L = dx
w = dz

k_{f
}= fracture permeability = constant (md)

C_{f}
= fracture conductivity = k_{f } w
= constant (md ft)

The discretized multiphase form
of Darcy's law (assuming laminar flow) gives the volumetric flow rate of phase j from block 1 to
block 2 as:

q_{j
}= (.001127 k_{f} A / L)
λ_{j}_{ }(P1 - P2)
= Tx λ_{j}_{ }(P1 - P2)
(rb/d), or

q_{j
}= (.001127 C_{f} dy / L)
λ_{j}_{ }(P1 - P2)
= Tx λ_{j}_{ }(P1 - P2)
(rb/d)

where λ_{j}_{
}is the phase mobility = relative permeability /
viscosity = k_{rj} / v_{j},
and Tx is the x-direction transmissibility between blocks 1 and 2.
Transmissibility is a function of gridblock dimensions dx,dy,dz (w) and
permeability, or conductivity. Turbulent flow can be modeled in
the well flow terms but usually can be neglected for block-to-block flow
inside the reservoirs, especially when the infinite permeability
assumption applies (all turbulence can be neglected in that case, as
increasing well PI or fracture permeability has no effect on results).

The
volumetric flux is

q_{j}* = q_{j}_{ }/ A = (.001127
k_{f}) λ_{j}_{ }[
(P1 - P2) / L ] (rb/ft2/d)

The last term in brackets is the
numerical approximation of potential
gradient. So,

**k**_{f}
= ratio of volumetric flux to the potential gradient for a unit-mobility
fluid.

Flux in a fracture is
independent of w, except as it may effect the value of permeability.

**k**_{f }
A
= ratio of flow rate to the potential gradient for a unit-mobility
fluid. A is the area in the direction of flow.

**So, the most appropriate
term characterizing fracture flow rate is k**_{f }
A, not C_{f}=k_{f }w.

Both permeability (or
conductivity) and fracture width affect fracture flow rate, and both are
represented along with gridblock geometry in the numerical model
transmissibilities.

The flux written in terms of conductivity is

q_{j}* = q_{j }/ A = (.001127
Cf / w) λ_{j}_{ }[
(P1 - P2) / L ] (rb/ft2/d)

with w in ft. So if we
wish we can address ideal fracture flow capacity in terms of the ratio
of conductivity to width (equals permeability), and the effects of
fracturing, propping, and closure on both of those variables, or we can
simply address those effects on the basic meaningful variables,
permeability and width, and on the
resulting numerical model transmissibilities that reflect both of them.

Transmissibility T = 0.001127 k
A / L = ratio of
volumetric flow rate to the potential difference
between two gridblocks across area A (= w dy) for a unit-mobility fluid. It is usually considered constant for unfractured and for naturally fractured reservoirs (no stress dependence
of either matrix or fracture permeability) unless extreme pressure is
applied, as in hydraulic fracturing.

The
simulator will predict the correct flow rate and flux if dx is small
enough to prevent significant error (numerical dispersion) due to
numerical finite-difference approximation of the pressure gradient.
Permeability is a meaningful measure of fracture flow capacity as a
function of pressure gradient. Conductivity by itself is not
(unless divided by w to obtain permeability). That is why we
measure permeability in the lab, rather than conductivity.

Dimensionless fracture conductivity
is usually
referred to as F_{cd}=k_{f}w/(k_{m}x_{f}),
conductivity divided by matrix permeability times fracture half length.
None of those parameters are a constant in any real naturally and/or
hydraulically fractured system. But for the ideal homogeneous and
isotropic single ideal planar fracture in matrix, Fcd is equivalent to
k_{f}A_{f}/(k_{m}A_{mf}), which
indicates the ratio of fracture to matrix flow for identical pressure
gradients and fluid mobility. A_{f} is the fracture area
in direction of flow along the fracture, and A_{mf} is the area
of contact between matrix and fracture.

All flows are functions of
pressure gradient distributions (including gravity and capillary
forces) and fluid mobilities in matrix and fracture, and flow from
matrix to fracture is additionally a function of matrix permeability and
the area in contact with fractures., while flow rates along fractures
are additionally a function of fracture permeability and area in
direction of flow.

The simple ratio of permeabilities has physical meaning as the
ratio of flux through the fractures to that through the matrix for a
given pressure gradient and fluid. But that doesn't mean much. and
neither does F_{cd}, since
potential gradients are very large in tight matrix and very small in
fractures. In general, all of the governing equations must be
solved in a reservoir simulator to make any conclusions of how flow
through matrix and fractures and production will behave as functions of
the input variables (distributions of permeability and porosity and rocktype/relperm/Pc, pvt, initial and boundary conditions, frac models,
...). Those conclusions will generally be case-dependent. No
single dimensionless quantity can possibly characterize the description
or behavior of any heterogeneous system.

The "infinite conductivity"
assumption or level is equivalent to the "infinite permeability"
assumption that we may check with our reservoir models. That means
that the conductivity or permeability (for given fracture distributions
and assumed behavior) is high enough such that the pressure drop in the
fractures is negligible. This is the solution that should always
be found first. Then, sensitivities to reduced values of fracture
permeability, along with the effects of any assumptions or
representations regarding fracture dimensional or volumetric behavior
and distributions, can be examined with the simulator.

For an open planar fracture of
width w (in), permeability is^{1}:

k_{f}
= 54 * 10^{9} w**2 (md)

and the flux of phase j through an ideal open fracture is

q_{j}*
= .001127 k_{f} λ_{j}_{ }[
(P1 - P2) / L ] = 6.0858 * 10^{7}
w**2 λ_{j}_{ }[
(P1 - P2) / L ] (rb/ft2/d)

If a fracture is packed with
proppant, then the fracture permeability will be equal to that of the
packed proppant bed, if particle size is small compared to w.

If the proppant is crushed or
degrades over time due to increased stress as pressure falls below
fracture pressure during production, then w will change and that will
have the indicated effect on numerical model fracture gridblock transmissibility
(Tx = k w dy / L). Permeability of the packed proppant may
or will also decrease due to these effects and will also have the
indicated effect on fracture gridblock transmissibilities. Any relative
decrease in permeability or conductivity or w causes flow rate and flux
to decrease by the same relative amount, for a given potential gradient.

In a dual porosity or a dual
permeability model, input values of fracture permeability are
effective values, scaled to apply over the entire area of the dual
porosity gridblock A_{d}. The
effective dual fracture perm k_{xf}*
is equal to k_{xf} A_{f
}/ A_{d}, where A_{f
}/ A_{d} is the ratio of
the total fracture area to total area in the x-direction. Fractures are represented in the
dual gridblock by their spacings in the x,y, and z directions as Lx, Ly,
and Lz. These are affected or caused by fracturing, but do not
change with stress and closure below fracture pressure.

Dual matrix-fracture exchange
transmissibilities are not a function of fracture permeability or
fracture width. They depend only on matrix permeability and
fracture spacings (See equation given for Tex in the Sensor Manual).

**So how can we compute
theoretically optimal well and fracture spacings in tight systems?
Sensor has two methods.**

The simplest and fastest method
is to assume constant identical well and fracture spacings LW and LX for
areally parallel and optionally stacked horizontal wells, orthogonal
planar fractures of known geometry, frictionless wellbores, identical
operation of all wells (boundary conditions) in a field-wide depletion,
and to include an economic model so that input parameters LW, LX, and
Tend can be easily adjusted to maximize NPV, either by manual adjustment
or by application of an optimization program. Sensor's HWELEMENT
and ECONOMICS options provide the ability to quickly determine optimal
well and fracture spacings and depletion end time, and examine their
dependencies for idealized systems. Any heterogeneities may be
included but they are symmetrical across planes of symmetry. The
assumptions allow the use of a discrete post-frac model of a horizontal
well pattern element for simulation of interior well performance,
minimizing model size and runtime and the number of variables, and
maximizing our abilities in iterative predictive optimization, and even
allow probabilistic optimizations and predictions using advanced
workflows, given uncertainties in the input data, if permeability is
high enough (very small permeability requires very small matrix
gridblocks in a single-porosity discrete fracture model to adequately
control numerical dispersion). See SensorFrac_HWE.dat for an
example.

Sensor's more advanced
dual-porosity/permeability Fracture Model can model stress
dependence of conductivity (transmissibilty) in matrix and fractures.
This is a basic component of Sensor's empirical fully-coupled
geomechanical model of hydraulic fracture network formation,
propagation, propping, and closure, including stimulation below fracture
pressure. Sensor provides the ability to simulate the fracturing
and production processes in a single model. See spe1_hydfrac.dat
for an example.

^{1}
Aguilera, R., __Naturally Fractured Reservoirs__, PennWell Books,
1995, p.16.