Geothermal brine invasion in oil reservoirs : a 3 d generalization of the buckley-leverett model using non-linear finite elements

In the southern coast of the Gulf of Mexico some deep geothermal aquifers are associated to hydrocarbon reservoirs. Some of their wells are invaded by geothermal brine, producing a variable mixture of hot water and oil. This water, at temperatures of 150°C and having a density of 1150 kg/m; flows vertically through a fault from the aquifer located 6000 m depth. The non isothermal conditions affect the effective saturations and the relative permeabilities of the immiscible phases. The relative permeability of oil is increased by the increase of temperature produced by the geothermal water. This effect reduces the residual saturation of heavy oils. At the same time the dynamic viscosities of water and oil are diminished, affecting the displacement of both fluids. Although the oil is extracted in wells finished upper the aquifer, the total volume of produced water, in some cases, equals or exceeds the oil production. The handling of this extra hot water becomes a practical serious problem. We introduce a numerical original model able to predict the critical oil rate for which the wells can be totally invaded by geothermal brine. For the construction of the model we apply classic laws and equations. We use standard published formulas for both relative permeabilities and capillary pressure. We obtain a single non linear partial differential equation (PDE) which depends only on water saturation, space and time. This PDE is a 3D generalization of the classical 1D BuckleyLeverett model. To solve the new PDE we use non linear finite elements. The numeric simulation could reproduce the effect of water invasion: After some time elapsed, the original oil volume diminishes abruptly, displacing the boundary of the water-oil contact and the transition zone in the vertical direction. Our objective is to estimate the optimum mass rate for producing wells in order to minimize the production of water or to achieve a mixture oilwater extraction where oil always prevails. INTRODUCTION The production of petroleum together with connate water is a common phenomenon in oil and gas reservoirs. This water is unusable, although its operation is, in general, quite expensive. The magazine Oilfield Review (Arnold, 2004) reported that only in the USA there are extracted 10 barrels (1.6 m) of water for each barrel of oil. In the whole world three barrels of water for each one of oil are produced. The cost of this water disposal is between 5 and 10 thousand million dollars in the USA and approximately of 40 thousand million dollars in the whole world. Even using the most advanced disposition techniques, water can represent 90% of the total volume of liquids at field’s surface, impacting seriously the commercial feasibility of the field. Due to its null commercial utility, this water should be reinjected into the formation to maintain reservoir pressure. Another possible future use is its treatment to make it potable and usable in the hydraulic nets of cities close to the oil field. GEOTHERMAL AQUIFERS AND OIL FIELDS Geothermal areas related to hydrocarbons reservoirs also exist in different parts of the world. The presence of interstitial hot water in the pores alters several parameters of the reservoir. The non isothermal conditions affect the effective saturations and the relative permeabilities of both immiscible phases. The relative permeability of oil is increased by the increase of temperature originated by geothermal water. At the same time the dynamic viscosities of water and oil diminish, affecting the displacement of both fluids. The Bellota-Jujo hydrocarbon complex, located in the southern coast of the Gulf of Mexico (Fig. 7), is a remarkable example of this type of coupled processes. The Port Ceiba reservoir, which is part of this system, is associated to an aquifer located 6000 meters under the surface of the field. For this reason it contains brine and hydrocarbons. The water in this reservoir flows vertically toward the production wells, through conductive faults, which connect the oil zone with the deep aquifer. The water of the aquifer is geothermal brine at 150°C, having a density of 1150 kg/m. Port Ceiba’s wells are oil producers, but some of them are invaded by brine, producing a variable mixture of water and oil. Although the oil is extracted at the upper zone of the oilwater contact (COW), the total volume of produced water equals or exceeds the oil production. The effect of water invasion, together with oil extraction, produces a gradual decrease of the original volume of oil and a vertical displacement of the COW. In this way the well receives more and more water until it becomes completely invaded. The handling of this water in the formation is a serious practical problem costing millions of pesos to the company every year. The main goal of this research is the understanding of the water invasion mechanism and the estimation of the critical volumetric rate in oil wells for which the invasion begins to happen. The model should allow predicting with precision this critical rate and, consequently, to be able to reduce the extraction rates in wells just on time, maximizing its productive life. In this work a numerical original model is developed, able to perform this task. GENERAL DESCRIPTION OF THE PROBLEM Hypothesis and Qualitative Information Available The brine in the formation has different physical behavior compared to hydrocarbons. Water conducts as a substance having a molecular weight larger than 18. This behavior is due to the fact that intramolecular forces of water are more intense than those of petroleum (Pedersen and Christensen, 2006). Due to superficial tensions, a great amount of oil is caught into the pores, in such a way that the mobility of the invasion water prevails. For heavier and more viscous oils, the mobility of water will dominate in the immiscible mixture of both fluids. This phenomenon is described by the total mixture rate q = qw + qo and by the quotient qw /qo = λw /λo > 1; λj = κj /μj is the phase mobility, κj its permeability and μj its dynamic viscosity (j = water, oil). If the volumetric rate of the well is very high, the produced fluid would be predominantly water. We call Bw = ρwS /ρwR the volume factor of water in the formation (density of water at standard conditions divided by density of water at reservoir conditions). This factor represents the expansion of the volume of water between the formation and the surface of the field. Assuming this expansion small, we will take the value Bw ≈ 1. The following information is available: • Geothermal water invades the oil reservoir through a fault that penetrates an aquifer at 6000 m of depth and 150°C of temperature. • The geothermal aquifer and the oil reservoir form a geologic unit system, delimited at their boundaries by impermeable rocks forming a profound closed and isothermal volume. • Water flows from the deep aquifer to the reservoir because of pressure variations at the COW. • Darcy’s Law and Continuity equation are valid in both phases. • Relative permeabilities and capillary pressure only depend on saturations. • The following parameters are constants: Rock permeability, viscosities and densities of both phases. FIELD DATA Available numerical data are summarized in Table 1 (Suarez & Samaniego, 2006). Average pressure pa = 940 kg/cm 2 Bottom flowing pressure pwf = 700 kg/cm 2 Volumetric rate q0 = 11000 Bce/D Oil Density ρo = 770 kg/m Water Density ρw = 1145 kg/m Pressure difference ∆ pw pa-pwf = 240 kg/cm 2 Vertical distance between The well and the COW ∆H = 375 m. Temperature of brine 150°C Capillary pressure Pc (Sw ) = po pw Saturations Sw + So = 1 Table 1.Numerical information from well PC-115 of the Puerto Ceiba Reservoir (PEMEX – PEP, 2004). A fundamental formula relating capillary pressure and capillary height is: 2 ( ) , , 9.8 / c w c w o P S h g g m s ρ ρ ρ ρ = ∆ ∆ = − = (1) Where hc is the height over the plane of capillary pressure pc = 0. This surface is the boundary of the oil-water contact (COW) where Sw ~ 1, So ~ 0. The transition area is the place where both phases coexist. The residual saturation of water Swi is reached at the point of the reservoir where So ~ 1. Relative Permeabilities and Capillary pressure For the capillary pressure the experimental values reported by Aziz (1999) were used, together with equation (1). The relative permeabilities for water and oil we used are the correlations proposed by Brooks and Corey in 1964 and verified experimentally in a recent publication (Cunha et al., 1999). The analytic expression of these correlations are as follows: