Evaluación experimental producción de biofertilizante

pwf pt q Modeled section in proxy model x y z Fractures Shale matrix blocks

= direction of gas flow

Ls ∆z

∆yf

Fig. 4.3: Illustration of reservoir and proxy model.

Hydraulically fractured shale and tight-formation gas reservoirs consist of tight, low permeable rock matrix blocks intersected by highly conductive fractures as il- lustrated in Fig. 4.3. These systems are mainly modeled either as a so-called dual- porosity system, see e.g. Bello and Wattenbarger (2010); Nobakht et al. (2012), or as fully PDE-discretized single-porosity dual-permeability models (Cipolla et al., 2010). The former, idealized modeling scheme is widely used to derive static long- term production forecasting tools and assumes steady-state operations, while the latter scheme leads to complex, numerically demanding models that take into ac- count transient effects dictated by the differential flow equations, transients that are important to the modeling of short-term shut-in periods as promoted in this pa- per. A reduced-order shale well and reservoir proxy model was derived in Knudsen et al. (2012); Knudsen and Foss (2013), using first-principal physics of the storage and transport mechanisms in the well and the reservoir. The proxy model was de- veloped for efficiently optimizing short cyclic well shut-ins to prevent co-produced liquids to accumulate in the wellbore, and shown through a tuning scheme to give a good transient fit. Knudsen et al. (2014b) developed a similar but slightly modi- fied shale-well proxy model based on the Cartesian geometry shown in Fig. 4.3. By extending the static pressure model of the wellbore with a quadratic friction term (Katz and Lee, 1990) and using a similar formulation of the frequency-selective

4.2. Shale well modeling

tuning scheme described in Knudsen et al. (2014a), the modified proxy model was shown to further improve the transient fit and in particular the steady-state fit compared to the model used in Knudsen and Foss (2013).

Consider again the idealized shale-well geometry shown in Fig. 4.3. By assuming equal spacing and distribution of the fractures, and that the fractures have infi- nite conductivity, i.e. with no pressure loss, and penetrate the entire organic-rich formation, a reduced-order proxy model can be constructed by only considering a quarter section of the matrix-fracture system as illustrated by the orange frame in Fig. 4.3 (Knudsen et al., 2014b). The dominating direction of the gas-flow in the shale-matrix is orthogonal to the fractures (Bello and Wattenbarger, 2010; Nobakht et al., 2012), that is, in the x-direction shown in Fig. 4.3. The proxy is hence constructed as a one-dimensional flow model, using a single layer, a spatially dependent permeability k(x) and an integral transformation from pressure p to pseudopressure m(p) (Al-Hussainy et al., 1966),

m(p) := 2 Z p pb p0 µ(p0_)Z(p0₎dp 0_{, (4.1)}

where µ(p) is the gas viscosity, Z(p) is the gas compressibility factor and pb is a

low base pressure, rendering the semi-linear initial-boundary value problem (IBVP) (Knudsen et al., 2012, 2014b) φµc∂m ∂t = ∂ ∂x  k(x)∂m ∂x  , (4.2a) ∂m ∂x     0 = q 2T psc Tsc∆z∆yfkf, (4.2b) ∂m ∂x    Ls 2 = 0, (4.2c) m(x, 0) = minit. (4.2d)

In (4.2), φ is porosity, c(p) is total compressibility, q is the gas rate at standard conditions, 1 bar and 15.6◦_{C, T is temperature and p is pressure. Spatial refer-}

ences are shown in Fig. 4.3. An I-dimensional spatial discretization of (2.6) is constructed using central difference approximations, while time discretization ap- plies the backward Euler approximation using a timestep ∆k. This leads to the discretized reservoir proxy model

Amk+1= mk+ Bqk+1, (4.3a)

m0= minit, (4.3b)

where k is the discrete time index. The gas rate flowing from the reservoir into the wellbore is given by

qk= β (mk1− mwf,k) , (4.4)

where mk1is the pseudopressure in the gridblock adjacent to the fracture, mwf is

pt, the gas rate the well can deliver in a timestep k is found by the intersection of

(4.4) and the static tubing-model (Katz and Lee, 1990), p2t,k= 1 C2 t qk2+ e−Sp2wf,k, (4.5a) mwf,k: = ˜a1p2wf,k+ ˜a2, (4.5b)

where (˜a1, ˜a2) in (4.5b) are regression parameters used to convert the square of

bottomhole pressure pwf to mwf (Knudsen et al., 2014b). The first term on the

right-hand side of (4.5a) corresponds to the pressuredrop caused by the tubing friction, while the term eS _{yields the hydrostatic head of the gas column. C}

t is

a tubing specific constant, see Katz and Lee (1990). Note that pt serves as the

boundary condition of the aggregated single well and reservoir proxy model.

4.2.1

Shale reservoirs as gas storage

Shale-gas reservoirs have as mentioned in Section 4.1 the special property that production may be shut in for short periods without losing significant long-term recovery. That is, the magnitude of the subsequent peak rate after a shut-in largely recovers for the temporary loss in production during shut-in, hence avoiding any significant reduction in net present value (NPV). This property is in contrast with conventional high-permeability gas reservoirs. For these reservoirs, considering a fixed life-span of the well, the temporary loss in production during a shut-in would not be recovered before after the end of the prediction horizon, thus reducing the NPV of the well. Applying shut-ins of shale-gas wells was initially developed and studied by Whitson et al. (2012) as a cyclic production scheme for eliminating so-called liquid loading in gas-wells (Turner et al., 1969). The basis for the shut-in property of shale-gas wells is the high pressure gradients between the low-permeable shale matrix and the interconnected network of fractures, causing the shale matrix to act merely as a source term feeding the fractures with gas. During shut-ins, these pressure gradients will cause the gas to continue to flow into the fractures, recharging the fractures with gas and thereby increasing the near-wellbore pressure. Once the well is reopened, the gas recharged in the fractures will cause a high peak in the gas rate (Knudsen and Foss, 2013). The volume of the gas recovered after the shut-in compared to if the well is continuously producing, depends on the length of the shut-in and the formation permeability, the latter being the main parameter controlling the ability of the gas to flow through the tight formation (Cipolla et al., 2010).

In Fig. 4.4, we demonstrate the effects of applying intermittent shut-ins on shale- gas wells by simulating a high-fidelity shale reservoir model for different shut-in lengths and different formation permeabilities. The figure shows the total recovery after a fixed 10 year simulation horizon, normalized against the cumulative rate obtained if the well is continuously produced, i.e. with no shut-ins, displayed as a function of the formation permeability and the % of the 10 years operation time the well is shut-in. For each simulated formation permeability, km, 0% shut-in

time hence corresponds to continuous production, giving a normalized recovery of 1 as shown in the upper left corner of Fig. 4.4. Each time the gas rate hits the

4.2. Shale well modeling 1 1.5 2 x 10−4 0 ₁₀ 20 ₃₀ 40 ₅₀ 60 ₇₀ 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 1.005 Permeability [mD] [%] time the well is shut-in

No rm a li ze d re co v er y [- ]

Fig. 4.4: Normalized relative recoveries as a function of the formation permeability km and the % time of total operation time the well is shut-in. Each total recovery rate is withdrawn after running the simulation with a fixed 10 year horizon, and normalized against the maximum recovery within this horizon obtained by applying no shut-ins.

lower (critical) rate, qgc, a rate needed to ensure continuous removal of co-produced

liquids Turner et al. (1969); Coleman et al. (1991); Lea and Nickens (2004), we shut in the well for a predefined, minimum time, and reopen the well once the pressure is sufficiently built-up to regain a flowrate above qgc. Observe that a relative recovery

of 1 is as such only a theoretical value, as co-produced liquids and possible liquid loading in practice always will require shut-ins throughout the life of a well. The simulations are performed using a finely gridded realization of the model shown in Fig. 4.3 implemented in the state-of-the-art reservoir simulation software SENSOR (SENSOR, 2011), assuming 10 equally spaced fractures, 200 bar initial pressure and using the correlation in Coleman et al. (1991) for computing qgc .

For this given shale-well realization, it is clear from Fig. 4.4 that low formation permeabilities permit the well to be shut-in a high percentage of the total opera- tion time while still recovering close to 100% of the maximum recovery obtained by producing the well continuously. Low permeabilities cause a long shut-in time compared to the length of the subsequent production period, giving more than 50 % total shut-in time for the lower range of the simulated formation permeabilities. Note that a lower value of the given shut-in rate qgcwould as such reduce the total

% time the well is shut-in of the fixed 10 year time horizon. Notwithstanding, it is evident that fractured shale-gas wells with very low permeabilities are particularly suitable to intermittent shut-in schemes. If the shut-ins are scheduled optimally

with respect to varying demands and prices, this property may then translate into increased profit for the operators. This utilization of shale-gas reservoirs is equiv- alent to a type of gas storage, since very little recovery is lost. It is important to note that above a certain percentage total shut-in time the recovery suddenly drops, eventually leading to loss in NPV and as such reducing the economic viability of such shut-in schemes.