Sulfur Deposition Model
The affect of sulfur deposition on the loss of well performance in some sour-gas reservoirs has been reported by Roberts (1997). As pointed out by Roberts (1997), sour-gas contains large quantities of elemental sulfur. Therefore, decrease of pressure and temperature during production of sour-gas can lead to the elemental sulfur dissolved in sour-gas to separate and deposit as solids and cause a decline of the well performance. Sulfur deposition can occur both in the well and reservoir formation (Hyne, 1968, 1980, 1983; Kuo, 1972; Roberts, 1997). In this section, a brief discussion of the simplified analytical modeling effort by Roberts (1997) for prediction of the formation damage by sulfur deposition is presented. Hyne (1968, 1980, 1983) has considered the possibility of formation of hydrogen polysulfides at high pressure and temperature conditions by reaction of the hydrogen sulfide and sulfur according to the following equation:
Roberts (1997) points out that two mechanisms may be considered for the solubility of sulfur in the sour-gas:
1 physical and
Hyne (1980, 1983) considered that sulfur may dissolve at high temperature and pressure conditions because of the production of hydrogen polysulfides by a reaction of the hydrogen sulfide with sulfur according to the following equation:
Thus, when the pressure and temperature are lowered, this reaction should reverse itself to form solid elemental sulfur. However, Roberts (1997) argues that its affect can be neglected because the reverse reaction is slow compared to the high flow rate conditions prevailing in the near-wellbore formation. Therefore, for all practical purposes, Roberts (1997) assumes and verifies that sulfur is physically dissolved in the sour-gas and separates instantaneously as solids when the pressure declines to below the saturation conditions. Roberts (1997) draws attention to the field experience (Chernik and Williams, 1993) that sulfur deposition in liquid form does not create many problems, whereas sulfur deposition in solid form may cause severe formation damage problems.
Solubility of Sulfur in Natural Gas
Roberts (1997) states that elemental sulfur freezes at 119°C at atmospheric pressure, but the freezing point decreases by increasing H2S concentration in the sour-gas. Because experimental measurement of the sulfur solubility is expensive and tedious, Roberts (1997) uses the simplified thermodynamic equation given by Chrastil (1982) based on the ideal solution theory for estimation of the sulfur solubility as:
In Eq. 21-3, cr represents the concentration of the solid sulfur dissolved in the gas expressed as mass per unit reservoir gas volume [g/reservoir m³), p is the density of the gas (&g/m³), T is the reservoir gas temperature, and k, A, and B are some empirically determined parameters. As shown by Roberts (1997), using the data by Brunner and Woll (1980, 1988), the plots of Eq. 21-3 at given temperatures are fairly linear on the log crvs. log p. Also, the plot of logcr,vs.(l/T) given by Roberts (1997) shows a linear trend. Therefore, Roberts (1997) concludes that Eq. 21-3 can be used for prediction of the sulfur solubility in gas and presents the following correlation:
Note that Eq. 21-4 implicitly includes the affect of the reservoir gas pressure because the gas density is given by:
Thus, invoking Eq. 21-5 and then differentiating Eq. 21-4 yields the following expression for the variation of the sulfur solubility by pressure at a prescribed reservoir temperature as:
Modeling Near-Wellbore Sulfur Deposition
Consider the radial flow model of an areal drainage region around a well. Roberts (1997) considered a sour-gas reservoir operating under isothermal and semi-steady state radial flow conditions and expressed the pressure gradient as:
where q is the constant gas production rate (m³/s), u is the gas viscosity (Pa.s), B is an empirical constant, the formation volume factor, h is the thickness of the net pay zone of the formation (m), r is the radial distance from the center of the well (m), k is the permeability of the formation at the initial water saturation (m²), and kr is the relative permeability of the gas (dimensionless). Roberts (1997) assumes a relative permeability function as (Kuo, 1972):
where "a" is an empirical constant. The volume of the sulfur separating as solid precipitates for an infinitesimal reduction of pressure, dp, within an infinitesimally small time interval, dt, is given by:
and the volume fraction, Ss, of the hydrocarbon pore volume occupied by the sulfur deposits within an infinitesimally small cylindrical element of dr width is given by:
where ɸ, and Swi denote the initial porosity and irreducible water saturation, respectively. Consequently, combining Eqs. 21-7 through 10 leads to the following expression for the rate of sulfur deposition:
Eq. 21-11 can be numerically integrated subject to the initial condition that
Note that the amount of sulfur deposits can also be expressed as the fraction of the bulk volume occupied by the deposits as:
However, assuming constant u, B, and (ðcr/ð)r values, Roberts (1997) integrates Eq. 21-11 analytically to obtain the following approximate expression:
Figure 21-1 by Roberts (1997) shows the sulfur saturation of the pore volume determined by solving Eq. 21-14 using the assumed parameter values given as: a = -6.22, 5 = 0.004583, u = 2.2SxW~5 Pa's, ka=l.0md, h = 3Qm, ɸ = 0.04 in fraction, Swi = 0, and del dp = 4.0 x 10~15 m³/m³ - Pa.
Prediction of Sulfur Deposition by Reservoir Simulation
Eq. 21-14 is only an approximate solution. Therefore, for realistic predictions, Roberts (1997) resorted to reservoir simulation. Roberts (1997) used a conventional black-oil reservoir simulator. For this purpose, Roberts (1997) considered that the oil phase with zero relative permeability can represent the sulfur deposits, and the condensate-to-gas ratio represents the partitioning of the elemental sulfur between the gas and oil phases. Roberts (1997) neglected the effect of the sulfur on the viscosity and formation volume factor of the sour-gas. Figure 21-2 developed by Roberts (1997) shows a successful history matching of field data using the two-dimensional radial model for a layered formation. However, Roberts (1997) warns that, "The match obtained in Figure 21-2 is unlikely to be unique. Other reservoir descriptions
(e.g., use of a layered model with different formation permeability) may yield equivalent results."
Calcite Deposition Model
Adapting Roberts' (1997) sulfur deposition model and the specific phase behavior of the CO2-brine systems, Salman et al. (1999) simulated the near-wellbore calcite deposition and the reduction of the geothermal well performance. The problem solved by Satman et al. (1999) and their results obtained using this model are presented in the following. The operating chemical reaction relevant to this problem is given by
Satman et al. (1999) used the calcite solubility data available from Segnit et al. (1962). Using q = Q.Q642m³/s, B = lrb/stb, r = 200°C, u= 1.4 x 10^ Pa • s, ka = 9.869 x 1(T13m², h = 100m, a = -6.22, ɸ = 0.1 in fraction ,and dc/dp = 1.135 x 10~12 m³/m³ - Pa in the deposition model, they constructed Figure 21-3, showing the variation of the calcite saturation as a function of time at various radial distances from the well. Satman et al. (1999) carried out a number of case studies. The results are presented in Figures 21-4 through 21-7. Figure 21-4 shows the variation of the wellbore flowing pressure at constant flow rates during calcite deposition. Figure 21-5 shows the variation of the flow performance of wells at constant
average reservoir fluid pressure during calcite deposition. Figure 21-6 shows the effect of the absolute permeability on the flow performance of wells during calcite deposition. Figure 21-7 shows the variation of the flow performance of wells during calcite deformation with different skin factors achieved by acid-stimulation of wells.
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