Boundary-Dominated Flow in Solutions-Gas-Drive Reservoirs

Abstract

Summary The performance of wells in solution-gas-drive reservoirs during the boundary-dominated flow period is examined. Both constant-wellbore-pressure and constant-oil-rate production modes in close systems are considered. For the constant-wellbore-pressure production mode. Arps' performance-prediction equations are examined, and predictions of future performance are shown to be strong functions of well predictions of future performance are shown to be strong functions of well spacing, well condition, and fluid properties. The parameters b (the decline exponent) and di (the initial decline rate) in the Arps equations are expressed in terms of physical properties. The conditions under which these equations can be used are specified. An empirical procedure to predict production rates is also presented. In the case of procedure to predict production rates is also presented. In the case of constant-oil-rate production, an expression to correlate the pressure distribution in the reservoir is presented. The correlating function permits us to extend the definition of pseudosteady-state flow to permits us to extend the definition of pseudosteady-state flow to solution-gas-drive systems. Its use also allows the simultaneous computation of average properties (pressure and saturation) during boundary-dominated flow from wellbore information. Introduction This work documents some theoretical results that are useful for predicting well performance from production data in predicting well performance from production data in solution-gasdrive reservoirs during the boundary-dominated flow period. In the process of documenting these results, we also furnish theoretical support for empirical observations that exist on this subject. Both constant-wellbore-pressure and constant-oil-rate production modes in circular closed systems are considered. The outcomes presented are based on the theoretical results presented in Refs. presented are based on the theoretical results presented in Refs. 1 and 2. Specifically, the results given here follow from our ability to correlate responses of solution-gas-drive systems with the response of a slightly compressible liquid flow during the boundary-dominated flow period for constant-oil-rate and constant-wellbore-pressure production modes. This work is divided into three parts. First, the theoretical results related to boundary-dominated flow given in Refs. 1 and 2 are outlined to establish a framework for findings presented in this paper. Second, the case of a well flowing at a constant pressure paper. Second, the case of a well flowing at a constant pressure during the boundary-dominated flow period is analyzed. For this case, Fetkovich showed that Arps' empirical family of curves can be combined with the slightly compressible liquid flow solution (exponential decline response) to obtain a family of curves that can be used to predict future performance and to estimate the reservoir PV. Refs. 1 and 6 report that during the boundary-dominated period, the rate response vs. time does not match a fixed value of the decline exponent, b, in the type curves of Ref. 3. An explanation for this observation is presented. Refs. 3 and 7 emphasize that b must be less than or equal to unity, but note that if transient data are used, the value of b in the Arps solution can be greater than unity. The development for transient flow given in Ref. 8 is used to provide a theoretical justification for this observation. An empirical procedure to predict production rates of wells produced at a constant pressure over short time spans is also produced at a constant pressure over short time spans is also presented. Third, the situation when production is held at a constant oil presented. Third, the situation when production is held at a constant oil rate is considered. For this case, it is known from Refs. 1, 2, and 9 that the reservoir does not achieve the pseudosteady-state condition; i.e., the derivative of pressure with respect to time is not constant and is not independent of position in the reservoir. In this work, a correlation for the pressure distribution in the reservoir during the boundary-dominated flow period is developed. This correlating function allows an extension of the pseudosteady-state concept to solution-gas-drive reservoirs. Furthermore, this function allows simultaneous computation of the values of average pressure, P, and average saturation, So, from wellbore information. The numerical results presented in this paper were obtained with a finite-difference model described in Ref. 1. Procedures followed to ensure the accuracy of the solutions are given in Refs. 1 and 2. Mathematical Model A homogeneous closed cylindrical reservoir with a fully penetrating well located at its center is considered. The well is penetrating well located at its center is considered. The well is capable of producing at either a constant oil rate or a constant wellbore pressure. An annular region concentric with the wellbore. with a pressure. An annular region concentric with the wellbore. with a different permeability from the formation is used to include the effect of a skin region. The effects of gravity, capillary pressure, and non-Darcy flow are not considered. Figs. 1 and 2 show the PVT properties of fluids used in this work. Fig. 3 presents relative permeability data. The data sets shown in Figs. 1 through 3 are identical to the data sets considered in Refs. 1, 2, 8, and 11 and are used here mainly to preserve continuity. The conclusions derived in this work do not depend on the specific data used in the simulations. Table 1 presents information on the range of variables examined. Background Following the development in Ref. 1. the dimensionless pseudopressure is defined as pseudopressure is defined as kh r delta p PpD(r, t)= --------------- (p, So) ------------ dr' PpD(r, t)= --------------- (p, So) ------------ dr' 141.2q (t) r delta r' t t delta p + (p, So)----------- dt' .......................(1) 0 delta t' r Here, (p, So)=kro(So)/[ o(p)Bo(p)],............................(2) and r=radius corresponding to the position in the reservoir at which p(r)=p. During boundary-dominated flow period, r 0.54928re (see Fig. 5 of Ref. 2). Ref. 1 establishes that during boundary-dominated flow, the following results are valid: 3 K 1 r4sD r3sD ppD (r, t)=PpD(t)+ 1n reD- - + s + --- 1 ---------------------- ppD (r, t)=PpD(t)+ 1n reD- - + s + --- 1 ---------------------- Ks 4 r4eD r3eD 2 k 1 (rD -1) 1n rD------------------ ........................(3) 2 ks 2 reD for 1 less than rD less than rsD, where rD=r/rw, and reD 3 k 1 rs4D r2sD PpD(r, t)- PpD(t)+ 1n----------+ ---1 ----------------- PpD(r, t)- PpD(t)+ 1n----------+ ---1 ----------------- rD 4 ks 4 r4eD r3eD 1 (r2sD-1) 1 (r2D-1) + --- ------- +----------...........................(4) 2 r3eD 2 r3eD for rsD less than rD less than reD. Here rsD=dimensionless radius of the skin zone and PpD(t) = volumetric average of the pseudopressure. By using the Muskat material-balance equation, we show in Ref. 1 that kh Pi PpD(t) = ----------------- (p', r) dp'=2 tAD,............(5) PpD(t) = ----------------- (p', r) dp'=2 tAD,............(5) 141.2q0(t) p(t) SPERE P. 503