturation gradients due
to gravity.
[6.2.3.1] Finally, a new protocol, introduced in [5], is used for application to experiment in the next section. [6.2.3.2] In [5] the cylindrical sample was oriented vertically, parallel to the direction of gravity in contradistinction to the conventional setup. [6.2.3.3] The wetting fluid was injected from the bottom against the direction of gravity. [6.2.3.4] The sample was always wetted by a water reservoir at the top. [6.2.3.5] The water pressure in the top reservoir was increasing during the experiment due to water accumulation. [6.2.3.6] A period of water injection was followed by a period of imaging the fluid distributions. [6.2.3.7] The new injection protocol resulting from these procedures is defined as
![]() |
(39a) | ||||
![]() |
(39b) | ||||
![]() |
![]() |
(39c) | |||
![]() |
![]() |
(39d) | |||
![]() |
![]() |
(39e) | |||
![]() |
![]() |
(39f) | |||
![]() |
![]() |
(39g) | |||
![]() |
![]() |
(39h) |
[page 7, §0]
where is chosen subject to condition (37).
[7.1.0.1] This protocol will be referred to as
DO/IWI/G standing for discontinuous oil/interrupted
water injection/gravity.
[7.1.0.2] Note, however, that the oil configuration was
typically percolating at
.
[7.1.0.3] During the imaging intervals
resaturation and
relaxation processes may have changed the
original fluid configuration and saturation
as compared to the instant when the pump
was switched off.
[7.1.1.1] This section applies concepts and results
from the preceding sections to recent
highly advanced capillary desaturation
experiments with simultaneous fast X-ray
computed microtomography [5].
[7.1.1.2] The experiments in [5] used
the DO/IWI/G-protocol defined in (39).
[7.1.1.3] The experiment had steps with
![]() |
(40) | ||
![]() |
(41) |
as injection rates, respectively phase velocities.
[7.1.1.4] After reaching stationary water flow without oil production,
the nonwetting phase saturations remaining inside the sample
were measured and found to be
,
,
,
,
.
[7.1.2.1] The experiments were performed on sintered borosilicate glass commercially available as VitraPOR P2 from ROBU Glasfilter Geräte GmbH (Hattert, Germany). [7.1.2.2] A quadratic cross section of this porous medium with a sidelength of 2.6 mm is shown in Figure 1 to illustrate its pore structure. [7.1.2.3] The pore structure is less homogeneous than that of certain natural sandstones often used for pore scale and core scale studies. [7.1.2.4] A cylindrical specimen
![]() |
(42) |
of this porous medium with diameter
![]() |
(43a) |
length
![]() |
(43b) |
and total volume
was measured to have
a pore volume of
and a grain volume of
.
[7.1.2.5] Its porosity and Klinkenberg corrected air permeability
![]() |
(44a) | ||
![]() |
(44b) |
correspond to a well permeable, medium to
coarse grained sandstone.
[7.1.2.6] Mercury injection porosimetry was performed on this sample.
[7.1.2.7] It showed a breakthrough
pressure of resulting in
a typical pore size of roughly
if
![]() |
(45a) | ||
![]() |
(45b) |
are used for the surface tension and contact angle of mercury.
[7.2.0.1] The capillary desaturation experiments in
[5] were performed using
-decane as the nonwetting fluid
and water with CsCl as contrast agent
as the wetting fluid
.
[7.2.0.2] The mercury pressures can be rescaled with
![]() |
(46a) | ||
![]() |
(46b) |
to the water/-decane system according to
![]() |
(47) |
if Leverett-J-function scaling is assumed to to be valid.
[7.2.0.3] The rescaled mercury drainage pressure function in
the range up to Pa is shown in the upper part of
Figure 2 with crosses.
[7.2.0.4] For subsequent computations
the imbibition curve and the relative permeabilties
shown in Figure 2 had to be assumed
theoretically, because experimental data were not available.
[7.2.0.5] The particular choice for their functional form
will influence the numerical results, but is not
important for our theoretical argument.
[page 8, §0]
[8.1.0.1] The fluid viscosities were
![]() |
(48a) | ||
![]() |
(48b) |
for water denoted and
-decane denoted as
.
[8.1.1.1] The capillary desaturation experiments in [5] were
performed not on the full sample ,
but on a small subset of
.
[8.1.1.2] That cylindrical subsample had
![]() |
![]() |
(49a) | |
![]() |
![]() |
(49b) | |
![]() |
![]() |
(49c) |
where denotes the cross
sectional area
.
3: Assuming perfect isotropy
the dimensionless
aspect ratio matrix becomes diagonal with
according to eq. (28) in [40].
Because of the ratio of
it should be kept in mind that
geometric factors can change the force balance by
an order of magnitude.
[8.1.2.1] The resulting microscopic capillary
numbers were
![]() |
(50) |
with .
4: These capillary numbers differ from those shown in
Figure 1 of [5] by a factor .
[8.1.2.2] To compute the macroscopic capillary number from
(28) the characteristic pressure
is taken from the rescaled drainage
curve in the upper part of Figure 2 as
![]() |
(51) |
[8.1.2.3] With this the macroscopic capillary numbers
for the -sample are
![]() |
(52) |
for , while
![]() |
(53) |
for the -sample.
[8.1.2.4] Note, that the width
of the capillary fringe of water
![]() |
(54) |
is around , where
is the density of water and
is the acceleration of gravity.
[8.1.3.1] Figure 3 compares the experimental
observations to the theoretical predictions.
[8.1.3.2] Assuming to be fixed, the
theoretically predicted capillary desaturation curve
for fixed force balance
is obtained from
the solution
of eq. (27)
![]() |
(55) |
as .
[8.1.3.3] Figure 3 shows two
capillary desaturation curves
for water injection into
continuous oil according to the CO/WI-protocol (36).
[8.1.3.4] One curve (crosses) represents drainage,
while the solid curve represents imbibition.
[8.2.0.1] Crosses are computed using the rescaled
mercury drainage pressures and relative
permeabilities for drainage shown in Fig. 2.
[8.2.0.2] The solid curve without symbols is computed from the
imbibition curves in Fig. 2.
[8.2.0.3] The values of
and
are indicated by
dashed horizontal lines.
[8.2.1.1] If all assumptions underlying the traditional equations
and the derivations of hold true,
then the experimental results are expected to fall
in between the two limiting drainage and imbibition curves.
[8.2.1.2] To test this expectation Figure 3
shows three experimental
capillary desaturation correlations.
[8.2.1.3] The experimental values
with
are plotted as squares against
from eq. (50),
as triangles against
from eq. (53),
and as circles against
from eq. (52).
[8.2.1.4] This comparison between theory and experiment rules out
the use of microscopic capillary number
as
abscissa in capillary desaturation curves.
[8.2.1.5] The misleading use of this number is still widely
spread in current literature although it has
been criticized already in [9, 32].
[8.2.1.6] The comparison with
confirms the predictions
of traditional two phase flow theory as far as orders
of magnitude are concerned.
[8.2.1.7] However, it must be emphasized that
the comparison uses the CO/WI-protocol for theory, but
the DO/IWI/G-protocol for experiment.
[8.2.1.8] The theoretical predictions restrict capillary
desaturation curves to the region
.
[8.2.1.9] This prediction is a consequence of the
fact that the traditional theory cannot account for
disconnected nonpercolating fluid parts.
[page 9, §0]
[9.1.0.1] Figure 3 represents, to the best of our knowledge,
the first example in which bounds for
capillary desaturation curves have been predicted based solely on the
constitutive functions of the traditional two phase flow theory.
[9.1.1.1] This subsection introduces for the first time continuous mode capillary saturation experiments in analogy to capillary desaturation experiments. [9.1.1.2] The new saturation protocol is defined as
![]() |
(56a) | ||||
![]() |
(56b) | ||||
![]() |
![]() |
(56c) | |||
![]() |
![]() |
(56d) |
where .
[9.1.1.3] For each fixed
the time
is chosen such that
![]() |
(57) |
holds, i.e. such that the water production has ceased. [9.1.1.4] The saturation protocol (56) will be referred to as CO/OI-protocol (continuous oil/oil injection). [9.1.1.5] To the best of our knowledge such capillary saturation experiments with CO/OI-protocol have not been performed.
[9.1.2.1] During the CO/OI-protocol the water phase is kept at a sufficiently high ambient pressure so that water can enter the sample while oil is injected. [9.1.2.2] If the ambient pressure is sufficiently high and the oil injection rates are small, the resulting displacement process is expected to show strongly interacting mesoscopic cluster dynamics with numerous breakup and coalescence processes of mesoscopic clusters.
[9.1.3.1] Applying the theoretical prediction from eq. (27)
yields capillary saturation curves
for fixed force balance
from solutions
of the equation
![]() |
(58) |
as
analogous to capillary desaturation curves shown in
Figure 3.
[9.1.3.2] The theoretically predicted bounding capillary
saturations curves
for drainage
(crosses) and imbibition (solid curve) are
displayed in Figure 4 using again
the function
shown in
Figure 2.
[9.1.3.3] Experiments following the CO/OI-protocol are
expected to fall in between these two limiting curves.
[9.1.3.4] Figure 4 shows that the
region between the curves becomes narrow for
or
for the chosen parameters.
[9.1.3.5] In this region strongly interacting mesoscopic
clusters are expected to arise from strongly
fluctuating breakup ond coalescence of oil ganglia.
[9.1.3.6] This expectation is consistent with
theoretical network modeling in [44] and
with recent experimental observations of two temporal
regimes of percolating and nonpercolating fluid flow
during imbibition into Gildehauser sandstone in [45].