Controls of pore throat radius distribution on permeability

1. Introduction

For two phase flow through granular media, capillary pressure can be viewed as the pressure required for driving the non-wetting fluid through a pore throat. Capillary pressure increases as the pore throats become smaller. The size and distribution of pore throats within a host rock controls the capillary pressure characteristics, which in turn controls the fluid behavior in the pore system(Jennings, 1987). Babadagli and Al-Salmi (2002) reviewed the equations used to predict permeability from different rock parameters. These authors did not consider pore throat radii.

Early studies focusing on the relationship between porosity, grain size and permeability were presented by Carman (1937) and Scheidegger (1974). Perhaps one of the most relationships between porosity and permeability is the Kozeny-Carman equation. The conventional form of the Kozeny-Carman equation formulate using the grain size to estimate permeability Freeze and Cherry (1979):(1) $k = \frac{{d_{50}}^{2}}{180} \frac{ϕ^{3}}{(1 - ϕ)}$ Where, $k$ is permeability (μm2), $d_{50}$ mean grain diameter (μm), $ϕ$ a porosity, fraction, S the specific surface area, τ a tortuosity and d is the grain diameter (μm).

However, this form assumes that the mean grain size can be related to surface area Bear (1972):(2) $d_{50} = 6 / S$ Where, S is the specific surface area.

Then the Kozeny-Carman equation becomes (Lala, 2013, Antonio, 2006):(3) $k = \frac{1}{2} ϕ^{3} / S^{2} τ^{2} = ({d_{50}}^{2} / 72 τ^{2}) \times (ϕ^{3} / 1 - ϕ^{2})$ Where, τ is the tortuosity.

The value of porosity is less important than the pore size distribution. A shortcoming of the Kozeny-Carman equation is that using grain size to estimate permeability will not account for diagenetic changes in pore radii. Also, it is impractical to measure tortuosity, shape factor and specific surface area for each rock type.

For constant porosity, permeability can be shown to be directly proportional to r2 (Lala, 2013):(4) $k = r^{2} \frac{ϕ}{8 τ^{2}}$

There have been many other attempts to relate permeability to petrophysical data since the Kozeny Carman relationship was first introduced. Timur, 1968, Coates and Denoo, 1981, EL Sayed, 1996 introduced new empirical equations to predict permeability using only porosity relations. Bethke et al. (1991) and Neuzil (1994) introduced a commonly used log relationship between the porosity and permeability. This relation shows a general trend of decreasing permeability. However, there is great variance using this relationship when applied to lithified reservoir rocks. Coates et al. (1997) developed an equation to estimate the permeability from the results of NMR logging tool as,(5) $k = (10^{4}) (ϕ^{4}) [{(ϕ_{m a c} / ϕ_{m i c})}^{2}]$ where, $ϕ_{m a c} i s t h e m a c$ ro-porosity and $ϕ_{m i c}$ is the micro-porosity.

This equation treats the porosity as two segments, macro-porosity (contributes to the fluid flow) and micro-porosity which does not contribute to the fluid flow (irreducible fluids).

More recent studies have focused on the relationships between pore throat radii and permeability. The pore throat radius at 35% pore volume (r35) of a given rock type both reflects its depositional and diagenetic fabric and influences fluid flow and reservoir performance (Hartmann and Coalson, 1990). Martin et al. (1997) attempted to divide the carbonate reservoir into different flow units, each with uniform pore throat size distribution and similar performance. Pore throat sorting (PTS) provides a measure of the degree of heterogeneity of pore throat radius through the thickness of reservoir and/or within a particular system in quantitative terms by applying a numerical value to the slope of the plateau found on a semi-log plot of the capillary pressure data. Values for PTS can be easily obtained by using the following equation, adopted from a sorting coefficient equation developed by Trask (1932):(6) $P T S = {[Q_{3} / Q_{1}]}^{1 / 2}$ where the first (Q1) and third (Q3) quartile pressures are obtained directly from the capillary pressure injection curves and reflect the 25 and 75% mercury saturation pressures adjusted for irreducible water saturation.

The capillary curve provides quantitative assessment of the reservoir rock, using such calculated value of the pore-throat sorting (PTS). It is argued below that this parameter can be used to identify a favorable reservoir rock.

A formation is well sorted when pore throat sorting coefficient equal to one. Poorly sorted formation has a PTS of eight or larger. Moderately sorted formations have PTS coefficients comprised between three and six. The nature of capillary pressure allows interpretive techniques such as PTS to be applied to both clastic and non clastic rocks irrespective of lithology. Laboratory studies have shown that the value of the PTS can have a marked influence of the ultimate recovery of oil (Tobin, 1997, Homaie et al., 2003, Tiab and Erle, 2004; and Dicus, 2008).

In this study a new relationship is presented relating permeability to pore throat radius size distribution parameters (r25, r35, r50, r75). We relate (PTS) to fluid flow permeability and efficiency of pore network.

Below we show that the log of permeability is mostly closely correlated to the log of the pore throat size (r35). The proposed logarithmic equation form of the new model of the form:(7) $l n (K) = 1.71 l n (r_{35}) + 1.9$

Then the estimated permeability using the new model was plotted against the measured permeability on additional new 123 samples (sandstone and limestone) to test the validity of the new obtained model to other reservoir rocks.

2. Materials and methods

A data set of laboratory measurements of porosity, permeability and pore throat radii at different mercury saturations in sandstone and limestone core samples were used to assess the relationship between pore throat radius and permeability. They represented different stratigraphic units of different geological ages, from different geographic areas and basins within Egypt and the Arabian Gulf.

Prior to measuring the petrophysical parameters in the laboratory, the rock samples were cleaned of pore fluid and other contaminants using the Soxhlet cleaning technique (Soxhlet, 1879) and dried in an electric oven by heating the samples to 98 C0, readying them for laboratory measurements.

The porosity of the samples was measured by the Frank Jones helium porosimeter (Lala and Nahla, 2015). A Ruska gas permeameter (cat. No. 6121) was used for measuring the permeability of the samples using nitrogen gas. The clean and dry core samples were pushed into a rubber stopper of the corresponding hole size. Delicate or fragile core samples are first imbedded with a sealing wax into a metal sleeve and placed in a core sleeve holder with the tapered end down.

Our samples were tested for the capillary pressure measurement using mercury injection method (Lala and Nahla, 2015). In this method, mercury is forced into the sample at the low pressure (1.169 kPa up to 779.44 kPa). The volume of mercury injected into the sample at each increasing pressure step is recorded after the stabilized condition has been achieved. The determination of capillary pressure was performed utilizing core lab mercury injection panel. The mercury injection capillary pressure data were converted from laboratory (air/Hg system) into reservoir conditions (oil/water system) using equations from EL Sayed, 1993a, EL Sayed, 1993b resulting in the derivation of pore throat size distribution, r25, r35, r50, r75-parameters.

3. Results

The petrophysical properties of the studied core samples cover a wide range of measured petrophysical parameters. Statistical summary of pore throat radii, porosity and permeability are listed in (Table 1).

Table 1. Statistical summary of pore throat radii, porosity and permeability.

All samples
Empty Cell	min	max	mean	SD
φ (%)	2.3	40	23	8.5
k (md)	0.003	5341	591.65	859.82
r25 (μm)	0.007	56	0.832	0.714
r35 (μm)	0.011	52	9.15	10.95
r50 (μm)	0.004	44	5.21	4.017
r75 (μm)	0.002	30	17.87	7.71

The pore throat sorting parameter (PTS) was determined using equation (3)corresponding to the 25% and 75% percentiles of mercury saturation from the mercury injection curves. It's computed values are ranged from 0.7 to 15.81.

The samples were arranged according to the four sorting classes. An un-sorted class is represented by only three samples. These samples had a PTS values between 8 and 15.8. A poorly sorted class is represented by two limestone and four sandstone samples had a PTS values between 5 and 7.1. A moderately sorted class comprised of five limestone and forty nine sandstone samples had a PTS values between 3.2 and 5. A well sorted class comprised of 33 limestone and 123 sandstone samples had a PTS values between 1.1 and 2.9. The size of the sample suite coupled with the wide range in porosity and permeability, the diverse composition and variable texture of the sandstone and limestone studied samples suggest that our analysis could be a representative of oil reservoir rocks in other geologic settings. The obtained petrophysical parameters and sorting coefficients values obtained from this study for some of the studied core samples are listed in (Table 2).

Table 2. Pore throat size sorting parameters.

no	K, md	Phi, %	Pres. 1st Q	r25	r35	r50	r75	Pres. 3 rd Q	PTS	PTS class
Well:112-82, zone IV, Belayim Formation, Gulf of Suez, Egypt, (Sandstone)
1	1010	34.3	13.1	8.2	6.5	4.2	0.8	134.6	3.2	moderate
4	101	25.7	14.9	7.2	5.4	2.8	0.4	269.3	4.2	moderate
16	265	29.1	16.5	6.5	5	2.3	0.5	215.4	3.6	moderate
26	142	19.4	11.9	9.0	6.5	3.1	0.7	153.9	3.6	moderate
35	18	21.8	8.9	12.0	10.3	7	2.5	43.0	2.2	well
100	191	25	13.1	8.2	6.4	4	1	107.7	2.9	well
112	1249	25.8	5.7	18.9	15.9	12	4	26.9	2.2	well
118	906	25.5	7.4	14.5	13	10	3	35.9	2.2	well
122	196	21.2	13.4	8.0	6.5	4.3	2.5	43.0	1.8	well
125	541	24.1	8.6	12.5	11	8	2	53.8	2.5	well

Well:113-81, Rudies Formation, Belayim land field, Gulf of Suez, Egypt, (Sandstone)
5	2637	25.7	7.5	14.2	13.6	13	8.5	12.6	1.2	well
12	408	18.6	12.2	8.8	8	6.4	1.6	67.3	2.3	well
45	2014	25.1	8.6	12.4	12.1	11.2	7.6	14.1	1.2	well
47	1068	22.9	9.2	11.6	11	9.8	5.6	19.2	1.4	well
66	875	23.8	10.3	10.4	9.8	8.6	4.6	23.4	1.5	well

Well:113-m-81, Rudies Formation, Belayim marine field, Gulf of Suez, Egypt, (Sandstone)
1	508	22.3	7.1	15.0	12.1	10	4.1	26.2	1.9	well
2	1844	25.3	11.3	9.5	8.3	6.5	3.5	30.8	1.6	well
4	103	14.9	16.0	6.7	6	5	2.4	44.9	1.7	well
5	2345	29.2	7.1	15.0	11.5	8.2	4.2	25.6	1.9	well
6	2384	25.9	8.2	13.0	11	8	4	26.9	1.8	well
7	2186	25.9	5.6	19.0	14.3	11	8	13.4	1.5	well
11	523	18.9	6.3	17.0	13	10	5	21.5	1.8	well
12	2094	27.7	6.7	16.0	13	9	4	26.9	2	well
12a	553	29.2	11.3	9.5	8	6.3	4	26.9	1.5	well
13	1731	24.9	4.5	23.5	17	12	7	15.3	1.8	well
15	2286	26.8	2.6	41.0	29.5	16.3	7	15.3	2.4	well
16	282	18.7	9.7	11.0	8.6	7	3	35.9	1.9	well

Well:BM-85, Matullah Formation, Belayim marine field, Gulf of Suez, Egypt, (Sandstone)
1	19.3	23.4	25.6	4.2	2.4	1	0.1	1077	6.5	poor
2	662	30.3	16.8	6.4	5.6	4	0.3	359	4.6	moderate
3	724	26.3	10.6	10.1	8.2	4.8	0.2	538	7.1	poor
4	1515	26.1	5.35	20	16.5	13.0	6.5	16.6	1.7	well
5	1093	25.3	8.4	12.8	11.6	9.2	1.6	67.3	2.8	well
6	619	27.9	9.9	10.8	10.0	8.8	3.2	33.6	1.8	well
7	618	28.2	9.2	11.6	10.4	8.9	2.0	53.8	2.4	well
9	478	25.1	10.7	10.0	9.2	6.9	0.4	269.3	5.0	moderate
10	355	22.1	21.5	5.0	3.7	2.0	0.5	215.4	3.2	moderate

Hadahid, surface section, Rudies Formation, Gulf of Suez, Egypt, (Sandstone)
1	156	11.5	15.3	7.0	4.5	2.3	0.4	269.3	4.2	moderate
5	670	14.5	8.2	13.0	9.5	5.5	2.5	43.0	2.3	well
6	7.2	11.6	59.8	1.8	1.5	1.0	0.15	718.2	3.5	moderate
7	5.7	13.9	46.8	2.3	2.1	1.6	0.2	538.7	3.4	moderate
9	12.1	10.8	25.6	4.2	2.6	1.9	0.2	538.7	4.6	moderate
14	24	22.2	25.6	4.2	3.2	2.2	0.4	269.3	3.2	moderate
16	370	17.9	7.6	14.0	12.9	9.3	2.5	43.1	2.4	well
18	101	27.3	16.5	6.5	5.5	3.5	1.5	71.8	2.1	well
19	634	18.4	8.6	12.5	10.3	7.0	2.5	43.0	2.2	well
21	9.1	17.8	44.8	2.4	2.2	1.4	0.2	538.7	3.5	moderate
23	43	7.3	19.5	5.5	4.0	2.3	0.3	359.1	4.3	moderate
24	870	16.8	4.3	25.0	21.0	14.0	2.2	48.9	3.4	moderate
25	183	10	17.9	6.0	5.0	3.5	1.5	71.8	2.0	well
29	58	13.1	23.9	4.5	3.3	2.4	1.0	107.7	2.1	well
31	6.1	8.1	71.8	1.5	1.1	0.7	0.3	359.1	2.2	well

Well:BP-1, off shore, Mediterranean sea, Egypt, (Sandstone)
1	2906	35.3	5.9	18.0	17.0	15.5	9.5	11.3	1.4	well
3	2770	34.6	6.1	17.5	17.0	16.0	12.0	8.9	1.2	well
5	2332	33.6	5.6	19.0	18.0	15.5	8.0	13.4	1.5	well
10	2885	29.9	4.3	25.0	23.0	21.0	15.0	7.2	1.3	well
14	1296	29.2	3.9	27.5	21.0	13.0	3.0	35.9	3.0	well
25	5341	34.5	4.7	23.0	22.0	21.0	18.0	5.9	1.1	well
45	3624	35.1	5.0	21.5	20.0	18.0	12.0	8.9	1.3	well
47A	26.2	29.6	43.0	2.5	1.5	0.6	0.01	10,773	15.8	no
61	3706	29.6	4.4	24.0	22.5	20.0	14.0	7.6	1.3	well
101	0.06	19.8	1346	0.08	0.03	0.02	0.01	15,390	3.4	moderate
105	25.4	23.7	35.9	3.0	2.4	1.15	0.02	5386	12.2	no
114	0.04	21.8	1197	0.09	0.03	0.02	0.01	13,466	3.4	moderate

3.1. Permeability prediction from pore throat parameters

Fig. 1 shows the relationship between porosity and log permeability for all sorted classes studied sandstone and limestone samples. The linear regression equations and correlation coefficients are represented in the graph. A weak relationship exists between log (perm.) and porosities of the data set analyzed in this paper. The deviation from the predicted value using this relationship spans two orders of magnitude.

The relationships between r25, r35, r50, r75 and permeability for all sorted classes samples are plotted in (Fig. 2). The regression coefficients for the four parameters are excellent and they demonstrate the high dependence of measured permeability on pore throat size. The Correlation coefficients for r25, r35, r50, r75 are 0.96, 0.96, 0.95 and 0.91 respectively.

We fit the following empirical models to this data:(8) $\ln (k) = 1.76 \ln (r_{25}) + 1.47$ (9) $l n (K) = 1.71 l n (r_{35}) + 1.9$ (10) $l n (K) = 1.7 l n (r_{50}) + 2.56$ (11) $l n (K) = 1.57 l n (r_{75}) + 4.4$ where k is gas permeability in (md) and r is the pore throat radius in (μm).

While any of the four equations can be used to predict permeability from pore throat size, the r35 has the best fit to the data.

The measured and estimated permeability from the r35 all sorted classes samples were plotted in (Fig. 3). The regression coefficient between computed and measured values was 0.96. The variation from the best fit line is only about one order of magnitude.

3.2. The effect of PTS

3.2.1. For well sorted samples pore network

A well sorted sample implies a very narrow range of pore throats which control a large volume of pore spaces. We suggest defining the macro-porosity based on a pore throat cutoff which separates the volumes of pores which are controlled by larger pore throats than the cutoff and contributes to 98% of the flow capacity of the sample, and volumes of pores which are controlled by smaller pore throats than the cutoff and doesn't contribute to the flow of the sample. The pore throat cutoff and macro-porosity were determined graphically. In Fig. 4, we start at 98% on the y-axis and move horizontally to interest flow capacity curve, at the intersection, a line is drawn perpendicular to the X-axis. The intersection of the vertical line with the X-axis represents the value of pore throat cutoff. About 89% of the pore volume or storage capacity (macro-porosity of larger throats than cutoff) (read on solid curve) is contributing to 98% of the flow capacity (free fluid) of the sample (read on dashed curve). The 98% of the flow capacity are controlled by a pore throat range from 14 down to cutoff pore throat of 1 μm which in turn control 89% of the pore volume. The incremental frequency of curve is unimodal with its mode at 10.1 μm (Fig. 5). We assume here that pore throats with a radius greater than 1 μm contributes to the flow capacity.