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Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
EXPERIMENTAL TECHNIQUE An Experimental Technique for Obtaining Permeability-Porosity Relationships in Acidized Porous Media Conwell C. McCune Chevron Oil Field Research Go., La Habra, California
H. Scott Fogler' and William E. Kline Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48 109
A novel technique has been developed for obtaining permeability-porosity relationships in naturally consolidated porous media. The results show that the local change of permeability with porosity is an order of magnitude greater than previously reported. The method may also be used for determining mass transfer coefficients in naturally consolidated systems where the acidization rate is diffusion controlled.
Introduction Matrix acidization is an important method of improving the productivity of oil and gas wells. This is a procedure in which HC1 or HF-HC1 mixtures are pumped down the well bore and out into the pores of the formation rock. The resultant dissolution of material from the pores gives rise to an increase in permeability. Laboratory studies are often carried out prior to field tests in an attempt to determine the amount and in what proportions acid is to be injected into the reservoir (Farley et al., 1970). Figure 1 shows a sequence for predicting the productivity changes resulting from a given acid treatment. The first step is to establish the dimensionless concentration of dissolvable minerals as a function of position and time, o ( x , t ) . As acid flows through the porous rock, dissolution occurs within a narrow zone which moves as a reaction front. The movement of this front has been modeled by Lund and Fogler (1976). The model requires only two dimensionless parameters- a Damkohler number and an acid capacity number. The Damkohler number, related to the shape of the dissolution front, is a function of the rate of reaction. The acid capacity number reflects the velocity of the front and is dependent on the reaction stoichiometry. These parameters are readily determined by only two experiments in which acid is pumped through core samples a t high and low flow rates, respectively. The porosity as a function of position and time, $ ~ ( x , t )then , follows easily, since the porosity change is a linear function of the amount of mineral dissolved, q ( x , t ) . If it is to be applied in determining changes in reservoir production rates, the flow and reaction model must further be extended to predict the permeability as a function of position and time, K(x,t). The critical final step, therefore, is the precise determination of the relationship between porosity and permeability. This paper describes a new laboratory technique which allows experimental determination of the permeability-porosity relationships in 0019-7874/ 79/ 1018-0188$01.OO/O
naturally consolidated rock, thus enabling the entire procedure in Figure 1 to be completed with actual data from the formation to be acidized. Tests were conducted on limestone and sandstone. The experimental results show that previously used permeability-porosity correlations predict permeabilities which are at least an order of magnitude low. Experimental Apparatus and Procedure The high pressure, high temperature permeameter used in this study has been described previously (Farley et al., 1970). The equipment is designed to handle the flow of hot acids, including HF-HC1 mixtures. Cores may be treated under simulated reservoir conditions of up to 250 O F , 4000 psi injection pressure and 10000 psi overburden pressure. Those equipment parts contacted by corrosive acid are constructed of Hastelloy B. Temperature, pressure, and flow data are automatically scanned and recorded for computer input. A schematic diagram of the apparatus is shown in Figure 2. The test cores for the runs reported here were in the form of disks, l l / zin. in diameter and 0.25 in. or less in length. A very thin core such as this may be considered analogous to a differential reactor, in that a t any time during acidization the variation of permeability, porosity, and acid concentration over the length of the disk is small. The disk mounting assembly is illustrated schematically in Figure 3. For limestone tests the disk was mounted between two 3/16-in.sintered stainless steel flow distributors. In the sandstone tests this purpose was served by six 100-mesh Hastelloy screens mounted on either side of the disk. The entire assembly was held together by endplates and sealed around the circumference with shrinkable Teflon tubing. The overburden pressure was applied with oil to the outside of the tubing. The fluids injected through the disks were driven by oil displacement. This study recorded pressure drops across the test disks a t constant volumetric flows, so a con1979 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979 IBUFFER
1;
bk%:,z
A
1
DISSOLUTION
BUFFER
DISSOLUTION
BUFFER
DISSOLUTION
189
BUFFER
1; ,"a
z,R ,E PERMEABILITY FROM A C I D INJECTION
--+?-- I M O V E M E N T OF DISSOLUTION F R O h T
TM O V E M E N T OF POROSITY F R O N T
Figure 1. Prediction of production rate changes following acidization.
+ J
TIME t OIL
Figure 4. Fluid injection sequence in permeability-porosity tests. Table I. Composition of Phacoides Sandstone wt%
quartz microcline feldspar albite feldspar dolomite illite
u WASTE
Figure 2. Schematic diagram of acid permeameter. PRESSURE OIFFERENTIAL
I
L
l O IN
U
I
O
~
u
SlNlLREOJ OISI OR SCREEhS
'ROCK OISU
A
LIOUlO OUT
'SHRINKABLF TEFLOM
Figure 3. Disk mounting assembly.
stant-flow diaphram pump was used to pump the oil. Small bore tubing (0.086 in. id.) was used for the acid lines in order to minimize liquid holdup. Before each run the test disk was vacuum saturated with water and mounted as described above. The overburden pressure was applied a t 3000 psig and the temperature was controlled a t slightly above room temperature. A permeability fluid, distilled water in the limestone tests and 1% HC1 in the sandstone tests, was pumped through the disk until a stable pressure drop was achieved. From this the initial permeability was calculated from Darcy's law. K AP 4=---
F
L
During the entire test the fluids were pumped at a constant volumetric flow rate against a constant downstream pressure of 1000 psig. Therefore, permeability varied inversely as the pressure differential. The porosity and permeability of the disk were then increased in a stepwise fashion through a series of injections of small volumes of dissolving acid (0.01 N HC1 for limestone disks and HF-HC1 mixtures for sandstone
78.4 12.5 6.6 1.8 1.7
disks). Each acid injection was followed by the permeability fluid to measure the resultant permeability and to provide a buffer between successive dissolutions. Figure 4 depicts the flow sequence schematically. The corresponding porosities were determined by measuring the dissolved mineral in the effluent from the start of each acid injection until the end of the subsequent permeability fluid step. In the limestone tests, the effluent was analyzed for calcium by atomic absorption spectrophotometry. From this the mass of CaC03 was calculated. The fractional porosity change was assumed to equal the fractional weight loss. In a similar manner, effluent from the sandstone disks was analyzed for Al, Ca, K, Na, and Si. Since all of the negative valence in silicate lattices is provided by oxygen, the dissolved mass can be taken as the mass of the corresponding oxides. This assumption has been verified by X-ray analysis of acidized cores. The sequence of dissolution followed by permeability measurement was repeated until the pressure differential was below the range of the instrumentation. Between three and seven cycles per disk were usually possible. Results and Discussion The experimental technique described above was used to determine the permeability-porosity relationships for Spergen limestone and Phacoides sandstone. Spergen limestone is a highly consolidated carbonate with a porosity of about 15% and a permeability of 0.5-1 millidarcy. It contains 96-98% CaC03. A series of rotating disk experiments (Lund et al. 1975) has previously shown that the dissolution of CaC03 in HC1 is mass transfer limited. The composition of Phacoides sandstone is listed in Table I. The dolomite fraction (CaMg(C0,)J was removed in each sandstone run by the initial injection of the 1%HC1 permeability fluid. The remaining quartz and aluminosilicate components are virtually inert to room temperature HC1 (Fogler et al., 1975). Rotating disk studies have further shown that the consumption of H F during sandstone acidization is due almost entirely to the reaction rate limited dissolution of the aluminosilicate feldspars and clays (Fogler et al., 1975).
Ind. Eng. Chem. Fundam., Vol. 18, No. 2, 1979
190
i
0
0
0 6
1
P A &
I O
1005
P'
'
1
1010
1
'
1
' 1
'
1
'
'
I ' '
'
1015 1020 POROSITY R A T I O p/po
1
I IO25
" "
I
1
1030
'
"
I
1035
10
20
15 P O R O S I T Y RATIO
25
do,
Figure 5. Local permeability as a function of porosity (Spergen limestone).
Figure 6. Local permeability as a function of porosity (Phacoides sandstone).
An initial permeability was measured for the test disk a t the start of each experimental run. The permeability and porosity of the disk were then increased in a sequential fashion by successive injections of the dissolving acid. Porosity changes were calculated from the dissolved mineral in the effluent, while permeabilities were measured by flowing the permeability fluid through the disk after each acid injection. The experimentally determined permeability-porosity relationships for Spergen limestone and Phacoides sandstone are shown in Figures 5 and 6. Small increases in porosity result in permeability increases which are orders of magnitude greater than those previously reported. The shaded band in Figure 6 represents the permeability porosity relationship predicted by the pore collision model (Schechter et al., 1971, 1972), while the dashed line is obtained by the Ergun Equation (Bird et al., 1960).
conducted primarily with manufactured sintered media having a much narrower pore size distribution than is found in naturally consolidated porous rock. One would expect that the sintered disks would be dissolved in a much more spatially uniform manner due to acid penetration into a large fraction of the pores. The reactive surface area within the Phacoides disks was calculated from eq 3
E = ( : ) ( = )1'
- 40
KO The much larger magnitude of the permeability increases observed in these experiments is probably due to two factors. First, natural porous media contains small particles which can become lodged a t pore openings. Permeabilities can be increased substantially simply by removing these constrictions. Secondly, it is probable that almost all the dissolution in natural rock takes place within only a small fraction of the total pore space. Swift and Fogler (1977) measured the pore size distributions in acidized and unacidized Phacoides cores using a mercury injection technique. They observed that while the Phacoides pore sizes ranged from 0.03 to 30 pm, dissolution took place only in those pores larger than about 1 pm. Previous experimental work, on the other hand, has been
A =
disk
($>,iff, "
(3)
'a
where ra is the rate of dissolution per unit area from rotating disk studies (Lund et al., 1975). The reactive area was found to be about 30 cm2/g. This is much smaller than the total surface area of 1to 2 m2/g as measured by BET nitrogen adsorption. It agrees well, however, with the macropore surface areas reported by Swift and Fogler (1977). In longer cores, of course, the porosity and permeability cannot be considered invariant with distance along the core. The permeability as a function of axial distance changes as the dissolution front moves through the core. The movement of this permeability front is a function of the ratio of the rate at which the acid reacts to the rate a t which it is transported by convection (Damkohler number), the ratio of the acid present in the void fraction to the acid necessary to consume the dissolvable minerals in the core (acid capacity number), and the functionality between porosity and permeability. When the porositypermeability relationships determined from the differential disk experiments were used with the previously described flow and reaction model to predict the permeability fronts in longer sandstone cores, excellent agreement was achieved between measured and calculated permeability
Ind. Eng.
t
4a00
PORE VOLUMES INJECTED
Figure 7. Permeability of a sandstone core vs. volume of injected acid: HF/HCl concentration, M, 2.05/1.25; space time T , min, 0.134; acid capacity no., 0.010; Damkohler no., 1.07.
ratios as a function of the acid injection time (Figure 7). Application of Experimental Method for Determination of Mass Transfer Coefficients This method has further application in determining mass transfer coefficients for diffusion-controlled dissolution in porous media. Rotating disk studies, for instance, have shown that the rate of dissolution of limestone in hydrochloric acid is determined by the transport of hydrogen ions to the solid surface (Lund et al., 1975). The convective flux, NA, is given by NA= ~ ( C -B cs) (4) where k is the mass transfer coefficient and CB - Cs is the acid concentration difference between the bulk and the solid surface. Since the surface reaction is extremely rapid (Lund et al., 19751, the surface concentration Cs may be neglected. This flux is rapid enough so that the differential reactor assumption of negligible variation in the bulk acid concentration is not valid, even for disks as thin as 0.0625 in. Assuming a mean mass transfer coefficient k a , the variation of acid concentration within the disk is described by
(5) where CA = C, a t z = 0. Integrating ka =-Q In C,/CA (6) V In terms of the measured concentration of dissolved calcium, [Ca2+]
The flow rate Q must be sufficiently large to ensure that the acid is not entirely depleted during flow through the disk, Le., so that the breakthrough acid concentration is nonzero. For the HC1 dissolution of a 3/32-in.Spergen disk a t room temperature, the lowest flow rate a t which this requirement was satisfied was 5.4 cm3/s. For several runs between 5.4 and 24.6 cm3/s, the calculated mass transfer coefficients k a were of the magnitude 0.1-1.0 These
Chem. Fundam.,
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191
values are in the range one would calculate from first principles for straight, cylindrical pores with radii between 5 and 50 pm. Summary An experimental technique has been developed for the determination of permeability-porosity functionalities of acidized porous media. Results were obtained for Spergen limestone and Phacoides sandstone. For limestone the increase in permeability resulting from an increase in porosity was found to be larger than an order of magnitude greater than that reported previously. In the case of sandstone the permeability was not found to increase with porosity as sharply as with limestone, but it does increase a t a rate approximately one order of magnitude greater than previously reported. The technique described also has application to the determination of mass transfer coefficients in naturally consolidated porous media. Preliminary measurements of the diffusion-controlled dissolution of Spergen limestone by HC1 give mass transfer coefficients k a with magnitudes between 0.1 and 1 s-'. Nomenclature A = surface area a = surface area per unit volume A , = acid capacity number A,, = superficial cross sectional area CB = acid concentration in bulk fluid C, = concentration of dissolved mineral in effluent Cs = acid concentration at solid surface D = diameter Da = Damkohler number K = permeability k = mass transfer coefficient L = length m = mass of mineral N A = acid flux to reacting surface P= pressure differential Q = volumetric flow rate q = fluid flux through superficial core area r. = dissolution rate per unit area t-= time V = superficial disk volume x = axial position Greek Letters fi
4
= fluid viscosity =
porosity
= dimensionless concentration of dissolvable minerals u = stoichiometric coefficient
Literature Cited Bird, R. B., Stewart, W. E., Liqhtfoot, E. N., "Transport Phenomena", Wiley, New York, N.Y., 1960. Farley, J T., Miller, B. M.,Schoettle, V., J . Pet. Technot., 433 (Apr 1970). Fogler, H. S.,Lund, K., McCune, C. C., Chem. Eng. Sci., 30, 1325 (1975). Guin, J. A., Schechter, R. S.,SOC. Pet. Eng. J . , 11, 390 (1971). Lund, K., Fogler, H. S.,McCune, C. C., Auk, J. W., Chem. Eng. Sci., 30, 825 (1975). Lund, K.,'Fogler, H. S.,Chern. Eng. Sci., 31, 1381 (1976). Schechter, R. S.,Gidley, J. L., AIChE J., 15, 339 (1969). Sinex, W. E., Schechter, R. S.,Silberberg, I . H., Ind. Eng. Chem. Fundam., 11, 205 (1972). Swift, S. T., Fogler, H. S.,Chern. Eng. Sci., 32, 392 (1977).
Received for review February 21, 1978 Accepted December 14,1978
