*Oil & Gas Science and Technology - Rev. IFP*Vol. 51 (1996), No.4, pp. 527-558

## Méthode analytique généralisée pour le calcul du coning. Nouvelle solution pour calculer le coning de gaz, d'eau et double coning dans les puits verticaux et horizontaux

### Generalized Analytical Method for Coning Calculation. New Solution to Calculation Both the Gas Coning, Water Coning and Dual Coning for Vertical and Horizontal Wells

Beicip-Franlab

Une nouvelle méthode analytique d'évaluation du coning d'eau par bottom water drive et/ou de gaz par gas-cap drive dans les puits horizontaux et verticaux a été développée pour les réservoirs infinis [1]. Dans cet article, une généralisation de cette méthode est présentée pour les réservoirs confinés d'extension limitée dont le toit est horizontal. La généralisation proposée est basée sur la résolution des équations différentielles de la diffusivité avec prise en compte des effets de drainage par gravité et des conditions aux limites pour un réservoir confiné. La méthode est applicable aux réservoirs isotropes ou anisotropes. L'hypothèse de pression constante à la limite de l'aire de drainage dans l'eau et/ou dans le gaz a été adoptée. Les pertes de charge dans l'aquifère et dans le gas-cap sont donc négligées. Les principales contributions de cet article sont : - L'introduction de la notion de rayon de cône, différent du rayon de puits. La hauteur du cône et le débit critique dépendent du rayon de cône alors qu'ils sont indépendants du rayon du puits. - Une nouvelle corrélation pour le calcul du débit critique sous forme adimensionnelle en fonction de trois paramètres : le temps, la longueur du drain horizontal (nulle pour un puits vertical) et le rayon de drainage. - Des corrélations pour le calcul du rapport des débits gaz/huile (GOR) ou de la fraction en eau (fw), pendant les périodes critique et postcritique, qui tiennent compte de la pression capillaire et des perméabilités relatives. - Des corrélations pour le calcul des rapports de débits gaz/huile et eau/huile pendant les périodes pré, post et supercritique en double coning. - Des critères pour le calcul du temps de percée au puits en simple coning de gaz ou d'eau, ou en double coning de gaz et d'eau.

Abstract

A new analytical method for assessing water and/or gas coning in horizontal and vertical wells has been developed for infinite reservoirs [1]. This article gives a generalized description of this method for confined reservoirs. The case corresponding to a horizontal top is considered (bottom water drive and gas-cap drive). The generalization proposed is based on the solving of differential equations for diffusivity, while taking gravity drainage effects and boundary conditions into consideration for a confined reservoir. The method can be applied to isotropic or anisotropic reservoirs. The principal contributions of this article are :- Introduction of the concept of coning radius, differing from the well radius. The coning height and critical flow rate depend on the coning radius, whereas they are independent of the well radius. - A new correlation for calculating the critical flow rate in an adimensional from as a function of three parameters :. time,. length of the horizontal drain hole (zero for a vertical well), . and drainage radius. - Correlations for calculating the gas/oil ratios (GOR) or the water fraction (fw) during critical and postcritical periods, which take capillary pressure and relative permeabilities into consideration. - Correlations for calculating the gas/oil and water/oil ratios during the pre, post and supercritical periods of dual coning. - Criteria for calculating the breakthrough time at the well for simple gas or water coning, or for dual gas or water coning. The critical flow rate varies with time. It depends on the geometric and physical parameters of the reservoir, such as the thickness of the layer or the drainage area of the well, and on an important parameter, the coning radius, which will be defined hereunder. On the contrary, the critical flow rate does not depend on the parameters characterizing the well or the near-well vicinity, i. e. the radius and damage (skin).N.B. No confusion must be made between the critical flow rate due to coning and other flow rates associated with the way the well operates, such as the potential flow rate of the well defined by the productivity index, or again the potential pumping flow rate.The height of coning is given by Equations (1.1) and (1.3). For dual coning, the coning heights depend on the same pressure drop DeltaPo in the oil zone between the drainage boundary and the coning radius and are inversely proportional to the differences in specific gravity.As shown in Figure 2, the equipotential passing via the coning apex cuts through the top of the reservoir, which is assumed to be horizontal here, at point A2, situated at distance re from the axis of the well. We have called this distance the coning radius(see Eq. 1. 4). The coning radius is a very important concept since it defines the origin for calculating the pressure drop governing the critical coning amplitude. This coning radius is involved in the correlations given in Section 3. The new correlations proposed are given in an adimensional form (Eq. 2. 1 and Figures I and II outside the text). The geometric parameters involved in the equations are given in Figure 4 for the vertical well and in Figure 5 for the horizontal well. A horizontal well with length L is considered to be the equivalent of a well made up of a horizontal drainage hole with length Lr and two vertical semiwells with radius 2 rc (Eq. 2. 12 and 2. 13). Correlations have been determined by integrating the diffential equations (see Ref. [1] for two types of boundary conditions) : constant flow rate or constant pressure. We have given a new solution for the horizontal well by assuming that the flow rate in the sum of the flow rates for a drain hole with lenght Lr and two vertical semiwells situated at the ends of the horizontal drain hole (Fig. 6). The correlations are used in three forms :- Calculation of the flow rate (Eq. 2. 22) for a given geometry and for a given breakthrough time;- Calculation of the coning height (Eq. 2. 23) for a given breakthrough time and a constant flow rate in time;- Calculation of the breakthrough time (Eq. 2. 24) for a constant flow rate in time. The dual coning pattern is shown in Figure 8. From solely the standpoint of coning, the optimum position is the one corresponding to the simultaneity of the two water and gas breakthroughs (Eq. 3. 5 and 3. 6). Several horizontal planes are involved in the characterization of double coning. First of all, there is the dual coning convergence plane (DCCP) at depth Zdccp (Fig. 9). In the extreme case, the two gas and water cones cut across the level of this plane, and the oil flow rate becomes zero (Eq. 3. 7). The optimal up perforation (OUP) and optimal down perforation (ODP) boundaries, the dual breakthrough upper limit (DBTUL) above which the dual breakthrough is not possible, only gas can break through, and the dual breakthrough down limit (DBTDL) beyond which only water breakthrough is possible are defined. The five characteristics areas for breakthrough for dual coning are given in Figure 9 :- gas breakthrough alone,- dual gas breakthrough followed by water,- simultaneous double breakthrough of gas and water,- dual breakthrough of water followed by gas,- and breakthrough of water alone. Capillarity plays a different role depending on the phases present. In the water/oil system (w/o), this role is negative. Water breakthrough is faster when capillarity is involved. On the contrary, in the gas/oil system (g/o), it is positive. The gas breakthrough is delayed by capillarity. The new correlations have been determined by assuming the hypothesis that fluid flow rates (gas, oil and water) depend on the perforation height covered by these fluids (Fig. 10). The correlations for calculating the water fraction (fw) were determined by Pietraru and Cosentino [1] as a function of coning height (hcw or how), perforation height ho, capillary pressure (Fig. 10 and 11) and the water-oil mobility ratio (Eq. 4. 7 and 4. 8). The correlations given above, with the equations for functions F, derived in the following sections, can be used to solve the following water-coning problems :- Calculating the watercut (fw) for a given coning height (hcw). First, how must be calculed using Equation 4. 2. - Calculating the height of the coning peak (hcw) for a given watercut (fw). In a similar way, the GOR is calculated for gas coning. The correlations given above were developed while considering that the production flow rate is constant. However, application of the superposition principle can be used to apply the proposed methods to cases of variable flow rate, hence the following options that may be analyzed with this method. Constant oil flow rate : This option has a theoretical nature. It can be applied in practice only up to breakthrough. The watertut, which is insignificant up to breakthrough, effectively increases very quickly after breakthrough. Constant total liquid flow rate : This is the simplest variant in practise, especially when the wells are pumped. The oil flow rate decreases in time. Constant watercut : With a continuous decrease in the oil flow rate in time. This option can be used to optimize the cumulative oil produced. It is difficult to apply in practice, but it is the ideal variant. Production with decrease of the liquid flow rate in stages : This is an interesting combination, in practice, of the preceding two flow regimes. This regime consists of a succession of constant liquid flow ratestages with a decrease in the flow rate each time the fixed acceptable limit of the fw (or GOR) is reached. The method proposed in this article is approximate because of the simplifying hypotheses introduced and because of :- the coning heights are calculated without taking the existence of perforations into consideration;- for high watercuts, the zero pressure-drop hypothesis in the aquifer has been retained, although it is not very realistic. The method has been checked, at least in the range between 1 and 10 to the power of 4 in adimensional time td, by comparison with :- production data,- laboratory experiments, - and numerical simulations. Data published for four wells [9, 10, 11 and 12] were used for the comparison. The well parameters as well as the breakthrough time and the critical flow rates are given in Appendix F. The results of the comparison are shown in Figure 14. The comparison of parameters td, qd, and Ld, both calculated and observed, shows satisfactory agreement of correlations with production data with differences of about ý 20%. The only published results of laboratory experiments with fairly complete data are those by Sobocinski [3] and Bournazel [12]. The empirical correlations proposed after examining the correlations proposed by these authors have been transcribed into adimensional parameters (§ 3). The experiments by Sobocinski and Bournazel covery two partially superposed fields that are limited in time, hardly two cycles in log scale (Fig. 15). The differences may be as much as 50%. After nonexhaustive investigations, we found fairly complete results in articles by Papatzacos [8], Tiefenthal [12] and Arbadi [14]. After computing the adimensional times and flow rates, td and qd, the results found by these authors are given in Figure 14. The results of simulations by the three authors are quite different. Arbabi's simulations are in agreement with the correlations. Papatzacos's results give flow rates twice as large as those provided by the correlations, whereas Tiefenthal's results are 30% lower. We do not know the specific conditions under which the simulations were performed, and thus a systematic study for the validation of the new method is desirable.

*© IFP, 1996*