**1. Introduction**

Many of the chalk formations in the North Sea contains large volumes of oil. In some areas, the porosity is higher than 45% [1,2]. During production, the reservoir pressure is decreased, the effective stress is increased and the reservoir subsequently compacts. Compaction is a significant driver for oil expulsion, and, as an example, for the Valhall field in the North Sea, the compaction is estimated to contribute to 50% of the total recovery [2]. For the Ekofisk field, reservoir compaction has led to seabed subsidence of approximately 9 m, where about 50% of the 9 m is due to pore pressure decline, and about 50% is due to the injected water [3]. The difference between physical and physicochemical effects is important to include when simulating the reservoir oil production and compaction.

In field-scale simulations of water-induced compaction in chalk, one uses the water saturation as an indicator of enhanced compaction [4]. In this paper, we argue that the water weakening effect is more related to changes in the water chemistry than changes in the water saturation, meaning that it is the changes in the concentration of specific ions that induce enhanced compaction, not only the ratio between oil and water. It is important to include a mechanistic understanding of fundamental processes of deformation, to model fluid flow and compaction in basins or reservoirs over observable timescales. Incorporating these insights on water weakening will produce reliable and more applicable basin models. In the next sections, we will review some of the relevant core experiments, and interpret

them by simple models. Thereafter we discuss how these changes may propagate on the field scale, by the use of streamline models. The practical consequence of this is that one can, to some extent, control the fluid chemistry inside the pores, control compaction and thereby impact the oil recovery. Finally, we present discussions, summary and recommendations for the future.

### **2. Interpreting Triaxial Core Deformation Experiments in Terms of the de Waal Model**

Hydrostatic tests are usually performed in triaxial cells, where the axial (length of the core) and, to some degree, the radial strain can be monitored during the experiment. Fluids are continuously passed through the cores, of approximately 7 cm in length, allowing for measurement of permeability and e ffluent fluid sampling. Back-pressure is applied to maintain a uniform pore pressure as the core compacts, and to avoid fluids boiling at elevated temperatures. A pressure di fference of a few kPa is enough to drive fluids through the sample. Thus, the stress–strain development is mainly dictated by the microstructural properties of the solid (matrix) framework.

Usually, the triaxial test data are reported for two stages of testing: i) The hydrostatic loading stage, where the confining pressure and pore pressure are increased linearly, and the axial strain is measured as the axial stress is increased until the core has passed its yield point. ii) The creep stage, during which the core subsequently deforms slowly under a small constant load [5]. The yield point can be considered a measure of the "strength" or "compaction strength" of the core, while the time-dependent creep that occurs after the yield point has been exceeded might be considered as a kind of "viscosity".

Figure 1 shows an example where a chalk core from the Aalborg outcrop was heated to 130 ◦C, loaded up to 12 MPa, and then left to creep for 61 days at a constant stress of 12 MPa, while a solution of 0.657M NaCl was passed through at a flooding rate of 1PV (Pore Volume) per day [6]. As can be seen from Figure 1a, the axial strain is proportional to the axial stress (e.g., the core deformation is elastic) up to the yield point at ~8.2 MPa (the solid black line). The yield point is not a single point, but rather a zone of transition between 7.5 and 8.5 MPa that separates the elastic and plastic (black dotted line) phases of deformation. The elastic bulk modulus is defined as the ratio of the change in axial stress to the change in volumetric strain up to the yield point [7] (the slope of the solid line in Figure 1a divided by 3). The plastic bulk modulus is defined as the ratio of the change in axial stress to the change in volumetric strain in the phase after the core has reached the yield point (the slope of the dashed line in Figure 1a divided by 3). Assuming that the material compresses equally in all directions, the axial strain can be converted to volumetric strain by multiplying by factor of 3, and the e ffective viscosity can be defined as the ratio between the axial stress and axial strain rate (η*eff* = σ *<sup>d</sup>*<sup>ε</sup>*V*/*dt*) [7]. The e ffective viscosity increases strongly with time in a creep experiment and as the stress is kept constant, the strain rate is reduced.

Numerous rock mechanical laboratory experiments have been carried out with a large degree of reproducibility between experiments. These experiments can be understood with reference to the grain contact model suggested by de Waal [8]. Here we describe the de Waal model and show how it provides a good basis for understanding experiments involving rock–fluid interactions and the various chemical factors that a ffect reservoir compaction. During the creep stage, compaction usually follows a log t trend as shown in Figure 1b. The following creep model captures this trend [8]:

$$\varepsilon\_V(t) = B \log \left( 1 + \frac{\varepsilon\_V(0)t}{B} \right) \tag{1}$$

*B* is a constant related to the friction between the grains, and . <sup>ε</sup>*V*(0) is the initial strain rate, which can be seen by di fferentiating the above equation with respect to *t*:

$$\frac{d\varepsilon\_V(t)}{dt} = \frac{\dot{\varepsilon\_V(0)}}{1 + \frac{\dot{\varepsilon\_V(0)}t}{B}} \stackrel{t \to 0}{\rightarrow} \dot{\varepsilon\_V(0)}\tag{2}$$

The model suggested by de Waal was derived from the basis of earlier work by [9] where it was shown that the friction coefficient between two blocks of rock, μ, depends on the time, *t*. The blocks were in stationary contact before sliding was initiated, and the friction coefficient was found to follow the empirical law:

$$
\mu = \mu\_0 + C \log(1 + Dt) \,\text{,}\tag{3}
$$

where *C* and *D* are constants, and μ0 the friction coefficient at t = 0. In [9,10] experiments were conducted over a time frame of one day, where the materials were kept in contact at a constant normal and shear stress for time t. When the static contact time was reached, the shear stress was rapidly raised to the level required to produce a slip. The reason for the dependence of friction on the stationary contact time is that the contact area between the surfaces increases as the static contact time increases [9].

De Waal argued that Equation (3) can be used to understand creep [8] if t in the equation is replaced with a time constant *ta* that describes the average lifetime of grain–grain contact, and this contact time is related to the inverse strain rate. The physical explanation for the macroscopic time-dependent creep of rocks is thus, according to de Waal, that the average lifetime of contact points affects the real contact area which determines the sliding friction.

**Figure 1.** (**a**) Hydrostatic loading stage. The elastic bulk modulus is 720 MPa (slope of solid line divided by 3). The plastic bulk modulus is 210 MPa (slope of the dashed line divided by 3). (**b**) The creep stage deformation. The core is left to deform at a constant stress 12 MPa. The dashed line is the de Waal model (text equation 1) with parameters . <sup>ε</sup>*V*(0) = 0.012%/hour and *B* = 0.43%. Data are from [6].

Westwood, Goldheim et al. [11] and Macmillan, Huntington et al. [12] extended Dieterich's [10] perspective by pointing out that the friction coefficient is also dependent on the fluid chemistry. Based on the experiments conducted, they argued that ion interactions modify the surface microhardness, and that it is greatest when the zeta potential is close to zero for, in this case, MgO. This implies that the friction is at its lowest level when the zeta potential is close to zero because grain contact is minimized by the hardness. A low value of the zeta potential should thus increase the strain rate. Westbrook and Jorgensen [13] also pointed out that water adsorption from the air affected the time dependence of microhardness observed in nonmetallic materials. They found up to a 10% decrease in the microhardness of wet calcite compared to dry, depending on the crystal plane.

The de Waal model has also been used to capture the dependence of creep strain rate on loading rate. The loading rate is known to a ffect the strain rate in the laboratory [14], and the laboratory strain rate is usually much larger than the strain rate in a reservoir where the loading results from the slow drawdown of fluid pressure by petroleum and water extraction. The challenge is to translate the higher strain rates observed in the laboratory to the lower field strain rates. Andersen, Foged et al. [14] extended the de Waal model from constant stress rate to constant strain rate and showed how to apply it to North Sea chalk. They found that the de Waal friction factor for the Valhall field was more or less constant over the stress range of interest, which reduced the number of tests required to estimate compaction at field depletion rates.

### **3. Water Activity Weakens Chalk in Loading Phase**

If the friction coe fficient, thus the strain rate, is dependent on the fluid chemistry, it follows that the mechanical properties of chalk also depend on the pore fluid [15–17]. The physical mechanism for water weakening has been attributed to capillary forces and di fferences in "wettability" between fluids [16,17], but for later publications [18], at least in part, water weakening was explained by the relations between physical and chemical properties of the chalk.

Risnes [19] found that the hydrostatic yield point of chalk (the full elastic deformation) increased by 50%, 70%, and 80% if, instead of water, the pore fluid was switched to methanol, oil and glycol, or air, respectively. The hydrostatic yield point was estimated to be 18 MPa for glycol and 10 MPa for water. These results have been confirmed by [20,21], where the chalk was saturated with supercritical CO2, and by experiments where the activity of water was changed systematically by changing the fraction of glycol mixed into water [20]. Chalk strength increases as the activity of water is decreased.

A completely di fferent approach by Røyne et al. [22] gives further insight into the water weakening effects. They used a double torsion method, wherein a calcite crystal with an initial crack is immersed in a fluid. The crack is extended by bending the calcite, and the crack velocity is measured. The crack velocity is considered to be a measure of surface energy, where a lower surface energy produces a greater crack velocity. By immersing the sample in a mixture of glycol and water, the surface energy of calcite was determined for di fferent water activities. Assuming that the hydrostatic yield point is related to pore collapse and bond breaking between the individual grains in the sample, a lower surface energy should mean a weaker chalk and a lower yield point. In the Røyne et al. [22] experiment, the surface energy was 0.32 J/m<sup>2</sup> for glycol and 0.15 J/m<sup>2</sup> for water. This factor of ~2 reduction is very close to the factor of 1.8 reduction determined by Risnes [20]. The activity of water can also be changed by changing the concentration of salts. Risnes et al. [23] were able to reduce the activity of water to 0.48 by the adding of CaCl2. Figure 2 plots against the activity of water, the surface energy of calcite immersed in fluids with di fferent water activity normalized to the surface energy of calcite immersed in pure water, or the yield point of cores saturated with fluids with di fferent water activities normalized to their yield point when saturated with pure water (activity = 1). It shows that the decrease in chalk strength with water activity is similar, regardless of how the activity of water is modified. The figure shows that not only does the yield point change by similar amounts with changes in water activity for all the experiments, but also the trend of the change with water activity is similar. Water weakens chalk, and the activity of water is a measure of the amount of weakening.

**Figure 2.** The normalized surface energy of calcite immersed in fluids with different water activity, or the normalized hydrostatic yield point of cores saturated with fluids with different water activity, are plotted against water activity. Normalization is achieved by dividing by the yield point or surface energy measured when the pore or emersion fluid is pure water. Blue triangular points show changes in calcite surface energy from [22]. Green, black and red square points show core tests where water activity was changed by mixing water and glycol [20]. Purple points show the changes in normalized yield point when the activity of water is modified by additions of CaCl2 [23]. Experimental uncertainties are not included in the figure.

### **4. Water Surface Chemistry A**ff**ects Chalk Deformation and Creep**

However, chalk strength depends on water *chemistry* as well as water *activity*. When comparing chalk deformed at 130 ◦C with seawater as the pore fluid, to chalk deformed at 130 ◦C with seawater without sulphate as the pore fluid, the yield point is 74% lower. The total dissolved solids concentration was kept constant by adding NaCl [24,25]. The activity of water, *aw*, for a brine is [26]

$$a\_w = 1 - 0.018 \sum\_{i} m\_{i\prime} \tag{4}$$

where *mi* is the molality of other ions in the solution. For seawater the water activity is ~0.98; thus, the activity of water is not a factor in this comparison. Sulphate ions weaken chalk and Figure 3 shows how sulphate lowers the yield point. Notice that although the yield point is dramatically lowered (the elastic bulk modulus is strongly decreased) by the presence of sulphate, the plastic bulk modulus (the slope of the dashed lines) is not changed.

**Figure 3.** Loading stage deformation for Liège cores saturated with different fluids and tested at 130 ◦C. NaSO4 concentration is varied while the amount of NaCl is adjusted to keep the ionic strength constant at 0.657 M. Pore fluids for the distilled water (DW) curve contain neither NaSO4 nor NaCl. The yield point is depressed and the elastic bulk modulus decreased by sulphate (solid lines), but the creep deformation and plastic bulk modulus are not affected by sulphate.

Megawati et al. [27] hypothesized that the electrostatic interactions that give rise to an osmotic pressure (disjoining pressure) near the grain contacts could explain the enhanced weakening. Sulphate adsorbed onto the calcite grains changes the surface and zeta potentials. The surface becomes more negatively charged. The variations in hydrostatic yield for all the chalk samples was proportional to the maximum peak of the disjoining pressure. Nermoen et al. [28] performed additional experiments and sugges<sup>t</sup> the electrochemical interactions that give rise to the disjoining pressure can be incorporated into an effective stress equation.

Sulphate in the brine also affects the creep rate [27]. Cores from different outcrops, which have different mineralogical compositions, have a very similar creep rate and creep rate development when the brine contains sulphate. This is in contrast to cores from the same outcrops flooded with NaCl (inert) brine, which shows greater variance in the strain rate development. This might be an indication that the grain-to-grain friction between the calcite grains is to some extend controlled by the diffusive layer counter ions, and not only the local mineralogical properties of the grains. If sulphate is introduced after the loading phase, i.e., in the creep phase, there is a sudden increase in the creep rate [29]. This effect is transient, and when the pore surfaces of the calcite grains are saturated with sulphate, the creep rate is reduced.

### **5. Core Deformation Related to Mineral Alteration**

Additional core deformation can be the result during the creep stage when the fluid passed through the deforming cores is out of equilibrium with the minerals in the core. Figures 4 and 5 compare the axial creep strain for two sets of experiments. In one set, a wide range of chalk cores was flooded with inert (e.g., nonreactive) NaCl brine. In the second set, cores from the same outcrops are flooded with reactive MgCl2. In both sets of experiments, the temperature is 130 ◦C. The axial creep for all the cores flooded with MgCl2 is greater than those flooded with inert NaCl, and is commonly more than three times greater. The chalk cores with few impurities (>99.7% calcite, the Stevns Klint and Mons (Trivières Fm)) flooded with MgCl2 show a temporary plateau in creep strain, see Figure 5. The chalk cores with lower carbonate content (the Aalborg, Liège, and Kansas) show no such plateau.

**Figure 4.** (**a**) Axial creep for cores flooded at 130 ◦C with nonreactive NaCl, and (**b**) a corresponding fit to de Waal model. AA=Aalborg, KA=Kansas, LI=Liège, MO=Mons, SK=Stevns Klint. The numbers in the parenthesis gives the porosity and permeability. All data from [6].

**Figure 5.** (**a**) Axial creep for cores flooded at 130 ◦C with MgCl2, the numbers in the parenthesis gives the porosity and permeability, and (**b**) corresponding fit to de Waal model and model parameters in the parenthesis. AA=Aalborg, KA=Kansas, LI=Liège, MO=Mons, SK=Stevns Klint. All data from [6].

The creep strains experienced by all the cores under NaCl flooding are well described by the de Waal model (Equation (1)), as shown in Figure 4b. The de Waal model does not match well with the cores flooded with MgCl2 brine, as seen in Figure 5b. We can now use the de Waal model to estimate the additional strain induced by the chemical reactions. If we simply subtract the "expected" creep from the cores flooded by MgCl2, by using the results in Figure 4, Figure 6 shows the excess creep strain that results when the cores are flooded with MgCl2. The excess creep strain takes a while to develop for some cores (the plateau mentioned above), but once established the excess strain rate is linear and ~0.04%/day in magnitude.

**Figure 6.** (**a**) Excess axial creep measured in cores flooded with reactive MgCl2 obtained by subtracting the de Waal deformation for the NaCl flooding, and (**b**) the creep rate.

Post-experimental investigations by scanning electron microscopy (SEM) show mineralogical alterations already after weeks of flooding. Dissolution of calcite takes place, and is commonly observed together with precipitation of magnesite. For the chemical alteration to be fully appreciated, reactive fluids must be passed through the chalk cores at elevated temperatures for very long periods of time. Figure 7 shows the results when a Liège chalk core was flooded for ~3 years (1072 days). There is a significant flux of Ca2+ out of the core but no flux in, and there is an almost exactly counterbalancing flux of Mg<sup>2</sup>+ into the core with no Mg<sup>2</sup>+ flux out. The sum of the Mg<sup>2</sup>+ in and Ca2+ out always almost exactly matches the injected MgCl2 concentration. This indicates that Ca2+ in the carbonate core is being replaced mole for mole with Mg<sup>2</sup>+. Mass balance indicates that 93 to 98% of the initial calcite is replaced by magnesite over the course of this experiment [28]. The initial porosity of the core was 41%, and the final porosity 40%. The pore volume is thus conserved and the 10% axial (30% volumetric) strain resulted from the mineralogical alteration of the matrix. Mineralogical alteration can cause substantial compaction. Note that the de Waal model fits the MgCl2 core flooding well in this case (see Figure 7); however, in this case, the flooding rate is changed during the experiment. At ~380 days, the observed creep is deviating from the de Waal model, but at this point, the flooding rate is decreased (less chemical reactions) and the axial creep rate is lowered. After approximately 700 days, the observed creep is below the de Waal model, but at ~800 days, the flooding rate increased and axial deformation is increased. Thus, in this case, the match with the de Waal model is a coincidence, due to changes in the flooding rate.

**Figure 7.** (**a**) Axial creep, permeability, and influent and effluent chemistry for a carbonate core flooded at 130 ◦C for ~3 years with reactive MgCl2. Line style indicates changes in flooding rate from 1 PV/day to 3 PV/day. Solid black line in (**b**) is the specific surface area development estimated from the experimental permeability and porosity using the Kozeny–Carman equation.

As the porosity did not change in the experiment shown in Figure 7, the dramatic and immediate drop in permeability is unexpected. The drop in permeability is most likely caused by magnesite crystallization in the large, permeable pores (see Figure 8a). In other chalk types, crystallization of magnesium-bearing clay minerals may also reduce permeability (Figure 8b). Minerals crystallized in pores increases the surface area per unit pore volume. Since permeability is directly related to porosity and inversely related to the square of surface area per unit of pore volume and pore tortuosity (c.f., the Carmen-Kozeny equation), an increase in surface area per unit pore volume strongly decreased permeability. The slight recovery in permeability between 200 and 300 days (Figure 6a) could sugges<sup>t</sup> either that precipitated magnesite had subsequently dissolved or that pore throats subsequently opened such that the tortuosity was reduced. Textural changes related to alteration can clearly and dramatically change permeability.

At the end of the experiment flooding with MgCl2, (as shown in Figure 7) had completely replaced the original calcite with magnesite. After ~800 days, injected Mg<sup>2</sup>+ was no longer reacting, but passing unchanged in concentration through the core. A similar experiment that was terminated at 516 days flooded a Liège chalk core under Ekofisk field conditions with 0.219 M MgCl2. Mapping by energy dispersive spectroscopy (EDS) by the use of mineral liberation analyzer (MLA) of the core from inlet to outlet after flooding, shows two regimes of mineralogical change. Toward the outlet, the core is only partly altered. The mineralogy is mainly calcite, the primary mineralogy of chalk, but precipitates of clay and magnesite are found within the calcite matrix. At the inlet, a total transformation from calcite to magnesite is observed. Only magnesite and clay are observed, and no calcite is left. For cores flooded for an even longer period, the transition between the two regimes has moved further into the core, indicating that this transformation of mineralogy moves like a front through the cores [30,31].

**Figure 8.** (**a**)Newly precipitatedmagnesite crystalsin alarge pore (edited from [30]). (**b**)Magnesium-bearing clay minerals crystallized in Aalborg chalk, which contains large amounts of primary opal-CT, after being flooded with MgCl2.
