## Abstract

The main focus of the paper is to introduce a new approach at studying and modelling the relationship of initial water saturation profile and capillarity in water-wet hydrocarbon reservoirs, and describe the available measurement methods and possible applications. As a side track it aims to highlight a set of derivable parameters of mercury capillary curves using the Thomeer-method. Since the widely used mercury capillary pressure curves themselves can lead to over-, or underestimations regarding in-place and technical volumes and misinterpreted reservoir behaviour, the need for a proper capillary curve is reasonable. Combining the results of mercury and centrifuge capillary curves could yield a capillary curve preserving the strengths of both methods, while overcoming their weaknesses. Mercury injection capillary curves were normalized by using the irreducible water saturations derived from centrifuge capillary pressure measurements of the same core plug, and this new, combined capillary curve was applied for engineering calculations in order to make comparisons with other approaches. The most significant benefit of this approach is, that all of the measured data needed for a valid drainage capillary pressure curve represents the very same sample piece.

## 1 Introduction

Capillarity (originating from the Latin word *capillus*, meaning hair) bears a central role in a series of disciplines among natural sciences from agriculture, environmental protection, through hydrogeology to petroleum engineering. Capillary characteristics of reservoir and seal rocks bear a huge effect from geological modelling throughout history matching to planning of EOR/IOR procedures that is to say from exploration through appraisal and development stages until the mature phase of a hydrocarbon field.

One can simply ascertain that real fluid surfaces do not behave as they should if merely based on hydrostatics [1]. Everyday examples are like a needle floating on the surface of a glass of water, some insects walking on the surfaces of rivers or lakes, although their densities are higher, than that of water. If placing (as shown in Figure 1) a row of decreasing diameter open, glass needles (capillary tubes) in a bowl of water one can observe, that the water will rise the higher in the tubes the smaller the diameter is. The latter example supposes that in a water-air-glass system the glass prefers to be wetted by water rather than air, the glass is preferably covered by water on molecular level against air. The same phenomenon is responsible for paper towels “soaking” up poured water, or fading of lacmus above the surface of water-based acids or lies.

(If repeating the same experiment with mercury instead of water, no rise can be observed, but the level of mercury will tend to sink. The latter example shows that mercury is not wetting the surface of glass. This phenomenon clearly indicates that capillary behaviour is highly dependent on wettability and also on the changes of wettability.)

### 1.1 Theory of capillarity

To summarize the most important physical, chemical reasons behind the phenomenon and processes described above give a short overview of laws of physics explaining them is given hereby. The topic is summarized in a detailed and substantial series of whitepapers and books [3–9].

The main reasons behind the capillary rise are the cohesion force among the molecules of fluids (*e.g* hydrogen bonds in water, or van der Waals forces in methane), and the adhesion forces between the fluid and the solid phases [1]. Equilibrium is achieved when adhesion and cohesion forces become equal, as they are in Figure 1. Adhesion forces tend to lift the water, while cohesion forces are acting against it. The rise of the wetting fluid’s level can be calculated by Equation 1. (All of the abbreviations and units are indicated in a table named Units and Abbreviations after Acknowledgements, in the end of the whitepaper.)

where *h* is height, *σ* stands for interfacial tension, *Θ* for contact angle, *r* for radius, *g* for gravitational acceleration and *ρ* for absolute density. Equation 1 is a general equation for defining the capillary rise above FWL in an *e.g* water-hydrocarbon system.

According to Equation 1 the factors influencing capillary rise of water in a water-wet system are listed below, indicating the direction of proportion and source of information:

radius of the capillary tube (inverse)

From a rock fabric point of view these tubes are the equivalents of pore throats that are measureable. Basically the drainage capillary curve is the cumulative distribution function of pore volume belonging to decreasing diameter pore throat’s as capillary pressure increases.

acceleration of gravity (inverse)

The Earth’s average value for

*g*is 9.81 m/s^{2}[1].

contact angle (inverse proportion, direct with its cosine)

Contact angle can be measured by several procedures, like USBM, Amott, imbibition, microscope, floatation methods [5]. It has a high effect on the dynamic behavior of the reservoir, and it can change over time by aging due to the fact that some minerals tend to change their wettability on molecular level [3, 4].

Based on the workof Young and Laplace the pressures on two sides of a curved fluid surface are non-equal [21, 22]. Applying this fact to a capillary system it can be proven that in equilibrium the capillary pressure equals to the difference of measured pressures in the non-wetting and wetting phases, Equation 2 expresses this relation [7, 23–27].

where capillary pressure (*P _{c}*) (bar) is expressed as the pressure difference of non-wetting (

*nw*) and wetting (

*w*) phases (cross-linked to Equation 1),

*ρ*is absolute density (g/cm

^{3}),

*g*is gravitational acceleration (m/s

^{2}), h is height (m),

*σ*is interfacial tension (dynes/cm),

*Θ*is contact angle (rad) and r is radius of the capillary tube (μm).

According to Equation 2, *P _{c}* (capillary pressure) can be explained as the pressure (or force) that would be sufficient to drive out the wetting fluid from the given size of pore throat. Supposing a theoretical rock type that has uniform pore throat size the capillary pressure can be calculated by solving Equation 2. That has to be overcome by buoyancy force to let the non-wetting phase enter the pore system and start draining the wetting one. The buoyancy force is provided by the difference of absolute densities, the higher the difference the lower the hydrocarbon column needed to reach the displacement or entering pressure. It means in practice that the same hydrocarbon saturation conditions can be acquired at lower heights in case of gas compared to oil reservoirs. Figure 2 gives a quick overview of the most important elements of a natural system and its tally from the laboratory.

Figure 2 shows the superposition of physical processes resulting in equilibrium of capillary and gravitational (buoyancy) forces in a natural, single rock type, two-phase system. That equilibrium defines the initial saturation profile of a hydrocarbon reservoir. The initial saturation profile is a key parameter at reservoir geological and engineering calculations *e.g* in-place volume, technical volumes, recovery factors, and via them significantly affecting CAPEX, OPEX, NPV, revenue.

Free water level (FWL) is a flat, horizontal surface if the hydrodynamic system is in equilibrium and no significant underground water flow occurs. FWL is the surface, where the capillary pressure is zero, in other words where the water level would set in an infinite diameter water-wet capillary tube.

The phase contact (100% water level) is dependent on the lithology, to be more precise, on the largest pore throat of a given rock type. Phase contact’s (100% water level) height above the FWL (HAFWL) can be defined by the displacement pressure (*p _{d}*) of the capillary curve. It is the minimum pressure that must be achieved to let the non-wetting phase enter the pore system through the largest pore throat. In the interval between the FWL and 100% water level of a given rocky type, the wetting phase is held by capillary forces against buoyancy even in the largest pore throats.

The producing water level (PWL) can be defined empirically to identify a proposed depth of the lowermost perforation producing with close to zero or zero water-cut. It must be emphasized, that this is a completely empirical approach.

Transition zone is the interval on the capillary curve from the phase contact to the starting point of the irreducible saturation (*i.e* the steep slope).

The irreducible water saturation (*Sw _{irr}*) is the portion of formation water that cannot be lowered by conventional production; it will not flow under depressions occurring while producing a reservoir. Although it has a significant role being the horizontal asymptote of the capillary curve. It shall not be confused with clay-bound, interstitial or connate water [30]. One of the main goals of this paper is to introduce a new method of coupling irreducible water saturation with mercury injection capillary pressure (MICP) data.

Capillary and buoyancy forces play a significant role in numerous subsurface processes, and since they can be described by the rocks’ capillary curves, these data are crucial and extremely useful at several aspects of reservoir characterisation from petrophysics to reservoir engineering [16, 23–25]:

differentiating rock types,

estimating net-to-gross,

determining initial water saturation profile,

defining in-place volumes’ spatial distribution,

estimating fluid contacts,

determining the maximum retainable hydrocarbon column (seal capacity),

predefining dynamic behaviour of reservoirs,

field development concept selection,

estimating the efficiency of EOR/IOR methods,

injection of fluids,

lifetime and processes of underground gas storages.

Drainage capillary curves reflect the process when non-wetting phase drives out the wetting phase (*e.g* secondary migration of hydrocarbon into a water-wet reservoir, water injection into oil-wet reservoir, gas injection into water-, or oil-wet reservoirs). While the opposite process *i.e* wetting phase’s saturation increase is called imbibition (*e.g* formation water(re)-entering a water-wet reservoir during production). Figure 3 is illustrating an example of drainage and imbibition curves, and capillary hysteresis. The latter phenomenon occurs mainly because of contact angle-hysteresis, a deviation of advancing and receding contact angles [7].

### 1.2 Goals of the work

This paper focuses on water-wet systems, keeping in mind that purely water-wet systems are usually only granted by homogenous clastic reservoir rocks (*e.g* siltstones, sandstones), by the minerals (*e.g* quartz, feldspars, calcite, limonite, clay minerals, gypsum) building them up. These are usually referenced as classical and simple reservoirs at least from wettability point of view. Differing geology, tectonic history, fluid compositions, or age can and will result in more complex behaviour regarding capillarity, wettability, saturation profile. The main reason of focusing on water-wet rocks and drainage capillary curves was to put the approach to a simple test, whether it was doable, applicable and reliable at all. To achieve these goals the simplest and most well-known conditions were studied so far, more complex and more unique systems, as oil, intermediate-, fractional-, or mixed-wet are not investigated in this study (Figure 4). Possession of an abundant dataset and professional experience on water-wet systems made the validation and comparison more reliable; especially with drainage curves.

A summary of the main goals of the work is listed below:

Introduce a new approach to classical SCAL measurement procedure and interpretation targeting a valid irreducible water saturation value attachable to MICP curves.

Combine data from centrifuge and MICP measurements into one capillary curve preserving the information of both.

Validate the viability of this method, and applicability of the data derived.

Re-visit the steps of a comprehensive workflow targeting to derive all data from capillary curves applying the Thomeer-method.

## 2 Applied methods

A scheme of current work’s main steps and their technical order – the applied workflow – is shown in Figure 5. The description of the work done, applied methods and the interpreted results follow the logic of Figure 5, it serves as a guideline.

### 2.1 Core sampling (Figure 5-Part 1)

The core samples of the study were drilled in an exploration well located in the Pannonian-basin close to the Derecske Trough on the Great Hungarian Plain (Figure 6–7). The reservoir rock is Pannonian-aged belonging to the Úfalui Formation that was deposited in delta plain environment, built up of clastic sediments ranging from shale to pure sandstones [31]. The reservoir is situated in a reference depth of 890 meters TVDSS, with hydrostatic initial reservoir pressure of 98 bars and reservoir temperature of 65^{°}C. The Úfalui Formation is well-defined, regionally extended deposit often serving as reservoir rock in the Pannonian Basin, numerous hydrocarbon fields were discovered belonging to it throughout Hungary.

The sampling was conducted in an interval previously identified as gas-bearing. The sample pieces for the investigation were defined focusing on representativity, trying to set up the sample set in a ratio identified on the total core section. All together 30 plugs were taken for capillary measurements, with an average bulk volume of 10–15 cm^{3}. Since the goal was to conduct twin (CCP and MICP) measurements on the very same pieces of rock, the samples must have been regular shaped due to the requirements of centrifuge measurements, though no such pre-conditions are present in the case of MICP. Asummary of both CCP and MICP methods’ main characteristics are enlisted in Table 1.

Method | |||

CCP | MICP | ||

Main characteristics | Time | hours | hours |

Size of sample | 4–32 cm^{3} | ∼1–32 cm^{3} | |

Shape of sample | standard | amorph/standard | |

Wetting phase | brine/water | - | |

Non-wetting phase | standard oil/soltrol/air | mercury (Hg) and air | |

phase | oil/soltrol/air | and air | |

Drainage curve | available | available | |

Imbibition curve | available | available | |

Extraction | yes | yes | |

Initial state of the sample | saturated by wetting fluid | extracted | |

Max. pressure | 90 bar (12 bar) | 4100 bar | |

Simulation of overburden p. | available | available | |

Reservoir fluid | applicable | not applicable | |

Closure-effect | no | yes |

### 2.2 Preparation of the sample piece (Figure 5-Part 2)

Since the first of the capillary measurements is the CCP the main pre-conditions shall be emphasized:

the samples must be standard/regular shaped due to the requirements of CCP,

the sample is saturated with brine at the beginning of the measurement in case of CCP (possibility of

*Sw*measurement), while it is completely extracted in case of MICP,_{irr}in this laboratory the maximum simulated pressure obtainable by CCP is 12 bars, while when using MICP it is 4000 bars,

no wetting phase is present in MICP,

overburden pressure could be simulated in both cases, but in our study it was neglected.

The average dimensions of plugs (cone-shaped) drilled from the core taken is *d* = 2.5 cm, *l* = 5–6 cm, giving an average total bulk volume of 27 cm^{3}. All of these samples are cut into half perpendicular to their long axis, one half is used for actual measurements, and the other is used as reference specimen for adsorbent tests. The plugs were cleaned physically by heating; since it is a dry gas reservoir no chemical cleaning was necessary like it would be in case of oil. The plugs were heated to a maximum of 105^{°}C in a laboratory oven. After geometrical and weight measurements the samples were saturated with brine water having an equivalent NaCl-concentration of 25 g/l.

After saturation they were weighted again for further use and porosity calculations. Afterwards the plug was carefully placed in the sample-holder and a computer-assisted centrifuge (Heraeus Cryofuge 8500i) rotated it with increasing revolutions per minute for at least 1.5 hours per step. In our case six steps were available (400, 900, 1400, 1900, 2500, 3300 RPM (revolutions per minute)). Each step simulates higher buoyancy force by the increasing centripetal force. The drained amount of wetting phase was defined by weight-measurements after each step. The non-wetting phase was air in our case. Since the maximum simulated pressure of our device was 12 bars the *Sw _{irr}* was only measurable in good reservoir quality rocks that we had. In poorer reservoir quality rocks (

*e.g*more heterogeneous, small average pore throat) it cannot be reached with pressures being so low. In the latter case no irreducible water saturation could be defined but only residual water saturation.

### 2.3 Centrifuge capillary pressure curve (CCP) (Figure 5-Part 3)

The dataset handed over by the laboratory included all of the measured data, but for the current experiment the highest priority was the centrifuge capillary pressure curve, meaning six pressure steps with corresponding saturation values as per sample piece. The narrowed dataset is shown in Figure 8. Rock typing was possible, four well-distinguished rock types were identified type 1 having excellent reservoir properties worsening towards type 4. Irreducible water saturation values were chosen as the last points of each curve, since it can be determined by visual inspection that each of the curves ends close to its horizontal asymptote, in case of type 4 this could lead to a slight overestimation.

### 2.4 Re-preparation of the sample piece (Figure 5-Part 4)

During MICP measurement buoyancy force is simulated by the pressure put on mercury surrounding the air-saturated core sample [33, 34]. The procedure was invented by RobertW. Purcell in the 1950’s. The most important properties of MICP is that neither of the fluids are wetting and extremely high pressures can be reached, up to 4000 bars, revealing the complete pore system including the smallest pore throats. The MICP measurements demands a plug completely cleaned, saturated with air. Since we used the same samples as during the CCP measurements the possible salt precipitations were dissolved and they were re-heated to 105^{°}C in order to clean it. After the MICP the samples must not be re-used, they have to be handled as hazardous.

### 2.5 Mercury injection capillary pressure curve (MICP) (Figure 5-Part 5)

Prior to the measurements the device (PASCAL 140/240/440 Porosimeter) was checked and the compressibility of mercury was defined in order to calibrate the measurement (calibration of dilatometer). The sample was placed in the dilatometer (140) that raised the pressure to 3–4 bars in order to enter the macro-pores. Afterwards the sample was placed into another dilatometer (240) that was capable of maintaining pressures up to 2000 bars. The pressure steps were automatized, each step was sustained until no change in the level of mercury was detected by a capacitive electrode. It was highly important to wait for equilibrium in each step in order to get a valid MICP curve.

Figures 9 and 10 illustrate the MICP drainage capillary curves. The same four rock types can be identified in these curves as in Figure 8. Each of the samples fall into the same rock type of the four in both curve sets. This fact suggests that the measurement procedure is achievable, at least when working with conventional clastic rocks. One of the curves had to be excluded from further investigations, because its rock type was under-sampled, reflected by a single curve, consequently its pair in the CCP was also excluded in order to keep the investigation valid.

### 2.6 Integration and interpretation of data (Figure 5-Part 6, 6a)

The first step of any data processing is quality checking (QC) on the acquired data, from the dataset only one MICP curve had to be excluded due to under-representation. So 29 curves were valid used for further investigations.

In line with Young’s and Laplace’s physics a conversion expression can be set, that makes re-calculation of laboratory capillary pressure available and result pore throat radius. One should take care that Equations 4–5 are valid for mercury-air system and a solid phase of quartz, in other words for simplified clastic environments. Equation 3 is the general expression for conversion, called the simplified Washburn-equation [35–37].

where the Washburn equation is solved for mercury-air-quartz system. Units are indicated in Equation 4, the final conversion expression is Equation 5.

Contact angle and interfacial values were derived from widely used literature data shown in Table 2.

Literature data | |||

System | Contact angle (Θ) (^{°}) | Interfacial tension (σ) (dynes/cm) | |

Laboratory | Air-brine | 0 | 72 |

Oil-brine | 30 | 48 | |

Air-mercury | 140 | 480 | |

Air-oil | 0 | 24 | |

Reservoir | Brine-oil | 30 | 30^{[*]} |

Brine-gas | 0 | 50^{[*]} |

In the 1950’s Thomeer J. H. attempted to find an analytical description of the relation between MICP (injection pressure) and bulk volume occupied by mercury [3, 32]. He found that when plotted on log-log scale in a Cartesian coordinate system a hyperbola can be fitted on the measured points (Equation 6).

where *S _{b}*

_{∞}refers to effective porosity (-),

*S*to ratio of injected mercury and total bulk volume of the sample (-),

_{b}*G*for G-factor (-),

*p*for displacement pressure (bar)

_{d}*P*for capillary pressure (bar), e is the Euler-number. Moreover each of the parameters defining the fitted Thomeer-hyperbola carries a geological meaning, making the method unique. Namely effective porosity, largest pore throat diameter, and pore geometrical factor:

_{c}*G*-factor (Figures 2 and 11) [32, 38].

*G*-factor is meant to describe the heterogeneity of the pore throat sizes of the sample, the smaller the G-factor the higher thehomogeneity [32, 38].Approximate limits for the Thomeer-parameters’ interval are shown in Table 3.

p_{d} (bar) | S_{b∞} (-) | G (-) | |

Excellent reservoir | 0.1 | 0.30 | 0.1 |

Poor reservoir | 7.5 | 0.08 | 0.6 |

Prior to fitting the Thomeer-hyperbola all of the MICP curves has to be closure-corrected. Closure is a phenomenon linked to MICP measurements *i.e* at low pressures (∼0.1–5 bar), in the first steps of pressure ladder the mercury tends to fill the pits of the surface of samples, but not entering the pore system. This section of the curve has to be cut because the information shown by them has no connection with the pore system so it is misleading, distorting. A general rule is that the smaller the volume of the sample the larger the closure-effect due to the increased ratio of surface to volume. It can be achieved manually or applying automatic algorithms [39]. This case the closure-correction was performed manually curve-by-curve using Microsoft Excel; an example for closure correction is shown in Figure 12.

The fitting of Thomeer-hyperbola of each curve was conducted using Microsoft Excel’s Solver add-in, setting the target parameter as root-mean-square error (RMS) between the measured points and fitted curve to be minimized by changing the Thomeer-parameters in a given, estimated interval using evolutive solving method. During fitting the Thomeer-hyperbola three of the curves had to be excluded since no proper hyperbola could be fitted within the allowed error range, probably due to measurement errors. The remaining 26 curves were used for further investigation. Each of the fitted hyperbolas yielded a set of Thomeer-parameters that permitted us to calculate the median capillary curves for each of the rock types. In order to be able to model initial water saturation the *P _{cl}* needs to be converted to

*P*(capillary pressure at reservoir conditions) if using Schlumberger’s Petrel reservoir modelling software and to HAFWL if using Roxar Irap RMS.

_{cr}The first step of conversion can be achieved by Equation 7, but some additional parameters are needed as inputs, that can be measured and/or calculated analytically.

where *P _{cr}* is capillary pressure at reservoir conditions (bar),

*P*is capillary pressure at laboratory conditions (bar),

_{cl}*σ*stands for interfacial tension (dynes/cm), and

*Θ*for contact angle (rad) at reservoir (r) and laboratory (l) conditions.

For laboratory conditions values indicated in Table 2 were used. For reservoir conditions literature value of Table 2 was applied for contact angle, since its cosine has much lower effect on the target value, than interfacial tension’s [12, 14, 15, 18, 48–50]. The interfacial tension was calculated using equations of Sutton P. R. [10]. The necessary PVT and compositional data of formation water and dry gas were available from the laboratory and all equations were chosen from widely used literatures and authors [23, 40–47], so all of the necessary data (absolute density at reservoir conditions, pseudo reduced and pseudo critical pressures and temperature, formation volume factor for gas and water, z-factor) became available. The dataset, measured and calculated, used in capillary pressure conversion is shown in Table 4.

PVT data of applied reservoir fluid | |||||||||

Component | C_{1} | C_{2} | C_{3} | i-C_{4} | n-C_{4} | i-C_{5} | n-C_{5} | ||

V/V% (m^{3}/m^{3}) | 0.97017 | 0.00798 | 0.00266 | 0.00182 | 0.00019 | 0.00044 | 0.00004 | ||

Component | i-C_{6} | n-C_{6} | C_{7} | C_{8} | C_{9} | N_{2} | CO_{2} | ∑ | |

V/V% (m^{3}/m^{3}) | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.01182 | 0.00487 | 1.00000 | |

γ_{g} (-) | ρ_{gr} | z-factor | B_{gi} | σ_{l} | σ_{r} | Θ_{l} (^{°}) | Θ_{r} (^{°}) | ρ_{vr} | B_{w} |

(g/cm^{3}) | (-) | (m^{3}/nm^{3}) | (dynes/cm) | (dynes/cm) | (g/cm^{3}) | (m^{3}/nm^{3}) | |||

0.574 | 0.065 | 0.896 | 0.011 | 480 | 55 | 140 | 0 | 0.99 | 1.02 |

The second step of capillary pressure’s conversion is transforming *P _{cr}* to HAFWL. The conversion is attainable by solving Equation 8 with the inputs from Table 4 and Equation 7.

where *HAFWL* is height above free water level (m), *P _{cr}* is capillary pressure at reservoir conditions (bar)

*ρ*and

_{gr}*ρ*are gas and water absolute densities at reservoir conditions (g/cm

_{gw}^{3}). The depth of FWL cannot be identified based on capillary curves

*i.e. HAFWL*is a relative depth, the FWL has to be derived from well-log, seismic, pressure gradient or well-test results.

## 3 Results

### 3.1 Integration and interpretation of data (Figure 5-Part 6b)

All of the pieces of this combined capillary curve puzzle became available by measurements and calculations introduced formerly. Rock types were identified on both CCP and MICP curves, the complete, filtered and quality checked set of curves counts for 26 CCP and 26 MICP drainage capillary curves. The identified rock types are visualized in Figures 8 and 10. The irreducible water saturation values were derived from CCP curves and their statistics are indicated in Table 5 as per rock type.

Irreducible water saturion | ||||||

Rock type | N | Minimum | Maximum | Arithmetic average | Median | Std. Dev |

1 | 15 | 0.10 | 0.16 | 0.14 | 0.14 | 0.02 |

2 | 6 | 0.17 | 0.23 | 0.21 | 0.21 | 0.02 |

3 | 3 | 0.29 | 0.30 | 0.30 | 0.30 | 0.00 |

4 | 2 | 0.49 | 0.54 | 0.52 | 0.52 | 0.04 |

Total | 26 | 0.10 | 0.54 | 0.20 | 0.16 | 0.11 |

The main results of fitting the Thomeer-hyperbola are shown in Table 6, namely effective porosity, G-factor, and displacement pressure. Their statistical analysis as per rock type made it possible to define a median capillary curve for each rock type from the MICP curves, by solving Equation 6 with median Thomeer-parameters for each rock type. (By solving for other percentiles *e.g* P90, P10 different cases can be derived for uncertainty analysis.)

Exploratory data analysis for Thomeer-parameters | |||||||

Rock type | N | Minimum | Maximum | Arithmetic average | Median | Std. Dev. | |

Thomeer-porosity | 15 | 0.22 | 0.25 | 0.22 | 0.22 | 0.01 | |

G-factor (-) | 1 | 15 | 0.18 | 0.32 | 0.22 | 0.22 | 0.04 |

p_{d} (bar) | 15 | 0.29 | 0.45 | 0.41 | 0.42 | 0.04 | |

Thomeer-porosity (S_{b∞}) | 6 | 0.18 | 0.21 | 0.19 | 0.19 | 0.01 | |

G-factor (-) | 2 | 6 | 0.11 | 0.23 | 0.27 | 0.18 | 0.04 |

p_{d} (bar) | 6 | 0.76 | 1.12 | 0.93 | 0.93 | 0.13 | |

Thomeer-porosity (S_{b∞}) | 3 | 0.18 | 0.19 | 0.18 | 0.18 | 0.01 | |

G-factor (-) | 3 | 3 | 0.34 | 0.45 | 0.39 | 0.38 | 0.05 |

p_{d} (bar) | 3 | 0.85 | 1.19 | 0.96 | 0.86 | 0.19 | |

Thomeer-porosity (S_{b∞}) | 2 | 0.14 | 0.15 | 0.14 | 0.14 | 0 | |

G-factor (-) | 4 | 2 | 0.66 | 0.67 | 0.66 | 0.66 | 0 |

p_{d} (bar) | 2 | 1.41 | 2.29 | 1.85 | 1.85 | 0.62 |

The reference drainage capillary curves as per rock type before normalizing them with *Sw _{irr}* are shown in Figure 13.

Permeability values were derived from MICP curves prior to generating the combined capillary curves, since numerous authors defined empirical equations between Thomeer-parameters and permeability [33, 38, 51–61]. Permeability was estimated for all of the 26 samples using Thomeer-, and Swanson-methods. The results of the calculations are summarized in Table 7 along with core measured, log interpreted and well-test calculated permeability for comparison.

Permeabilities | ||||||||

Rock type | Thomeer A-type k_{air} | Swanson k_{air} | Thomeer B-type k_{air} | Swanson k_{water} | Klinkenbeg | Well-log data | Well-test (BUP) | |

Arithmetic (mD) | 410 | 403 | 498 | 257 | 473 | |||

1 | Harmonic (mD) | 390 | 379 | 473 | 242 | 461 | ||

Geometric (mD) | 400 | 390 | 485 | 249 | 467 | |||

Arithmetic (mD) | 93 | 110 | 113 | 70 | 92 | |||

2 | Harmonic (mD) | 75 | 93 | 96 | 59 | 53 | ||

Geometric (mD) | 83 | 100 | 103 | 64 | 72 | |||

Arithmetic (mD) | 24 | 27 | 28 | 17 | 24 | |||

3 | Harmonic (mD) | 23 | 26 | 27 | 17 | 24 | ||

Geometric (mD) | 24 | 26 | 28 | 17 | 24 | |||

Arithmetic (mD) | 2 | 1 | 2 | 1 | 1 | |||

4 | Harmonic (mD) | 2 | 1 | 1 | 1 | 1 | ||

Geometric (mD) | 2 | 1 | 1 | 1 | 1 | |||

Arithmetic (mD) | 261 | 261 | 317 | 166 | 297 | 283 | 251 | |

Harmonic (mD) | 20 | 14 | 16 | 9 | 16 | |||

Geometric (mD) | 134 | 135 | 156 | 86 | 138 |

### 3.2 Combined capillary curve (Figure 5-Part 7)

The applied approach was to normalize MICP curves with Swirr values measured by CCP (referenced as Method A from this point). There are several methods aiming to link irreducible water saturation to MICP curves, some of the most important one (not including Method A) are:

Using and normalizing the MICP curves to a “virtual”

*Sw*derived from theMICP curve itself by slitting it in the HAFWL equal to maximum HC column of the given reservoir. (Referenced as Method B from this point.)_{irr}Analog data

*e.g*based on Worldwide Rock Catalog.Mathematical conversion of saturations [64].

A simple normalization equation without any weight factors or higher order functions was used for combining MICP curves and *Sw _{irr}*. It is shown in Equation 9 (before: Figure 14, after: Figure 15).

where *Sw* stands for water saturation (-), *n* stands for normalized and *min* and *max* as minimum and maximum.

A quick comparison was conducted between Method A and Method B. Since in Method B maximum HC columnhas a great significancewe supposed four different oil reservoirs with differing maximum oil columns (Table 8). Average initial water saturation was calculated for each of these reservoirs (Table 8) using both methods. As a consequence all of the capillary curves were re-converted using oil parameters instead of gas (refer to Table 2). The absolute densities of fluids at reservoir conditions were 0.71 g/cm^{3} for oil and 0.99 g/cm^{3} for water.

Method A | |||||

Res. F1 | Res. F2 | Res. F3 | Res. F4 | ||

V_{bulk} | 15.7 | 15.7 | 15.7 | 15.7 | MMm^{3} |

HC-col. | 8 | 26 | 51 | 201 | m |

Φ_{eff} | 0.21 | 0.21 | 0.21 | 0.21 | - |

NtG | 0.87 | 0.87 | 0.87 | 0.87 | - |

S_{wi} | 0.00 | 0.00 | 0.00 | 0.00 | - |

B_{oi} | 1.31 | 1.31 | 1.31 | 1.31 | m^{3}/nm^{3} |

OIIP_{1} | 2.19 | 2.19 | 2.19 | 2.19 | MMm^{3} |

Φ_{eff} | 0.21 | 0.21 | 0.21 | 0.21 | - |

S_{wi} | 0.00 | 0.00 | 0.00 | 0.00 | - |

B_{oi} | 1.31 | 1.31 | 1.31 | 1.31 | m^{3}/nm^{3} |

k | 0.298 | 0.298 | 0.298 | 0.298 | Darcy |

μ_{wi} | 0.27 | 0.27 | 0.27 | 0.27 | cP |

μ_{oi} | 0.51 | 0.51 | 0.51 | 0.51 | cP |

p_{a} | 120 | 120 | 120 | 120 | bar |

p_{i} | 202 | 202 | 202 | 202 | bar |

η_{1} | 40.9 | 45.7 | 47.7 | 50.7 | % |

Reserve | 0.32 | 0.59 | 0.69 | 0.83 | MMm^{3} |

Figure 16 is illustrating the different capillary curves of the same rock type according to the two methods and to the different maximum oil columns in case of Method B.

The average initial water saturations (*S _{wi}*) were calculated by weighted averaging where the weights were the ratios of a given rock type. The consequence was derived that the lower quality the reservoir rock the higher the deviation. The under-estimation of

*S*is increasing as the maximum oil column is increasing (Figure 17).

_{wi}Moving a step forward a comparison of influenced parameters was made, *e.g* in-place and technical volumes that is to say a sensitivity analysis for the uncertainty of *S _{wi}* belonging to different methods. The in-place volumes were calculated using the volumetric method, applying Equation 10.

where OIIP (m^{3}) stands for original oil initially in-place, *V _{bulk}* for total rock volume (m

^{3}), NtG for net-to-gross (-),

*φ*for effective (interconnected) porosity (-),

_{eff}*Sw*for initial water saturation (-),

_{i}*B*for initial formation volume factor (m

_{oi}^{3}/nm

^{3}).

The recovery factors were calculated using an empirical method, the Arps-equation for water-drive reservoirs (Equation 11).

where *η* is recovery factor (%), *φ _{eff}* for effective (interconnected) porosity (-),

*Sw*for initial water saturation (-),

_{i}*B*for initial formation volume factor (m

_{oi}^{3}/nm

^{3}),

*k*for absolute permeability (D),

*μ*and

_{wi}*μ*initial viscosities for water and oil (cP),

_{oi}*p*and

_{a}*p*are initial average reservoir pressures at abandonment and initially (bar). The Arps equation was derived based on data from 312 various oil reservoirs to promote a quick method of estimating oil recovery factors [25, 65]. The input parameters and results are shown in Table 8 and Table 9.

_{i}Method B | |||||

Res. F1 | Res. F2 | Res. F3 | Res. F4 | ||

V_{bulk} | 15.7 | 15.7 | 15.7 | 15.7 | MMm^{3} |

HC-col. | 8 | 26 | 51 | 201 | m |

Φ_{eff} | 0.21 | 0.21 | 0.21 | 0.21 | - |

NtG | 0.87 | 0.87 | 0.87 | 0.87 | - |

S_{wi} | 0.00 | 0.00 | 0.00 | 0.00 | - |

B_{oi} | 1.31 | 1.31 | 1.31 | 1.31 | m^{3}/nm^{3} |

OIIP_{1} | 2.19 | 2.19 | 2.19 | 2.19 | MMm^{3} |

Φ_{eff} | 0.21 | 0.21 | 0.21 | 0.21 | - |

S_{wi} | 0.00 | 0.00 | 0.00 | 0.00 | - |

B_{oi} | 1.31 | 1.31 | 1.31 | 1.31 | m^{3}/nm^{3} |

k | 0.298 | 0.298 | 0.298 | 0.298 | Darcy |

μ_{wi} | 0.27 | 0.27 | 0.27 | 0.27 | cP |

μ_{oi} | 0.51 | 0.51 | 0.51 | 0.51 | cP |

p_{a} | 120 | 120 | 120 | 120 | bar |

p_{i} | 202 | 202 | 202 | 202 | bar |

η_{1} | 40.3 | 45.7 | 48.6 | 54.6 | % |

Reserve | 0.28 | 0.59 | 0.74 | 0.99 | MMm^{3} |

The deviations of Table 8 and Table 9 reflect the target value’s sensitivity to the applied method of determining *S _{wi}*. The differences of these two tables are indicated in Table 10. The difference shown by

*S*is ± 5 absolute%, Method B resulting in more optimistic values as the HC-column is increasing.

_{wi}*OIIP*is naturally in inverse proportion with

*S*having a ± 10% change in different cases and methods. The differences shown by the recovery factor are in the range of −1−+5 absolute%, reflecting an undesired optimism by Method B. Method A yields more conservative values compared to Method B and the gap is growing as HC column is increasing, regarding

_{wi}*Sw*, OIIP and recoverable volumes as well as recovery factors.

_{i}Method A & B (differences) | ||||||

Res. F1 | Res. F2 | Res. F3 | Res. F4 | Method A | ||

HC-col. | 8 | 26 | 51 | 201 | m | |

Swi | 0.65 | 0.41 | 0.34 | 0.25 | - | |

OIIP_{1} | 0.77 | 1.29 | 1.45 | 1.64 | MMm^{3} | |

η_{1} | 40.9 | 45.7 | 47.7 | 50.7 | % | |

Rec. | 0.32 | 0.59 | 0.69 | 0.83 | MMm^{3} | |

Res. F1 | Res. F2 | Res. F3 | Res. F4 | Method B | ||

HC-col. | 8 | 26 | 51 | 201 | m | |

Swi | 0.68 | 0.41 | 0.31 | 0.17 | - | |

OIIP_{1} | 0.69 | 1.30 | 1.52 | 1.81 | MMm^{3} | |

η_{1} | 40.3 | 45.7 | 48.6 | 54.6 | % | |

Rec. | 0.28 | 0.59 | 0.74 | 0.99 | MMm^{3} | |

Deviations (B-A) | ||||||

HC-col. | 8 | 26 | 51 | 201 | m | B-A |

Swi | 0.04 | 0.00 | −0.03 | −0.08 | - | |

Swi | 5.16 | −0.71 | −10.08 | −44.56 | relative% | |

OIIP_{1} | −0.08 | 0.01 | 0.07 | 0.17 | MMm^{3} | |

OIIP_{1} | −11.16 | 0.49 | 4.43 | 9.32 | relative% | |

η_{1} | −0.59 | 0.07 | 0.97 | 3.91 | absolute% | |

η_{1} | −1.47 | 0.16 | 2.00 | 7.16 | relative% | |

Rec. | −0.04 | 0.00 | 0.05 | 0.16 | MMm^{3} | |

Rec. | −12.79 | 0.64 | 6.34 | 15.81 | relative% |

## 4 Conclusion

A step-by-step workflow (Figure 5) was followed throughout the study in order to present a series of data derivable from CCP and MICP curves, focusing on Thomeer-parameters of water-wet, clastic rocks and their drainage capillary curves.

A new approach to routinely used core measurements was described for clastic reservoir rocks (more precisely: for reservoir rocks with petrophysical parameters indicated in Tables 5–7), *i.e* conducting both centrifuge and mercury injection measurements on the very same core plug. The laboratory measurements were achievable and seem to be validated, but an increased number of measurements would be needed to cross-check the workflow and increase the domain of data. The output data were quality checked and interpreted by applying the Thomeer-method; procedures for hyperbola fitting, rock typing, reference capillary curve estimation as well as permeability calculations were described. As a result of dual measurement and data interpretation we got irreducible water saturation values from CCP and detailed and total capillary curves from MICP. The two data were coupled by normalizing, and a new, combined capillary curve was introduced.

A comparison was made of our combined capillary curve (Method A) and another applicable method (Method B). The two methods yielded slight to medium differences in *Sw _{i}*,

*OIIP*, recovery factor and technical volumes, Method B being the more optimistic from a technical volume point of view. The geological settings, ratio of rock types have a significant effect on the deviations, since the poorer quality from a petrophysical point of view the reservoir rock the higher the differences can be due to underestimation of

*Sw*.

_{irr}These deviations were proved theoretically, not including all of the uncertainties regarding either methods, but it was shown that Method B has an error of underestimation of *Sw _{i}* and as a consequence overestimation of

*OIIP*and technical volumes. This error can be overcome by Method A, but at least a more realistic

*Sw*and (when 3D modelling) a more reliable 3D distribution of initial water saturation based on Look-Up functions (drainage capillary curves) could be achieved.

_{i}A regression was tested for Thomeer-parameters and *Sw _{irr}*, but due to the low number of samples the derived equation is not reliable. A higher number of samples are needed to establish a possible coherent connection between

*e.g*Thomeer-parameters and

*Sw*in order to become able to apply Method A to older MICP curves, where no centrifuge measurements were and can be conducted.

_{irr}The main conclusion of the work done is a feedback that this dual measurement technique can be applied. A drainage capillary curve along with irreducible water saturation can be derived targeting to capture the returning problem of having MICP drainage capillary curves, but no reliable horizontal asymptote. Several methods are available to overcome this challenge, but neither of them is based on laboratory measured data, but all of them are calculated and/or estimated except for the approach described in this work.

Every project demands a unique approach and a series of decisions on numerous subtasks affected by input data, quality, timeframe, risks, priorities and the goals of the project, so in each work it has to be decided which methods to use, which has the best revenues compared to the work done and what parameters are the most significant to the outcome. There are multiple scenarios for way-forward: increased number of samples, various rock types, non-water-wet rocks, imbibition curves.

## Acknowledgements

I would like to thank the possibility granted by MOL Group to conduct the laboratory measurements in the core, fluid and PVT laboratories. I am grateful for all support and help from my colleagues and professors and reviewers, without their advices and guidance this work would never have come to reality.

## Nomenclature

*η*recovery factor [%]

*γ*(-)_{g}specific gravity gas [-]

*μ*_{oi}viscosity of the oil at initial reservoir pressure [cP]

*μ*_{wi}viscosity of the water at initial reservoir pressure [cP]

*Φ*_{eff}effective porosity [-]

*ρ*absolute density [g/cm

^{3}]

*σ*interfacial tension [dynes/cm]

*Θ*contact angle [rad]

*B*_{gi}formation volume factor gas (initial) [m

^{3}/nm^{3}]

*B*_{w}formation volume factor water [m

^{3}/nm^{3}]

*BV*bulk volume [cm

^{3}]

*G*G-factor [-]

*HAFWL*height above free water level [m]

*OIIP*oil initially in place [m

^{3}]

*p*_{a}abandonment reservoir pressure [bar]

*p*_{d}displacement pressure [bar]

*p*_{i}initial reservoir pressure [bar]

*S*_{b}_{∞}effective porosity [-]

*S*_{b}ratio of injected mercury and total bulk volume [-]

*Sw*_{irr}irreducible water saturation [%]

*Sw*_{i}average initial water saturation [%]

- c
capillary [-]

- CAPEX
capital expenditure [-]

- CCP
centrifuge capillary pressure

- e
Euler-number [-]

- EOR
enchanced oil recovery [-]

- FWL
free water level [m]

- g
gravitational acceleration [m/s

^{2}]

- h
height [m]

- HC
hydrocarbon [-]

- IFT
interfacial tension [dynes/cm]

- IOR
improved oil recovery [-]

- k
absolute permeability [D]

- l, r
laboratory, reservoir

- MICP
mercury injection capillary pressure [-]

- N
number of samples [piece]

- NPV
net present value [-]

- nw
non-wetting [-]

- OPEX
operational expenditure [-]

- P
pressure [bar]

- PVT
pressure-volume-temperature [-]

- PWL
producing water level [m]

- QC
quality-check [-]

- r
radius [mm; μm]

- Rm
average pore throat radius [mm; μm]

- RMS
root-mean-square error [-]

- std. dev.
standard deviation [-]

- TVDGL
true vertical depth ground level [m]

- TVDSS
true vertical depth subsea [m]

- w
wetting [-]

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**Received:**2015-8-17

**Accepted:**2015-11-9

**Published Online:**2016-2-3

**Published in Print:**2016-1-1

© I. Nemes, published by De Gruyter Open.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.